Research Article Finite Element Modeling of Multiscale ...downloads.hindawi.com/journals/jnm/2015/482698.pdf · Finite Element Modeling of Multiscale Diffusion in Intercalated Nanocomposites

Research ArticleFinite Element Modeling of Multiscale Diffusion inIntercalated Nanocomposites

Antonio Greco and Alfonso Maffezzoli

Department of Engineering for Innovation University of Salento Via per Monteroni 73100 Lecce Italy

Correspondence should be addressed to Antonio Greco antoniogrecounisalentoit

Received 23 March 2015 Accepted 21 June 2015

Academic Editor J De Sousa

Copyright copy 2015 A Greco and A Maffezzoli This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This work is aimed to study the diffusion in 3D nanocomposites obtained with stacks of lamellar nanofillers characterized by thepresence of permeable galleries by finite element (FE) analysis To this purpose a geometricmodel based on a randomdistributionof noninterpenetrating stacks with each one beingmade of regularly spaced lamellae was developedThedevelopedmodel is able toaccount for diffusion between stacks (interstack diffusion) as well as diffusion inside stacks (intrastack diffusion) Simulation resultsshowed that intrastack diffusion related to flow inside galleries can be quite relevant particularly at high values of gallery thicknessComparison of the simulation results with literature models shows that when intrastack diffusion is not taken into account thediffusion behavior in intercalated nanocomposites is not well predicted Therefore intrastack permeability of nanofillers such asorganic modified clays cannot be neglected Such intrastack diffusivity is shown to depend on the morphological features of thenanofiller requiring the development of a proper mathematical model

1 Introduction

The substantial decrease of permeability to mass transportbrought about by the addition of nanofiller is a major advan-tage of polymermatrix nanocompositesThe improvement ofthe barrier properties is mainly attributed to the increase ofthe tortuous path that diffusing molecules must travel insidethe diffusing medium [1 2] On the other hand it is generallyaccepted that such increase depends on different parametersamong which are volume fraction orientation and aspectratio of the nanofiller [3 4]

Themass diffusion process in polymer nanocomposites isvery often studied by finite elements simulation andormath-ematical modeling accounting for the effects of different fac-tors some of which are very difficult to quantify andmeasure(ie orientation of nanofiller) Due to the difficulties in char-acterizing the structure of polymer nanocomposites by wideangle X-ray diffraction (WAXD) [5] and transmission elec-tron microscopy (TEM) [6] simulation and modeling arevery often used together with experimental data to obtainindirect measurements of the morphologic features of nano-composites

Some mathematical models developed in the literatureare aimed to predict the tortuous path of diffusing particles inpolymer nanocomposites Different models were developedwhich are able to account for the high aspect ratio of thenanofillers and are valid in 2D or in 3D problems [1 7] Allthe developed models assume that the nanofiller is perfectlyimpermeable to the diffusing species Only recently the effectof increased volume of impermeable phase strictly correlatedto this assumption has been introduced [8] Consequently2D [9] and 3D [10] analytical models have been developedwhich are able to account for diffusion inside the galleries

Alternatively and in some cases complementarily thediffusion process in polymer nanocomposites has been stud-ied by finite element analysis Inmany cases simulationswererun on perfectly aligned particles perpendicular or parallelto the direction of diffusion modeling the nanofiller as discs[11 12] or prisms [13] However perfect alignment of nano-particles is an ideal situation not commonly found in realnanocomposites system Only recently the effect of randomdistribution of particles [14] or of particles oriented with acertain degree with respect to the direction of diffusion [15]has been introduced In addition 2D [9] and 3D [10] FEM

Hindawi Publishing CorporationJournal of NanomaterialsVolume 2015 Article ID 482698 11 pageshttpdxdoiorg1011552015482698

2 Journal of Nanomaterials

(a) (b)

Figure 1 Sketches for modeling of diffusion in multiphase systems (a) permeable spherical particles and (b) impermeable high aspect ratioplatelets

models able to account for diffusion inside the galleries havebeen recently developed

Accounting for diffusion inside stacks is of primaryimportance in order to simulate and model the mass trans-port behavior in intercalated nanocomposite which retainthe basal order of the nanofiller The structure of the nano-composite is characterized by the presence of lamellar stackswhose aspect ratio depends on both the number of plateletswhich are on the average present in each stack and the dis-tance between individual plateletsThe twodifferentmorpho-logical features of the nanofiller are defined as degree of dis-persion and degree of intercalation respectively A stronglyintercalated weakly dispersed nanocomposite is character-ized by the presence of thick galleries with each stack beingcomposed of many platelets in contrast a strongly dis-persed weakly intercalated nanocomposite is characterizedby thin lamellar galleries but the stacks are composed of afew number of platelets

In such situation diffusion should be considered at twodifferent scales the first one associated with diffusion amongstacks (interstack diffusion) and the second one associatedwith diffusion inside stacks (intrastack diffusion) [16] Thedual scale approach has been recently introduced in order tomodel the mechanical behavior of polymer nanocomposites[17 18]The reported results show the relevance of the degreeof dispersion on the mechanical behavior of polymer nano-composites The dual scale diffusion problem is similar tothe problem of polymer flow during fiber impregnation ininfusion processes used in composite materials fabricationIn this case two mechanisms of impregnation have beenclearly identified the first one occurring at lower time scalescharacterized by a higher value of permeability as defined bythe Darcy law is associated with interbundle flow whereasthe second one occurring at higher time scale and thereforecharacterized by a lower value of permeability is associatedwith intrabundle flow [19]The first mechanism involves flow

of the molten polymer between fiber bundles (macroimpreg-nation) whereas the second one involves polymer flow insideeach bundle (microimpregnation)

The aim of this paper is to study the 3D diffusion behaviorof nanocomposites in the presence of permeable lamellarstacks by finite elements (FE) analysis To this purpose a geo-metric model made of randomly and uniformly distributedstacks was developed by the use of a Matlab routine Tothe best of our knowledge this is the first time that such amultiscale approach is introduced to study 3D diffusion inpolymer nanocomposites The results from simulation arecompared with different analytical models showing the rele-vance of intrastack diffusion Finally a proper model whichis able to predict the intrastack diffusion coefficient is devel-oped

2 Modeling of Diffusion in Nanocomposite

The first paper about modeling of diffusion in multiphasesystems most commonly quoted is by Maxwell who solvedthe problem of a suspension of spheres sparsely distributed inthe continuum and therefore characterized by a reasonablysmall volume fraction V119891 The model accounts for thepresence of permeable particles as reported in the schemeof Figure 1(a) characterized by a diffusivity 119863119904 dispersed ina permeable matrix characterized by a diffusivity119863119898 [20]

119863norm = 119863119898

(119863119904119863119898) (1 + 2V119891) + 2 minus 2V119891(119863119904119863119898) (1 minus V119891) + 2 + V119891

(1)

in which 119863norm the diffusivity of the multiphase systemis defined as the diffusion coefficient of a homogeneousmedium exhibiting the same steady-state behavior as thetwo-phase composite Differentmodifications of theMaxwellmodel have been proposed in order to account for the shapeof dispersed particles Among these the model developed by

Journal of Nanomaterials 3

3

2

1

5

4

W

TP

TGTS

Figure 2 Scheme of lamellasr stack morphology

Fricke [21] is valid for oblate spheroids characterized by twoaxes 119886 and 119887 with 119887 gt 119886 and 119886 being the rotation axis of theellipse

119863norm = 119863119898

(119863119904119863119898) (1 + 120588V119891) + 120588 minus 120588V119891(119863119904119863119898) (1 minus V119891) + 120588 + V119891

(2)

where

120588 =

(119863119891119863119898) (120573 minus 1) + 1

(119863119891119863119898) minus 1 minus 120573

120573 =1

3[

4

2 + ((120601 minus (12) sin 2120601) sin3120601) cos120601 (119863119904119863119898 minus 1)

+1

1 + (1 minus ((120601 minus (12) sin 2120601) sin3120601) cos120601) (119863119904119863119898 minus 1)]

sdot (119863119904

119863119898

minus 1)

120601 = cosminus1 ( 1

AR)

(3)

with AR the aspect ratio of the nanofiller 119887119886On the other hand modeling the diffusion in polymer

nanocomposites must be able to account for the high aspectratio of the nanofiller The most commonly quoted approachwas developed by Bharadwaj [1] whose model is able toaccount for the orientation angle of the nanofiller and in thecase of a random distribution can be written as

119863norm =1

1 + (AR6) V119891 (4)

The Bharadwaj model assumes that nanofiller is perfectlyimpermeable to the diffusing gas as reported in the schemeof Figure 1(b)

Nevertheless a perfect dispersion of the nanofiller is verydifficult to attain Single platelets of nanofiller can only befound in perfectly exfoliated nanocomposites In most casescorresponding to intercalated nanocomposites stacks ofplatelets characterize the structure of the nanocomposite asreported in the scheme of Figure 2 which represents a 2Dscheme of the nanofiller In 3D the rectangle becomes a cyl-inder of diameter119882

The morphological features of the stacks are the width ordiameter of the platelets 119882 the average number of platelets

in each stack 119873119875 the thickness of platelets 119879119875 the numberof galleries 119873119866 = 119873119875 minus 1 and the thickness of galleries 119879119866The number of platelets in each stack defines the degree ofdispersion whereas the gallery thickness defines the degreeof intercalationThe total thickness of the stack 119879119878 is given by

119879119878 = 119879119875 [119873119875 + (119873119875 minus 1)119879119866

119879119875

] (5)

According to the scheme of Figure 2 it is possible to definetwo different values for the aspect ratio The first one is theaspect ratio of platelets AR119875 defined as

AR119875 =119882

119879119875

(6)

whereas the second one is the aspect ratio of a stack AR119878 =119882119879119878 which also accounting for (5) is

AR119878 = AR1198751

119873119875 + (119873119875 minus 1) (119879119866119879119875) (7)

This equation highlights that the aspect ratio of the stack isdependent on both the degree of intercalation through theparameter 119879119866119879119875 and the degree of dispersion through theparameter119873119875 [9]

Further two different values of volume fraction can bedefined The first one V119891 is defined as the ratio between thevolume of the nanofiller and the total volume of the nano-composite If 119899119904 is the number of stacks per unit volume of thenanocomposite then the volume fraction V119891 is

V119891 = 119899119904

1205871198822

4119873119875119879119875

(8)

The second one V119878 is defined as the ratio between the totalvolume of lamellar stacks including the galleries volume andthe total volume of the nanocomposite which is

V119878 = 119899119904

1205871198822

4(119873119875119879119875 + (119873119875 minus 1) 119879119866)

(9)

Taking the ratio between (8) and (9) the following correlationcan be obtained between V119878 and V119891

V119878 =V119891119873119875

(119873119875 + (119873119875 minus 1)119879119866

119879119875

) (10)

Recently Nazarenko et al [22] recognized that in assumingimpermeability of the nanofiller the volume fraction of inor-ganic phase must be substituted by the total volume fractionof impermeable phase also including lamellar galleries Insuch case the Bharadwaj model is corrected to

119863norm =1

1 + (16) V119891 (AR119875119873119875) (11)

where according to (7)ndash(10)

V119891AR119875119873119875

= V119878AR119878 (12)


(a) (b)

Figure 3 Scheme for modeling of diffusion in intercalated polymer nanocomposites (a) impermeable stacks and (b) permeable stacks

Therefore the Nazarenko model can be considered an exten-sion of the Bharadwaj model in which the volume fractionand aspect ratio of impermeable phase are considered asreported in the scheme of Figure 3(a) Nazarenko model isbased on the assumption that at relatively low values of119879119866119879119875 diffusion inside the nanofiller can be neglected Nev-ertheless as reported in the scheme of Figure 3(b) diffusion

can occur either between stacks (interstack diffusion) orinside stacks (intrastack diffusion)This is a typical dual scaleproblem which has been solved to yield the following modelfor the diffusivity [10]

119863norm =

1 minus V119891

[1 + (43120587) (1radic3) (1 minus radic23) (V119891AR119875119873119875) (1 + (6radic2120587) (1AR119878))]4 (13)

Comparison between (11) and (13) shows that the lattermodel allows to account for the interstack diffusion (throughthe term V119891AR119875119873119875 homologous to that found in theNazarenkomodel) and intrastack diffusion (through the term(V119891AR119875119873119875)(6radic2120587)(1AR119878))

3 Simulation of Dual Scale Diffusion

Finite element analysis Comsol Multiphysics software (ver-sion 35 Comsol AB Sweden) was used to study the 3Ddiffusion in nanocomposites characterized by the presence ofintercalated lamellar nanofillers characterized by permeablestacksThe nanocomposite was modeled as an array of stacksof impermeable lamellae uniformly and randomly dispersedin the integration domain The position and orientation ofstacks inside the cubic domain were randomized by meansof a properly developed subroutine of Matlab 65 whichis based on a Monte Carlo stochastic approach and includesnoninterpenetrating conditions between stacks [9] Asreported in a previous work [10] the average distance bet-ween the centers of the stacks is in the range of 05ndash1 timesthe diameter of the stacks depending on volume fractionnumber of platelets in each stack and aspect ratio of theplatelets

Each stack is composed of a regular arrangement of par-allel and equidistant platelets represented as cylinders Asketch of the domains obtained is reported in Figure 4 Eachstack can intercept the domain boundary surfaces parallel tothe direction of diffusion In such case the actual volume ofthe stack which falls inside the domain is considered

The problem is solved by means of the mass trans-port module diffusion submodule in stationary conditionsassuming that the diffusivity inside the galleries is equal to thediffusivity of the matrix

This assumption does not take into account the anoma-lous effects which could arise at low values of 119879119866 At least twofactors can contribute to such anomalous diffusion

(a) The existence of a rigid amorphous fraction [23] at thelamellar sheet interface which can drastically reducethe diffusivity inside galleries

(b) The presence of low molecular weight organic modi-fier inside galleries which can increase the diffusivity

However changes of diffusivities inside the galleries can bereadily implemented in the simulation as will be discussedin the following section


x

z

y

A1

A2

L0

Figure 4 FEM simulation domain in 3D for modeling of diffusion in intercalated nanocomposites

The software solves the mass balance equation

nabla2119888 = 0 (14)

coupled with the proper boundary conditions

120597119888

120597 119899= 0 (15)

at each platelet surface with normal 119899 (impermeability)

120597119888

120597119899= 0 (16)

at the boundaries of the domain (transparent boundaries inFigure 4) parallel to the direction of diffusion (119911 axis)

119888 = 1198881 (17)

at the upper boundary of the domain perpendicular to thedirection of diffusion (upper gray boundary 1198601 in Figure 4)and

119888 = 0 (18)

at the lower boundary of the domain perpendicular to thedirection of diffusion (lower gray boundary 1198602 in Figure 4)

The concentration 1198881 was chosen in order to havefor all the simulations a concentration gradient 11988811198710 =

103molm4 with 1198710 being the length of the integration

domain along the direction of diffusion (119911 axis in Figure 4)Some simulations were also run using 2D geometry Insuch case each platelet was represented as rectangle 119887 lowast 119886

dimensions Randomization of the position was obtainedusing the same procedure described for 3D simulations

TheMatlab routine also includes a subroutine for the cal-culation of the average path length of massless particles dif-fusing due to concentration gradientThe subroutine is basedon the scheme reported in Figure 5 for a two-dimensionalproblem

The diffusion path is made by a series of points (119909119901 119910119901119911119901) and its slope can be calculated according to Fickrsquos lawaccording to the scheme of Figure 5 as

119869119909 = minus119863119898

120597119888

120597119909= 119888

119889119909119901

119889119905

119869119910 = minus119863119898

120597119888

120597119910= 119888

119889119910119901

119889119905

119869119911 = minus119863119898

120597119888

120597119911= 119888

119889119911119901

119889119905

997904rArr

119889119909119901

119889119911119901

=120597119888120597119909

120597119888120597119911

119889119910119901

119889119911119901

=120597119888120597119910

120597119888120597119911

(19)

where 119869 is the flux and 119888 is concentration Therefore theposition of each point can be calculated as

119911119901 (119894) = 119911119901 (119894 minus 1) + Δ119911

119909119901 (119894) = 119909119901 (119894 minus 1) +119889119909119901

119889119911119901

Δ119911 = 119909119901 (119894 minus 1) + 120597119888120597119909

120597119888120597119911Δ119911

119910119901 (119894) = 119910119901 (119894 minus 1) +119889119910119901

119889119911119901

Δ119911 = 119910119901 (119894 minus 1) +120597119888120597119910

120597119888120597119911Δ119911

(20)

Finally the diffusion path length is calculated as

119871nano =119873

sum

119894=2

radic[119909119901 (119894) minus 119909119901 (119894 minus 1)]2+ [119910119901 (119894) minus 119910119901 (119894 minus 1)]

2+ [119911119901 (119894) minus 119911119901 (119894 minus 1)]

2 (21)


Δx

Δz

i

i minus 1

tan120572 =120597c120597x

120597c120597z

Figure 5 Procedure used for calculation of the path length ofdiffusing particles

The normalized path length 119871norm was obtained as the ratiobetween the average path length of diffusing particles insidethe nanocomposite and 1198710 which corresponds to the diffu-sion length in the absence of lamellae

119871norm =119871nano1198710

(22)

The normalized diffusion coefficient 119863norm was obtained asthe ratio between the normal flow rate on the two boundariesperpendicular to the direction of diffusion (1198601 and 1198602) andthe flow rate in the neat matrix

119863norm =119863nano119863mat

=12

(100381610038161003816100381610038161003816int1198601

119863 (120597119888120597119911) 1198891198601100381610038161003816100381610038161003816+100381610038161003816100381610038161003816int1198602

119863 (120597119888120597119911) 1198891198602100381610038161003816100381610038161003816)

119863 (Δ119888119871119911) 1198601

(23)

Being in stationary conditions int1198601

119863(120597119888120597119911)1198891198601 =

minusint1198602

119863(120597119888120597119911)1198891198602 The set of equations used in (19)ndash(23)were also used for 2D simulations

In order to account for the morphology of the nanofillersimulations were run adopting three different nanofillervolume fractions (0015 00225 and 003) a number of lamel-lae in each stack changing from 3 to 5 and three differentgallery thicknesses (15 25 and 35 times the lamellar thick-ness)The aspect ratio of platelets was held at a constant valueof 50 The values chosen for simulation are in the typicalrange of montmorillonite nanofilled polymers which arecharacterized by a lamellar thickness of about 0883 nm [24]and separated by lamellar galleries of thickness between 15and 4 nm [25] Each result reported is obtained as the averageof 15 simulations each one runs in a domain comprising 10stacks

4 Results of Numerical Analysis

The results of 2D simulations are reported in Figure 6showing the diffusion paths calculated according to (20)

Figure 6 Results from 2D simulation highlighting the intrastackdiffusion

The blue ellipses evidence some of the intrastack diffusionzones characterized by the fact that the diffusing particletravels inside the lamellar galleries

The normalized coefficient of diffusion obtained by 3DFE simulation at different values of gallery thickness andV119891 = 0015 and 119873119875 = 3 is reported in Figure 7 The two setsof data are relative to simulationwith impermeable stacks andsimulation with permeable stacks (diffusion inside galleries)For comparison purposes the coefficient of diffusion corre-sponding to null gallery thickness 119879119866 = 0 is also reportedBoth models (either accounting or neglecting intrastackdiffusion) converge at the same value when 119879119866 = 0 thatis impermeable galleries On the other hand neglectingintrastack diffusion yields a significant underestimation ofthe coefficient of diffusion The error associated with theimpermeable stack approximation increases with increasinggallery thickness

Initially a sensitivity analysis was performed runningsimulations using a variable intrastack diffusivity The plot ofFigure 8 reports the coefficient of diffusion of the nanocom-posite as a function of the ratio between intrastack and inter-stack diffusivity for V119891 = 0015 119879119866119879119875 = 15 and 119873119875 = 3One-way analysis of variance (ANOVA)was performed usingthe intrastack diffusivity as the factor level Considering theset of 9 data ranging between 119863intrastack119863interstack = 06 and119863intrastack119863interstack = 14 an 119865 value equal to 184 was calcu-lated This value was compared with the critical 119865CV(8 126)= 201 with 8 being the degree of freedom of the factor leveland 126 = 9 lowast 14 the degree of freedom associated to theerrorThe estimated 119865 value is lower than the critical 119865 valueindicating that the different simulations provide an almostconstant value of the coefficient of diffusion On the otherhand when the ANOVA is performed on the set or 11data ranging between119863intrastack119863interstack = 05 and119863intrastack119863interstack = 15 the calculated 119865 = 263 is higher than thecritical119865CV(10 154)= 189 which indicates that the intrastack


0 1 2 3 4088

089

090

091

092

093

094

095

TGTP

TG = 0

No intrastack diffusionDual scale diffusion

Dno

rm

Figure 7 Normalized coefficient of diffusion as a function of gallerythickness at V119891 = 0015 and119873119875 = 3 calculated form 3D simulations

02 04 06 08 10 12 14 16 18091

092

093

094

095

096

DintrastackDinterstack

Dno

rm

Figure 8 Coefficient of diffusion as a function of intrastack dif-fusivity

diffusivity has a significant influence on the simulated coef-ficient of diffusion Therefore the ANOVA analysis indicatesthat the assumption of intrastack diffusivity equal to inter-stack diffusivity can be considered valid in the range 06 le

119863intrastack119863interstack le 14The normalized coefficient of diffusion is reported as a

function of the gallery thickness for different values of thenanofiller volume fraction at constant value of 119873119875 = 3in Figure 9 For comparison purposes the normalized coeffi-cient of diffusion is also reported for a null gallery thicknessthat is 119879119866 = 0

Due to the quite high error bars one-way analysis ofvariance (ANOVA) was initially performed on the simulatedcoefficient of diffusion Considering 3 different gallery thick-nesses as factor levels and 15 simulations on each level acritical value of the 119865 ratio (119865CV(2 42)) equal to 322 at a

0 1 2 3 4

084

087

090

093

096

099

TGTP

SimulationModel eq (11)Model eq (9)



f = 0015 f = 00225 f = 003

Dno

rm

Figure 9 Normalized coefficient of diffusion as a function of gallerythickness at different V119891 and119873119875 = 3

significance level 120572 = 005 is obtained The 119865 ratio valuescalculated for the coefficient of diffusion simulated at V119891 =

0015 V119891 = 00225 and V119891 = 003 are 685 735 and 567respectively Therefore in any case 119865 ratio is much higherthan 119865CV which indicates that the differences between theresults of the simulations obtained at different gallery thick-ness are relevant at the 5 significance level

For comparison purposes the simulated normalizeddiffusion coefficient is reported for a gallery thickness equalto 0 (corresponding to impermeable stack with aspect ratioequal to 167 according to (7)) The normalized coefficient ofdiffusion decreases with increasing lamellar spacing whichis in agreement with the generally accepted concept andmany experimental results indicating that improved barrierproperties are achieved when a higher amount of polymerchain segments is intercalated As expected for each value ofgallery thickness the coefficient of diffusion decreases withincreasing volume fraction of the nanofiller The predictionsaccording to the Nazarenko model (11) also reported inFigure 9 are not able to capture the dependence of thecoefficient of diffusion on the gallery thickness In contrastmodel prediction according to (13) accounting for intrastackdiffusion shows an excellent agreement with FE simulationresults as confirmed by the coefficient of determination1198772 =0972

The normalized coefficient of diffusion is reported inFigure 10 as a function of the number of platelets in eachstack at different values of the volume fractions keepingconstant 119879119866119879119875 = 25

Even in this case in order to account for the effectof the wide error bars ANOVA analysis was performedconsidering the number of lamellae in the stacks as the anal-ysis factor An 119865 value equal to 130 56 and 32 was found forthe three different volume fractions which again indicatesthat the differences between simulations run at different


3 4 5084

086

088

090

092

094

096

098

100




f = 0015 f = 00225 f = 003

NP

Dno

rm

Figure 10 Normalized coefficient of diffusion as a function of thenumber of platelets at different V119891 and 119879119866119879119875 = 25

number of lamellae in each stack are relevant at the 5significance level The simulation results are in good agree-ment with the expected behavior of nanocomposites sincea decrease of the number of lamellae in each stack which isequivalent to an increase in the aspect ratio and hence to animprovement of the degree of dispersion involves a decreaseof the diffusion coefficient of nanocomposite In this case themodel of (11) despite being able to capture the dependenceof the coefficient of diffusion on the number of plateletsprovides a very poor fitting of the simulation data In contrastmodel of (13) provides an excellent fitting of simulation databeing 1198772 = 0988

The data in Figure 11 show the normalized coefficient ofdiffusion as a function of the volume fraction of the stacksgiven by (10) for constant values of the aspect ratio of thestacks given by (7) Therefore the Fricke model (2)-(3)was used in order to calculate the normalized intrastackdiffusivity 119863119904119863119898 According to the Fricke model 119863119904 rep-resents the diffusivity of the stacks or more correctly theequivalent coefficient of diffusion of a homogeneousmediumexhibiting the same transport properties of the stack Theexcellent fitting reported in Figure 11 by the continuous lineis evidenced by the value of 1198772 = 0961

The values calculated for 119863119904119863119898 is reported in Figure 12as a function of the aspect ratio of the stacks As it can beobserved the intrastack diffusivity increases with decreasingaspect ratio of the stacks In order to explain this behaviorthe domain surrounding each stack can be considered asthat reported in Figure 13(a) The elliptic cylinder is tangentto the stack and is therefore characterized by the dimen-sions reported in Figure 13(a) The angle 120579 is the angleformed between the direction normal to stack surface and

003 005 007 009 011 013075

078

081

084

087

090

093

096

S

ARS = 833

ARS = 625ARS = 500

ARS = 435ARS = 333

Dno

rm

Figure 11 Normalized coefficient of diffusion as a function ofvolume fraction of stacks at different values of the aspect ratio ofthe stacks and Fricke model fitting

3 4 5 6 7 8 9

002

004

006

008

010

012

014

ARS

DsD

m

Figure 12 Coefficient of diffusion of the stacks as a function of thestack aspect ratio

the direction of diffusion For a random orientation of stacksas that investigated in this work in 3D 120579 = cosminus1(1radic3) andin 2D 120579 = 1205874

The 2D visualization of the domain is reported inFigure 13(b) In the domains of Figures 13(a) and 13(b) dif-fusion can only occur inside the stacks Therefore due to theimpermeability of the single platelets it can be assumed thatin 1198811 and 1198812

120597119888

120597119911= 0 (24)

whereas in the volume 1198813 the average path length of a diffu-sing particle is [10]

119871diff =2120587119882+119879119878 cos 120579 (25)


120579

n

t

z

W

Wsin

120579+TSco

s120579

Wcos120579 + TSsin120579

(a)

120579

n

t

z

Wsin

120579+TSco

s120579

Wcos120579 + TSsin120579

V1

V2

V3

Δc

(b)

Figure 13 Schemes for prediction of diffusive flow rate in the presence of stack with permeable galleries (a) 3D and (b) 2D

Therefore due to a concentration difference between the twohorizontal surfaces Δ119888 the average gradient in the directionparallel to the platelet surface 119905 is given as

120597119888

120597119905=

Δ119888

(2120587)119882 + 119879119878 cos 120579(26)

and with 120579 being the angle between the 119905 and the 119911 directionthe concentration gradient in the 119911 direction is

120597119888

120597119911=

Δ119888

(2120587)119882 + 119879119878 cos 120579sin 120579 (27)

The flow rate is given by the concentration gradient mul-tiplied by diffusivity and by the area crossed by diffusingparticles which is (119873119875 minus 1)119882119879119866

119865119911 = 119863119898

Δ119888

(2120587)119882 + 119879119878 cos 120579sin 120579 (119873119875 minus 1)119882119879119866 (28)

The same domain of Figure 13(a) made of stacks of imperme-able platelets can be schematized as reported in Figure 14(a)which is made of a single permeable inclusion which has thesame geometric characteristic (diameter and total thickness)of the stack and has a diffusivity equal to119863119904 In such situationfor 119863119904 ≪ 119863119898 the diffusion paths follow parallel to the stackthickness Further the concentration gradient in the matrixis negligible compared to the concentration gradient in thestack This is shown in Figure 14(b) where the concentrationgradient and the diffusion path are shown in the correspond-ing 2D simulation Therefore the upper surface of the stackcan be assumed to be at a constant concentration equalto that of the upper boundary of the domain whereas thelower surface of the stack can be assumed to be at a con-stant concentration equal to the concentration of the lowerboundary of the domain Therefore for a 3D problemthe concentration gradient inside the stack is parallel to

the direction 119899 (perpendicular to lamellar stack surface) andis given as

120597119888

120597119899=Δ119888

119879119878

(29)

The gradient in the 119911 direction is therefore given as

120597119888

120597119911=Δ119888

119879119878

cos 120579 (30)

And the total flow rate in the 119911 direction can be obtained bymultiplying the gradient by the diffusivity of the stack and theflow passage 12058711988224

119865119911 = 119863119904

Δ119888

119879119878

1205871198822

4cos 120579 (31)

By combining (28) and (31)

119863119904

119863119898

=4 (119873119875 minus 1) (119879119866119879119878) sin 120579

120587AR119878 cos 120579 ((2120587)AR119878 + cos 120579) (32)

The values of stack diffusivity calculated according to (32) arealso reported in Figure 12 showing a very good agreementwith the data calculated from FE simulation According to(32) the diffusivity of the stack is mainly dependent on theaspect ratio of the stack itself For a fixed value of the aspectratio of the stack the diffusivity also depends on the numberof platelets and the thickness of the galleries

5 Conclusions

In this work a finite elementsrsquo model was developed in orderto simulate the 3D diffusion into polymer nanocompositeswith nanofillers made of lamellar stacks characterized bypermeable galleries as can occur when organic modified


120579

n

t

z

2

W

Wsin

120579+TSco

s120579Wcos120579 + TSsin120579

(a)

002

004

006

008

01

012

014

016

018

02

0

(b)

Figure 14 Schemes for prediction of diffusive flow rate in the presence of stacks regarded as homogenous inclusions and diffusivity 119863119904 (a)3D and (b) 2D solution

montmorillonites are dispersed in a polymer matrix Thedeveloped method is able to introduce a random distribu-tion of nonoverlapping stacks Each stack is composed ofan arbitrary number of platelets separated by galleries ofarbitrary thickness To the best of our knowledge this is thefirst time that this dual scale approach has been used tosimulate the diffusion in polymer nanocomposites occurringeither between lamellar stacks or within lamellar stacks

The results obtained from numerical simulation showthat intrastack diffusion that is diffusion inside galleriescan be quite relevant particularly at the higher thicknessof the galleries Therefore the assumption usually made inmodeling and simulating diffusion in polymer nanocompos-ites based on the impermeability of the nanofiller cannot beconsidered valid

As a consequence of this the literature models whichdo not account for the intrastack diffusivity are not able tocorrectly predict the coefficient of diffusion of intercalatednanocomposites In contrast literature models (the proposedone) which account for intrastack diffusion show an excel-lent agreement with the simulation data

The dependence of the coefficient of diffusion on thevolume fraction of the stack has been used to estimatean equivalent coefficient of diffusion of a homogeneousmediumexhibiting the same transport properties of the stackThis equivalent diffusivity was shown to be significantlydependent on the aspect ratio of the stack An analyticalmodel which is able to represent such property of the stackwas proposedThis model shows a very good agreement withthe data obtained from simulation

Conflict of Interests

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

References

[1] R K Bharadwaj ldquoModeling the barrier properties of polymer-layered silicate nanocompositesrdquo Macromolecules vol 34 no26 pp 9189ndash9192 2001

[2] M Bhattacharya S Biswas and A K Bhowmick ldquoPermeationcharacteristics andmodeling of barrier properties ofmultifunc-tional rubber nanocompositesrdquo Polymer vol 52 no 7 pp 1562ndash1576 2011

[3] G H Fredrickson and J Bicerano ldquoBarrier properties of ori-ented disk compositesrdquoThe Journal of Chemical Physics vol 110no 4 pp 2181ndash2188 1999

[4] C Lu and Y-W Mai ldquoPermeability modelling of polymer-layered silicate nanocompositesrdquo Composites Science and Tech-nology vol 67 no 14 pp 2895ndash2902 2007

[5] S Sinha Ray and M Okamoto ldquoPolymerlayered silicate nano-composites a review from preparation to processingrdquo Progressin Polymer Science vol 28 no 11 pp 1539ndash1641 2003

[6] T D Fornes and D R Paul ldquoModeling properties of nylon6clay nanocomposites using composite theoriesrdquo Polymer vol44 no 17 pp 4993ndash5013 2003

[7] G Choudalakis and A D Gotsis ldquoPermeability of polymerclaynanocomposites a reviewrdquo European Polymer Journal vol 45no 4 pp 967ndash984 2009

[8] S Nazarenko P Meneghetti P Julmon B G Olson and SQutubuddin ldquoGas barrier of polystyrene montmorillonite claynanocomposites effect of mineral layer aggregationrdquo Journal ofPolymer Science Part B Polymer Physics vol 45 no 13 pp 1733ndash1753 2007

[9] A Greco ldquoNumerical simulation and mathematical modelingof 2D multi-scale diffusion in lamellar nanocompositerdquo Com-putational Materials Science vol 90 pp 203ndash209 2014

[10] A Greco and A Maffezzoli ldquoNumerical simulation and mod-eling of 3-D multi scale diffusion in nanocomposites withpermeable lamellar stacksrdquo Computersamp Chemical EngineeringSubmitted


[11] A A Gusev and H R Lusti ldquoRational design of nanocompos-ites for barrier applicationsrdquo Advanced Materials vol 13 no 21pp 1641ndash1643 2001

[12] M Minelli M G Baschetti and F Doghieri ldquoA comprehensivemodel for mass transport properties in nanocompositesrdquo Jour-nal of Membrane Science vol 381 no 1-2 pp 10ndash20 2011

[13] C Swannack C Cox A Liakos and D Hirt ldquoA three-dimen-sional simulation of barrier properties of nanocomposite filmsrdquoJournal of Membrane Science vol 263 no 1-2 pp 47ndash56 2005

[14] A Greco and A Maffezzoli ldquoTwo-dimensional and three-dimensional simulation of diffusion in nanocomposite witharbitrarily oriented lamellaerdquo Journal of Membrane Science vol442 pp 238ndash244 2013

[15] A Greco ldquoSimulation and modeling of diffusion in orientedlamellar nanocompositesrdquo Computational Materials Sciencevol 83 pp 164ndash170 2014

[16] M AOsman VMittalMMorbidelli andUW Suter ldquoEpoxy-layered silicate nanocomposites and their gas permeation prop-ertiesrdquoMacromolecules vol 37 no 19 pp 7250ndash7257 2004

[17] P E Spencer and J Sweeney ldquoModeling of polymer claynanocomposites for a multiscale approachrdquo in Nano- andMicro-Mechanics of Polymer Blends and Composites J Karger-Kocsis and S Fakirov Eds Carl Hanser Munich Germany2009

[18] M Fermeglia and S Pricl ldquoMultiscale molecular modeling innanostructured material design and process system engineer-ingrdquo Computers and Chemical Engineering vol 33 no 10 pp1701ndash1710 2009

[19] R Gennaro A Greco and A Maffezzoli ldquoMicro- and macro-impregnation of fabrics using thermoplastic matricesrdquo JournalofThermoplastic CompositeMaterials vol 26 no 4 pp 527ndash5432013

[20] J Crank The Mathematics of Diffusion Clarendon PressOxford UK 1975

[21] H Fricke ldquoA mathematical treatment of the electric conductiv-ity and capacity of disperse systems I The electric conductivityof a suspension of homogeneous spheroidsrdquo Physical Reviewvol 24 no 5 pp 575ndash587 1924

[22] S Nazarenko P Meneghetti P Julmon B G Olson and SQutubuddin ldquoGas barrier of polystyrene montmorillonite claynanocomposites effect of mineral layer aggregationrdquo Journal ofPolymer Science B Polymer Physics vol 45 no 13 pp 1733ndash17532007

[23] C E Corcione G Mensitieri and A Maffezzoli ldquoAnalysis ofthe structure and mass transport properties of nanocompositepolyurethanerdquo Polymer Engineering and Science vol 49 no 9pp 1708ndash1718 2009

[24] O L Manevitch and G C Rutledge ldquoElastic properties of asingle lamella of montmorillonite by molecular dynamics sim-ulationrdquo Journal of Physical Chemistry B vol 108 no 4 pp1428ndash1435 2004

[25] D S Homminga B Goderis V B F Mathot and G Groen-inckx ldquoCrystallization behavior of polymermontmorillonitenanocomposites Part III Polyamide-6montmorillonite nano-composites influence ofmatrixmolecular weight and ofmont-morillonite type and concentrationrdquo Polymer vol 47 no 5 pp1630ndash1639 2006

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of



CeramicsJournal of


CompositesJournal of

NanoparticlesJournal of



International Journal of

Biomaterials


NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research



MetallurgyJournal of


BioMed Research International

MaterialsJournal of


Nano

materials


Journal ofNanomaterials


(a) (b)

Figure 1 Sketches for modeling of diffusion in multiphase systems (a) permeable spherical particles and (b) impermeable high aspect ratioplatelets

models able to account for diffusion inside the galleries havebeen recently developed

Accounting for diffusion inside stacks is of primaryimportance in order to simulate and model the mass trans-port behavior in intercalated nanocomposite which retainthe basal order of the nanofiller The structure of the nano-composite is characterized by the presence of lamellar stackswhose aspect ratio depends on both the number of plateletswhich are on the average present in each stack and the dis-tance between individual plateletsThe twodifferentmorpho-logical features of the nanofiller are defined as degree of dis-persion and degree of intercalation respectively A stronglyintercalated weakly dispersed nanocomposite is character-ized by the presence of thick galleries with each stack beingcomposed of many platelets in contrast a strongly dis-persed weakly intercalated nanocomposite is characterizedby thin lamellar galleries but the stacks are composed of afew number of platelets

In such situation diffusion should be considered at twodifferent scales the first one associated with diffusion amongstacks (interstack diffusion) and the second one associatedwith diffusion inside stacks (intrastack diffusion) [16] Thedual scale approach has been recently introduced in order tomodel the mechanical behavior of polymer nanocomposites[17 18]The reported results show the relevance of the degreeof dispersion on the mechanical behavior of polymer nano-composites The dual scale diffusion problem is similar tothe problem of polymer flow during fiber impregnation ininfusion processes used in composite materials fabricationIn this case two mechanisms of impregnation have beenclearly identified the first one occurring at lower time scalescharacterized by a higher value of permeability as defined bythe Darcy law is associated with interbundle flow whereasthe second one occurring at higher time scale and thereforecharacterized by a lower value of permeability is associatedwith intrabundle flow [19]The first mechanism involves flow

of the molten polymer between fiber bundles (macroimpreg-nation) whereas the second one involves polymer flow insideeach bundle (microimpregnation)

The aim of this paper is to study the 3D diffusion behaviorof nanocomposites in the presence of permeable lamellarstacks by finite elements (FE) analysis To this purpose a geo-metric model made of randomly and uniformly distributedstacks was developed by the use of a Matlab routine Tothe best of our knowledge this is the first time that such amultiscale approach is introduced to study 3D diffusion inpolymer nanocomposites The results from simulation arecompared with different analytical models showing the rele-vance of intrastack diffusion Finally a proper model whichis able to predict the intrastack diffusion coefficient is devel-oped

2 Modeling of Diffusion in Nanocomposite

The first paper about modeling of diffusion in multiphasesystems most commonly quoted is by Maxwell who solvedthe problem of a suspension of spheres sparsely distributed inthe continuum and therefore characterized by a reasonablysmall volume fraction V119891 The model accounts for thepresence of permeable particles as reported in the schemeof Figure 1(a) characterized by a diffusivity 119863119904 dispersed ina permeable matrix characterized by a diffusivity119863119898 [20]

119863norm = 119863119898

(119863119904119863119898) (1 + 2V119891) + 2 minus 2V119891(119863119904119863119898) (1 minus V119891) + 2 + V119891

(1)

in which 119863norm the diffusivity of the multiphase systemis defined as the diffusion coefficient of a homogeneousmedium exhibiting the same steady-state behavior as thetwo-phase composite Differentmodifications of theMaxwellmodel have been proposed in order to account for the shapeof dispersed particles Among these the model developed by


3

2

1

5

4

W

TP

TGTS



119863norm = 119863119898

(119863119904119863119898) (1 + 120588V119891) + 120588 minus 120588V119891(119863119904119863119898) (1 minus V119891) + 120588 + V119891

(2)

where

120588 =

(119863119891119863119898) (120573 minus 1) + 1

(119863119891119863119898) minus 1 minus 120573

120573 =1

3[

4


+1


sdot (119863119904

119863119898

minus 1)

120601 = cosminus1 ( 1

AR)

(3)



119863norm =1

1 + (AR6) V119891 (4)





119879119878 = 119879119875 [119873119875 + (119873119875 minus 1)119879119866

119879119875

] (5)


AR119875 =119882

119879119875

(6)


AR119878 = AR1198751

119873119875 + (119873119875 minus 1) (119879119866119879119875) (7)



V119891 = 119899119904

1205871198822

4119873119875119879119875

(8)


V119878 = 119899119904

1205871198822

4(119873119875119879119875 + (119873119875 minus 1) 119879119866)

(9)


V119878 =V119891119873119875

(119873119875 + (119873119875 minus 1)119879119866

119879119875

) (10)


119863norm =1

1 + (16) V119891 (AR119875119873119875) (11)


V119891AR119875119873119875

= V119878AR119878 (12)


(a) (b)




119863norm =

1 minus V119891












x

z

y

A1

A2

L0



nabla2119888 = 0 (14)


120597119888

120597 119899= 0 (15)


120597119888

120597119899= 0 (16)


119888 = 1198881 (17)


119888 = 0 (18)








119869119909 = minus119863119898

120597119888

120597119909= 119888

119889119909119901

119889119905

119869119910 = minus119863119898

120597119888

120597119910= 119888

119889119910119901

119889119905

119869119911 = minus119863119898

120597119888

120597119911= 119888

119889119911119901

119889119905

997904rArr

119889119909119901

119889119911119901

=120597119888120597119909

120597119888120597119911

119889119910119901

119889119911119901

=120597119888120597119910

120597119888120597119911

(19)


119911119901 (119894) = 119911119901 (119894 minus 1) + Δ119911

119909119901 (119894) = 119909119901 (119894 minus 1) +119889119909119901

119889119911119901

Δ119911 = 119909119901 (119894 minus 1) + 120597119888120597119909

120597119888120597119911Δ119911

119910119901 (119894) = 119910119901 (119894 minus 1) +119889119910119901

119889119911119901

Δ119911 = 119910119901 (119894 minus 1) +120597119888120597119910

120597119888120597119911Δ119911

(20)


119871nano =119873

sum

119894=2


2+ [119911119901 (119894) minus 119911119901 (119894 minus 1)]

2 (21)


Δx

Δz

i

i minus 1

tan120572 =120597c120597x

120597c120597z



119871norm =119871nano1198710

(22)


119863norm =119863nano119863mat

=12

(100381610038161003816100381610038161003816int1198601

119863 (120597119888120597119911) 1198891198601100381610038161003816100381610038161003816+100381610038161003816100381610038161003816int1198602

119863 (120597119888120597119911) 1198891198602100381610038161003816100381610038161003816)

119863 (Δ119888119871119911) 1198601

(23)


119863(120597119888120597119911)1198891198601 =

minusint1198602










0 1 2 3 4088

089

090

091

092

093

094

095

TGTP

TG = 0


Dno

rm


02 04 06 08 10 12 14 16 18091

092

093

094

095

096


Dno

rm






0 1 2 3 4

084

087

090

093

096

099

TGTP




f = 0015 f = 00225 f = 003

Dno

rm








3 4 5084

086

088

090

092

094

096

098

100




f = 0015 f = 00225 f = 003

NP

Dno

rm





003 005 007 009 011 013075

078

081

084

087

090

093

096

S

ARS = 833

ARS = 625ARS = 500

ARS = 435ARS = 333

Dno

rm


3 4 5 6 7 8 9

002

004

006

008

010

012

014

ARS

DsD

m




120597119888

120597119911= 0 (24)


119871diff =2120587119882+119879119878 cos 120579 (25)


120579

n

t

z

W

Wsin

120579+TSco

s120579

Wcos120579 + TSsin120579

(a)

120579

n

t

z

Wsin

120579+TSco

s120579

Wcos120579 + TSsin120579

V1

V2

V3

Δc

(b)



120597119888

120597119905=

Δ119888

(2120587)119882 + 119879119878 cos 120579(26)


120597119888

120597119911=

Δ119888

(2120587)119882 + 119879119878 cos 120579sin 120579 (27)


119865119911 = 119863119898

Δ119888

(2120587)119882 + 119879119878 cos 120579sin 120579 (119873119875 minus 1)119882119879119866 (28)



120597119888

120597119899=Δ119888

119879119878

(29)


120597119888

120597119911=Δ119888

119879119878

cos 120579 (30)


119865119911 = 119863119904

Δ119888

119879119878

1205871198822

4cos 120579 (31)


119863119904

119863119898

=4 (119873119875 minus 1) (119879119866119879119878) sin 120579

120587AR119878 cos 120579 ((2120587)AR119878 + cos 120579) (32)


5 Conclusions



120579

n

t

z

2

W

Wsin

120579+TSco

s120579Wcos120579 + TSsin120579

(a)

002

004

006

008

01

012

014

016

018

02

0

(b)








References


































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




3

2

1

5

4

W

TP

TGTS



119863norm = 119863119898

(119863119904119863119898) (1 + 120588V119891) + 120588 minus 120588V119891(119863119904119863119898) (1 minus V119891) + 120588 + V119891

(2)

where

120588 =

(119863119891119863119898) (120573 minus 1) + 1

(119863119891119863119898) minus 1 minus 120573

120573 =1

3[

4


+1


sdot (119863119904

119863119898

minus 1)

120601 = cosminus1 ( 1

AR)

(3)



119863norm =1

1 + (AR6) V119891 (4)





119879119878 = 119879119875 [119873119875 + (119873119875 minus 1)119879119866

119879119875

] (5)


AR119875 =119882

119879119875

(6)


AR119878 = AR1198751

119873119875 + (119873119875 minus 1) (119879119866119879119875) (7)



V119891 = 119899119904

1205871198822

4119873119875119879119875

(8)


V119878 = 119899119904

1205871198822

4(119873119875119879119875 + (119873119875 minus 1) 119879119866)

(9)


V119878 =V119891119873119875

(119873119875 + (119873119875 minus 1)119879119866

119879119875

) (10)


119863norm =1

1 + (16) V119891 (AR119875119873119875) (11)


V119891AR119875119873119875

= V119878AR119878 (12)


(a) (b)




119863norm =

1 minus V119891












x

z

y

A1

A2

L0



nabla2119888 = 0 (14)


120597119888

120597 119899= 0 (15)


120597119888

120597119899= 0 (16)


119888 = 1198881 (17)


119888 = 0 (18)








119869119909 = minus119863119898

120597119888

120597119909= 119888

119889119909119901

119889119905

119869119910 = minus119863119898

120597119888

120597119910= 119888

119889119910119901

119889119905

119869119911 = minus119863119898

120597119888

120597119911= 119888

119889119911119901

119889119905

997904rArr

119889119909119901

119889119911119901

=120597119888120597119909

120597119888120597119911

119889119910119901

119889119911119901

=120597119888120597119910

120597119888120597119911

(19)


119911119901 (119894) = 119911119901 (119894 minus 1) + Δ119911

119909119901 (119894) = 119909119901 (119894 minus 1) +119889119909119901

119889119911119901

Δ119911 = 119909119901 (119894 minus 1) + 120597119888120597119909

120597119888120597119911Δ119911

119910119901 (119894) = 119910119901 (119894 minus 1) +119889119910119901

119889119911119901

Δ119911 = 119910119901 (119894 minus 1) +120597119888120597119910

120597119888120597119911Δ119911

(20)


119871nano =119873

sum

119894=2


2+ [119911119901 (119894) minus 119911119901 (119894 minus 1)]

2 (21)


Δx

Δz

i

i minus 1

tan120572 =120597c120597x

120597c120597z



119871norm =119871nano1198710

(22)


119863norm =119863nano119863mat

=12

(100381610038161003816100381610038161003816int1198601

119863 (120597119888120597119911) 1198891198601100381610038161003816100381610038161003816+100381610038161003816100381610038161003816int1198602

119863 (120597119888120597119911) 1198891198602100381610038161003816100381610038161003816)

119863 (Δ119888119871119911) 1198601

(23)


119863(120597119888120597119911)1198891198601 =

minusint1198602










0 1 2 3 4088

089

090

091

092

093

094

095

TGTP

TG = 0


Dno

rm


02 04 06 08 10 12 14 16 18091

092

093

094

095

096


Dno

rm






0 1 2 3 4

084

087

090

093

096

099

TGTP




f = 0015 f = 00225 f = 003

Dno

rm








3 4 5084

086

088

090

092

094

096

098

100




f = 0015 f = 00225 f = 003

NP

Dno

rm





003 005 007 009 011 013075

078

081

084

087

090

093

096

S

ARS = 833

ARS = 625ARS = 500

ARS = 435ARS = 333

Dno

rm


3 4 5 6 7 8 9

002

004

006

008

010

012

014

ARS

DsD

m




120597119888

120597119911= 0 (24)


119871diff =2120587119882+119879119878 cos 120579 (25)


120579

n

t

z

W

Wsin

120579+TSco

s120579

Wcos120579 + TSsin120579

(a)

120579

n

t

z

Wsin

120579+TSco

s120579

Wcos120579 + TSsin120579

V1

V2

V3

Δc

(b)



120597119888

120597119905=

Δ119888

(2120587)119882 + 119879119878 cos 120579(26)


120597119888

120597119911=

Δ119888

(2120587)119882 + 119879119878 cos 120579sin 120579 (27)


119865119911 = 119863119898

Δ119888

(2120587)119882 + 119879119878 cos 120579sin 120579 (119873119875 minus 1)119882119879119866 (28)



120597119888

120597119899=Δ119888

119879119878

(29)


120597119888

120597119911=Δ119888

119879119878

cos 120579 (30)


119865119911 = 119863119904

Δ119888

119879119878

1205871198822

4cos 120579 (31)


119863119904

119863119898

=4 (119873119875 minus 1) (119879119866119879119878) sin 120579

120587AR119878 cos 120579 ((2120587)AR119878 + cos 120579) (32)


5 Conclusions



120579

n

t

z

2

W

Wsin

120579+TSco

s120579Wcos120579 + TSsin120579

(a)

002

004

006

008

01

012

014

016

018

02

0

(b)








References


































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




(a) (b)




119863norm =

1 minus V119891












x

z

y

A1

A2

L0



nabla2119888 = 0 (14)


120597119888

120597 119899= 0 (15)


120597119888

120597119899= 0 (16)


119888 = 1198881 (17)


119888 = 0 (18)








119869119909 = minus119863119898

120597119888

120597119909= 119888

119889119909119901

119889119905

119869119910 = minus119863119898

120597119888

120597119910= 119888

119889119910119901

119889119905

119869119911 = minus119863119898

120597119888

120597119911= 119888

119889119911119901

119889119905

997904rArr

119889119909119901

119889119911119901

=120597119888120597119909

120597119888120597119911

119889119910119901

119889119911119901

=120597119888120597119910

120597119888120597119911

(19)


119911119901 (119894) = 119911119901 (119894 minus 1) + Δ119911

119909119901 (119894) = 119909119901 (119894 minus 1) +119889119909119901

119889119911119901

Δ119911 = 119909119901 (119894 minus 1) + 120597119888120597119909

120597119888120597119911Δ119911

119910119901 (119894) = 119910119901 (119894 minus 1) +119889119910119901

119889119911119901

Δ119911 = 119910119901 (119894 minus 1) +120597119888120597119910

120597119888120597119911Δ119911

(20)


119871nano =119873

sum

119894=2


2+ [119911119901 (119894) minus 119911119901 (119894 minus 1)]

2 (21)


Δx

Δz

i

i minus 1

tan120572 =120597c120597x

120597c120597z



119871norm =119871nano1198710

(22)


119863norm =119863nano119863mat

=12

(100381610038161003816100381610038161003816int1198601

119863 (120597119888120597119911) 1198891198601100381610038161003816100381610038161003816+100381610038161003816100381610038161003816int1198602

119863 (120597119888120597119911) 1198891198602100381610038161003816100381610038161003816)

119863 (Δ119888119871119911) 1198601

(23)


119863(120597119888120597119911)1198891198601 =

minusint1198602










0 1 2 3 4088

089

090

091

092

093

094

095

TGTP

TG = 0


Dno

rm


02 04 06 08 10 12 14 16 18091

092

093

094

095

096


Dno

rm






0 1 2 3 4

084

087

090

093

096

099

TGTP




f = 0015 f = 00225 f = 003

Dno

rm








3 4 5084

086

088

090

092

094

096

098

100




f = 0015 f = 00225 f = 003

NP

Dno

rm





003 005 007 009 011 013075

078

081

084

087

090

093

096

S

ARS = 833

ARS = 625ARS = 500

ARS = 435ARS = 333

Dno

rm


3 4 5 6 7 8 9

002

004

006

008

010

012

014

ARS

DsD

m




120597119888

120597119911= 0 (24)


119871diff =2120587119882+119879119878 cos 120579 (25)


120579

n

t

z

W

Wsin

120579+TSco

s120579

Wcos120579 + TSsin120579

(a)

120579

n

t

z

Wsin

120579+TSco

s120579

Wcos120579 + TSsin120579

V1

V2

V3

Δc

(b)



120597119888

120597119905=

Δ119888

(2120587)119882 + 119879119878 cos 120579(26)


120597119888

120597119911=

Δ119888

(2120587)119882 + 119879119878 cos 120579sin 120579 (27)


119865119911 = 119863119898

Δ119888

(2120587)119882 + 119879119878 cos 120579sin 120579 (119873119875 minus 1)119882119879119866 (28)



120597119888

120597119899=Δ119888

119879119878

(29)


120597119888

120597119911=Δ119888

119879119878

cos 120579 (30)


119865119911 = 119863119904

Δ119888

119879119878

1205871198822

4cos 120579 (31)


119863119904

119863119898

=4 (119873119875 minus 1) (119879119866119879119878) sin 120579

120587AR119878 cos 120579 ((2120587)AR119878 + cos 120579) (32)


5 Conclusions



120579

n

t

z

2

W

Wsin

120579+TSco

s120579Wcos120579 + TSsin120579

(a)

002

004

006

008

01

012

014

016

018

02

0

(b)








References


































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




x

z

y

A1

A2

L0



nabla2119888 = 0 (14)


120597119888

120597 119899= 0 (15)


120597119888

120597119899= 0 (16)


119888 = 1198881 (17)


119888 = 0 (18)








119869119909 = minus119863119898

120597119888

120597119909= 119888

119889119909119901

119889119905

119869119910 = minus119863119898

120597119888

120597119910= 119888

119889119910119901

119889119905

119869119911 = minus119863119898

120597119888

120597119911= 119888

119889119911119901

119889119905

997904rArr

119889119909119901

119889119911119901

=120597119888120597119909

120597119888120597119911

119889119910119901

119889119911119901

=120597119888120597119910

120597119888120597119911

(19)


119911119901 (119894) = 119911119901 (119894 minus 1) + Δ119911

119909119901 (119894) = 119909119901 (119894 minus 1) +119889119909119901

119889119911119901

Δ119911 = 119909119901 (119894 minus 1) + 120597119888120597119909

120597119888120597119911Δ119911

119910119901 (119894) = 119910119901 (119894 minus 1) +119889119910119901

119889119911119901

Δ119911 = 119910119901 (119894 minus 1) +120597119888120597119910

120597119888120597119911Δ119911

(20)


119871nano =119873

sum

119894=2


2+ [119911119901 (119894) minus 119911119901 (119894 minus 1)]

2 (21)


Δx

Δz

i

i minus 1

tan120572 =120597c120597x

120597c120597z



119871norm =119871nano1198710

(22)


119863norm =119863nano119863mat

=12

(100381610038161003816100381610038161003816int1198601

119863 (120597119888120597119911) 1198891198601100381610038161003816100381610038161003816+100381610038161003816100381610038161003816int1198602

119863 (120597119888120597119911) 1198891198602100381610038161003816100381610038161003816)

119863 (Δ119888119871119911) 1198601

(23)


119863(120597119888120597119911)1198891198601 =

minusint1198602










0 1 2 3 4088

089

090

091

092

093

094

095

TGTP

TG = 0


Dno

rm


02 04 06 08 10 12 14 16 18091

092

093

094

095

096


Dno

rm






0 1 2 3 4

084

087

090

093

096

099

TGTP




f = 0015 f = 00225 f = 003

Dno

rm








3 4 5084

086

088

090

092

094

096

098

100




f = 0015 f = 00225 f = 003

NP

Dno

rm





003 005 007 009 011 013075

078

081

084

087

090

093

096

S

ARS = 833

ARS = 625ARS = 500

ARS = 435ARS = 333

Dno

rm


3 4 5 6 7 8 9

002

004

006

008

010

012

014

ARS

DsD

m




120597119888

120597119911= 0 (24)


119871diff =2120587119882+119879119878 cos 120579 (25)


120579

n

t

z

W

Wsin

120579+TSco

s120579

Wcos120579 + TSsin120579

(a)

120579

n

t

z

Wsin

120579+TSco

s120579

Wcos120579 + TSsin120579

V1

V2

V3

Δc

(b)



120597119888

120597119905=

Δ119888

(2120587)119882 + 119879119878 cos 120579(26)


120597119888

120597119911=

Δ119888

(2120587)119882 + 119879119878 cos 120579sin 120579 (27)


119865119911 = 119863119898

Δ119888

(2120587)119882 + 119879119878 cos 120579sin 120579 (119873119875 minus 1)119882119879119866 (28)



120597119888

120597119899=Δ119888

119879119878

(29)


120597119888

120597119911=Δ119888

119879119878

cos 120579 (30)


119865119911 = 119863119904

Δ119888

119879119878

1205871198822

4cos 120579 (31)


119863119904

119863119898

=4 (119873119875 minus 1) (119879119866119879119878) sin 120579

120587AR119878 cos 120579 ((2120587)AR119878 + cos 120579) (32)


5 Conclusions



120579

n

t

z

2

W

Wsin

120579+TSco

s120579Wcos120579 + TSsin120579

(a)

002

004

006

008

01

012

014

016

018

02

0

(b)








References


































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




Δx

Δz

i

i minus 1

tan120572 =120597c120597x

120597c120597z



119871norm =119871nano1198710

(22)


119863norm =119863nano119863mat

=12

(100381610038161003816100381610038161003816int1198601

119863 (120597119888120597119911) 1198891198601100381610038161003816100381610038161003816+100381610038161003816100381610038161003816int1198602

119863 (120597119888120597119911) 1198891198602100381610038161003816100381610038161003816)

119863 (Δ119888119871119911) 1198601

(23)


119863(120597119888120597119911)1198891198601 =

minusint1198602










0 1 2 3 4088

089

090

091

092

093

094

095

TGTP

TG = 0


Dno

rm


02 04 06 08 10 12 14 16 18091

092

093

094

095

096


Dno

rm






0 1 2 3 4

084

087

090

093

096

099

TGTP




f = 0015 f = 00225 f = 003

Dno

rm








3 4 5084

086

088

090

092

094

096

098

100




f = 0015 f = 00225 f = 003

NP

Dno

rm





003 005 007 009 011 013075

078

081

084

087

090

093

096

S

ARS = 833

ARS = 625ARS = 500

ARS = 435ARS = 333

Dno

rm


3 4 5 6 7 8 9

002

004

006

008

010

012

014

ARS

DsD

m




120597119888

120597119911= 0 (24)


119871diff =2120587119882+119879119878 cos 120579 (25)


120579

n

t

z

W

Wsin

120579+TSco

s120579

Wcos120579 + TSsin120579

(a)

120579

n

t

z

Wsin

120579+TSco

s120579

Wcos120579 + TSsin120579

V1

V2

V3

Δc

(b)



120597119888

120597119905=

Δ119888

(2120587)119882 + 119879119878 cos 120579(26)


120597119888

120597119911=

Δ119888

(2120587)119882 + 119879119878 cos 120579sin 120579 (27)


119865119911 = 119863119898

Δ119888

(2120587)119882 + 119879119878 cos 120579sin 120579 (119873119875 minus 1)119882119879119866 (28)



120597119888

120597119899=Δ119888

119879119878

(29)


120597119888

120597119911=Δ119888

119879119878

cos 120579 (30)


119865119911 = 119863119904

Δ119888

119879119878

1205871198822

4cos 120579 (31)


119863119904

119863119898

=4 (119873119875 minus 1) (119879119866119879119878) sin 120579

120587AR119878 cos 120579 ((2120587)AR119878 + cos 120579) (32)


5 Conclusions



120579

n

t

z

2

W

Wsin

120579+TSco

s120579Wcos120579 + TSsin120579

(a)

002

004

006

008

01

012

014

016

018

02

0

(b)








References


































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




0 1 2 3 4088

089

090

091

092

093

094

095

TGTP

TG = 0


Dno

rm


02 04 06 08 10 12 14 16 18091

092

093

094

095

096


Dno

rm






0 1 2 3 4

084

087

090

093

096

099

TGTP




f = 0015 f = 00225 f = 003

Dno

rm








3 4 5084

086

088

090

092

094

096

098

100




f = 0015 f = 00225 f = 003

NP

Dno

rm





003 005 007 009 011 013075

078

081

084

087

090

093

096

S

ARS = 833

ARS = 625ARS = 500

ARS = 435ARS = 333

Dno

rm


3 4 5 6 7 8 9

002

004

006

008

010

012

014

ARS

DsD

m




120597119888

120597119911= 0 (24)


119871diff =2120587119882+119879119878 cos 120579 (25)


120579

n

t

z

W

Wsin

120579+TSco

s120579

Wcos120579 + TSsin120579

(a)

120579

n

t

z

Wsin

120579+TSco

s120579

Wcos120579 + TSsin120579

V1

V2

V3

Δc

(b)



120597119888

120597119905=

Δ119888

(2120587)119882 + 119879119878 cos 120579(26)


120597119888

120597119911=

Δ119888

(2120587)119882 + 119879119878 cos 120579sin 120579 (27)


119865119911 = 119863119898

Δ119888

(2120587)119882 + 119879119878 cos 120579sin 120579 (119873119875 minus 1)119882119879119866 (28)



120597119888

120597119899=Δ119888

119879119878

(29)


120597119888

120597119911=Δ119888

119879119878

cos 120579 (30)


119865119911 = 119863119904

Δ119888

119879119878

1205871198822

4cos 120579 (31)


119863119904

119863119898

=4 (119873119875 minus 1) (119879119866119879119878) sin 120579

120587AR119878 cos 120579 ((2120587)AR119878 + cos 120579) (32)


5 Conclusions



120579

n

t

z

2

W

Wsin

120579+TSco

s120579Wcos120579 + TSsin120579

(a)

002

004

006

008

01

012

014

016

018

02

0

(b)








References


































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




3 4 5084

086

088

090

092

094

096

098

100




f = 0015 f = 00225 f = 003

NP

Dno

rm





003 005 007 009 011 013075

078

081

084

087

090

093

096

S

ARS = 833

ARS = 625ARS = 500

ARS = 435ARS = 333

Dno

rm


3 4 5 6 7 8 9

002

004

006

008

010

012

014

ARS

DsD

m




120597119888

120597119911= 0 (24)


119871diff =2120587119882+119879119878 cos 120579 (25)


120579

n

t

z

W

Wsin

120579+TSco

s120579

Wcos120579 + TSsin120579

(a)

120579

n

t

z

Wsin

120579+TSco

s120579

Wcos120579 + TSsin120579

V1

V2

V3

Δc

(b)



120597119888

120597119905=

Δ119888

(2120587)119882 + 119879119878 cos 120579(26)


120597119888

120597119911=

Δ119888

(2120587)119882 + 119879119878 cos 120579sin 120579 (27)


119865119911 = 119863119898

Δ119888

(2120587)119882 + 119879119878 cos 120579sin 120579 (119873119875 minus 1)119882119879119866 (28)



120597119888

120597119899=Δ119888

119879119878

(29)


120597119888

120597119911=Δ119888

119879119878

cos 120579 (30)


119865119911 = 119863119904

Δ119888

119879119878

1205871198822

4cos 120579 (31)


119863119904

119863119898

=4 (119873119875 minus 1) (119879119866119879119878) sin 120579

120587AR119878 cos 120579 ((2120587)AR119878 + cos 120579) (32)


5 Conclusions



120579

n

t

z

2

W

Wsin

120579+TSco

s120579Wcos120579 + TSsin120579

(a)

002

004

006

008

01

012

014

016

018

02

0

(b)








References


































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




120579

n

t

z

W

Wsin

120579+TSco

s120579

Wcos120579 + TSsin120579

(a)

120579

n

t

z

Wsin

120579+TSco

s120579

Wcos120579 + TSsin120579

V1

V2

V3

Δc

(b)



120597119888

120597119905=

Δ119888

(2120587)119882 + 119879119878 cos 120579(26)


120597119888

120597119911=

Δ119888

(2120587)119882 + 119879119878 cos 120579sin 120579 (27)


119865119911 = 119863119898

Δ119888

(2120587)119882 + 119879119878 cos 120579sin 120579 (119873119875 minus 1)119882119879119866 (28)



120597119888

120597119899=Δ119888

119879119878

(29)


120597119888

120597119911=Δ119888

119879119878

cos 120579 (30)


119865119911 = 119863119904

Δ119888

119879119878

1205871198822

4cos 120579 (31)


119863119904

119863119898

=4 (119873119875 minus 1) (119879119866119879119878) sin 120579

120587AR119878 cos 120579 ((2120587)AR119878 + cos 120579) (32)


5 Conclusions



120579

n

t

z

2

W

Wsin

120579+TSco

s120579Wcos120579 + TSsin120579

(a)

002

004

006

008

01

012

014

016

018

02

0

(b)








References


































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




120579

n

t

z

2

W

Wsin

120579+TSco

s120579Wcos120579 + TSsin120579

(a)

002

004

006

008

01

012

014

016

018

02

0

(b)








References


































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials


























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



Documents

Research Article Finite Element Modeling of Multiscale ...downloads.hindawi.com/journals/jnm/2015/482698.pdf · Finite Element Modeling of Multiscale Diffusion in Intercalated Nanocomposites