Modeling of rheological behavior of nanocomposites by … · 2005. 1. 11. · Modeling of rheological behavior of nanocomposites by Brownian dynamics simulation Korea-Australia Rheology

Korea-Australia Rheology JournalVol. 16, No. 4, December 2004 pp. 201-212

Modeling of rheological behavior of nanocomposites byBrownian dynamics simulationYoung Seok Song and Jae Ryoun Youn*

School of Materials Science and Engineering, Seoul National University,56-1, Shinlim-Dong, Gwanak-Gu, Seoul 151-744, Korea

(Received December 1, 2004)

Abstract

Properties of polymer based nanocomposites depend on dispersion state of embedded fillers. In order toexamine the effect of dispersion state on rheological properties, a new bi-mode FENE dumbbell model wasproposed. The FENE dumbbell model includes two separate ensemble sets of dumbbells with different fric-tion coefficients, which simulate behavior of well dispersed and aggregated carbon nanotubes (CNTs). Anew parameter indicating dispersion state of the CNT was proposed to account for degree of dispersionquantitatively as well as qualitatively. Rheological material functions in elongational, steady shear, andoscillatory shear flows were obtained numerically. The CNT/epoxy nanocomposites with different dis-persion state were prepared depending on whether a solvent is used for the dispersion of CNTs or not. Dis-persion state of the CNT in the epoxy nanocomposites was morphologically characterized by the fieldemission scanning electronic microscope and the transmission electron microscope images. It was foundthat the numerical prediction was in a good agreement with experimental results especially for steady stateshear flow.Keywords : nanocomposites, FENE dumbbell, carbon nanotubes, rheology

1. Introduction

Polymer based nanocomposites filled with carbon nan-otubes (CNTs) have been studied to achieve superiorelectrical, thermal, and mechanical properties comparedwith composites filled with other carbon nanomaterials(CNMs) like carbon blacks (CBs) (Allaoui et al., 1993).The outstanding properties are attributed to the highaspect ratio which is generally higher than 1000. In orderto achieve optimal enhancement in the property of theCNT/polymer composites, there are several key issues tobe resolved, i.e., improved dispersion of CNTs, align-ment of CNTs in the polymer resin, and functionalizationof the CNT surface for good adhesion. In order to dis-perse CNTs in the polymer homogeneously, modificationof pristine CNT surface such as chemical or plasmatreatments has been carried out (Shaffer et al., 1998).

It is necessary to understand rheological behavior of theCNT/polymer mixture so as to improve manufacturingprocedure of the nanocomposites. There are a few reportson rheological behavior of CNT/polymer composites.Pötschke et al. (2002) investigated rheological propertiesof the CNT/polycarbonate (PC) composites. It was found

that increase in the viscosity of the nanocomposites filledwith CNTs was much higher than increase in the viscosityof polymer composites filled with carbon fibers or CBs.The rheological behavior of aqueously dispersed CNTswas studied by Kinloch et al. (2002) under consideration ofthe interaction between the nanotubes. They reported thatdispersion state of the CNT was highly sensitive to thestrain applied in the linear viscoelastic region and the stor-age and loss moduli were independent of frequency. It wasshown that viscosity increased to the highest point whenfibrous fillers such as carbon fibers were added and to thelowest point when spherical fillers were added. Agglom-erates of the fillers caused higher viscosity (Shenoy, 1999)since the presence of agglomerates leads to apparent higherfiller loading. It was found that the nanocomposites filledwith functionalized CNTs had better dispersion of the CNTand showed higher storage modulus and complex viscosityat low frequency. On the other hand, it hasn’t been numer-ically investigated how the dispersion state of the CNTaffects various properties of the nanocomposites. In thisstudy, the effect of the CNT dispersion on rheological behav-ior was examined based on a proposed bi-mode FENE(finitely extensible nonlinear elastic) dumbbell model.

Dynamic behavior of polymeric solutions is an area ofgreat interest in the field of polymer science (Larson,1988). However, the size and complexity of polymer mol-

*Corresponding author: [email protected]© 2004 by The Korean Society of Rheology

Korea-Australia Rheology Journal December 2004 Vol. 16, No. 4 201

Young Seok Song and Jae Ryoun Youn

ecules prohibit use of realistic polymer models. Therefore,molecular theories of polymers have concentrated on thesimple models that include only the most basic charac-teristics of polymer molecules. The model used frequentlyis a dumbbell model which consists of beads and springs.This study also adopted the dumbbell model to describerheological behavior of the CNT/polymer compositesquantitatively as well as qualitatively. Microstructuralmodeling of the nanocomposites was conducted usingBrownian dynamics simulation which is a very powerfultool for understanding the dynamics of polymers both atequilibrium and during undergoing flow. Originally, thesimulation is based on numerical integration of the equa-tions of motion for the beads of a single chain molecule,where presence of Brownian or random forces implies thatthe equations of motion become stochastic differentialequations. The diffusion equations for suitable configura-tional distribution of the dumbbells, which are oftenreferred to as Fokker-Planck equations, are equivalent tostochastic differential equations. Numerical integration ofthe stochastic differential equations leads to stochastic sim-ulation, which is known to be so simple that even com-plicated problems can be treated without much efforts.

Warner (1972) proposed a FENE dumbbell model andcalculated steady shear flow and small amplitude oscil-latory flow by using the model. He found that variousmaterial functions varied between Hookean spring andrigid dumbbell. Christiansen and Bird (1977; 1978) com-pared experimental results for intrinsic viscosity and intrin-sic complex viscosity with values predicted by the FENEdumbbell and showed that the FENE dumbbell adequatelypredicted the intrinsic viscosity, but less successfully por-trayed the behavior in small amplitude oscillatory motion.Armstrong and Ishikawa (1980) adopted nearly-Hookeandumbbells and showed substantial improvements overHookean dumbbell predictions. They used perturbationscheme to obtain an expression for the stress tensor. Birdand Deaguiar (1983) employed an encapsulated FENEdumbbell model for concentrated solutions and melts in thecase where the Brownian motion and hydrodynamic forcesacting on the beads were made anisotropic. Bird and Wiest(1985) modified kinetic theory for suspensions of elasticdumbbells by introducing anisotropic hydrodynamic dragand anisotropic Brownian motion. Fan (1985) improvedrheological features of FENE dumbbells in dilute suspen-sion undergoing steady-state shear flow by assuming a dis-tribution function to remove the singularity and improvebehavior of the weight function in the Galerkin expression.Schieber (1993) proposed a noninertial Hookean dumbbellwith internal viscosity in transient and steady shear flowsby using Gaussian closure on the second moment equationof the dumbbell configuration, which greatly overestimatedthe relative stretching of the polymer coil. Herrchen andÖttinger (1997) compared three dumbbell models, i.e.,

original FENE model with the Warner spring force, FENE-P model based on the Peterlin approximation, and FENE-CR model as suggested by Chilcott and Rallison. Vaccaroand Marrucci (2000) developed a model for the imper-manent network of associating telechelic chain whichincorporated the basic molecular mechanisms and obtainedan analytical expression for the viscosity at low shear rates.Hernández Cifre et al. (2003) employed two kinds ofensembles of dumbbells which consist of active and dan-gling chains, to model reversible network of associativechains.

As mentioned above, there have been many studies onmodeling rheological properties and outstanding resultsusing Brownian dynamic simulations for pure polymersolutions or melts. However, few studies on theoreticalmodeling of rheological properties of nanocomposites havebeen reported. In this study, rheological properties of CNT/epoxy composites were characterized by experimentalmeasurements and a new bi-mode FENE dumbbell modelwas proposed to predict rheological behavior of the nano-composites.

2. Experimental

2.1. Materials and preparation of CNTs/epoxycomposites

Multiwalled carbon nanotubes (MWNTs) used in thisstudy were supplied by Iljin Nanotech Co. The CNTs syn-thesized by the chemical vapor deposition (CVD) processhave the average diameter of 13 nm and the length of 10µm. Epoxy resin was selected as the polymer matrixbecause it is known that the CNTs are dispersed well in theepoxy resin compared with other polymer resins. Epoxyresin (YD 128) and hardener (TH 432) were obtained fromthe Kukdo Chemical, based on diglycidyl ether of bisphe-nol-A and modified aromatic amine, respectively. TheCNTs prepared were composed of many aggregates of dif-ferent sizes. The aggregates would be obstacles to uniformdispersion of the MWNTs and were hardly broken intoindividual tubes in the epoxy resin. The epoxy compositescontaining well dispersed CNTs were prepared by the fol-lowing procedures. The CNTs of 0.5, 1.0 and 1.5 wt%were first dispersed in ethanol, under sonication for 2hours. The CNT/ethanol solutions with different CNTweight fractions were then mixed with the epoxy resin.The mixture was also sonicated for 1 hour at 80oC and keptin a vacuum oven for 5 days to remove air bubbles and theethanol. After adding the hardener, the mixture was stirredby using a magnetic bar for 15 minutes under sonication.On the other hand, the poorly dispersed CNT/epoxy com-posites were prepared under sonication for 3 hours withoutusing the solvent. It was expected that relatively well dis-persed CNT nanocomposites could be obtained when thesolvent was employed in the CNT dispersion process.

202 Korea-Australia Rheology Journal

Modeling of rheological behavior of nanocomposites by Brownian dynamics simulation

However, without using the solvent, the aggregates of pris-tine CNTs could remain in the polymer matrix. The qualityof dispersion was verified by FESEM and TEM obser-vations.

2.2. Rheological measurementsDynamic rheological measurement was carried out by

using a stress controlled rotational rheometer (C-VOR fromBohlin instrument Ltd). The measurement was conducted inan oscillatory shear mode using a parallel plate geometry at25oC. Stress sweep tests were performed to identify linearviscoelastic regime. Shear stress was fixed at 1 Pa during allfrequency sweep tests and sample thickness was set to be1 mm. Steady shear test was also performed with anotherstress controlled rotational rheometer (AR2000). Cone andplate geometry was employed to obtain steady-state shearviscosity and first normal stress coefficient.

2.3. Morphological characterizationFESEM images were obtained with JEOL JSM-6330F

operating at 5 kV to examine surfaces of the specimensfractured during tensile test. They were coated with Pt for5 minutes prior to the observation. In addition to FESEM,TEM observation was carried out with JEOL JEM-2000EXII at 100 kV for more exact characterization of themorphology. The TEM specimens were microtomed to anultra thin section with the thickness of about 80 nm andcoated with carbon for 7 minutes to prevent the specimensfrom the degradation caused by the irradiation of electrons.

3. Microstructural modeling of nanocomposites

3.1. FENE dumbbellsFENE dumbbell model is appropriate for modeling dilute

polymer solutions. It was assumed that the CNT/epoxynanocomposites prepared in this study included CNTs ofso low weight fraction that they could be thought as adilute suspension of the CNTs in the epoxy matrix. In thebi-mode FENE dumbbell model, the FENE dumbbells areregarded as the CNTs instead of polymer chains. The mainsimplification of the bi-mode FENE dumbbell model isthat aggregated and free FENE dumbbells can simulaterheological behavior for the CNT/epoxy composites thatconsists of the mixture of aggregated and well dispersedCNTs. It is proposed that the aggregated and free FENEdumbbells imitate the aggregated CNTs and well dispersedCNTs, respectively. The aggregated CNTs within polymermatrix have lower mobility than well dispersed CNTswhich are also called free CNTs. The aggregated and freeCNTs are modeled by two separate ensemble sets of FENEdumbbells as shown in Fig. 1. It is assumed that the aggre-gated and free FENE dumbbells do not interact each otherand the total number of aggregated and free dumbbells isconstant.

3.2. Bi-mode FENE dumbbellThe simplest method to simulate the polymer chain is to

employ Hookean elastic dumbbell. Because the dumbbellmodel can be solved analytically, it has been used todevelop constitutive equations. However, there are manylimitations to the dumbbell model such as infinite elon-gational viscosity under a finite elongational rate. A solu-tion to overcome these problems is to replace the Hookeanspring by a non-linear elastic spring to limit the dumbbellextension to a maximum value, i.e., FENE spring intro-duced by Warner. In the FENE dumbbell model, connectorforce is written as below.

(1)

where Q0 denotes the maximum possible spring extension.It is known that the model can describe the shear ratedependence of the shear viscosity well but not the complexviscosity quantitatively since it does not include sufficientinternal motions of the macromolecules. A major draw-back of the FENE model is that it does not yield a closedform constitutive equation for the polymer stress. TheFENE kinetic theory involves a relaxation time λH = ζ/4Hand a dimensionless finite extensibility parameter b =

In this study, b is fixed as 50, which is within therange of values that are consistent with the underlyingkinetic framework. In order to simplify numerical calcu-lations the following dimensionless quantities are employed.

(2)

By setting na as the number of aggregated dumbbells perunit volume and nf as the number of free dumbbells perunit volume, total number density n is given as below.

n = na + nf (3)

A new parameter, c, is defined such that it representsdegree of dispersion state of CNTs as follows.

(4)

The model built under the above assumptions is governedby a set of two Fokker-Planck equations. Let ψa(Qa, t) andψf (Qf, t) be distribution functions describing the probabilityof finding aggregated and free dumbbells. Relationsbetween these distribution functions and c are found to be

(5)

where dV denotes the volume element in Q space. The dif-fusion equation for the distribution function of aggregateddumbbells is given as

(6)

Fc HQ1 Q( 2– Q0

2)⁄---------------------------=

HQ02 kT⁄ .

Q̂ Q kBT H⁄( )1 2⁄⁄= , t̂ t λH⁄= , κ κλH= , ψ̂ ψ kBT( ) H⁄( )3 2⁄=

na

n----- c=

nf

n---- 1 c–=,

c ψa Vd∫= 1 c– ψf Vd∫=,

∂ψa

∂t--------- ∂

∂Qa---------– κ Qaψa⋅ 2

ζa----Fa

cψa– 2kBT

ζa----------- ∂

∂Qa--------- ∂

∂Qa---------ψa⋅+⋅=



where ζa is the friction coefficient of aggregated dumbbells,kB is the Boltzmann constant, and T is the temperature. Sincehomogeneous flows are considered, the velocity field can bewritten as v = κ · r where The evolution equationfor the distribution function of free dumbbell is written as

(7)

where ζf is the friction coefficient of free dumbbells, which islower than that of aggregated dumbbells. The Fokker-Planckequations are expressed in the following dimensionless formsbased on the assumption that λH is equal to ζf /4H.

(8)

(9)

It is found that Eq. (9) is equal to that of original FENEdumbbell model. When the Brownian motion is isotropic,the above pair of equations are equivalent to the Ito sto-chastic differential equations (SDE) for a three dimensionalMarkov process

(10)

(11)

where W is the three dimensional Wiener process. Sincethe non-linear SDEs cannot be solved analytically, numer-ical methods must be used to solve them. The second-orderpredictor-corrector algorithm suggested by Öttinger (1996)is adopted to calculate Eq. (10) and Eq. (11) as below.

(12)

(13)

(14)

(15)

where ∆Wj = Wj+1 − Wj. The magnitude of connect vectorsdetermined by Eq. (12) to (15) is between 0 and b1/2. Forany given velocity gradient κ, contribution of the dumb-bells to the instantaneous stress tensor is calculated bymeans of weighted average over both the aggregated andfree dumbbells based on the Kramers form expression.

(16)

Total stress tensor is given by the following equation.

(17)

here τs is the contribution by the solvent of the viscosity ηs.In order to obtain more precise solutions in the stochastic

simulations, it is necessary to implement variance reductiontechniques. Recently, it has been reported that variancereduction techniques are applied to simulations of polymerdynamics (Öttinger, 1994). Fluctuation of the averagedstress tensor may occur at low flow rate. Parallel equilib-rium simulation employing the same initial configurationsand the same stochastic displacements is carried out. Inorder to solve the ζf /ζa Eq. (12) to (15), a ratio of frictioncoefficient, must be determined. The translational diffusiv-ity Dtr defined by the Nernst-Einstein equation is related tothe friction coefficient Z for the entire molecule as follows.

(18)

According to phase-space theory (Bird et al., 1987), whenhydrodynamic interaction is neglected for the dilute solu-tion, the translational diffusivity is also written as below.

(19)

κ ∇ v( )T.=

∂ψf

∂t-------- ∂

∂Qf--------– κ Qfψf⋅ 2

ζ f----Ff

cψf– 2kBT

ζ f----------- ∂

∂Qf-------- ∂

∂Qf--------ψf⋅+⋅=

∂ψ̂a

∂t̂--------- ∂

∂Q̂a---------– κ̂ Q̂aψ̂a⋅ 1

2---ζ f

ζa----F̂a

cψ̂a– 1

2---ζ f

ζa---- ∂

∂Q̂a

--------- ∂∂Q̂a

---------ψ̂a⋅+⋅=

∂ψ̂f

∂t̂-------- ∂

∂Q̂f

--------– κ̂ Q̂fψ̂f⋅ 12---F̂f

cψ̂f– 1

2--- ∂∂Q̂f

-------- ∂∂Q̂f

--------ψ̂f⋅+⋅=

Q̂.

dQ̂a κ̂ Q̂a⋅ 12---ζ f

ζa---- 1

1 Q̂a2– b⁄

-------------------Q̂a– dt̂ ζ f

ζa----dWa+=

dQ̂f κ̂ Q̂f⋅ 12--- 1

1 Q̂f2– b⁄

-------------------Q̂f– dt̂ dWf+=

Q'ˆa j 1+, Q̂a j, κ̂ j Q̂a j,⋅ 1

2---ζ f

ζa---- 1

1 Q̂a j,2– b⁄

---------------------Q̂a j,– + ∆t̂ ζ f

ζa----∆Wj+=

Q̂a j 1+, Q̂a j,

κ̂ j 1+ Q̂a j 1+,⋅ 12---ζ f

ζa---- 1

1 Q̂a j 1+,2– b⁄

--------------------------Q̂a j 1+,–

κ̂ j Q̂a j,⋅ 12---ζ f

ζa---- 1

1 Q̂a j 1+,2– b⁄

--------------------------Q̂a j,–+

+ ∆t̂=

ζ f

ζa----∆Wj+

Q'ˆf j 1+, Q̂f j, κ̂ j Q̂f j,⋅ 1

2--- 1

1 Q̂f j,2– b⁄

---------------------Q̂f j,– + ∆t̂ ∆Wj+=

Q̂f j 1+, Q̂f j,

κ̂ j 1+ Q̂f j 1+,⋅ 12--- 1

1 Q̂f j 1+,2– b⁄

--------------------------Q̂f j 1+,–

κ̂ j Q̂f j,⋅ 12--- 1

1 Q̂f j 1+,2– b⁄

--------------------------Q̂f j,–+

+ ∆t̂ ∆Wj+=

τp na QaFac⟨ ⟩ nf Qf Ff

c⟨ ⟩ f nkBTδ–+=

τ τ s τp ηsγ· τp+=+=

DtrkBTZ

--------=

DtrkBTNζ--------=

Fig. 1. Modeling of two kinds of dumbbell sets, (a) aggregatedFENE dumbbell which has lower mobility and (b) freeFENE dumbbell which has higher mobility.



For polymer melts or concentrated polymer solutions, thetranslational diffusivity is expressed as

(20)

where N is 2 in the case of elastic dumbbell. Eq. (20)means that concentrated polymer solutions have lower dif-fusion, i.e., lower mobility than dilute solutions. Becausethe aggregated and free dumbbells copy the concentratedand dilute polymer solutions, it is assumed that ζf /ζa isequal to 0.5.

5. Results and discussion

5.1. Experimental results5.1.1. Rheological propertiesFig. 2(a) and (b) show the storage modulus, G', for the

epoxy composites filled with well dispersed CNTs andpoorly dispersed CNTs. It is found that the storage mod-ulus of the nanocomposite dramatically increases with

increase in the CNT loading. The increase in the storagemodulus can be explained by the fact that the CNTs gen-erate the percolated structure due to their high aspect ratioand high surface area. Fiber reinforced composites withhigh fiber volume fraction show the same result as theCNT nanocomposites. The epoxy composites filled withCNTs dispersed by using the solvent have much higherstorage modulus than those filled with CNTs dispersedwithout the solvent. Terminal slopes of G' and G'' are listedin Table 1. As the CNT loading increases, the storage mod-ulus exhibits a frequency independent solid like behavior atlower frequencies. Fig. 2 and Table 1 indicate that thenanocomposites do not show terminal behavior. Thepower-law dependence for G' is much smaller than theexpected 2. In the case of poor dispersion of CNTs, moregradual decrease is observed in the slope of G' with respectto the CNT loading. It is attributed to the fact that theagglomerates in the poorly dispersed CNT nanocompositesact as large particles as if the filler loading were higher.The agglomerates trap polymer resin in the void betweenCNTs and the nanocomposites behave as if it had highervolume fraction of fillers. Differences in the terminalslopes are closely related to the internal structure of thenanocomposites which is affected by particle-particle inter-action of the CNTs in the polymer matrix. Dynamic lossmodulus, G'', of the nanocomposites is plotted in Fig. 3(a)and (b). It is known that the G' is more sensitive to mor-phology of particles than the G''. The loss modulus of theCNT/epoxy composite also shows different behavior withrespect to the nanotube loading and the degree of disper-sion. G' and G'' have the similar rheological behavior, i.e.,Fig. 3 and Table 1 show non-terminal behavior of the lossmodulus and the non-terminal behavior is more dominantin the case of poorly dispersed CNT/epoxy composites.Complex viscosity, |η*|, of the CNT/epoxy composites isshown in Fig. 4. The epoxy composites filled with thepoorly dispersed CNTs exhibit stronger non-Newtonian

DtrkBTN 2ζ---------=

Fig. 2. Storage moduli of epoxy nanocomposites embedded with(a) well dispersed CNTs and (b) poorly dispersed CNTs asa function of angular frequency.

Table 1. Terminal slopes of G' and G'' for CNTs/epoxy nano-composites prepared (a) with the solvent and (b) withoutthe solvent

(a)

Samples Slope of G' Slope of G''0.5 wt% CNT 0.83 0.881.0 wt% CNT 0.67 0.801.5 wt% CNT 0.61 0.76

(b)

Samples Slope of G' Slope of G''0.5 wt% CNT 0.59 0.761.0 wt% CNT 0.43 0.701.5 wt% CNT 0.35 0.61



behavior, close to the power law fluid, than those with thewell dispersed CNTs. Viscosity of the former increasesmore rapidly with increase in the CNT loading. As thefiller loading increases, interaction between the CNTs andthe polymer resin becomes larger due to the high aspectratio of the CNTs. The increase in the complex viscositywith respect to the CNT loading is primarily caused by thelarge increase in the storage modulus, G'. Steady state shearviscosity and the first normal stress coefficient of the nano-composites with different dispersion states are plotted inFig. 5 for 1 wt% CNT loading. The poorly dispersed CNT/epoxy composites have higher rheological material func-tions than the well dispersed ones both in the steady-stateshear flow and in the oscillatory shear flow.

5.1.2. MorphologyTEM images of the CNTs dispersed in the epoxy resin

are shown in Fig. 6 and Fig. 7. After conducting the tensileexperiment of the CNT/epoxy composites, FESEM imagesof the fracture surface are shown in Fig. 8 and Fig. 9. TheFigs. 6 and 8 show that the CNTs are relatively well dis-persed in the epoxy resin. The CNTs are not broken but

pulled out due to the weak interfacial bonding between theCNT and the polymer resin. The CNTs dispersed in thepolymer without using the solvent exist in the form ofagglomerates as shown in Fig. 7 and Fig. 9. The whiteregions in Fig. 9(a) represent the aggregated CNTs and theregion is magnified as shown in Fig. 9(b) in order to verifythe existence of CNTs. The agglomerates reduce reinforc-ing effects of the CNTs because they are acting as flaws inthe resin. This is why the poorly dispersed CNT/epoxycomposites have lower mechanical properties than the welldispersed ones.

5.2. Numerical results5.2.1. Material functions in steady state shear flowCalculated shear viscosity and first normal stress coef-

ficient are shown in Figs. 10 and 11. Total 200000 FENEdumbbells and dimensionless time interval of 0.00001were employed in the computation of bi-mode FENEdumbbell model. In order to verify the numerical data theresults calculated when c = 0, which means that the pro-posed bi-mode FENE dumbbell is identical to the original

Fig. 3. Loss moduli of epoxy nanocomposites filled with (a) welldispersed CNTs and (b) poorly dispersed CNTs withrespect to angular frequency.

Fig. 4. Complex viscosities of epoxy nanocomposites filled with(a) well dispersed CNTs and (b) poorly dispersed CNTs.



FENE dumbbell model, were compared with thoseobtained by means of perturbation method. It is found thatthe aggregated FENE dumbbells with high c values, whichrepresents the presence of aggregated fillers, have highershear viscosities and first normal stress coefficients. Stressgrowth after inception of shear flow was analyzed by

assuming that the fluid was at rest before start-up and thenthe constant shear rate was applied. Growth of shear vis-cosity and first normal stress function is plotted withrespect to dimensionless time in Figs. 12 and 13. The mate-rial functions in the steady state shear flow exhibit over-shoot phenomena and have different steady state valueswith respect to the degree of dispersion. The overshootoccurs at the same time regardless of the CNT dispersion.It is found that the shear viscosity is more sensitive to theCNT dispersion than the first normal stress coefficient.

5.2.2. Material functions in elongational flowElongational viscosities are plotted as a function of elon-

gation rate in Fig. 14. It is shown that the elongational vis-cosity goes up with increase in the elongation rate and theelongational viscosity of aggregated FENE dumbbells ishigher than that of free FENE dumbbells. Numericalresults of bi-mode FENE dumbbells calculated when c = 0are in a good agreement with analytic solutions in the lim-

Fig. 5. (a) steady shear viscosity and (b) first normal stress coef-ficient of the CNTs/epoxy composites with respect toshear rate.

Fig. 6. TEM images of epoxy nanocomposites embedded withrelatively well dispersed CNTs of 1.5 wt%.

Fig. 7. TEM images of epoxy nanocomposites filled with poorlydispersed CNTs of 1.5 wt%.

Fig. 8. FESEM images of the fracture surface for CNTs/ epoxycomposites prepared by using a solvent.




Fig. 9. FESEM images of CNTs/epoxy nanocomposites preparedwithout the solvent.

Fig. 10. Calculated shear viscosity as a function of shear rate forvarious dispersion states of the filler.

Fig. 11. Calculated first normal stress coefficient as a function ofshear rate for various dispersion states of the filler.

Fig. 12. Growth of shear viscosity after inception of shear flowwhen dimensionless shear rate is λHγ· 10.=

Fig. 13. Growth of first normal stress coefficient after inceptionof shear flow when dimensionless shear rate isλHγ· 10.=


iting range of elongation rate. Growth of the viscosity afterinception of elongational flow for the various dispersionstates is shown in Fig. 15. As the number of aggregatedFENE dumbbells increases, higher steady elongational vis-cosity is obtained. As seen in Fig. 16, the elonational vis-cosities increase until they reach steady state. Higher strainhardening behavior is observed as the elongation rateincreases. Figs. 17 and 18 show radial distributions ofaggregated and free FENE dumbbell lengths in the start-upof elongational flow. After the flow is started from theequilibrium state, the radial distribution of the dumbbelllength evolves toward steady state with respect to timetoward steady state. Since the aggregated FENE dumbbellshave a higher friction coefficient, they are subject to largerfriction force against the applied flow field. As a results,they have higher radial dumbbell length and narrower dis-tribution of length than free dumbbells. It is also verified

Fig. 14. Elongational viscosity calculated as a function of elon-gation rate by using bi-mode FENE dumbbells.

Fig. 15. Time evolution of elongational viscosity after inceptionof elongational flow when dimensionless elongation rateis λHε· 10.=

Fig. 16. Time evolution of elongational viscosity after inceptionof elongational flow when c = 0.5.

Fig. 17. Change in distribution of aggregated FENE dumbbelllength after start-up of elongational flow when dimen-sionless elongation rate is 1.0.

Fig. 18. Change in distribution of free FENE dumbbell lengthafter start-up of elongational flow when the number ofdumbbell is 10,000 and dimensionless elongation rate is1.0.



that the radial dumbbell length in the FENE dumbbellmodel can not exceed that is the upper limit (in thisstudy, b = 50).

5.2.3. Material functions in small amplitude oscilla-tory flow

The predicted storage modulus increases with increase inthe degree of dispersion as shown in Fig. 19. Generally, itis known that the polymer solution filled with aggregatedparticles has higher storage modulus than that with welldispersed particles and the storage modulus has a solid likebehavior at low frequency [19]. The bi-mode FENE dumb-bell model can account for the increment of storage mod-ulus in polymer systems containing the aggregated fillers.However, the model fails to depict solid like behavior ofstorage modulus, which is caused by original limitation ofthe FENE dumbbell. Fig. 20 shows the effect of dispersionstate on the loss modulus. The loss modulus of free FENEdumbbells is higher than that of aggregated FENE dumb-

bells in the range of high frequency, i.e., high deformationrate.

5.3. Comparison between numerical and experi-mental results

The results predicted and measured in the case of steadystate shear flow are compared as shown in Fig. 21. Thepredicted shear viscosity has a good agreement with exper-imental data as shown in Fig. 21(a). That is, the aggregatedFENE dumbbells and CNT nanocomposites exhibit highershear viscosities than free FENE dumbbells and well dis-persed CNT nanocomposites. The calculated data were fit-ted based on the assumption that relaxation time λH of theaggregated FENE dumbbell is twice as long as that of thefree FENE dumbbell since ζf /ζa = 0.5. It is indicated thatthe parameter c successfully represents degree of CNT dis-persion for nanocomposites in the steady state shear flowquantitatively. The first normal stress coefficients plotted inFig. 21(b) show that there is a little discrepancy between

b

Fig. 19. Storage modulus calculated as a function of frequencyby using bi-mode FENE dumbbells.

Fig. 20. Loss modulus calculated as a function of frequency byusing bi-mode FENE dumbbells.

Fig. 21. Comparison between predicted and measured (a) steadyshear viscosities and (b) first normal stress coefficients.The dumbbell data are fitted based on λH = 2.48 s andnkBT = 8.3 Pa⋅s.



the predicted and experimental slopes. However, the aggre-gated FENE dumbbells or aggregated CNTs still havehigher first normal stress coefficients than the free ones. Asa result, the bi-mode FENE dumbbell model can success-fully describe the influence of the CNT dispersion on thematerial functions in the steady state shear flow quanti-tatively as well as qualitatively. On the other hand, it isknown that original FENE dumbbells depict the depen-dence of shear viscosity on the shear rate well but do notprovide material functions quantitatively in the oscillatoryshear flow.

Fig. 22(a) and (b) present storage and loss moduli. Thecalculated results are different from the experimental data.Especially, the storage modulus obtained by using the bi-mode FENE dumbbell is overestimated in the range of lowfrequency. However, the bi-mode FENE dumbbells areable to capture convergence of the modulus valuesbetween aggregated and well dispersed CNT nanocom-posites with respect to frequency and explain dependenceof storage and loss moduli on the CNT dispersion.

6. Conclusions

In order to examine the influence of CNT dispersion onthe rheological behavior of the CNT filled composites, twodifferent specimens with the well dispersed and the poorlydispersed CNTs were prepared depending on whether sol-vent was used during the CNT dispersion process. Therheological properties of the nanocomposites were exper-imentally examined with respect to the weight fraction ofthe CNTs. Dispersion state of the CNT in the epoxy nano-composites was characterized morphologically by the fieldemission scanning electronic microscope and the trans-mission electron microscope. It was found by the exper-iment that the nanocomposites containing the poorlydispersed CNTs exhibit higher storage modulus, loss mod-ulus, and complex viscosity than those with the well dis-persed CNTs. To describe the effect of the CNT dispersionstate on the rheological properties numerically, Browniandynamics simulation based on stochastic differential equa-tion was performed. A new bi-mode FENE dumbbellmodel was proposed. Material functions in the elonga-tional, steady shear, and oscillatory shear flows were com-puted numerically and compared with measured ones. Itwas shown that the model depict rheological behavior ofthe CNT nanocomposites successfully and the new param-eter c can represent the degree of dispersion of filled par-ticles quantitatively.

Acknowledgements

This study was partially supported by the Korea Scienceand Engineering Foundation through the Applied Rheol-ogy Center (ARC) and by the Ministry of Science andTechnology through the National Research Laboratory.The authors are grateful for the support.

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Modeling of rheological behavior of nanocomposites by … · 2005. 1. 11. · Modeling of rheological behavior of nanocomposites by Brownian dynamics simulation Korea-Australia Rheology