Modelling of the chemo-rheological behavior of thermosetting polymer nanocomposites

Modelling of the Chemo–Rheological Behavior ofThermosetting Polymer Nanocomposites

Govanni Lelli, Andrea Terenzi, Jose M. Kenny, Luigi TorreDepartment of Civil and Environmental Engineering, University of Perugia,Loc. Pentima Bassa, 21, 05100 Terni, Italy

Epoxy/amine/montmorillonite nanocomposite systemsare studied in this article. Both a thermo2kinetic analy-sis (performed using a differential scanning calorime-ter) and a chemorheological characterization were car-ried out. The comparison of DSC thermograms hasshown that the addition the nanofiller does not changethe mechanism of crosslinking from a qualitativestandpoint, but the nanoreinforcement seemed to pro-duce an evident hindrance on the molecular mobility,which in turn influences the cure reactions. As none ofthe kinetic models available in literature was able todescribe the cure behavior of the aforementionedmaterials, a new phenomenological model is proposedin this work, which considers the activation energy ofthe networking process a function of the degree ofcure (rising exponentially towards infinity whenthe system approaches vitrification). The effects of thepresence of the clay on the chemorheology of thecomposites was resumed as follows: the viscosity ofthe nanocomposite was higher at any temperature, fur-thermore the composite viscosity showed an higherheating sensitivity before networking and gelationoccurred at lower degrees of cure, thus determining anarrower shape of the chemoviscosity behavior. Amodified version of the classical Williams–Landel–Ferry(WLF) equation that took into account the gelation andthe effects of crosslinking was uses as chemorheologi-cal model. Once the characteristic parameters of boththe neat resin and the nanocomposite were found, thechemoviscosity models were integrated using a numer-ical algorithm, to check their ability to foresee thebehavior of the systems during a dynamic cure pro-cess. A very good correspondence between the resultsand the experimental data was obtained. POLYM.COMPOS., 30:1–12, 2009. ª 2008 Society of Plastics Engineers

INTRODUCTION

The increased interest towards polymer/clay nanocom-

posites can be ascribed not only to the chance they offer

to get a noticeable improvement of mechanical, physical,

and chemical properties with small amounts of nanorein-

forcement, but also to the fact that the processing behav-

ior of the composite material is quite similar to that of

the neat matrix. Nonetheless, the interaction at a nano-

scale level between an organic and an inorganic phase

determines the need for a slight modification of process

parameters, independently from the nature (i.e., either

thermoplastic or thermosetting) of the polymeric matrix.

In particular, in the case of thermosetting nanocompo-

sites, the chemical reactions taking place during the cure

process affect resin morphology, which in turn determines

the behavior of the cured system. For this reason, the

relationships between processing, morphology, and prop-

erties of a wide range of epoxy systems have been deeply

investigated [1, 2] in the last few years. Such researches

pointed out that a deep understanding of curing mecha-

nisms and cure kinetics is the key for obtaining an opti-

mal cure, which allows one to model the reaction proc-

esses as accurately as possible. In this framework, the

elaboration of an appropriate kinetic equation is of funda-

mental importance. As the behavior of a thermosetting

polymer is strongly affected by the curing process, poorly

controlled cure can produce unexpected variations in the

thermal and rheological properties of the system during

the manufacturing processes and of the final product.

Therefore, an accurate model not only allows to monitor

the evolution of the curing system (to plan and/or to con-

trol single processes), but it can also be used for material

design, to compare the behavior of different formulations.

Although several studies have been carried out on the

modeling of the kinetic behavior of thermosetting resins,

the wide variety of resin formulations available, implies a

continuous study and introduces new experimental diffi-

culties and complexities. Moreover, switching to nano-

composites, introduces a further variable in the complex-

ity of the modelling problem, as the effects of the nanofil-

lers and the compatibilizer somehow affect the reaction

kinetics.

A better way to assure high quality and low manufac-

turing cost of the products is to create mathematical mod-

els of the different processing stages. In the last decade,

several studies focused on the analysis of the fundamental

Correspondence to: Luigi Torre; e-mail: [email protected]

DOI 10.1002/pc.20517

Published online in Wiley InterScience (www.interscience.wiley.com).

VVC 2008 Society of Plastics Engineers

POLYMER COMPOSITES—-2009

aspects of processing conditions [3–7]. With the develop-

ment of more powerful computers and software, there are

now cheap and robust tools to perform rapidly this other-

wise complex and time-consuming analysis. It is currently

possible to build up simulation algorithms that are able to

foresee how the material behaves during the manufactur-

ing and cure processes.

The different submodels that form a general nanocom-

posite processing model are shown in Fig. 1. The kineticmodel is needed to predict the reaction rate and the

degree of cure as a function of time and temperature. The

rheological model also describes the viscosity as a func-

tion of time and temperature. The viscosity is also de-

pendent on the degree of cure, so the rheological model

combined with the kinetic model forms a chemorheologi-cal model. Lastly, a heat transfer model is required to

describe the nonisothermal conditions in thicker compo-

sites: together with the kinetic and chemorheological

models, it allows to predict the degree of cure, viscosity,

and temperature as a function of time and position in the

composite.

This work proposes a model based on thermal analysis

and rheological experiments to describe the cure mecha-

nisms and the cure kinetics of a nanocomposite based on

epoxy matrix (DGEBA)/phyllosilicate (montmorillonite)

cured with cycloaliphatic amines, as well as the effects of

matrix networking on the flow behavior of the nanocom-

posite.

The Cure Kinetics Model

As mentioned earlier, epoxy resins are widely used in

both the automotive field and in electric/electronic appli-

cations by virtue of their unique combination of mechani-

cal properties and thermal stability, which may be even

enhanced by the presence of nanofillers dispersed within

the polymer matrix.

The conversion from the liquid (viscous) state of

monomers or oligomers having low molecular weight to

three-dimensionally networked rigid materials is carried

out by means of a very exothermic process known as

cure reaction. At the end of this process, the system can

be seen as an infinite network extending to all space

directions, in which any molecular movement is hindered.

In the case of epoxy resins, the networking process

occurs when one or more epoxy groups are opened in

presence of hydroxyl groups or of molecules of curing

agents like amines or anhydrides. Depending on the hard-

ener chemical properties, the cure reaction can be driven

at room temperature or require the application of external

heat. The stoichiometry of the cure reaction and the

amount of hardener functions and epoxy groups can also

be used to modify the crosslink density and the crosslink

distribution [8], which, in turn, determines the final

behavior of the cured product [9].

In particular, this work is focused on the analysis of a

plasticizer-free epoxy resin containing a bifunctional reac-

tive aliphatic diluent, cured with a mix of cycloaliphatic

amine hardeners [10]. The chemical formulas and function-

alities of the two matrix precursors are reported in Fig. 2.

FIG. 1. Modelling philosophy of the processing of nanocomposite

materials.

FIG. 2. Chemical formulas of polymer matrix precursors used in the present work. [Color figure can be

viewed in the online issue, which is available at www.interscience.wiley.com.]

2 POLYMER COMPOSITES—-2009 DOI 10.1002/pc

When an epoxy ring reacts with an amine, the three

reactions showed in Fig. 3 are possible However, several

authors [8] found that the only significant reaction, in the

absence of an accelerator, is that of the amine hydrogens

(i.e., the first and second reactions). The tertiary amine

formed in the second reaction was found to be too steri-

cally hindered to give the third reaction.

As in the case of all other thermosetting polymers, the

earlier reactions are exothermal. Therefore, it is possible

to relate the heat of reaction to the rate of reaction and

the degree of cure. The degree of cure (a) can be

described as the ratio between the heat evolved during

cure [DH(t)] and the total heat of reaction (DHR) as given

in Eq. 1:

aðtÞ ¼ DHðtÞDHR

¼R t0@H@t dtR end

start@H@t dt

ð1Þ

An expression of the reaction rate can be obtained differ-

entiating Eq. 1:

@a@t

¼ 1

DHTOT

@½DHðtÞ�@t

ð2Þ

The differential scanning calorimeter (DSC) is an ideal

tool to monitor these kinds of reactions. The DSC provides

the heat flow (qH/qt), or instantaneous heat, released or

absorbed by the reactive system as a function of the abso-

lute temperature or time. Dynamic and isothermal DSC

analyses are widely used to obtain an indirect estimation of

the degree of cure of a thermosetting system, assuming

that the heat released during polymerization is always pro-

portional to the consumption rate of reactive groups [1, 6,

7, 11] (which actually describes the extent of the reaction).

Calorimetric data can therefore be used to both create and

verify theoretical and empirical kinetic models and obvi-

ously to calculate kinetic parameters.

A combination of the dynamic and isothermal tests

makes possible to determine the variations of the degree

of cure and the rate of cure as a function of time and/or

temperature. In a polymerization process, the most gen-

eral kinetic model can be expressed as follows:

@a@t

¼ yðT; aÞ ¼ KðTÞ f ðaÞ ð3Þ

There are several ways to determine this relationship.

Unfortunately, none of the equations available in literature

[12] is able to describe the cure kinetics of the systems

analyzed in the present work (in each case, the nonlinear

fitting gave in fact very scattered parameters). This differ-

ence from the usual behavior of the epoxy systems can be

ascribed to several reasons. Firstly, the inhomogeneous

composition of the curing agent could determine a super-

position of networking reactions, which are activated at

different temperatures (and therefore they require different

kinetic parameters depending on the curing temperature).

Secondly, the epoxy resin contains a bifunctional reactive

aliphatic diluent, which could play an active role in the

networking process. The presence of the nanofiller (char-

acterized by the presence of organic terminations that

could in turn participate to the reaction) can only compli-

cate this situation.

For these reasons, a new kinetic equation was devel-

oped taking into account the fact that the activation

energy of the reaction is actually a function of the degree

of cure. Indeed, as the resin turns into a networked micro-

structure, the mobility of the active sites (i.e., epoxy

groups and primary and secondary amine groups) is pro-

gressively hindered, so an increasing energy is required to

keep the reaction active—which eventually becomes infi-

nite when the system reaches vitrification. To sum up,

this means that the kinetic parameter K(T) is actually a

combination between an Arrhenius function of T and a

growing exponential function of the degree of cure:

dadt

¼ Kða; TÞ am ð1� aÞn

Kða; TÞ ¼ K0 exp �Ea0

RT

� �exp �c

1� amax

amax � a

� �

amaxðTÞ ¼ 1

1þ expðbT0:5Þ expð�bTÞ

8>>>>>>><>>>>>>>:

ð4Þ

where amax is the final conversion degree reached by the

system (which was found to be a sigmoid function of the

cure temperature), T0.5 is the absolute temperature

(Kelvin) that allows the achievement of a half of the max-

imum networking degree, b is a temperature-independent

parameter that influences the slope of the sigmoid func-

tion, m and n are constant reaction orders and the cure-de-

pendent activation energy of the networking process can

be represented by the sum of the classical Arrhenius term

with a function of cure and temperature:

Eaða; TÞ ¼ Ea0 þ cRTð1� amaxÞðamax � aÞ ð5Þ

The Chemoviscosity Model

The increase of the system viscosity during isothermal

cure is a consequence of the molecular structural changes

FIG. 3. A scheme of the reactions between an epoxy ring and a pri-

mary or secondary amine.

DOI 10.1002/pc POLYMER COMPOSITES—-2009 3

induced by the polymerization reactions when crosslinked

polymer molecules form an infinite network, gelation

occurs. As the network has elastic properties, not present

in the low-molecular-weight linear or branched polymeric

chains, the gelation means the transformation of a viscous

liquid into a rubbery state solid. This significant increase

of viscosity is of critical importance from the processing

point of view, as the shaping must be completed before

gelation, i.e., while the polymer is still able to flow and

stresses can relax to zero.

The addition of reinforcing fillers to polymers is very

popular and common to realize requested properties, but

it also increases the complexity of the rheological behav-

ior. Rheological studies of the curing process indicate for

example that intragallery diffusion before curing is essen-

tial for clay exfoliation, before the morphology is frozen

by gelation and vitrification [13]. Therefore, the ability to

process these materials depends mostly on the control of

the interdependence among the material constituents

(resin, curing agent, nanoparticles, etc.), their effect on

the state of cure (cure kinetics, crosslink density) and the

morphology development of the dispersed nanoparticles.

All of this has to be accomplished within the period of

the resin gelation and vitrification, since after that point

the morphology will be essentially locked by the cross-

linked structure.

A lot of work was done to determine the appropriate

models that best describe the chemoviscosity of thermo-

sets. An advance was made from the batch-specific vis-

cosity versus time or temperature correlations to the mod-

els that evaluate the time-dependent chemoviscosity as a

function of the cure cycle and the cure kinetics of the

resin formulations [9, 14]. An extensive overview of the

chemorheology of thermosetting systems is reported by

Halley and Mackay [12].

The viscosity of a thermosetting system is influenced

by two main factors. Firstly, the increasing temperature

which increase the mobility of the molecules and there-

fore lower the viscosity, secondly the growing size of the

molecules because of the resin networking, which pro-

duces an increase of the viscosity. A commonly used

model to describe the viscosity dependence on tempera-

ture for the uncured system is the William–Landel–Ferry

(WLF) equation, derived from the free volume theory [9,

14–16]:

gðTÞ ¼ gg exp � C1ðT � Tg0ÞC2 þ T � Tg0

� �ð6Þ

where Tg0 is the glass transition temperature of the mono-

mer, gg is a constant (with an assigned value of 1012 Pa

s) representing the viscosity at the glass transition and C1

and C2 are material-dependent but temperature-independ-

ent parameters.

However, it is possible to see directly that a modifica-

tion of the WLF equation is necessary since the glass

transition of a curing resin is a function of the extent of

reaction and hence varies during the in situ polymeriza-

tion. In particular, the Tg increases because of the network

formation and must be evaluated as a function of the

degree of cure a.The gel point can be described as the point at which

the average molecular weight, Mw, of the system becomes

infinite. A consequence of Mw rising towards infinity is

that the shear viscosity also becomes infinite. To deter-

mine successfully the viscosity as a function of time and

temperature, the model must also consider this molecular

growth:

gðT; aÞ ¼ f1ðTÞ f2ðaÞ ð7Þ

FIG. 4. Dynamic tests at different heating rates for the neat resin (a) and the nanocomposite (b).

TABLE 1. A comparison among exothermic peaks and polymerization

enthalpies for the neat resin and the nanocomposite at different heating

rates.

Test

TS/Tpeak/TE (8C)Curing enthalpy

DHR (J/g)

Neat resin Nanocomp. Neat resin Nanocomp.

Dyn. 28C/min 31/83/105 58/88/124 2477.3 2522.5

Dyn. 58C/min 65/100/138 67/103/146 2480.2 2522.3

Dyn. 108C/min 75/117/167 77/117/159 2488.6 2499.6

Dyn. 158C/min 80/123/176 85/126/172 2470.4 2522.5

Avg. networking enthalpy? 2481.8 2516.5


Equation (7) shows a general equation for the viscosity.

Several works [7, 9, 17] showed that a modified WLF

equation was appropriate to describe the chemorheology

of epoxy-based systems:

gðTÞ ¼ gg exp � C1

�T � TgðaÞ

�C2 þ T � TgðaÞ

� �� ag

ag � a

� �nrheo

ð8Þ

where ag is the extent of reaction at the gel point, Tg(a) isa function describing the dependence of the Tg on the net-

working degree and nrheo is a constant to be determined

by regression analysis of the experimental data. This

model was taken as a reference for the characterization of

the materials analyzed in the present work.

EXPERIMENTAL

Materials

Every compound was based on a mixture of 30 parts

of amine (W131) for every 100 parts of epoxy resin

(EC157). Nanocomposites were obtained adding 2% wt of

organo-modified montmorillonites (Cloisite1 20A, pro-

duced by Nanocor, USA) to the resin.

The first stage of the procedure for the preparation of

the nanocomposite is a preintercalation of the resin. A

batch of 200 g of DGEBA monomer was mixed with the

Cloisite1 for 1 h using a mechanical stirrer (HEIDOLPH

RZR 2041) at a temperature of 708C and a speed of

3,300 rpm. In each case, the nanoreinforcement was

added to the resin with a mixing ratio of 2% wt to obtain

an overall content of about 1.53% wt of nanofiller in the

composite. Every compound, once cooled down to room

temperature, was immediately put in a freezer at a tem-

perature of –208C, to avoid any secondary effect due to

the environmental aging.

Differential Scanning Calorimetry

Of the three submodels treated in this work, the kinetic

model is the foundation of the other two and therefore

was studied first. The model was developed using a power

compensating Differential Scanning Calorimeter MET-

TLER DSC822e (controlled by a computer and connected

to an intracooler unit that allowed the rapid cooling of the

samples), to produce data related to the evolution of the

cure process both in isothermal and in dynamic tests.

Dynamic scans from 250 to 2508C at different heating

rates were performed on both the neat resin and the nano-

composite, to make a comparison between the different

cure reactions and consequently to determine possible var-

iations induced by the addition of a nanofiller. Thereafter,

the isothermal cure processes at different temperatures for

each aforementioned system were analyzed to calculate

the temperature dependence of the kinetic parameters and

to consider the effects of diffusion controlled phenomena

during the isothermal reaction. All tests were performed

in a nitrogen atmosphere. Every sample had a mass of

about 157 20 mg, which ensured a good signal of the

heat flux without losing the homogeneity of the tempera-

ture field within the DSC crucible.

Dynamic Viscosity Tests

The rheological characterization of the system was car-

ried out using a Rheometric Scientific model ARES

dynamic viscosity spectrometer. Steel parallel plates with

a diameter of 12.5 mm were used throughout, a rotation

FIG. 5. Thermograms of the isothermal scans of (a) the near resin and (b) of the nanocomposite.

TABLE 2. A comparison among isothermal polymerization enthalpies

for the neat resin and the nanocomposite at different temperatures.

Test

Curing enthalpy DHiso [J/g] amax ¼ DHiso/DHR

0Neat resin0 Nanocomp. Neat resin Nanocomp.

Iso. 658C 2360.2 – 0.747 –

Iso. 708C 2420.5 2368.4 0.873 0.713

Iso. 758C 2462.8 2426.7 0.961 0.826

Iso. 808C 2463.2 2427.4 0.961 0.827

Iso. 858C 2460.5 2444.3 0.956 0.860

Iso. 908C – 2481.3 – 0.932

Iso. 958C 2474.1 – 0.984 –


rate is imposed to the lower plate, to perform either

steady shear or oscillatory shear flow tests. A transducer

measures the viscous response of the material under the

form of a torque exerted by the fluid on the upper plate.

RESULTS AND DISCUSSION

Dynamic and Isothermal DSC Scans

A preliminary set of dynamic scans from 2508C to

2508C of both the resin and the nanocomposite were per-

formed at heating rates of 2, 5, 10, and 158C/min as

shown in Fig. 4.

The most prominent characteristic of the thermograms is

the exothermic peak due to resin networking. The shape of

the thermograms is almost the same for the two different

mixtures and could be interpreted as the effect of a linear

combination of two peaks associated to the first two reac-

tions shown in Fig. 3. The effect of the nanofiller on the

cure kinetics can be therefore estimated only on the basis of

some characteristic parameters, namely, the total heat

released (DHR), the onset (TS) and offset (TE) temperatures

of the polymerization process, and the peak temperature

(Tpeak). A complete resume of the aforementioned values for

both the resin and the nanocomposite is given in Table 1.

It must be noticed that for both the materials the peak

shifts towards lower temperatures at lower heating rates,

while for the same heating rate the presence of the nano-

filler produces a higher curing enthalpy and a shift of the

peak towards slightly higher temperatures. This effect

could be ascribed to the presence of clay particles, which

act as an obstacle against molecular mobility. In this way,

higher temperatures are required to complete the reaction,

which in turn advances more slowly and regularly carry-

ing eventually to higher networking degrees (and there-

fore to higher values of DHR).

The information given by the dynamic scans, however,

are not sufficient to describe completely the behavior of the

reacting systems. Indeed, the self-catalysis and control effects

from the diffusion phenomena on the last stages of the cure

process can be hardly analyzed during dynamic heating.

Therefore, for an exhaustive calorimetric characterization

(i.e., to calculate the rate constants and the reaction orders)

several isothermal tests at different temperatures are needed.

Seven isothermal scans were performed from 65 to

958C at steps of 58C for 2 h for both the neat resin and

the nanocomposite (reported in Fig. 5). The integral of

each peak (corresponding to the heat of polymerization at

that temperature) is reported in Table 2, together with the

final conversion degree aMAX reached by the system. It

must be noticed that the ‘‘obstacle’’ determined by the

presence of the clay is confirmed by the lower values of

the aMAX obtained at each given temperature. Figure 6

shows the evolution of the maximum degree of cure vs.

temperature at different isothermal cure conditions respec-

tively for the neat resin and the nanocomposite.

Determination of the Kinetic Constants

The kinetic parameters of the models expressed by

Eqs. 4 and 5 were determined using a nonlinear regres-

FIG. 6. Maximum degree of cure vs. temperature and relative trend-line for both the neat resin (a) and the nanocomposite (b).

TABLE 3. Results of the non linear fitting of amax vs. T based on

Eq. 4.

Parameter

Material

Neat resin Nanocomp.

T0.5 331.3 K � 588C 330.4 K � 578Cb 0.16 K21 0.07 K21

Linear correlation coefficient (R) 0.9975 0.9868

TABLE 4. Average kinetic parameters of the analyzed materials.

Parameter

Material

Neat resin Nanocomposite

ln (K0) 10.07 14.66

Ea0/R 5834.2 7504.7

Coefficient of determination (R2) 0.9694 0.9971

c 0.15 6 0.09 0.23 6 0.06

m 0.17 6 0.02 0.24 6 0.02

n 1.53 6 0.14 1.80 6 0.12


sion analysis of the experimental data obtained from DSC

test. The average values of the kinetic parameters

obtained for both the pure resin and the nanocomposite

are reported in Tables 3 and 4. All values are in good

agreement with the results obtained by several authors [1,

7, 16, 17–21] for epoxy-based systems. From those values

it was possible to evaluate the effect of the nanofiller on

the networking behavior. A 30% increase in activation

energy and a 20% increase of the overall reaction order

(m þ n) is observed: this confirms the considerations

made before about the hindrance produced by the clay, to

such an extent that, higher temperatures are required to

start and complete the cure process.

Previous considerations do not however suffice to con-

firm the validity of the proposed thermo2kinetic model.

A correct model should in fact be able to predict the

behavior of a given polymeric system not only in a partic-

ular combination of temperature and curing time, but also

in a wide range of conditions to which the system may be

exposed. For this reason, the model should describe the

evolution of the cure degree a(t) once both the properties

of the reactive system (e.g. activation energies, kinetic

constants, etc.) and the environmental conditions have

been defined. In other words, the correctness of the model

must be confirmed through the integration of the theoreti-

cal curves:

dadt

¼ Kða; TÞ am ð1� aÞn

ln½K� ¼ ln½K0� � Ea0

RT� c

1� amax

amax � a

amaxðTÞ ¼ 1

1þ expðbT0:5Þ expð�bTÞ

8>>>>>><>>>>>>:

ð9Þ

starting from a set of characteristic parameters of the ana-

lyzed compound for each reaction peak (c, m, n, ln [K0],

Ea0/R) and for user-defined curing conditions (temperature

and cure time).

Since an algebraic integration of the differential Eq. 9was impossible, it was necessary to implement an algo-

rithm for its numerical integration, based on the fourth-order Runge2Kutta method. Figure 7 shows a comparison

between the shapes of the numerically integrated curve

aRK(t) and the experimental curve a(t) for both the resin

and the nanocomposite system under isothermal condi-

tions. Good concurrence between the fitting curves and

the experimental data can be noticed (especially at high

temperatures), within the limits of experimental uncer-

tainty.

The model was also checked for the ability to predict

the behavior of the polymeric system during dynamic

cure at a constant heating rate. The same algorithm was

modified to describe the evolution of the degree of cure

FIG. 7. Integration of the kinetic model (isothermal conditions) for (a) the neat resin and (b) the nanocomposite.

FIG. 8. Integration of the kinetic model (dynamic conditions) for both the neat resin (a) and the nanocomposite system (b).


using the same characteristic parameters of the material,

allowing the possibility to choose test conditions, which

in this case were the initial and final temperatures and the

heating rate. The results of the application of these mod-

els and the comparison with the experimental results are

shown in Fig. 8 for both of the analyzed systems.

Relationship Between Tg and a

To determine the chemorheological model, it is neces-

sary to know the relationship between a and Tg. In gen-

eral, for phenomenological modelling purposes, it is suffi-

cient to make a linear or exponential fit in the temperature

region of interest. However, it is also possible make a

more theoretical approach. It is possible to show that, for

a partially cured thermosetting system, the glass transition

temperature up to an extent of reaction a is given by

[22]:

1

Tg¼ 1� a

Tg0þ a

Tg1þ q a ð1� aÞ ð10Þ

where Tg0 and Tg! are, respectively, the glass transitiontemperatures for the monomer and the fully cured system.To find the parameters of Eq. 10, a series of isothermalscans were performed at 70 and 808C. The samples werecured for a short period of time (3–12 min) and quicklyquenched to the room temperature at a rate of –1008C/min. This treatment was followed by a dynamic scanfrom 250 to 2508C at 108C/min to determine the Tg forthe undercured system. Tg! was calculated from adynamic heating scan at 108C/min on the fully cured sys-tems. All the data used in the analysis are reported in Ta-ble 5. The results of such analysis are plotted in Fig. 9,which show a very good curve fit. The obtained equationsare shown below for the neat resin and the nanocompo-site, respectively (temperatures are expressed in Kelvin):

TRESINg ðaÞ ¼ 1� a

251:12þ a

354:43

�

þ 12:4 3 10�4 a ð1� aÞ��1

½K� ð11Þ

TNANOC:g ðaÞ ¼ 1� a

285:04þ a

359:98

�

þ 5:9 3 10�4 a ð1� aÞ��1

½K� ð12Þ

The most evident difference between the resin and the

nanocomposite is represented by the gap in the Tg0, whichis more than 308C higher when clays are dispersed in the

resin. This effect is much less evident in the networked

structure, as in this case the difference between the Tg!of the neat resin and of the composite of only 58C.

The earlier results are of great interest, as the glass

transition temperatures of the unreacted systems strongly

affect the choice of processing conditions. These data

could be interpreted taking into consideration the state of

dispersion of the clay within the polymer: mechanical stir-

ring probably did not favor a complete intercalation of

the polymer, so clay particles remained dispersed at a mi-

crometer level. This produced an increase in the free vol-

ume of the polymer, lowering the Tg of the filled resin.

FIG. 9. Regression analysis of the relationship between a and Tg for both the neat resin and the nanocomposite.

TABLE 5. Degree of cure(a) and Tg for the partially cured systems.

Test

a/Tg (8C)

Neat resin Nanocomp.

3 min at 708C 0.089 218.2 0.042 14.6

6 min at 708C 0.187 220.2 0.097 14.4

9 min at 708C 0.276 219.7 0.154 14.2

12 min at 708C 0.354 218.2 0.208 14.2

4 min at 808C 0.205 217.8 0.132 14.6

8 min at 808C 0.383 216.9 0.272 15.6

2 h at 658C 0.741 39.7 – –

2 h at 708C 0.867 52.4 0.685 47.5

2 h at 758C 0.935 58.3 0.758 57.0

2 h at 808C 0.966 62.8 0.817 60.4

2 h at 858C 0.980 74.9 0.864 65.6

2 h at 908C – – 0.900 71.5

2 h at 958C 0.991 78.2 – –

Dyn at 108C/min. 1 82.6 1 84.0


Viscosity Measurements

Oscillatory shear flow measurements were conducted

under isothermal and constant heating rate conditions.

Preliminary frequency and strain sweeps were carried out

to determine the optimum experimental conditions. Figure

10 shows a comparison among isothermal tests performed

at 808C with different values of the applied strain: it can

be clearly seen how the shape of the curves, and therefore

the calculated gel time, is strongly dependent on the strain

imposed, making necessary to choose high values of de-

formation to observe gelation in a reasonable time. This

variation can be attributed to the fact that at higher strain,

the elastic response become higher, therefore, since the

complex viscosity is influenced both by the elastic modu-

lus G0 and by the loss modulus G00 the curve of the com-

plex viscosity have a steeper inflection at shorter times.

However steady shear flow measurements were not per-

formed, relying on the empirical Cox–Merz rule [9, 23]

which predicts the close similarity between the steady shear

viscosity with a shear rate of 10 s–1 and the complex viscosity

measured at a frequency of 10 rad/s [i.e., gð _cÞ ¼ g�ðxÞ]. Astrain of 150% and an oscillating frequency of 1 Hz (corre-

sponding to an angular frequency x of about 10 rad/s) were

chosen eventually, representing the best compromise

between measurement accuracy and test efficiency.

Isothermal experiments were performed under the

same thermal conditions as the DSC measurements. The

test fixture was preheated at the isothermal cure tempera-

ture and the plate spacing was zeroed. The chamber was

then opened, the plates separated, and the resin sample

rapidly inserted. The plates were then quickly brought

back together to a gap of approximately 0.6 mm. The

chamber surrounding the plates was closed and the

experiment initiated. The viscoelastic properties of the

sample during cure, including the complex dynamic vis-

cosity (g*), shear storage modulus (G0), and shear loss

modulus (G00), were monitored. The isothermal scans

were used to determine the constant nrheo in Eq. 8.

Several dynamic curing experiments at heating rates of

2, 5, and 108C/min were done as well. The dynamical

scans were performed by first cooling the plates to 208C(a temperature above the Tgs of both the neat resin and

the nanocomposite). Once inserted a sample of the mate-

rial to be cured, the plates were closed to 0.6 mm and

newly cooled to 208C. The scan interval was set from 20

to 2508C (but the tests were interrupted before reaching

the upper temperature limit, i.e., just after an evident ge-

lation of the system).

Figure 11 shows a dynamic scan at 58C/min for both

the neat resin and the nanocomposite. It can be clearly

seen that the presence of the nanofiller causes an increase

of the complex viscosity of about one order of magnitude,

together with an anticipation of the gel point.

In the present work, the first part of each curve (where

the degree of cure is negligible) was used to calculate C1

and C2. The last part was used to determine ag and to ver-

ify the calculated n from the isothermal scans.

FIG. 11. A comparison between the rheological behavior of the neat

resin and the nanocomposite under dynamical heating at 58C/min.

FIG. 12. Determination of C1 and C2 for the neat resin at 28C/min.

FIG. 10. Complex viscosity of the neat resin versus time at 808C dif-

ferent levels of strain.


The extent of reaction during the first part of the

dynamic scans was assumed so low that it could be

ignored therefore Tg behaves as a constant in this region.

This assumption was also confirmed by the application of

the kinetic model. According with Eq. 8, when a ¼ 0 the

term (ag/(ag 2 a))n goes to 1 and the viscosity depends

only on the temperature therefore C1 and C2 can be easily

determined by regression analysis (Fig. 12 shows an

example of regression for the neat resin). The viscosity at

the glass transition temperature has been assumed to be

g ¼ 1012 Pa s, and Tg(a) ¼ Tg0 for both of the materials

analyzed. The following equation was used in the regres-

sion analysis:

ln½gðTÞ� ¼ 12 ln½10� þ C1

�T � Tg0

�C2 þ T � Tg0

ð13Þ

C1 and C2 are taken as the mean value of the results

obtained in the three tests (Table 6).

The Determination of the Gel Point, ag

The gel point, which is characterized by an infinite

value of the steady shear viscosity, is not as easy to dis-

cern in case of oscillatory shear flow measurements. Dif-

ferent criteria have been proposed in the literature, such

as the inflection point of the G00 curve [9], the crossover

point between the G0 and G00 curves [24], or the point

where the loss tangent (tan d) becomes independent from

the frequency [25]. In this work, gelation was assumed to

occur when the dynamic complex viscosity diagrams have

their inflection points. Figures 13 and 14 show the effect

of gelation on the complex viscosity of the neat resin and

the nanocomposite respectively. The a values, which

appear in the abscissa of the diagram, were calculated

integrating the kinetic models for each material, based on

time and temperature arrays exported from the rheometer.

Combining the dynamic rheological data with the frac-

tional conversion data obtained by means of DSC, the

values for the fractional conversion at the gel point (ag)were determined and reported in Table 7. According to

Flory’s gelation theory [26], gelation occurs at a fixed

conversion as long as the reaction mechanism is not a

function of the temperature. The deviation between agvalues for different isothermal experiments is about 2%.

To reduce the number of parameters, an average ag value

was assumed.

An anticipation of the gel point can be noticed for the

nanocomposite, which can be explained with some kind

of ‘‘linking’’ effect produced by the presence of the clay,

which enhances the stress transfer ability of the material

(and therefore reduces the degree of networking required

to get a ‘‘nearly solid’’ behavior).

To complete the chemorheological model, the constant

nrheo was determined from the isothermal scans. As the

software used in the regression analysis allowed to use

only one variable, Eq. 8 had to be rewritten in a form

that describes the viscosity as a function of the extent of

reaction only. The equation obtained is shown below.

FIG. 13. Determination of the gel point for the neat resin.

FIG. 14. Determination of the gel point for the nanocomposite.

TABLE 7. Degree of cure at the gel point of the analyzed materials.

ag


0.762 6 0.006 0.695 6 0.012

TABLE 6. The results of the regressions for the determination of C1

and C2.

C1 C2 (K)

Neat resin Nanocomp. Neat resin Nanocomp.

235.17 6 1.41 229.05 6 0.58 15.38 6 2.87 2.13 6 0.40


ln½gðaÞ� ¼ 12 ln½10�

þC1 Tiso � 1�a

Tg0þ a

Tg1þ q a ð1� aÞ

� ��1� �

C2 þ Tiso � 1�aTg0

þ aTg1

þ q a ð1� aÞ� ��1

þ nrheo lnag

ag � a

� �ð14Þ

The mean among the results of the linear regression is

reported in Table 8. After the calculation of the constants

of the chemorheological model, the model was verified

against the experimental data. With this aim, the integra-

tion algorithm described earlier was modified to take into

account also the chemoviscosity effects. Figures 15 and

16 show the logarithm of the complex viscosity versus

absolute temperature for all the heating rates and their

corresponding model simulations, respectively for the neat

resin and the nanocomposite. A very good correspondence

was obtained, so the model can be said to describe

adequately the process. Table 9 shows a resume of the

constants used in the chemorheological model.

Figures 15 and 16 allow one to make also some gen-

eral considerations about the effects of the clay on the

chemorheological behavior of the polymer matrix. It can

be clearly seen that the viscosity of the nanocomposite is

always higher than the one of the neat resin. A higher

‘‘heating sensitivity’’ of the composite viscosity before

networking can also be noticed from the higher slope of

the left horn (and obviously from the lower C1 and C2

values). Moreover, in the presence of clay, gelation

occurs at lower degrees of cure, thus determining a nar-

rower shape of the chemoviscosity diagrams (confirmed

by the higher value of nrheo).The chemorheological characteristics can be adequately

described with the following equations:

Neat Resin

dadt

¼ Kða;TÞ a0:17 ð1�aÞ1:54

Kða;TÞ ¼ expð10:07Þ exp �5834:2

T

� �

3 exp �0:151�amaxðTÞamaxðTÞ�a

� �

amaxðTÞ ¼ 1

1þ expð0:16 �331:34Þexpð�0:16 �TÞ

gða;TÞ¼ 1012 exp �35:17�T�TgðaÞ

�15:38þT�TgðaÞ

� �0:762

0:762�a

� �1:81

TgðaÞ¼ 1�a251:12

þ a354:43

þ 12:43 10�4 a ð1�aÞ� ��1

8>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>:

TABLE 9. Constants of the chemorheological model for the materials

analyzed.

Parameter

Material


gg 1012 1012

C1 235.17 229.05

C2 (K) 15.38 2.13

Tg0 (K) 251.12 285.04

Tg! (K) 354.43 359.98

q 12.4 3 1024 5.9 3 1024

ag 0.762 0.695

nrheo 1.81 2.53FIG. 15. Model verification at different heating rates for the neat resin.

FIG. 16. Model verification at different heating rates for the nanocom-

posite.

TABLE 8. Average chemo-rheological kinetic constant of the analyzed

materials.

nrheo


1.81 6 0.75 2.53 6 1.05


Nanocomposite

dadt

¼Kða;TÞa0:24 ð1�aÞ1:80

Kða;TÞ¼expð14:66Þ exp �7504:7

T

� �

3 exp �0:231�amaxðTÞamaxðTÞ�a

� �

amaxðTÞ¼ 1

1þexpð0:073330:39Þexpð�0:07 �TÞ

gða;TÞ¼1012 exp �29:05 ��T�TgðaÞ�

2:13þT�TgðaÞ� �

0:695

0:695�a

� �2:53

TgðaÞ¼ 1�a285:04

þ a359:98

þ5:9310�4 að1�aÞ� ��1

8>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>:

CONCLUSIONS

The neat resin and the nanocomposite underwent calo-

rimetric and rheological characterizations based on both

dynamic and isothermal tests, which allowed determining

the evolution of their viscosity during the cure process.

The main feature of the analyzed systems is represented

by their cure kinetics. As the behavior of both of them can-

not be represented by models available in literature, a new

kinetic model was developed, taking into account also the

fact that the activation energy of the networking process is

actually a function of the degree of cure (rising exponen-

tially towards infinity when the system approaches vitrifica-

tion). Moreover, the nanoparticles seem to hinder the mo-

lecular mobility, so that the cure temperature window is

broader and shifted towards higher temperatures than the

one of the neat resin. Obviously, this effect influences also

the cure kinetics, as confirmed by the higher values of the

activation energy and of the overall reaction order of the

nanocomposite.

The effects of the clay on the chemorheological behav-

ior of the polymer matrix can be resumed as follows:

higher viscosity of the nanocomposite at any temperature,

higher ‘‘heating sensitivity’’ of the composite viscosity

before networking and gelation occurring at lower degrees

of cure, thus determining a narrower shape of the chemo-

viscosity. The applied chemorheological model was a

combination of the classical Williams–Landel–Ferry

(WLF) equation and a conversion term originally used by

Castro and Macosko [27].

Once found the characteristic parameters of both the

neat resin and the nanocomposite, the chemoviscosity

models were numerically integrated using an algorithm

based on the 4th-order Runge–Kutta method, to check

their ability to foresee the behavior of the systems dur-

ing a dynamic cure process. The results fit the experi-

mental data with very good precision, so the model is

ready to be applied for the description of practical case

studies.

REFERENCES

1. J.M. Kenny, A. Trivisano, and L. Nicolais, Eds., ThermalCharacterization of the Cure Kinetics of Advanced Matricesfor High-Performance Composites, American Chemical So-

ciety, Washington, D.C. (1993).

2. C.I. Park, O.O. Park, J.G. Lim, and H.J. Kim, Polymer, 42,7465 (2001).

3. M. Alexandre and P. Dubois, Mater. Sci. Eng., 28, 1 (2000).

4. W.B. Xu, S.P. Bao, S.J. Shen, G.P. Hang, and P.S. He, J.Appl. Polym. Sci., 88, 2932 (2003).

5. W.B. Xu, Z.F. Zhou, P.S. He, and W.-P. Pan, J. Therm.Anal. Calorimetry, 78, 113 (2004).

6. J.M. Kenny, L. Torre, L. Valentini, J. Biagiotti, and D.

Puglia, ‘‘Processing Structure and Properties of Polymer Ma-

trix Nanocomposites for Industrial Applications,’’ in China–EU Forum on Nanosized Technology Proceedings, Beijing,China, 24–37 (2002).

7. J.M. Kenny, Compos. Struct., 27, 129 (1994).

8. J. Galy, A. Sabra, and J.P. Pascault, Polym. Eng. Sci., 26,1514 (1986).

9. M. Ivankovic, L. Incarnato, J.M. Kenny, and L. Nicolais, J.Appl. Polym. Sci., 90, 3012 (2003).

10. Baxxodur2: When curing makes the difference. Brochre &

Product portfolio overview from the BASF Group, 2006.

11. L. Torre, G. Lelli, and J.M. Kenny, J. Appl. Polym. Sci., 94,1676 (2004).

12. P.J. Halley and M.E. Mac Kay, Polym. Eng. Sci., 36, 593(1996).

13. D. Dean, R. Walker, M. Theodore, E. Hampton, and E.

Nyairo, Polymer, 46, 3014 (2005).

14. J. Mijovic and C.H. Lee, J. Appl. Polym. Sci., 37, 889 (1989).

15. J.M. Kenny, A. Maffezzoli, and L. Nicolais, Compos. Sci.Technol., 38, 339 (1990).

16. M. Opalicki, J.M. Kenny, and L. Nicolais, J. Appl. Polym.Sci., 61, 1025 (1996).

17. J.M. Kenny and M. Opalicki in Makromol. Chem., Macro-mol. Symp., 68, 41 (1993); Huthig & Wepf Verlag, Basel.

18. L.A. Berglund and J.M. Kenny, SAMPE J., 27, 27 (1991).

19. J.M. Kenny and A. Trivisano, Polym. Eng. Sci., 31, 1426 (1991).

20. J.M. Kenny, A. Trivisano, M.E. Frigione, and L. Nicolais,

Thermochimica Acta, 199, 213 (1992).

21. A. Trivisano, A. Maffezzoli, J.M. Kenny, and L. Nicolais,

Adv. Polym. Technol., 10, 251 (1990).

22. I. Havlicek, K. Dusek, S. Stokrova, J. Biros, J.M. Kenny,

and L. Nicolais, ‘‘Theoretical Modelling of the Rheological

Behavior of an Epoxy-amine System During Curing,’’ in

Proceeding of European Symposium on Polymeric Materi-als, Lyon, France, September 14–18, 1987, pp CA03.

23. W.P. Cox and E.H. Merz, J. Polym. Sci., 28, 619 (1958).

24. C.Y.M. Tung and P.J. Dynes, J. Appl. Polym. Sci., 27, 569(1982).

25. H.H. Winter, Polym. Eng. Sci., 27, 1698 (1987).

26. P.J. Flory, Principles of Polymer Chemistry, Cornell Univer-sity Press, Ithaca, NY (1953).

27. J.M. Castro and C. Macosko, AICHE J., 28, 251 (1982).


Modelling of the chemo-rheological behavior of thermosetting polymer nanocomposites

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