Quantification of layered silicate dispersion in polymer nanocomposites

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Polymer 55 (2014) 4216e4225

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Polymer

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Quantification of layered silicate dispersion in polymernanocomposites

Qian Gou a, Mark D. Wetzel b, Babatunde A. Ogunnaike a, *

a University of Delaware, Newark, DE, USAb E. I. du Pont de Nemours and Co., Inc., Wilmington, DE, USA

a r t i c l e i n f o

Article history:Received 26 February 2014Received in revised form9 May 2014Accepted 17 May 2014Available online 24 May 2014

Keywords:NanocompositesDispersion quantificationGamma probability distribution

* Corresponding author.E-mail address: [email protected] (B.A. Ogunnai

http://dx.doi.org/10.1016/j.polymer.2014.05.0470032-3861/© 2014 Elsevier Ltd. All rights reserved.

a b s t r a c t

The dispersion of layered silicates strongly influences the performance of polymer/clay nanocomposites,making the ability to assess clay dispersion quantitatively essential to any effective product qualitycontrol effort. However, a quantitative description of dispersion that can be related directly to productperformance remains a challenge. Here, we propose a methodology for quantifying clay dispersion, basedon describing particle length distribution data obtained from transmission electron microscopy (TEM)micrographs with a gamma probability distribution model, g(a,b), where the parameter a, (which rep-resents the mean number of platelets per particle), and b, (which represents the mean “effective plateletlength”), are shown to be related to the degree of dispersion. Published data sets on a wide variety ofpolymer nanocomposites are then used to establish an explicit quantitative relationship between theseparameters and the extent of dispersion. Specifically, we found that dispersion characteristics arecaptured effectively by a characteristic dispersion parameter, d, defined along dispersion contours rep-resented by the empirical relation d ¼ a þ 0.05b, where d < 5 for exfoliated composites, 5 < d < 6.5 forless exfoliated composites with stacks containing several layers, and d > 6.5 for intercalated composites.The proposed method is validated experimentally by using it to quantify the dispersion of polymernanocomposites produced in our laboratory.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Compared to pristine polymers and conventionally modifiedcomposites, polymer nanocomoposites often show superior me-chanical properties, enhanced gas barrier traits, and improvedflame retardancy [1e3]. Such property improvements are generallyattributed to the presence of nanofillers used for reinforcing thepolymer matrix. Although a variety of nanofillers have been used inthe preparation of nanocomposites, layered silicate minerals (e.g.,montmorillonite clay) remain the most commonly used. In thenative state, individual clay layers (or, platelets), 1 nm in thicknessand from a few angstroms to possibly microns in the lateral di-mensions, are stacked on top of one another [4]. After integrationwithin the polymer matrix, these platelets do not usually dispersein the pristine polymer as individual platelets; instead, they tend toaggregate to form much larger and more complicated structures.Depending on the layered silicate structure, polymer/clay

ke).

nanocomposites are traditionally classified as: (1) immiscible e inwhich most platelets are packed together into structures oftenreferred to as “tactoids”; (2) intercalated e in which platelets arepartially stacked; and (3) exfoliated/delaminated e in which indi-vidual platelets are well-dispersed in pristine polymer [5].

Because the degree of clay dispersionwithin the polymermatrixis central to obtaining desired product properties, any rationalstrategy for manufacturing polymer nanocomposites that will meetprescribed product property targets consistently requires a meansof quantifying clay dispersion. In practice, it is customary to char-acterize the dispersion level of clay layers by such techniques as X-ray diffraction (XRD) [6,7], transmission microscopy (TEM) [7e12],rheological measurements [12e14], atomic forcemicroscopy (AFM)[15] and nuclear magnetic resonance (NMR) [16,17], with XRD andTEM being the most widely used. While XRD is effective in mostcases, it is often incapable of distinguishing between what shouldbe classified as immiscible and what is delaminated [7,18]. On theother hand, TEM, with its visual representation of clay dispersion,provides a much more direct and reliable method for determiningthe degree of dispersion than XRD. However, in principle, thesetechniques allow only qualitative evaluation of the degree of clay

Q. Gou et al. / Polymer 55 (2014) 4216e4225 4217

dispersion. To date, the few techniques available for quantitativeanalysis of the degree of layered silicates dispersion are based onTEM image analysis and fall into the following two categories:

1). Distance measurement-based methods. Eckel et al. [19] quan-tified the dispersion of clay particles with linear interceptdistances, determined by placing an array of parallel linesacross TEMmicrographs and then dividing the total length ofthe lines by the number of times the lines intersect the clayparticles. A smaller line intercept distance, indicating a largernumber of particles along the lines, indicates better disper-sion. However, this method is based on average distances, anaggregate metric not truly representative of the actual dis-tribution of the spacing between particles (since it is possiblefor differently distributed spacings to end up with identicalaverages); the metric is also highly dependent on clayloading and clay orientation. Luo and Koo [20] developed aquantification method based on relating the degree ofdispersion to the “free-path distance”, the distance betweensingle platelets along parallel lines superimposed on TEMimages. A histogram of the free-path distance was then ob-tained from TEM image analysis and fitted to a log-normalmodel. A dispersion parameter, D0.1, defined as the proba-bility of the free-path distance falling in the range of m ± 0.1m(where m is the mean value of the free-path distance), wasthen calculated from the log-normal model. They noted thatfor exfoliated composites, D0.1 > 8%, while 4% < D0.1 < 8% forintercalated composites. This method is effective for exfoli-ated and intercalated microstructures with small size tac-toids and provides critical values for systems with differentdispersion levels. However, for a microstructure containinglarger agglomerations, the method is ineffective because ofthe inherent difficulty in determining distances within thelarge agglomerations accurately. To rectify this problem sothat agglomerations can be adequately taken into consider-ation, Tyson et al. [21] modified the Luo and Koo's method byintroducing an agglomeration parameter A in addition toD0.1.The parameter A0.3, representing the total number of parti-cles in the range of m ± 0.3m, was then calculated from thelog-normal model fitted to the histogram of particle size data(including agglomerations). By its definition, a lower value ofA0.3 indicates lower agglomeration. The dispersion param-eter D0.1 and agglomeration parameter A0.3 are consideredsimultaneously in determining dispersion. This method wasshown to be effective when applied to carbon nanotubes(CNTs) within aqueous solutions; but it does not work wellwith agglomerations of uniform dimensions. Also, as notedin Bray et al. [22], there are no formal statistical assessmentof the validity of the probability model assumptions onwhich these methods are based.

2). Particle analysis-based methods. Dennis et al. [23] and Forneset al. [12] proposed clay particle density, the number ofplatelets or intercalates per unit area, as a quantitativemeansof comparing the degree of dispersion in different samples.According to the proposition, a higher clay density shouldindicate better dispersion. However, this metric is clearlydependent on clay loading and therefore cannot be useduniversally. An alternative method proposed by Nam et al.[24] involves measuring the clay particle length Lclay andthickness dclay of dispersed clay particles, and the correlationlength xclay between dispersed layers (the space distancesurrounded by the dispersed clay particles), and using themto estimate the average number of individual layers in a claystack. However, this method is also based on averages, whichare non-unique representations of full distributions of the

Lclay, dclay, and xclay measurements. Recent studies[8e11,25,26] characterize the dispersion of clay particleswith an additional parameter: the aspect ratio of the claystacks. For example, Paul and coworkers in Lee et al. [8], Shahet al. [9], Kim et al. [10], and Spencer et al. [11] used mea-surements of number-/weight-average particle lengthðln; lwÞ, number-/weight-average particle thickness ðtn; twÞ,and number-/weight-average aspect ratio ð⟨l=t⟩w; ⟨l=t⟩nÞ tocharacterize the dispersion levels of clay particles,concluding that higher aspect ratios indicate lower numberof platelets in a stack, and hence correspond to betterdispersion. The complete distributions of these quantities,while available, were nevertheless only used to providequalitative descriptions of clay dispersion. Carastan et al. [25]and Vermogen et al. [26] categorized clay particle formationsin nanocomposites into several discrete groups (such as“individual sheets”, “thinner tactoids”, “thicker tactoids”,and “agglomerates”) according to the number of platelets ineach particle. Each categorical group was then characterizedby such metrics as mean thickness, mean length, and aspectratio. These groups are subsequently used as the basis forcharacterizing the dispersion state of a polymer nano-composite sample in terms of the relative frequency ofoccurrence of each group in the sample in question. Forexample, the predominance of micron size agglomerates in aPNC sample indicates poor dispersion, as opposed to thepredominance of “individual sheets”. The methodology isthus based on a bar chart representation of the compositionof the particles in each sample, indicating the relativeamount of the “basis groups” represented in the sample.Such a representation is thus only visual and does not pro-vide an absolute, single quantitative characterization of theactual extent of dispersion in each sample.

Our objective in this paper is to propose an alternative quanti-fication method for evaluating layered silicate dispersion based ona probabilistic analysis of particle length distributions obtainedfrom TEM images. The approach is based on a formally validatedprobability model of particle length distribution and uses themodel parameters to quantify clay dispersion directly. The rest ofthe paper is organized as follows: first, we present fundamentalarguments to justify the appropriateness of a proposed probabilitymodel for characterizing the distribution of clay particle lengths inpolymer matrices. The resulting particle length distribution (PLD)model is then validated with experimental data before we discusshow to quantify dispersion with the PLD characteristic parameters.Finally, the proposed technique is demonstrated experimentally byusing it to quantify the dispersion of polymer nanocomposites withdistinct dispersion characteristics that were produced in ourlaboratory.

2. Quantification method development

2.1. Characterizing particle length distributions

In polymer nanocomposites, the particle length, defined as thelength of a single dispersed platelet or an agglomerate of platelets,as depicted in Fig. 1 [12], is not uniform for several reasons. First,the length of each platelet in a particle is non-uniform. Secondly, aparticle consists of one or more platelets, so that the number ofplatelets in each particle also varies randomly from one particle toanother (Fig. 1 (a)). Finally, how the platelets are stacked to form aparticle is also non-uniform, so that two particles with the samenumber of platelets may still have different effective lengths (Fig. 1(b)). Consequently, the lengths of layered silicate particles in a

Fig. 1. Definition of (a) single particles and platelets; (b) effective particle length inpolymer nanocomposites [12].

Q. Gou et al. / Polymer 55 (2014) 4216e42254218

polymer matrix are randomly distributed over a range of valueswhich in practice can be anywhere from several nanometers to afew hundred microns.

In order to describe the variability inherent in layered silicateparticle lengths adequately, we observe first that the number ofplatelets in a particle has Poisson-like characteristics (see Ref. [27])in the sense that it is an integer, typically with relatively low values.Thus, if z is the number of platelets in a particle, then the formationof each particle results from the occurrence of z Poisson events; andif the Poisson events are occurring at a mean “intensity” of 1/b perunit length, then x, the interval length over which a Poisson eventshave occurred is known [27] to follow a gamma distribution withthe following probability distribution function (pdf):

f ðxÞ ¼ 1baGðaÞx

a�1e�x=b (1)

Thus, we postulate that for truly randomly dispersed clayplatelets in a polymer matrix, the distribution of the particle lengthwill be described adequately by the indicated gamma g(a,b) pdf,where a represents themean number of platelets per particle. Now,m, the mean of the gamma distribution (in this case, the meanparticle length) by definition is given by:

m ¼ ab (2)

Consequently, the parameter b in this particular application,which, mathematically is obtained as:

b ¼ m=a (3)

implies that for this specific application of the gamma distribution,

b ¼ mean particle lengthmean number of platelets per particle

(4)

We may now observe that the parameter b physically corre-sponds to a mean effective platelet length. (See Fig. 1)

We propose that these gamma pdf parameters may be used toquantify layered silicate dispersion as follows. Because the disper-sion of silicate layers is related to the number of platelets stacked

together, technically, fewer platelets should imply better disper-sion. Consequently, smaller values of a will indicate better disper-sion. But this is not sufficient. The second parameter, b, bydefinition of the gamma distribution, is a scale parameter, indica-tive of the width of the PLD. Larger values of b are associated withbroader distributions, generally indicating poorer overall disper-sion. But since b represents the mean effective particle length, itsvalue can be large (within reason) either because the mean particlelength is large (not desired) or because the mean number ofplatelets per particle, a, is small (desired), or both. The implicationtherefore is that a relatively large a value by itself is not necessarilybad, so long as it is accompanied by a commensurably small valueof b; similarly, a large value of b does not automatically indicatepoor dispersion, so long as the accompanying a value is commen-surately small. Thus, neither a nor b in isolation is sufficient by itselfto ascertain the quality of dispersion; a combination of both isrequired. As we show later with actual data, quality of dispersionmay be quantified by the empirical expression:

aþ C1b ¼ C2 (5)

But first wewill validate the proposed gamma probability modelfor characterizing PLD with experimental data.

2.2. PLD model validation

To validate the postulated gamma probability model, we employit to describe PLD data, obtained from the open literature, and thencarry out formal statistical goodness-of-fit tests. The studies fromwhich these data sets were obtained involve polymer/clay nano-composites samples made by melt intercalation and PLD data ob-tained from TEMmicrographs of the samples. The lengths of severalhundred particles were determined from appropriate image ana-lyses of the TEM micrographs, from which the particle length dis-tribution data were generated. A representative example in Fig. 2(a) shows a histogram of raw particle length data (ranging from10 nm to 300 nm), for polyamide 6/organoclay nanocomposites[28]. The theoretical gamma probability model fit to the normalizedhistogram (gray bars) is shown in Fig. 2 (b) (blue line) along theleast squares estimates of the parameters a and b and the corre-sponding 95% confidence intervals.

While a visual inspection of Fig. 2 (b) indicates reasonableagreement between model and data, rigorous model validationrequires formal statistical tests of “goodness-of-fit” of which thetwo most widely employed are the Chi-square test and the Kol-mogoroveSmirnov test [27].

The Chi-square test is a formal hypothesis test based on the teststatistic C2 defined as:

C2 ¼Xm

i¼1

�f 0i � 4i

�2

4i(6)

where f 0i is the observed frequency from data, fi is the corre-sponding frequency from the candidate probability model underconsideration, and m is the number of bins in the histogram. Thenull hypothesis (that the data follow the postulated probabilitymodel), when tested at the traditional a¼ 0.05 significance level, isrejected if

C2 >c20:05ðyÞ (7)

where c20:05ðyÞ is the y-degrees-of-freedom Chi-square variate forwhich the right tail area probability is 0.05 (here y ¼m e p e 1, and

Fig. 2. (a) Particle length histogram for polyamide 6/clay nanocomposites [28]; (b) Particle length distribution (histogram) and the corresponding theoretical gamma distribution fit(Indicated estimates of parameters a and b along with corresponding 95% confidence intervals obtained via least squares).

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

noitcnuFnoitubirtsi

Devitalu

Q. Gou et al. / Polymer 55 (2014) 4216e4225 4219

p is the number of parameters in the candidate probability model)(see Chapter 17 of Ref. [27]).

The KolmogoroveSmirnov (KeS) test, on the other hand, is anon-parametric test based on a comparison of the empirical cu-mulative distribution function FE(x) (obtained from the data inquestion) to the theoretical cumulative distribution function F(x) ofthe postulated probability model. The null hypothesis (that the datafollow the postulated probability model) is rejected at the 0.05significance level when the value of the test statistic:

Dn ¼ supx2R

jFEðxÞ � FðxÞj (8)

exceeds the critical value D0.05, a value typically determined fromstatistical analysis software packages (see Chapter 18 of Ref. [27]).

For the illustrative example in Fig. 2 (b), the result of Chi-squaretest is shown in Table 1, where we observe that C2 <c20:05ð26Þ. Weconclude therefore that the gamma probability model providesadequate representation of the particle length distribution for theexample data.

Fig. 3 shows a comparison of the empirical cumulative distri-bution (from data) and the theoretical gamma model cumulativedistribution. The results of a formal KeS test returned a value ofDn ¼ 0.0371 which when compared with the critical valueD0.05 ¼ 0.0582 leads us to conclude, once again, that the gammaprobability model provides an adequate fit to the experimentaldata. In the rest of the paper, we will employ only the KeS test tovalidate our probability models.

The gamma pdf model g(a,b) with a ¼ 3.6 and b ¼ 25.65 hasthus been formally validated as providing a good representation ofthe particle length distribution in polyamide 6/clay nano-composites discussed in Ref. [28]. For now, we observe that thevalue a ¼ 3.6 indicates that the material in question has an average

Table 1Results of the Chi-square goodness-of-fit test of the gamma probability model fit tothe polyamide 6/clay nanocomposite PLD data.

Polymer nanocomposites Chi-square test

C2 c20:05ð26Þ Result

Polyamide 6/clay nanocomposites 35.29 38.90 Adequate

of 3.6 platelets stacked together per particle, indicating relativelygood layered silicates dispersion. The value b ¼ 25.65 (represen-tative of a relatively broad distribution of particle lengths as seenfrom the histogram), even though indicative of a fairly large meaneffective platelet length, taken in combination with the relativelysmall a value appears to suggest good clay dispersion. As we nowshow, it is possible to combine these parameters mathematically togenerate a quantitative and objective measure of dispersion.

2.3. Dispersion quantification

Dispersion states, indicative of how particles are distributedwithin the polymer matrix, are generally (and qualitatively) clas-sified as exfoliated, intercalated or immiscible. Having just estab-lished that such particle distributions (specifically of the particlelengths) are well represented quantitatively by the gamma pdf,g(a,b), with the parameters a and b fundamentally and jointlyindicative of the extent of dispersion but in a yet unspecifiedmanner, what remains now is for us to relate these parameters

0 50 100 150 200 250 3000

0.1

0.2

Particle Length (nm)

muC

Empirical CDFGamma CDF

Fig. 3. Comparison of theoretical and empirical cumulative distributions as part of theKolmogoroveSmirnov (KeS) goodness-of-fit test for the polyamide 6/clay nano-composite PLD data of [28].

Table 2Gamma probabilitymodel parameter estimates (and corresponding 95% confidence intervals) for various polymer nanocomposite PLD data, and corresponding goodness-of-fittest results (“A” indicates that the model is adequate, while “IN” indicates that the model's adequacy is indeterminate).

Polymer nanocomposites (NC) a with 95% confidence interval b with 95% confidence interval KeS test result No. of particles

PP/M2(HT)2 NC (1wt%) [8] 4.21 ± 1.71 72.71 ± 33.28 A 120PP/M2(HT)2 NC (2.8wt%) [8] 4.60 ± 0.82 48.23 ± 9.49 A 226PP/M2(HT)2 NC (6.8wt%) [8] 5.64 ± 1.07 42.26 ± 8.72 A 311EMAA-1/M2(HT)2 NC [9] 5.20 ± 1.56 24.59 ± 8.12 A 321PA-6/(HE)2M1R1-WY NC [29] 3.60 ± 0.69 25.65 ± 5.61 A 551PA-6/(HE)2M1R1-YM NC [29] 3.15 ± 0.52 32.40 ± 6.34 A 436PA-6/M2(C18)1 NC (240 �C) [31] 4.45 ± 0.45 10.87 ± 1.22 IN 2121PA-6/M2(C18)1 NC (270 �C) [31] 3.98 ± 0.53 11.90 ± 1.78 IN 2275PA-66/M2(C18)1 NC (270 �C) [31] 4.82 ± 0.55 11.11 ± 1.46 IN 1414Li ionomer/M2(HT)2 NC [30] 3.65 ± 0.38 47.02 ± 5.50 A 502Na ionomer/M2(HT)2 NC [30] 3.91 ± 0.49 24.16 ± 3.45 A 762Zn ionomer/M2(HT)2 NC [30] 3.58 ± 0.33 26.84 ± 2.84 A 703

Q. Gou et al. / Polymer 55 (2014) 4216e42254220

explicitly and quantitatively to dispersion states. For this, we adoptthe following empirical approach where (i) PLD data from a widevariety of polymer nanocomposites having different dispersioncharacteristics will be fit to the gamma pdf model, thereby (ii)generating a collection of a and b parameter values along with thedispersion states corresponding to each pair, from which finally,(iii) wewill then determine a quantitative expression relating a andb pairs to dispersion explicitly.

The specific PLD data sets chosen for this procedure all comefrom Paul and co-workers [8,9,29e31], i.e., from the same researchgroup, and ostensibly generated by the same experimental proto-col, thereby eliminating any potential and extraneous lab-to-labvariability. The relevant details about the various nanocompositematerials used in generating the data sets are as follows. PolymerMatrices: polyamide 6 (PA-6) [29,31], polyamide 66 (PA-66) [31],ethylene/methacrylic acid copolymer (EMMA) [9], sodium, zinc andlithium ionomers of EMMA (ionomer) [30], polypropylene (PP) [8];Organoclays: sodium montmorillonites (Na-MMT), which were ionexchanged with different quaternary ammonium surfactants; Sur-factants: (HE)2M1R1-YM and (HE)2M1R1-WY [29], M2(C18)1 [31],and M2(HT)2 [8,9,30] (where YM and WY stand for Na-MMT sour-ces located in Yamagata, Japan and Wyoming, USA, respectively,and some frequently used abbreviations are employed here todescribe the ammonium structures, i.e., HE for hydroxy-ethyl, R forrapeseed, M for methyl, H for hydrogen, and HT for hydrogenated-tallow oil). As discussed in the references, PA-6 and PA-66 basednanocomposites show fully exfoliated dispersion [29,31], and thenanocomposites prepared from EMMA, and sodium, zinc andlithium ionomers have thicker clay stacks compared to the onesprepared from PA-6 and PA-66 [9,30]; PP/organoclay nano-composites have a mixed morphological structure, a combinationof exfoliated particles and intercalated stacks [8]. These data setstherefore collectively provide representative samples from all thedispersion states of interest.

Table 3Validated gamma probability model parameter estimates for various polymer nanocomorder.

“�a-priori”dispersion rank

Polymer nanocomposites(NC) a with 95% confidenceinterval

1 PA-6/(HE)2M1R1-YM NC [29] 3.60 ± 0.692 PA-6/(HE)2M1R1-WY NC [29] 3.15 ± 0.523 Zn ionomer/M2(HT)2 NC [30] 3.58 ± 0.334 Na ionomer/M2(HT)2 NC [30] 3.91 ± 0.495 EMAA-1/M2(HT)2 NC [30] 4.19 ± 1.406 Li ionomer/M2(HT)2 NC [30] 3.65 ± 0.377 PP/M2(HT)2 NC (1wt%) [8] 4.21 ± 1.718 PP/M2(HT)2 NC (2.8wt%) [8] 4.60 ± 0.829 PP/M2(HT)2 NC (6.8wt%) [8] 5.64 ± 1.07

Upon subsequently characterizing the PLD data sets with thegamma pdf, the resulting least squares estimates of the character-istic parameters, a and b alongwith their respective 95% confidenceintervals, are shown in Table 2. Each resultant gamma probabilitymodel is validated using the Kolmogorov-Smirnoff (KeS) test; theresult is shown in Table 2, where “A” indicates that the gamma pdfmodel is sufficiently adequate to describe the experimental data(i.e., the associated p-value is greater than 0.05), while “IN” in-dicates that the model's adequacy is indeterminable (because0.01 < p < 0.05; see Ref. [27]). Observe that the three cases forwhich the gamma model adequacy is questionable are all fromRef. [31]. The reason for this anomaly is not clear, but it may beattributable to the nature of the data sets, which in each case in-volves a large number of total particles but which were availableonly in the form of normalized particle length histograms, not rawparticle length data. For consistency and to err on the side ofcaution, we excluded these data sets from subsequent analyses.

From well-established characteristics presented in the litera-ture, the dispersion state was determined for each nanocompositeallowing us to assign an �a-priori categorical rank to each material.For example, PA-6 based nanocomposites are often “fully exfoli-ated” (excellent dispersion); EMMA and ionomer based nano-composites are “less exfoliated” compared to PA-6 basednanocomposites; and PP based nanocomposites are “intercalated”(poor dispersion). Such an “�a-priori” dispersion rank is shown inTable 3, in descending order from 1 to 9, where 1 indicates the bestdispersion and 9 the worst. We observe that overall the degree ofdispersion worsens as a and b values increase. However, as notedearlier, the value of a or of b by itself in isolation is insufficient as anindicator of the degree of dispersion; instead, the dispersion statedepends on both a and b simultaneously.

A scatter plot of a versus b for each nanocomposite sample,shown in Fig. 4, reveals a few important features regarding howthese two parameters jointly encode dispersion characteristics.

posites, along with associated �a-priori organoclay dispersion ranking in descending

b with 95% confidenceinterval

d with 95% confidenceinterval

No. ofparticles

Clay wt%

25.65 ± 5.61 4.88 ± 0.74 436 3.232.40 ± 6.34 4.77 ± 0.61 551 2.626.84 ± 2.84 4.92 ± 0.36 703 524.16 ± 3.45 5.12 ± 0.52 762 531.47 ± 11.77 5.76 ± 1.52 321 2.547.02 ± 5.50 6.00 ± 0.46 502 572.71 ± 33.28 7.85 ± 2.39 120 148.23 ± 9.49 7.01 ± 0.95 226 2.842.26 ± 8.72 7.75 ± 1.16 311 6.8

20 30 40 50 60 70 803

3.5

4

4.5

5

5.5

6

β

α

δ

δ

= 5

= 6.5

Fig. 4. Scatter plot of a versus b for various polymer nanocomposites: for PA-6/(HE)2M1R1-YM NC; for PA-6/(HE)2M1R1-WY NC; for Zn ionomer/M2(HT)2 NC; :for Na ionomer/M2(HT)2 NC;< for EMAA-1/M2(HT)2 NC; ◄ for Li ionomer/M2(HT)2 NC;for PP/M2(HT)2 NC (1 wt%); six-pointed red star for PP/M2(HT)2 NC (2.8 wt%), for

PP/M2(HT)2 NC (6.8 wt%). The indicated lines are “dispersion contours” represented bya þ 0.05b ¼ d.

Q. Gou et al. / Polymer 55 (2014) 4216e4225 4221

First, the points represented by the blue symbols, which corre-spond to nanocomposites with exfoliated dispersion characteris-tics, lie in the bottom left hand corner of the aeb plane; the pointsrepresented by the black symbols, corresponding to less exfoliatednanocomposites, lie in the middle of the plane; while the remain-ing points represented by the red symbols, corresponding tocomposites consisting of a combination of tactoids and exfoliatedplatelets (indicative of poor dispersion), are located in the upper-most portion of the plane. Next, we observe that the three broadlyidentified regions in the aeb plane may be separated, to a firstapproximation, by the indicated parallel straight lines, representedmathematically as:

aþ 0:05b ¼ d (9)

A careful observation of the values computed for d, as shown inTable 3, shows that the dispersion for the PP/M2(HT)2 NC (1 wt%)sample appears worse than that for the other PP NC samples, con-trary to what one would expect from the “�a-priori” dispersion rankbased strictly on the claywt%. However, the apparent discrepancy isdue to the uncertainty associated with the data used to characterize

Fig. 5. TEM photomicrograph of (a) Na ionomer/M2(HT)2 nanocomposite with nominal 5 wt%

the sample with 1 wt% clay. First, note the 95% confidence intervalassociated with the estimate of b for this sample. It is about 4 timeswider than the corresponding confidence intervals for the estimatesof b for other PP/M2(HT)2 NC samples. This relatively high degree ofuncertainty is primarily due to the intrinsic nature of the originaldata set itself, in combinationwith the fact that, as shown in column6 of Table 3, the total number of particles used to carry out theanalysis (120) is much smaller than what was used for the others.Consequently, the corresponding confidence interval for thecomputed value of d is much wider. The end result is that the truevalue of d for the PP/M2(HT)2 NC (1wt%) sample is computed to lie inthe range 5.46 < d < 10.24, while for PP/M2(HT)2 NC (2.8 wt%), it ismuch narrower, 6.06 < d < 7.96. Statistically speaking, therefore, atthe 0.05 significance level, there is no “statistically significant dif-ference” between the dispersion characteristics of the PP/M2(HT)2NC 1 wt% and 2.8 wt% samples, since the two 95% confidence in-tervals for d overlap. The substantial uncertainty associatedwith the1 wt% sample data, which is responsible for the overly wide 95%confidence interval, may be mitigated by collecting more data. It isimportant to keep in mind, however, that the data set in questionwas obtained from the open literature. As such it is not within ourpower to collect more data on this sample.

Still, the results for the PP/M2(HT)2 NC samples may be inter-preted as follows: (i) The fact that there is no “statistically signifi-cant difference” between the dispersion computed for the twosamples may mean that the 1.8 wt% difference is not sufficient toaffect the dispersion significantly. (ii) Alternatively, wemay observethat the low end of the range of d for the PP/M2(HT)2 NC (1 wt%),which is 5.64, suggests that it is quite possible that the dispersion isbetter than that for the PP/M2(HT)2 NC (2.8 wt%) sample, whoselowest possible value is 6.06. Thus, without additional data to helpreduce the uncertainty, we conclude that the dispersion for the PP/M2(HT)2 NC (1 wt%) is at worst indistinguishable from that for thePP/M2(HT)2 NC (2.8 wt%) sample, and at best, possibly better. (iii)Finally, the “�a-priori” dispersion rank is just that: a subjectiveranking prior to computing the objective dispersion coefficient.Even though subject to significant uncertainty because of theoriginal data set, the information provided by the result is stillimportant in obtaining the empirical relationship. As this is a singledata point out of 9, the uncertainty should not adversely affect theempirical relationship any more than a single data point subject tofairly high measurement noise in a standard linear regression willadversely affect the resulting regression model.

The entire collection of parallel straight lines represented by Eq(9) may now serve as dispersion contours in the aeb plane, with

MMT; (b) Na ionomer/organoclay mixture nanocomposites with nominal 5 wt% MMT.

Fig. 6. Schematic representation of TEM micrograph image processing procedure.

Table 4Summary Statistics of particle length data for Sample 1 (Na ionomer/M2(HT)2nanocomposites) and Sample 2 (Na ionomer/organoclay mixture nanocomposites).

Samplenumber

Materials m (nm) Minimumx (nm)

Maximumx (nm)

No. ofparticles

1 Na ionomer/M2(HT)2 NC 114.25 30.26 350.22 3042 Na ionomer/organoclay

mixture NC138.70 48.60 426.43 332

Q. Gou et al. / Polymer 55 (2014) 4216e42254222

d as a quantitative measure of the dispersion state for each poly-mer/layered silicate nanocomposite in question, whose PLD datahas been characterized by gamma model parameters a and b. AsFig. 4 shows, lower d values indicate better dispersion while higherd values are associated with poorer clay dispersion. Finally we notein particular in Fig. 4, that the dispersion for the specific samplesused to generate this figure, d ¼ 5 indicates the boundary betweenexfoliated dispersion and less exfoliated dispersion, while d ¼ 6.5indicates the boundary between less exfoliated dispersion and thepoorer dispersion with combination of exfoliated particles andintercalated stacks. Consequently, if the data in Fig. 4 is represen-tative of general dispersion characteristics of polymer/layered sili-cate nanocomposites, then the implication is as follows: Fordispersion characteristics quantified according to the empiricalexpression in Eq (9), d < 5 for exfoliated composites, 5 < d < 6.5 forless exfoliated composites with stacks containing several layers,and d > 6.5 for intercalated composites. The next step is to confirmthis result experimentally with new materials that are not part ofthe data employed to develop the methodology and the resultsshown in Fig. 4.

3. Experimental confirmation

The methodology proposed above is validated experimentallyby using it to characterize and quantify the dispersion of twopolymer nanocomposite samples made in our lab as describedbelow.

3.1. Materials

The polymer matrix used in this study was a sodium ionomer ofpoly(ethylene-co-methacrylic acid), Surlyn® 8945, which, for brev-ity, will be referred to as “Na ionomer”. The organically modifiedclays, Cloisite® 20A and Cloisite® 93A, were used as reinforcement.Both organoclays were prepared by a cation exchange reaction be-tweenNa-MMTand ammonium surfactants. The surfactant used forpreparing Cloisite® 20A was a quaternary ammonium surfactantwith two long-alkyl tails (di-methyl bis(hydrogenated) tallow), i.e.,M2(HT)2 [32], while Cloisite® 93A was prepared using a ternaryammonium surfactant with two long-alkyl tails (methyl di(hydro-genated) tallow), i.e., M1H1(HT)2. One composite (Sample 1) wasreinforcedwithM2(HT)2, while the other (Sample 2) was reinforcedwith a clay mixture consisting of 80 wt% M2(HT)2 and 20 wt%M1H1(HT)2. A recent study [33] shows that improved claydispersionin Na ionomer matrix could be achieved through the use of surfac-tants with M2(HT)2 instead of M1H1(HT)2; therefore, Sample 1 isexpected to show better dispersion than Sample 2. The proposedmethod should be able to confirm this expected observation andalso quantify the degree of dispersion for each sample.

3.2. Nanocomposite preparation

A masterbatch method was used to synthesize nanocompositesby diluting clay rich masterbatches (25 wt% organoclay) intonanocomposites with 5wt% clay in a CoperionW&P ZSK-18mm co-rotating, intermeshing twin screw extruder (L/D¼ 41) operated at abarrel temperature of 200e210 �C, a screw speed of 700 rpm, and atotal feed rate of 15 lb/h. Tensile specimens were obtained via aDSM micro compounder and injection molding system.

3.3. TEM characterization

Samples for TEM analysis were cut from the central part of thetensile bars perpendicular to the flow direction using an ultrami-crotome with a diamond knife. The samples were then examined

by FEI TECNAI G2 12 TWIN TEM under an accelerating voltage of120 kV. The resulting TEM micrographs were obtained at magni-fications of 30e60 K, depending on the extent of exfoliation in thesample. Representative TEMmicrographs are shown in Fig. 5 (a) forSample 1 and Fig. 5 (b) for Sample 2. In both cases the stacks of clayparticles are broken up nicely and the clay particles are well exfo-liated overall, even though fewer and smaller aggregates areobserved in Sample 1 compared to Sample 2. A visual inspection ofthese TEM images therefore clearly suggests that Sample 1 showsbetter dispersion than Sample 2. We now employ the methodproposed in Section 2 to ascertain quantitatively the dispersioncharacteristics of each sample and the extent to which the disper-sion in one is better than the dispersion in the other.

Fig. 7. Particle length histogramwith gamma density model fit superimposed in blue along with least squares estimate of the parameters a and b and corresponding 95% confidenceintervals for (a) Na ionomer/M2(HT)2 nanocomposite; (b) Na ionomer/organoclay mixture nanocomposites. (For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.)

Q. Gou et al. / Polymer 55 (2014) 4216e4225 4223

4. Results and discussion

4.1. Image processing

Because TEM images typically consist of different shades of gray,it is very difficult for standard image analysis tools to detect anddistinguish the dispersed particles from the polymer matrix effec-tively, making it much more difficult to determine particle lengthsand PLDs [28]. This necessitates image pre-processing in order toconvert the low contrast TEM images to black and white imagesthat are more conducive to image analysis. For this specific study,the image processing procedure we employed, shown schemati-cally in Fig. 6, is as follows.

The original TEM micrographs, saved in .jpg or .tiff format, werefirst opened in GIMP (GNU Image Manipulation Program), wheretransparent layerswere superimposedon theTEM images. Then, thelengths of the layered silicate particles were tracedmanually on thetransparent layers, which were subsequently exported as black andwhite imageswhere the particles stand out in sharp contrast against

Fig. 8. Comparison of empirical cdf and theoretical gamma cdf as part of the KeS goodness-onanocomposites.

the white background. To avoid potential image analysis errorscaused by variable line thicknesses, the black lines were “skele-tonized” (rendering thick lines thin) with the software “Image J” toproduce images of particles rendered as uniformly thin lines.

4.2. Image analysis

The processed images were then imported into MATLAB® forimage analysis from where we obtained raw particle length datausingMATLAB's image analysis function regionprops. Particle lengthhistograms were subsequently generated from data sets obtainedfrom five to six TEM images in each case, for a total of at least 300particles. The summary statistical characteristics for both data setsare shown in Table 4.

4.3. Dispersion characterization

We quantified organoclay dispersion for these samples byapplying the proposed method to the resulting particle length

f-fit test: (a) Na ionomer/M2(HT)2 nanocomposites; (b) Na ionomer/organoclay mixture

Q. Gou et al. / Polymer 55 (2014) 4216e42254224

histograms shown in Fig. 7 (a) and (b). First, a least squares fit of agamma pdf to each PLD data produced the results shown in Fig. 7(a) and (b); the blue lines represent theoretical probability modelfit to the PLD data (gray bars), with estimated parameters alongwith 95% confidence intervals superimposed. The results of the KeStest employed to assess each probability model's goodness-of-fit eusingMATLAB@ hypothesis test function ksteste are shown in Fig. 8(a) and (b) for Sample 1 and Sample 2, respectively. (In each case,the formal statistical test, returned a value of h ¼ 0, indicating thatthere is no evidence to reject the null hypothesis, that the datafollow the gamma probability model, at the 0.05 significance level).We conclude therefore that the gamma probability model indeedprovides adequate theoretical representation of the particle lengthdistribution for each sample.

From the gamma distribution parameters and Eq (9), we nowobtain the dispersion characteristic parameters as shown inTable 5: for the Sample 1, d1 ¼ 5.85 ± 0.69 (between 5 and 6.5),indicating a mixed morphological structure, i.e., a combination ofexfoliated platelets and intercalated stacks, as expected; for Sample2, d2 ¼ 6.96 ± 0.82 (larger than 6.5) indicating that there are moreintercalated stacks in the polymer matrix, again, as expected. Thedispersion index d shows that the dispersion of Sample 1 is about20% better than the dispersion of Sample 2, which is consistentwith the findings in the reference [33]. We have thus established,with this set of new, independent experimental data obtained inour laboratory, that the proposed method is able to provideappropriate quantification of the extent of organoclay dispersion inpolymer nanocomposites.

5. Summary and conclusions

Effective quality control of polymer/organoclay nanocompositesis virtually impossible without a simple and appropriate methodfor quantitative dispersion characterization that is capable ofrelating clay dispersion directly to product performance. In thispaper, we have proposed a new method for quantifying layeredsilicates dispersion in polymer/organoclay nanocomposites. Themethod is based on a gamma probability model g(a,b) descriptionof particle length distribution data obtained from TEM micro-graphs, a postulate based on the fundamental mechanisms asso-ciated with clay particle formation in the polymer matrix. As aconsequence of linking the postulated probability model tofundamental mechanisms, we have been able to establish aconnection between the probability model parameters and phys-ical characteristics of the polymer nanocomposites: specifically, arepresents the mean number of platelets per particle, and b, themean “effective platelet length”. An empirical analysis of multipledata sets from a wide variety of polymer nanocomposite materialsshowed that these characteristic gamma distribution parametersare related to the degree of dispersion according to the expression:a þ 0.05b ¼ d, with d as a dispersion index able to distinguishamong different dispersion states. Specifically, from the data setsused in the analysis, we observed that: d < 5 for exfoliated com-posites, 5 < d < 6.5 for less exfoliated composites with stackscontaining several layers, and d > 6.5 for intercalated composites.

Table 5Dispersion characteristics of Sample 1 (Na ionomer/M2(HT)2 nanocomposites) andSample 2 (Na ionomer/organoclay mixture nanocomposites).

Samplenumber

Materials a b d

1 Na ionomer/M2(HT)2 NC 4.65 ± 0.66 24.56 ± 3.68 5.85 ± 0.692 Na ionomer/organoclay

mixture NC5.76 ± 0.80 23.97 ± 3.48 6.96 ± 0.82

The proposed method has been confirmed with independentexperiments involving two nanocomposites with different mor-phologies synthesized in our lab.

The methodology has the unique advantage that the proposeddispersion index arises as a combination of parameters derivedfrom mechanisms of clay particle formation, and a probabilitymodel that has been rigorously validated as appropriate for rep-resenting particle length distributions. All the parameters aretherefore related directly to physical characteristics of clay parti-cles, and hence provide theoretical insight into the structures ofthese particles in the polymer matrix. Furthermore, we believe thatthis method facilitates quantitative comparison of clay dispersionfor polymer nanocomposites prepared with different polymers,reinforced with different organoclays, or prepared under differentprocessing conditions. Applying this methodology to implementeffective property control of polymer nanocomposites requires (i)quantitative connection between processing conditions and orga-noclay dispersion index, d, (ii) the correlation of d with the mate-rial's property and performance in end-use, and (iii) the design andimplementation of an effective model-based control scheme basedon the resulting models. This is the subject of continuing work inour laboratory.

Acknowledgment

The authors would like to acknowledge Yingchao Chen of theUniversity of Delaware for technical assistance in acquiring TEMimages; Delmer Ray Pettitt and Robert Allan Lewis of DuPont forassistance with the PNC synthesis experiments; and ChristopherDavid Chan of DuPont for preparation of the TEM samples. Thiswork was supported by the National Science Foundation (NSF)through Grant No. CBET-1034213.

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