Composites Science and Technology 93 (2014) 106–113

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Composites Science and Technology

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Dependence of the Pukánszky’s interaction parameter B on the interfaceshear strength (IFSS) of nanofiller- and short fiber-reinforced polymercomposites

0266-3538/$ - see front matter � 2014 Published by Elsevier Ltd.http://dx.doi.org/10.1016/j.compscitech.2014.01.002

⇑ Corresponding author. Tel.: +39 050 2217807; fax: +39 050 2217903.E-mail address: a.lazzeri@ing.unipi.it (A. Lazzeri).

Andrea Lazzeri ⇑, Vu Thanh PhuongUniversity of Pisa, Department of Chemical Engineering, Via Diotisalvi 2, 56126 Pisa, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 26 August 2013Received in revised form 28 December 2013Accepted 2 January 2014Available online 9 January 2014

Keywords:B. Interfacial strengthA. Carbon nanotubesA. Nano compositesA. Short-fiber compositesB. Mechanical properties modeling

In this paper Pukánszky’s model for the tensile strength, originally developed for composites filled withquasi-spherical fillers, has been analyzed in terms of the Kelly–Tyson model for the prediction of mechan-ical strength of composites reinforced with short fibers, often used also for nanocomposites with organ-ically modified nanoclays and carbon nanotubes. In this way it was possible to establish a direct linkbetween Pukánszky’s interaction parameter B and fundamental material parameters such as tensilestrengths of the matrix and of the fibers, the aspect ratio of the fibers, and the orientation factor andthe interfacial shear strength IFSS. Also it was possible to determine the minimum value of B for whichit is possible to predict the tensile strength of the composite from the modified rule of mixtures, as wellthe maximum value that B can achieve in the case of continuous aligned fibers with the same type ofmatrix, fibers and interface shear strength. Moreover, a critical volume fraction, ucrit, was defined corre-sponding to the minimum amount of filler content necessary for the composite strength to be greaterthan the strength of the unreinforced matrix, i.e. corresponding to the case rc = rm. It was also shown thatfor this condition Bcrit ffi 3.

A few examples of calculations of the IFSS, s, from Pukánszky’s interaction factor B have been provided,using published literature values relating to nanocomposites with organically modified nanoclays andcarbon nanotubes, as well as composites reinforced with short natural fibers.

� 2014 Published by Elsevier Ltd.

1. Introduction

In recent years there has been a rapid growth in the develop-ment and application of nanofiller- and natural short fiber-rein-forced thermoplastic polymer composites. In correspondencewith this interest, increasing efforts have been devoted to betterunderstand and measure the micro-mechanical parameters, whichcontrol the structure–property relationships in such composites.The properties of filled thermoplastic composites result from bothnanoparticle/fiber and matrix properties and the ability to transferstresses across the nanoparticle/fiber–matrix interface. Variablessuch as the volume fraction, aspect ratio, tensile strength, orienta-tion of the nanoparticle or the short fiber as well as the interfacialstrength are of primary importance to determine the mechanicalproperties of these composites [1–5]. In particular, the ability totransfer stresses across the interface is often discussed in termsof ‘adhesion’ but is, in fact, related to a complex combination of fac-tors such as surface energy of the reinforcing phase, adhesion

strength of the interfacee and the thickness of the interphase layer,i.e. the zone around the filler where the chemical and physicalproperties of the matrix are substantially altered due to inter-and supra-molecular interactions with the surface of the filler[6–12]. Therefore, it is not surprising if the nature of ‘adhesion’ isstill a matter of debate in the literature and many techniques havebeen developed to measure it. For continuous fiber composites,‘adhesion’ is generally related to the interfacial shear strength (sor IFSS) or to the interlaminar shear strength (ILSS), which is some-how related to IFSS yet still a different parameter. Several experi-mental methods have been developed for their determination.These can be divided into two general categories: the single (directtesting) and the multiple fiber tests (indirect testing). The direct- orsingle fiber-testing methods aim at measuring the interfacial adhe-sion of individual fibers in a matrix (microcomposites), while theindirect- multiple fiber-methods consider the collective behaviorof fibers in a matrix (real composite) and estimate the interfacestrength via simplistic models. The experimental methods for sin-gle fibers are mainly the pull-out and the fragmentation tests andprovide measures of IFSS [13,14]. In the case of multiple parallel fi-bers, the most important experimental techniques for the determi-nation of ILSS are the Short Beam Interlaminar Shear and the

A. Lazzeri, V.T. Phuong / Composites Science and Technology 93 (2014) 106–113 107

Iosipescu Shear tests. There is no general consensus about which ofthese tests gives the most reliable measurements, but the situationis further complicated in the case of nanoparticle- or short fiber-reinforced thermoplastic composites since these techniques donot lend themselves to an easy extension for these types ofcomposites.

Several methods have been recently developed for deriving val-ues for s (the IFSS) and the Krenchel fiber orientation factor, go [15]from tensile stress–strain curve of the composite and the fiberlength distribution based on modifications of the Kelly–Tysonequation [16,17]. Bader and Bowyer, Fu and Lauke [18,19] pre-sented a method for deriving values for s and go from a simplecombination of the tensile stress–strain curve and the compositefiber length distribution. Recently, Thomason [21] improved thismethod and illustrated its application to injection molded glass-fi-ber-reinforced thermoplastic composites. Furthermore, he showedhow the analysis could be extended to obtain the average fiberstress at composite failure, ruf.

Although developed originally for composites containing spher-ical particles, the semi-empirical equation proposed by Pukánszky[22] to describe the effects of filler volume fraction, uf, and inter-face interactions on yield stress and tensile strength of particu-late-filled polymers has been recently successfully applied toanisotropic fillers such as layered silicate nanoparticles, multi-walled carbon nanotubes (MWCNTs), and wood fibers [23–25].

In Pukánszky’s method an interaction parameter B, that consid-ers the capacity of stress transfer between various components,can be easily calculated by knowing the yield stress of compositesfilled with different volume percentages of filler particles.

Despite its simplicity and widespread use to characterize nano-particle- and short fiber-reinforced composites, the adimensionalPukánszky B-factor is not related with interfacial shear strength,s, and other experimental variables like filler aspect ratio (ar) andorientation factor.

In this paper we explore the relation between Pukánszky’sinteraction parameter B and these parameters which are knownto have a strong effect on the mechanical properties of composites.We present a modification of Pukánszky’s approach by comparingit with the Kelly–Tyson equation to make a connection betweenthe B-factor and the relevant physical–mechanical parameters.We apply this new approach to nanoparticle- and short fiber-rein-forced thermoplastic composites. Values of s and go obtained usingthis improved version of the original model are presented and dis-cussed. We furthermore show how the analysis enables us toachieve values of the interfacial shear strength similar to those al-ready published in the literature.

2. Theoretical analysis

2.1. Analytical expression for parameter B

The reinforcing effect of a filler or a fiber is expressed quantita-tively by the following equation proposed by Pukánszky [21]:

rc ¼ rm1�uf

1þ 2:5ufexpðBuf Þ ð1Þ

In this equation, terms rc and rm are the yield stress of the compos-ite and of the matrix, respectively, while the (1–uf)/(1 + 2.5uf) termindicates the decrease of effective load-bearing cross section due tofiller introduction the decrease of effective cross section on filling.The term exp(Buf) considers the filler-matrix interactions, bymeans of the interaction parameter B that considers the capacityof stress transfer between various components. For composites with

quasi-spherical particulate fillers, the interaction parameter B hasbeen shown to depend on the thickness of the interphase, and thestrength of the interphase as shown in the following equation [22]:

B ¼ 1þ Af qf l� �

lnri

rmð2Þ

where Af, qf, l, ri are the specific surface area and the density of thefiller, the thickness of the interphase and the strength of interphase,respectively, that can be evaluated by knowing the tensile strengthof composites filled with different volume percentages of particu-late fillers [26]. Alternatively, the thickness of the interphase be-tween particles and matrix can be evaluated by using the Shen–Limodel which assumes the formation of a non-homogeneous inter-phase [27,28]. In the Shen–Li model, the mechanical properties ofthe medium at the microscopic scale do not change abruptly atthe interface between the spherical particle and the polymeric ma-trix, but a transition region (or interphase) exists, in which theproperties continuously relax until reaching those of the pure ma-trix at sufficiently long distances from the center of the filler parti-cle. Recently the Shen–Li model has been improved by Sevostianovand Kachanov [8,9]. Estimations of the interphase size and proper-ties in nanocomposites have been also carried out using MolecularDynamics (MD) methods [10,11].

A micromechanical analytical model was recently proposed formodeling the mechanical properties of polymer nanocompositesfilled with silica nanoparticles. It takes into account an interphase,introduced as a third phase, corresponding to a perturbed region ofthe polymer matrix around the nanoparticles [29].

Since the polymer matrix is expected to be constrained in thevicinity of the nanoparticle, it was assumed that the interphasepresents a graded modulus, ranging from that of the silica to thatof the polymer matrix. The change in elastic modulus in the inter-phase was described by a power law introducing a parameterlinked to interfacial features. It was shown that particle size andinterphase features are the dominant parameters controlling theoverall nanocomposite behavior [29].

We can write Eq. (1) in linear form:

logðrredÞ ¼ logrcð1þ 2:5uf Þrmð1�uf Þ

¼ Buf ð3Þ

and plotting the natural logarithm of Pukánszky’s reduced (adimen-sional) tensile strength, rred, against volume fraction (this graphwill be called Pukánszky’s plot in the following) should result in alinear correlation, the slope of which is proportional to the interac-tion parameter B [22].

According to the Kelly–Tyson model, the strength of the com-posite can instead be estimated by a simple modification of themixture rule. Two cases must be identified, depending on whetherthe average length of the fibers, L, is lower or larger than the criticallength, Lc, which is the minimum length necessary such that thestress is efficiently transferred from the matrix to the fibers, so thatthe center of the fiber reaches the ultimate (tensile) strength rf:

Lc ¼rf D2s¼ rf L

2sarð4Þ

where D, ar are the diameter and aspect ratio of the fibers, respec-tively. For layered fillers the filler aspect ratio ar can be defined asthe ratio between the average diameter d of a circle of the same areaas the larger face of the platelet and the mean thickness of theplatelet h: ar = d/h. Thus, it can be shown that h = d/ar can replaceD in Eq. (4).

For a composite containing more than a certain volume fraction,umin, the Kelly–Tyson model leads to the following equation(u > umin):

108 A. Lazzeri, V.T. Phuong / Composites Science and Technology 93 (2014) 106–113

rc ¼ gorf 1� Lc

2L

� �uf þ r0mð1�uf Þ

¼ gouf rf 1� rf

4sar

� �þ r0mð1�uf Þ ð5Þ

where r0m represents the stress borne by the matrix when the strainof the composite is such that the fibers are strained to their ultimatetensile strain, ec.

The orientation factor in Eq. (5), go, is the fiber orientation fac-tor, similar to the Cox–Krenchel model [15], to take off-axis fiberorientation into account and is equal to 1 for unidirectional discon-tinuous composites. The actual values of the fiber orientation fac-tor for strength differ from those used in the Cox–Krenchelequation for the determination of the modulus of short fiber com-posites [15] and can be estimated by the critical-zone model ofFukuda and Chou [30–32] and measured experimentally by themethod of Bowyer and Bader, later improved by Fu and Laukeand Thomason [18–21].

In general, comparing Eq. (1) with Eq. (5) leads to the followingexpression:

s ¼ rf

4ar 1� rmð1�uf Þgouf rf

expðBuf Þð1þ2:5uf Þ

� ah in o for Lf > Lc ð6Þ

Substituting rc from Eq. (5) into Eq. (3) we get

logðrredÞ ¼ loggouf rf 1� rf

4sar

� �þ r0m 1�uf

� �rm

þ logð1þ 2:5uf Þð1�uf Þ

ð7Þ

The second logarithmic term on the r.h.s. of Eq. (7) is approxi-mately linear with uf in the range 0 6uf 6 0.6:

logð1þ 2:5uf Þð1�uf Þ

ffi 3:04uf ð8Þ

The first logarithmic term on r.h.s. of Eq. (7) can be written as:

loggouf rf 1� rf

4sar

� �þ r0m 1�uf

� �rm

¼ logr0m �uf r0m � gorf 1� rf

4sar

� �h irm

¼ ð9Þ

¼ logr0mrm

1�uf 1� gorf

r0m1� rf

4sar

� �� �� �

¼ logr0mrmþ log 1�uf 1� go

rf

r0m1� rf

4sar

� �� �� �

The latter equation, when uf ? 0, can be approximated into:

logr0mrmþ log 1�uf 1� go

rf

r0m1� rf

4sar

� �� �� �

ffi logr0mrm�uf 1� go

rf

r0m1� rf

4sar

� �� �ð10Þ

We can thus rewrite Eq. (7) in the following way:

logðrredÞ ¼ logr0mrm�uf 1� go

rf

r0m1� rf

4sar

� �� �þ 3:04uf ð11Þ

We observe that a Pukánszky’s plot of Eq. (11) will give astraight line with slope

B ¼ 3:04� 1� gorf

r0m1� rf

4sar

� �� �

¼ 2:04þ gorf

r0m1� rf

4sar

� �ð12Þ

and intercept log r0mrm

for uf = 0. When r0mrm� 1 the straight line

passes from the origin of the axis, as in the original Pukánszky’splot. Therefore Eq. (11) provides the theoretical basis for the exten-sion of Pukánszky’s model for short fiber- and nanofiller-reinforcedcomposites because it makes a direct link between the Pukánszky’sfactor B and micromechanical parameters like aspect ratio, tensilestrength of the filler and matrix, orientation factor and interfacialshear strength of the matrix. In general, the plot of log (rred) forshort fiber-reinforced composites is nonlinear with uf but we canobtain B from the derivative of the log (rred) vs. uf in the limit ofuf ? 0.

From Eq. (12) we see that the interaction parameter B is depen-dent upon the tensile strengths of the matrix and of the fibers orfillers, the aspect ratio, the orientation factor and the interfacialshear strength. In particular, Eq. (12) shows a linear dependenceof parameter B from the fiber orientation factor, g0.

Eq. (12) enables us to calculate the maximum value for B, con-sidering continuous aligned fibers or perfectly aligned layerednanofillers (g0 = 1) in the limit of an infinite 2D layer structure(ar ?1):

Bmax � 2:04þ rf

r0mffi 2:04þ Ef ef

Emef¼ 2:04þ Ef

Emð13Þ

where a ¼ r0mrm

.Thus Eq. (12) also enables to estimate the IFSS from the value of

B:

s ¼ rf

4ar 1� B�2:04go

rfr0m

" # ð14Þ

2.2. Expressions for the minimum and critical values of the volumefraction

Eqs. 5, 11, and 12are valid only when the fibers exceed a certainminimum volume fraction, umin, see Fig. 1, that can be estimatedby considering that a composite will undergo immediate fractureif [33]:

rc � rmð1�uf Þ þ gorfLc

2Luf ð15Þ

where the last term takes into account the fact that fibers that haveends within Lc/2 of the cross-section at which the first fiber fails willnot be broken. Now, substituting for rc from Eq. (5),

gorf 1� Lc

2L

� �uf þ r0mð1�uf Þ > rmð1�uf Þ þ gorf

Lc

2Luf ð16Þ

we obtain umin, as follows:

umin ¼rm � r0m

gorf 1� LcL

� þ rm � r0m

¼ rm � r0mgorf 1� rf

2sar

� �þ rm � r0m

ðL > LcÞ ð17Þ

At volume fractions less than umin, the strength of the compos-ite is given by the equality in Eq. (15). Therefore, the strength of thecomposite will be always greater than the strength of the unrein-forced matrix, when:

rmð1�uf Þ þ gorfLc

2Luf > rm ð18Þ

that is:

rm < gorfLc

2L¼

r2f

4sarð19Þ

Fig. 1. Schematic representation of mechanical behavior for composites withu > umin (a), u < umin (b) and theoretical variation of composite strength, rc, withvolume fraction, uf, of nanofiller or short fiber reinforcement (c).

A. Lazzeri, V.T. Phuong / Composites Science and Technology 93 (2014) 106–113 109

If rm > gorfLc2L, this defines a critical volume fraction in the com-

posite, ucrit, necessary for composite strength to be greater thanthe strength of the unreinforced matrix, i.e. corresponding to thecase rc = rm.

The critical volume fraction is given by the following expression[33]:

ucrit ¼rm � r0m

gorf 1� Lc2L

� � r0m

¼ rm � r0mgorf 1� rf

4sar

� �� r0m

ð20Þ

Comparing Eqs. (17) and (20), we can see that umin = ucrit whenrm ¼ gorf

Lc2L ¼ go

r2f

4sar

From Eq. (12) we can write:

r0mðB� 2:04Þ ¼ gorf 1� rf

4sar

� �ð21Þ

and substituting into Eq. (11):

ucrit ¼rm � r0m

gorf 1� rf

4sar

� �� r0m

ffi rm � r0mr0mðB� 2:04Þ � r0m

¼rmr0m� 1

B� 3:04ð22Þ

The existence of a critical volume fraction of CNTs based nano-composites to get appropriate strengthening (as observed inmicrocomposites) has been suggested by Andrews and Weisenber-ger [34]: ‘‘Often, however, in a polymer/CNT composite a reductionin strength is observed suggesting that the CNTs may promotecrystalline defects in the matrix or, considering their size, act as de-fects themselves. This suggests a critical volume fraction of CNTs isrequired for hindering matrix strain allowing strengthening to oc-cur as typically observed in composites’’.

Similar reductions of strength occur in nanoclay based compos-ites. For example from the data published in [35] a critical volumefraction of 0.4% can be determined for halloysite nanotubes (HNT)/epoxy nanocomposites.

2.3. Expression for the critical value of parameter B

The last equation sets a condition for the value of B above whichthe nanofillers or the fibers show a reinforcing action, Bcrit:

Bcrit ffi 3:04þrmr0m� 1

ucrit� 3 ð23Þ

Comparing Eq. (1) with Eq. (15), we can calculate the value of Bcorresponding to umin:

rmð1�uminÞ þ gorfLc

2Lumin ¼ rm

1�umin

1þ 2:5uminexpðBminuminÞ ð24Þ

Solving for Bmin we get:

Bmin ¼1

uminlog

1þ 2:5umin

1�uminð1�uminÞ þ go

rf

rm

Lc

2Lumin

� � �ð25Þ

Considering the approximation given by Eq. (8), we can write:

Bmin ffi 3:04þlog ð1�uminÞ þ go

rf

rm

Lc2L umin

h iumin

ð26Þ

And with a suitable rearrangement:

Bmin ffi 3:04þlog 1þumin go

rf

rm

Lc2L� 1

� �h iumin

ð27Þ

For small values of umin, we can further approximate as follows:

Bmin � 2:04þ gorf

rm

Lc

2L¼ 2:04þ go

r2f

4srmarð28Þ

Bmin is the minimum value of B for which it is possible to predictthe tensile strength of the composite from the modified rule ofmixtures and that enables to estimate s from Eq. (14).

From Eq. (28) we see that the minimum interaction parameterBmin is dependent upon the tensile strengths of the matrix and ofthe fibers, the aspect ratio of the fibers, the orientation factor,and the interfacial shear strength.

Again, following Kelly and Davies [33], we now consider thecase when L = Lc. From Eq. (17) where it is evident that umin = 1,thus the failure of the composite will occur by flow of the matrix.

However, from Eq. (19), it is evident that a strengthening effectwill occur when:

rm < gorf

2that is when Bmin � 2:04þ go

rf

2rm> 3:04 ð29Þ

Fig. 2. Pukánszky’s plot for sepiolite/PA6 nanocomposites, using the data fromBilotti et al. [22].

110 A. Lazzeri, V.T. Phuong / Composites Science and Technology 93 (2014) 106–113

in this case, the strength of the composite will be given by

rc ¼ go

rf uf

2þ rmð1�uf Þ ð30Þ

In an analogous way, we get:

B ¼log 1þ2:5uf

rmð1�uf Þgo

rf uf

2 þ rmð1�uf Þh in o

ufð31Þ

For small values of uf we can approximate into:

B ffi 2:04þ gorf

2rm¼ Bmin ð32Þ

Finally, for the case when L < Lc, it is apparent from Eq. (17) thatuf is always <umin, so the failure of the composite occurs by plasticflow of the matrix. Thus the strength of the composite will be givenby [33]:

rc ¼gosaruf

2þ rmð1�uf Þ ð33Þ

Substituting rc from Eq. (1), we get

B ¼log 1þ2:5uf

rmð1�uf Þgosaruf

2 þ rmð1�uf Þh in o

ufffi 2:04þ gosar

2rmð34Þ

Solving this equation for the IFSS, we find:

s ¼ 2rmðB� 2:04Þgoar

ð35Þ

Eq. (35)shows that the interfacial shear strength is directly propor-tional to B. It is important to note that for values of B smaller than2.04, s < 0, which means that the fillers or the fibers do not showany reinforcing action on the matrix. Therefore, we can concludethat 2.04 is the lower limit for B for the application of Eqs. (14)and (35) in the estimation of the IFSS, s.

3. Discussion

Pukánszky’s equation for tensile strength, originally developedfor composites reinforced with quasi-spherical fillers [22], hasbeen recently extended to nanofibers, carbon nanotubes and shortnatural fibers [23–25], although no theoretical justification for theuse of such an equation for this type of composites has been pro-vided so far in the literature. In this paper we propose a new inter-pretation of Pukánszky’s model based on the classical Kelly–Tysonapproach for short fiber composites [16,33]. With this approach itwas possible to find a correlation between Pukánszky’s interactionfactor B and the IFSS, s. From Eq. (12) we can predict that B will in-crease with the tensile strengths of the fibers or fillers, their aspectratio, orientation factor and the interfacial shear strength, and de-crease upon increasing the stress reached by the matrix when thestrain of the composite is such that the fibers are strained to theirultimate tensile strain, r0m.

In the present section we will show some examples of applica-tions of this new approach with reference to some published datareported in the recent literature.

Table 1Values recalculated from original experimental data from [12,24] (a); (b) data from [25];

Material r0m (MPa) B ar

PA6/sepiolitea 81.76 12.6 50PA6/MMT-USAa 76.38 11.7 36PA6/MMT-Japana 80.13 11.4 45PP/MWCNTb 34.38 7.4 40PP/wood flour no MAPPc 17.04 3.8 12.PP/wood flour with MAPPc 17.04 5.8 12.

Bilotti et al. [23,36] showed the results of tensile tests carriedout on sepiolite clay/polyamide 6 nanocomposites obtained bymelt compounding and compared with similar nanocompositesprepared with two organically modified montmorillonite clays,identified by the geographic location from which they were mined,i.e., Yamagata, Japan (Kunipia-P hereafter MMT-JP) and Wyoming,USA (Cloisite hereafter MMT-USA) [37,38].

In Fig. 2, we replotted Pukánszky’s plot of the data from Bilotti,and we show the data regression lines, calculated without forcingthem to pass through the origin of the Cartesian axis, for the threeseries of nanocomposites. In Table 1 we report the values of B andr0m obtained by the Pukánszky’s model, as modified by Eq. (12). Itcan be observed that the B values are approximately constant forall types of nanocomposites, probably due to the small diffencein the aspect ratio of the nanofillers considered in this work. Alsothe values of r0m are similar but significantly larger than the tensilestrength of the unmodified matrix, rm = 69.7(MPa). This can be ex-plained by considering that, in the case of nanocomposites, theimmobilized layer of polymer chains that forms the interface willalter the polymer – filler interaction at the molecular level, sincethe different structural organization of the matrix at the interface,with respect to the original matrix, hinders the polymer chainmobility [39].

Starting from these data we can estimate the IFSS of these com-posites. The aspect ratio reported in Table 1 has been calculated byBilotti et al. [23,36] from experimental data of Young’s modulus,using ar as the fitting parameter of the Halpin–Tsai equationsand considering an elastic modulus of all the nanoclays of200 GPa [23,40]. A difficulty arises in the estimation of rf, sinceno published data are available – to the authors’ knowledge – onthe tensile strength of individual nanoclay platelets. By consideringthat these nanoclays may possess the same tensile strength ofindustrially-produced alumina-silica fibers with similar Young’smodulus (150–250 GPa), we assume a value of 2500 MPa in ourcalculations [41].

(c) data from [11].

rf (MPa) go s (MPa) Bmax

2500 0.6 29.5 32.62500 0.6 34.2 34.72500 0.6 28.7 33.21720 0.6 13.1 52.1

6 300 0.6 7.2 19.66 300 0.6 9.2 19.6

Fig. 3. Theoretical dependence of B on the aspect ratio according to Eq. (11), forsepiolite/PA6 and MWCNT/PP nanocomposites, using the data from Bilotti et al. [22]and from Satapathy et al. [24,42], as reported in Table 1.

Fig. 4. Pukánszky’s plot for MWCNT/PP composites, using the data from Satapathyet al. [24,42]. In the figure both Eqs. (6) and (10) are plotted to show thenonlinearity of the Kelly–Tyson equation in this type of graph. In the insert thePukánsky’s plot of the data by Satapathy et al. [24,42] is limited to low volumefractions and the experimental points can be fitted by a line, as in the originalPukánsky’s model.

A. Lazzeri, V.T. Phuong / Composites Science and Technology 93 (2014) 106–113 111

Bilotti reports a partial inplane orientation of sepiolite needle-like clays, thus we assume an orientation factor of 0.6 in our esti-mations. For random in plane orientation an orientation factor of 3/8 ffi 0.4 is often assumed [42], while for fully oriented fibers the ori-entation factor is 1. We assumed an orientation factor of 0.6, anintermediate value between 0.4 and 1, since the work of Bilotti re-fers to injection moulded samples.

Table 1 reports the results of the calculation of IFSS for the threesystems. Values of s in the range 28–35 MPa are obtained from Eq.(14), with insignificant differences among the three types of nanoc-lays. Also the calculated value falls well in the expected range.Fig. 3 shows the dependence of B on the aspect ratio, ar, calculatedfrom Eq. (12) using the data in Table 1. As we can see, B monoton-ically rises until a plateau value of about 20.3 is obtained for PA6/Sepiolite. In the case go = 1, we obtain from Eq. (13) Bmax = 32.6 asreported in Table 1. For PA6/MMT-USA and PA6/MMT-Japan, thesame calculations give 34.7 and 33.2. These figures compare wellwith the largest B values published according to Százdi et al.[43], which are below 25, although the same authors predict a va-lue of about 200 for a completely exfoliated silicate on the basis ofthe linear dependence of B on the specific surface area.

The specific surface area of a nanoclay platelet can be related toits aspect ratio. By definition, the specific surface is given by:

As ¼AM¼ A

qV¼ 2pr2

qpr2h¼ 2

qhð36Þ

As stated in Part 2.1, the aspect ratio for a platelet can be de-fined as the ratio of mean diameter of a circle of the same areaas the face of the plate to the mean thickness of the platelet,ar = d/h, thus:

As ¼2qd

ar ð37Þ

For a nanoclay, both q and d are constant for each specific type,while h can decrease with the degree of exfoliation. In clay rein-forced polymers, different filler morphologies can occur (immisci-ble, intercalated and exfoliated) which make the determination ofthe filler aspect ratio not straightforward. For these filler geome-tries it is possible to follow the approach used in reference [44].In the case of agglomerates of nanoparticles (nanoclusters) withdimensions of these clusters from dozens to hundred times morethe diameter of the individual particles, during the melt blending

stage, the shear forces cause the nanoclusters to stretch into theline shapes forming structures that can be modeled as ‘‘discontin-uous fibers’’ [45].

Thus, our model predicts that the relation between B and aspectratio and specific surface area cannot be linear until complete exfo-liation is attained. In fact, Eq. (13) related Bmax with the ratio of thetensile strength of the nanofiller to the stress reached by the ma-trix when the composites fails.

Before leaving Fig. 3, where a constant s is assumed, we alsonote that for values of ar < 20, B is negative which means that thefiller is actually detrimental for the tensile strength of thecomposites.

Polypropylene (PP) multi-walled carbon nanotube (MWNT)composites offer another interesting example of the model pro-posed in this paper. We consider the data published by Satapathyet al. [25,46], which we replot in Fig. 4. The inserted plot in this fig-ure is a magnification of the graph in the region of lower values ofvolume fraction, similar to the Pukánszky plot published in the ori-ginal paper. From this Pukánszky plot we obtain a value of B = 7.36and r0m ¼ 34:38 MPa. Extracting a value of IFSS from these data,using Eq. (14), requires knowledge of the tensile strength, aspectratio and orientation factor of the MWCNTs. The MWNTs used bythe group of Satapathy have been produced using chemical vapordeposition (CVD), and have a diameter in the range of 10–15 nm,lengths between 0.1 and 10 lm, giving an average aspect ratio of40 [25,46]. Similar to Satapathy et al. [25,46], here we do not con-sider the effect of CNT waviness and entanglements on themechanical properties. Recent studies [47] have modeled the effectof waviness on the interfacial shear stress of nanotube reinforcedpolymer composites and found that the maximum IFSS of a wavyCNT is higher than that of straight ones and increases with increas-ing waviness.

By using the value of tensile strength, 1.72 GPa, measured byPan et al. [48], on CVD-grown aligned MWCNTs we can estimatean IFSS of 13.1 MPa. Using these data in Fig. 3 we observe thatby increasing the aspect ratio an asymptotic value of B is reachedat 31.5. For aligned MWCNT Eq. (13) provides an estimate ofBmax = 52.1.

The data of Satapathy et al. [25,46] are also replotted in thePukánszky plot of Fig. 4 together with the theoretical predictionsof Eqs. (7) and (11). As it can be observed, the experimental data

Fig. 6. B as function of interface shear strength, according to Eq. (11), for sepiolite/PA6 and MWCNT/PP nanocomposites, and for wood flour/MAPP/PP compositesusing the data of Bilotti et al. [22], Satapathy et al. [24,42] and Pukánszky et al. [23].

112 A. Lazzeri, V.T. Phuong / Composites Science and Technology 93 (2014) 106–113

lay in the range where the curve obtained from the modified rule ofmixture – Eq. (7) – can be approximated by Eq. (11). Significantdeviations between the two curves only arise for volume fractionsabove 0.05.

Finally, we examine some data published by the group of Prof.Pukánszky on PP/wood flour composites [24]. In this work, an eth-ylene-propylene random copolymer ethylene-propylene randomcopolymer (Tipplen R 359 from TVK Plc. – Tiszai Vegyi Kombinát- Hungary) has been mixed with lignocellulosic fibers (Arbocel FT400 from Rettenmaier GmbH, Germany). To improve interfacialadhesion a maleic anhydride modified polypropylene PP (MAPP –Orevac CA 100 from Arkema Inc., Colombes Cedex, France) wasused. The MAPP/wood ratio was chosen as 0.1.

Fig. 5 shows the semi-log plot of reduced tensile strength as afunction of the volume fraction for the two series of PP/wood flourcomposites: with and without modification with PP-g-MA. Fromthe values of the interaction parameter obtained and reported inTable 1, we can calculate the IFSS. Again, we made some assump-tions since no information is provided in [24] to allow us to deter-mine the orientation factor; moreover, we do not know the tensilestrength of the wood fibers. Considering the literature on lignocel-lulosic fibers we find values between 130 and 600 MPa [49–53].Assuming rf = 300 MPa and an orientation factor go = 0.6, the IFSSfor the composites with PP/wood flour not treated with PP-g-MA isestimated to be 7.2 MPa. This is in agreement with the value of7.9 MPa found by Rodriguez et al. [54] for corn stalk fiber rein-forced PP. For PP/PP-g-MA/wood flour composites, the estimatedinterface shear strength rises to 9.2 MPa, with an increase of about30%. This value of s is substantially lower than that found fromRodriguez et al. [54] for composites containing 6% MAPP, corre-sponding to 15.5 MPa. This can be explained by the fact that theaddition of PP-g-MA has brought the IFSS very close to its maxi-mum predicted according to the von Mises criterion,s ¼ rm=

ffiffiffi3p¼ 9.8 MPa [55,56]. It has to be noted that all values of

IFSS in Table 1 are below the respective values predicted by thiscondition.

Finally, in Fig. 6 we show the relationship between the interac-tion parameter, B, and the interface shear strength, according to Eq.(14), using the data in Table 1 for PP/MWCNT, PA/Sepiolite and PP/Wood flour-MAPP. The two parameters, B and s, are clearly related,as often implied in the literature [22,24,26], but the relationship isnonlinear and also other material constants such as fiber tensilestrength, aspect ratio and orientation factor as well as the stressin the matrix when the composites breaks contribute in actual va-lue of B.

Fig. 5. Pukánszky’s plot for wood flour/MAPP/PP composites using the data fromPukánszky et al. [23].

4. Conclusions

In this paper the Pukanszky model for tensile strength of a com-posite is generalized to the case when the filler has a non-sphericalshape. The Pukanszky model enables to determine parameter B in astraightforward manner even for composite systems where thedetermination of micro-mechanical parameters required by theKelly–Tyson equation or newer variants, for fillers like layered sil-icate nanoparticles, multi-walled carbon nanotubes (MWCNTs),and natural fibers, is not an easy task. In fact, many elegant tech-niques have been developed for the quantification of these mi-cro-mechanical parameters in recent years. However, thesesophisticated methods are often time consuming, complex, ineffi-cient and labour intensive and have proven to be of limited useand even inapplicable in industrial environments, especially tothe purpose of optimization of new composite formulations inview of their commercial application.

In the original Pukanszky model for tensile strength, developedfor composites with quasi-spherical particulate fillers, the interac-tion parameter B takes into account the capacity of stress transferbetween various components and has been shown to depend onthe thickness of the interphase, and the strength of the interphase.

In this paper, we show that, when applied to composites withinhomogeneities of non-spherical shape, the Pukánszky’s interac-tion parameter B depends not only from fundamental materialparameters such as tensile strengths of the matrix and of the fibers,but also to the interfacial shear strength, s, and other geometricalvariables like filler aspect ratio and orientation factor.

From this analysis it was possible to express the interfacialshear strength in terms of B and other materials parameters, Eq.(14). This also lead to the determination of the minimum valueof B for which is possible to predict the determine the IFSS, as wellthe maximum value that B can reach in the case of continuousaligned fibers with the same type of matrix, fiber and interfaceshear strength. Moreover, a critical value, Bcrit ffi 3 was defined cor-responding to case where the composite strength is greater thanthe strength of the unreinforced matrix, i.e. corresponding to thecase rc = rm.

A few examples of calculations of the IFSS, s, from Pukánszky’sinteraction factor B have been provided, using published literaturevalues relating to nanocomposites with organically modifiednanoclays and carbon nanotubes, as well as composites reinforcedwith short natural fibers. All results obtained fall within the value

A. Lazzeri, V.T. Phuong / Composites Science and Technology 93 (2014) 106–113 113

expected from similar literature values and below the maximumpredicted according to the von Mises criterion, s ¼ rm=

ffiffiffi3p

.

Acknowledgments

The authors gratefully acknowledge the financial support of theFORBIOPLAST (Forest Resource Sustainability through Bio-Based-Composite Development) Project – Contract No. 212239-FP7-KBBE,funded by the European Commission under the 7th FrameworkProgramme (FP7) (http://www.forbioplast.eu).

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