Changes in Molecular Dynamics during Bulk Polymerization ofan Epoxide-Amine System As Studied by Dielectric RelaxationSpectroscopy

Jerome Fournier, Graham Williams,* Christine Duch, andGeorge Anthony Aldridge

Department of Chemistry, University of Wales, Swansea, Singleton Park,Swansea SA2 8PP, United Kingdom

Received December 4, 1995; Revised Manuscript Received July 24, 1996X

ABSTRACT: Dielectric relaxation spectroscopy (DRS) and differential scanning calorimetry (DSC) havebeen used simultaneously as a means of following the isothermal cure of the diglycidyl ether of BisphenolA with 4,4′-diaminodicyclohexylmethane in the temperature range 290-353 K. The dielectric permittivityand dielectric loss of the thermosetting mixture have been measured as a function of reaction time overthe frequency range 101.2-105 Hz. The evolution of the dielectric properties was studied as the curingtemperature was lowered to values close to the solidification of a sample. The kinetics of the cure havealso been determined, using calorimetry, for four reaction temperatures over the whole range of conversionup to the point where the system vitrifies and the reaction becomes diffusion-controlled. Correlationsbetween the changes in molecular dynamics and the chemical kinetics occurring during the thermosettingprocess have been made in some detail, and a theoretical working model has been developed that allowsDRS to predict the course of the reaction in the vitrification range. Previous interpretations of dielectricevents in this vitrification region, based on experimental kinetic and dielectric results, are reexamined.

1. Introduction

Glass-forming systems, such as molecular liquids oramorphous polymers, exhibit different relaxation phe-nomena in the glass transition range which arise fromthe translational and reorientational motions of themolecules. They have been studied for many yearsusing a variety of physical techniques including thoseof dynamic mechanical,1,2 nuclear magnetic resonance,and dielectric relaxation.1,3-7 In dielectric relaxationspectroscopy (DRS) the complex dielectric permittivityε(ω,tr) at a frequency f ) ω/2π and a reaction timetr, may be used to monitor the course of chemicalreactions.8-15 During the isothermal curing of a di-amine/diepoxide system, the epoxide groups open toform a linkage with the amine groups16,17 and thereaction proceeds relatively slowly toward vitrification.At long times, the diffusion control of the reactinggroups, arising from the densification and increasingviscosity of the network during the vitrification, retardssubstantially the polymerization process, and the reac-tion effectively “stops”. The dielectric properties of asystem change as the reaction progresses and thesechanges correspond to (i) a change in the dipole momentper unit of monomer in the mixture, resulting from thedisappearance of epoxide and primary amine groups andthe appearance of new groups and (ii) an increase inthe average dielectric relaxation time of the materialdue to a decrease in the molecular mobility in themedium as the network is forming. Dielectric experi-ments give information on the molecular chain dynam-ics at given reaction times and hence on the evolutionof the molecular mobility of the polymer moleculesduring the curing process but do not give direct indica-tions of the extent of reaction R. Dielectric and me-chanical properties of a thermoset cure are normallyeach used on their own to predict the approach ofvitrification, which involves several assumptions in theabsence of simultaneously obtained chemical data.

Sheppard and Senturia6 have reviewed the differentempirical approaches which attempted to link thekinetic and the dielectric data for epoxide-aminethermosetting systems. They concluded that most ofthese were based on unjustified assumptions and failedbecause of oversimplifications of the chemical kineticsor because of invalid interpretation of dielectric events.The kinetic characteristics of the thermosets (e.g. diep-oxide/diamine) are now well understood16,18 and a rathercomplex model was recently proposed by Cole,17 whichdescribes the cure kinetics of epoxide-amine thermo-setting resins over the entire range of R taking inaccount all the reactions, including etherification.In this work, we give extensive dielectric data for

epoxide-amine thermosetting systems at different reac-tion temperatures which were determined over a rangeof frequency and time cumulatively during the reaction.In addition, we have made simultaneous complementarystudies of the thermochemistry in real time for reactionsat different temperatures. By combination of the di-electric and thermochemical data, we have establisheda new comprehensive correlation model of dielectricproperties during a thermoset cure of the diepoxide-diamine system, including an improved model forchemical kinetics and an improved relation betweenstate of cure, temperature, and various dielectric pa-rameters that were contained in earlier studies. Therepresentation of such a model, built on a few clearworking assumptions, provides a general trend for thevariation of dielectric parameters (e.g. the frequencyof maximum dielectric loss) as the extent of reactionincreases. This work allowed us to reexamine the inter-pretation of dielectric events which were widely usedpreviously in the literature having basic assumptionswithout adequate verification.

2. Experimental Section2.1. Sample Preparation. Diglycidyl ether of Bisphenol

A (DGEBA) was provided by Shell Resins under the tradename Epikote 828, having MW ) 340. The 4,4′-diaminodicy-clohexylmethane (PACM) was obtained from Aldrich ChemicalCo. All chemicals were used as received. The sample was

X Abstract published in Advance ACS Abstracts, September 15,1996.

7097Macromolecules 1996, 29, 7097-7107

S0024-9297(95)01786-4 CCC: $12.00 © 1996 American Chemical Society

prepared by mechanically mixing 2 mol of DGEBA with 1 molof PACM during 5 min at 293 K.2.2. Dielectric Measurements. The liquid epoxy mixture

was simply sandwiched between the parallel-electrode dielec-tric cells which consist of two, strictly flat brass disks separatedby two 0.12 mm thick PTFE spacers. Then the cell was setup in an enclosed chamber and the electrodes were linked tothe Digibridge (see below). Simple contacts between the toppart of the cell and the top electrode allows the sample to beconnected to the dielectric assembly.The empty cell capacitance C0 was found to be approxi-

mately constant with the measurement frequency, at a valueequal to 36.5 pF. The theoretical value for C0 was 36.2 pF.The deviation between the measured and the calculated valuewas less than 1%, arising from the contributions of the spacersand the edge capacitance. The value of the loss index Gp(ω)/ωfor the empty cell was approximately equal to 0.01 pF for allthe measurement frequencies. The small variations wereassumed to result from the noise, and they do not affect thecalculation and the interpretation of the following results.The enclosed chamber was placed into a water bath (Techne,

Cambridge) during the experiment, and a Techne TU-160thermoregulator was used to control the temperature of thewater in the working range 288-353 K. The dielectricpermittivity and dielectric loss were measured at 20 frequen-cies in the range from 12 Hz to 200 kHz. The measurementsof equivalent sample parallel capacitance Cp(ω) and loss indexGp/ω, where Gp is the sample conductance and ω ) 2πf/Hz,were taken every 2 min across the entire frequency range, andthe times of measurement corresponding to each of themeasurement data points were stored for the 20 frequenciesscanned. The dielectric spectrometer was a 1693 RLC GenRadDigibridge interfaced with a Hewlett-Packard 915 3C computerfor automatic data acquisition. Such data were post-processedto yield 2D plots of ε′ and ε′′ vs time at fixed frequency ofmeasurement, ε′ and ε′′ vs log(f) at fixed time of measurement,and 3D plots of ε′ and ε′′ vs log(f) and time. Here ε′ and ε′′ arethe real and imaginary parts of the complex dielectric permit-tivity ε.2.3. Kinetic Measurements. A Stanton Redcroft STA 625

differential scanning calorimeter (DSC) was used in conjunc-tion with a PL Thermal Science data acquisition system forall the thermokinetic studies. The instrument was calibratedprior to the set of experiments, using the melting temperaturesof pure indium (156.60 °C) and tin (231.88 °C), and a baseline calibration was also accomplished by scanning without asample up to a desired temperature, to check that the scanwas free of any peaks or discontinuities due to impurities. Thethermal analysis consisted of an isothermal DSC run followedby a dynamic run. Approximately 50 mg of sample derivingfrom the same epoxy mixture used for the simultaneousdielectric measurements was placed in a standard aluminumcrucible and placed in the microbalance. The reference wasan empty aluminum crucible. The sample was heated in anair atmosphere from 288 to the experimental temperature atthe fixed rate of 10 K/min, and the calorimeter signal (heatflow ) dQ/dt) was recorded by the computer every 2 min duringthe isothermal polymerization. When the heat flow hadreached a constant value corresponding to the end of theexothermic reaction, the sample was then cooled to 288 K.Then a heating run was performed to 523 K at a suitabletemperature increase rate (5 K/min), allowing the determina-tion of the residual heat of polymerization Qr for all thesamples. The residual heat of cure Qr is evolved when a curedsample is heated above its glass transition temperature, which,for the samples prepared here, approximates to TR, thereaction temperature. In this type of cure, the heat evolvedis directly proportional to the number of epoxide groupsreacted,16,19 so the extent of reaction R(t) is given by thefollowing equation:

where Q(t) is the heat produced by the reaction at the time t:

Qi is the total heat evolved during the isothermal polymeri-zation at T ) TR, andQr is the residual heat of cure determinedas described above. R0 is the extent of reaction at thebeginning of the cure. R0 was found to be negligible and theexpressions of R and dR/dt as a function of time are reducedto

Equations 3 and 4 were used to determine R(t) and dR/dt forepoxide-amine reactions at four reaction temperatures.

3. Dielectric DataThe aim of our dielectric experiments was to monitor

the permittivity ε′(ω,tr) and loss factor ε′′(ω,tr) at dif-ferent measuring frequencies f ) ω/2π and reactiontimes tr for a fixed reaction temperature TR. Experi-ments were conducted at eight temperatures between290 and 353 K. As one example, Figures 1a and 1bshow ε′ and ε′′ vs log10(tr) at the fixed frequency of 105Hz for these eight temperatures. In the initial plateauregion ε′(ω,tr,TR) is the static permittivity ε0(TR), whichis approximately independent of tr but decreases with

R(t) ) R0 + [ 1 - R0

Qi + Qr]Q(t) (1)

Figure 1. (a) Permittivity ε′(ω,tr) against the reaction timetr, on a logarithmic scale, for the thermoset DGEBA-PACMat the measurement frequency f ) 105 Hz for seven differenttemperatures in the range 290-353 K. (b) Loss factor ε′′(ω,tr)against the reaction time tr, on a logarithmic scale, for thethermoset DGEBA-PACM at the measurement frequencyf ) 105 Hz for six different temperatures in the range 290-333 K.

Q(t) )∫0tdQdt dt (2)

R(t) )Q(t)

Qi + Qr(3)

dRdt

) [ 1Qi + Qr]dQdt (4)

7098 Fournier et al. Macromolecules, Vol. 29, No. 22, 1996

increasing temperature TR. At longer times, ε′(ω,tr)exhibits a fall from a liquid to a glassy polymer value,reaching the “unrelaxed” permittivity ε∞. As TR isincreased, this dispersion region moves to shorter times,reflecting the decreased times required for glass forma-tion with respect to this fixed measuring frequency. Theloss data ε′′(ω,tr,TR) are shown in Figure 1b and arecomplementary to the data of Figure 1a. The processmoves to shorter times and the peak height decreasesas TR is increased. The behavior shown in Figures 1aand 1b is similar to that observed for other epoxide-amine thermosetting systems.6-15 The ability of themeasuring system to acquire data of this kind over arange of frequencies enabled us to make similar plotsto those in Figure 1 at the other 19 frequenciesmeasured during an experiment. The plot of ε′′ vs tr(or log10(tr)) exhibits a maximum at tr ) tm for eachmeasuring frequency at a given reaction temperatureTR. Numerical analysis of the plots in Figure 1b allowedvalues of tm to be determined at each of the 20 frequen-cies in each run, allowing Figure 3 to be constructed.Our permittivity and loss data could also be presentedin 3D form, as shown in Figures 2a (for the permittivity)and 2b (for the loss factor), at 313 K as one example.Similar plots were obtained at the other reaction tem-peratures. In Figure 2a, the permittivity falls from itsshort-time plateau value through the relaxation regionas the material appears to become a glass, with respectto this frequency of measurement. The complementaryloss data in Figure 2b show a short-time low-frequencyregion of loss due to ionic species and a well-defined loss

peak whose frequency location changes systematically,moving to lower frequencies as the reaction proceeds.3D representations of the dielectric behavior duringreaction of a related epoxy-amine system have beenshown previously by Maistros and co-workers.15 Figure3 shows the values of the time of maximum loss factortm as a function of measuring frequency for differentreaction temperatures derived from Figure 2b. Only aportion of our data is shown, but even then, Figure 3contains the values of tm determined from 90 plots ofε′′(ω,tr,TR) vs tr at fixed ω and TR. Such data could noteasily be obtained before the advent of modern semi-automatic dielectric spectrometers of the kind used forthis study.The dielectric behavior seen in Figures 2a and 2b

arises from dipolar and ionic species whose concentra-tion and reorientational dynamics (for dipoles) andtranslational dynamics (for ions) change with time as aresult of chemical reaction. We may write

where ε′ and ε′′ contain contributions from dipolarspecies and ε′′i is the contribution from ionic species.For a simple conductivity process, ε′′i ) σ/ωε0, where ε0is the permittivity of free space. The rising loss at lowfrequencies and short times in Figure 2b is due to ionicspecies, but ε′′i follows a power law ε′′i∼ ω-n, which maybe rationalized in terms of hopping or dispersive modelsof ion transport.The short-time behavior of ε′ in Figure 1a gives the

static permittivity ε0(tr), which is expressed as a functionof the sum of the concentration and average squareddipole moments, ci and ⟨µi2⟩, respectively, of the differentspecies in the mixture at the time tr:

ε∞ is the limiting high-frequency permittivity of thematerial and F(ε0,ε∞) takes into account the relationbetween applied and local electric fields.1 ⟨µi2⟩ refersto the mean square dipole moment of amine (primaryand secondary), hydroxyl, and ether groups present inthe mixture. ε0(tr) shows only small variations with tr(see e.g. Figures 1a and 1b), so replacement of reactantdipoles with product dipoles during reaction gives onlya small variation in ∑ci⟨µi2⟩ in eq 6. Increase in reaction

Figure 2. (a) Three-dimensional plots of the dielectric permit-tivity ε′(ω,tr) against the time and the frequency, on alogarithmic scale, for the thermoset DGEBA-PACM at thecure temperature T ) 313 K. (b) Three-dimensional plots ofthe dielectric loss factor ε′′(ω,tr) against the time and thefrequency, on a logarithmic scale, for the thermoset DGEBA-PACM at the cure temperature T ) 313 K.

Figure 3. Time of maximum loss tm against the frequency,on a logarithmic scale. Here shown are tm for the thermosetDGEBA-PACM at the cure temperatures T ) 290 K (b), T )293 K (O), T ) 298 K (9), T ) 303 K (0), T ) 308 K (2), T )313 K (1), T ) 323 K (4), T ) 333 K (3), and T ) 353 K ()).

ε(ω,tr) ) ε′(ω,tr) - iε′′(ω,tr) - iε′′i(ω,tr) (5)

ε0(tr) ) ε∞ + F(ε0,ε∞)∑ici

⟨µi2⟩

kT(6)

Macromolecules, Vol. 29, No. 22, 1996 Molecular Dynamics during Polymerization 7099

temperature gives a decrease in the relaxation strength∆ε ) ε0(tr) - ε∞. In contrast, ε0(tr) in the short-timeplateau region in Figure 1a decreases with increasingreaction temperature. Equation 6 predicts that ∆ε )ε0 - ε∞ should be inversely proportional to temperature(K), and, taking ε∞ ∼ 3.5, this is true, approximately,for our data.The relaxation behavior seen in Figures 1 and 2 arises

from the changes in molecular mobility of the dipolesin the mixture as a liquid is transformed to a glassthrough the bulk-polymerization process. The dielectricparameters determined from our studies are sum-marized in Table 1. The values of ∆ε have a limitedaccuracy because ε∞ is difficult to evaluate for the lowercuring temperatures. At 290, 293, and 303 K the valueof ε∞ is determined by extrapolation as the average valueof ε∞ of the other experiments, for which the unrelaxedpermittivity is relatively constant and does not dependupon the final degree of conversion of the polymer. Thecharacteristic dipolar loss peaks are displayed in Figure1b for different experimental temperatures, arising fromthe energy loss principally entailed by dipole motionwhen the network becomes a glass. It can be describedsatisfactorily if the three following parameters areknown: tm, the time of maximum loss; ε′′max, the heightof the peak; HW, its half-width. These parameters aredisplayed in Table 1. The observed decrease in tm, ε′′max,and HW, obeying approximately the Fuoss-Kirkwoodequation,1 is a reflection of an irreversible thermallyactivated chemical reaction. The origin and the theo-retical analysis of these changes in the real and theimaginary parts of the complex permittivity have beenstudied in detail in several previous publications. Inour case, we are interested in studying the dielectricproperties of the epoxide-amine thermoset with the aimof establishing relations between dielectric parametersand chemical events and the vitrification or the degreeof polymerization.We chose to use the time of maximum loss ε′′ to define

an operational curing time with respect to measuringfrequency and we called it tm. The values of tm wereplotted against frequency for different experimentaltemperatures in Figure 3, and tm was found to belinearly dependent on log(f/Hz) in our range. FollowingLane and Seferis,20 we write

where τ0 is the relaxation time for the uncured material,k is a constant for a given reaction temperature, and tmis the time of maximum loss for a given frequency ofmeasurement. Insofar as the frequency of maximumloss fm can be related to the average relaxation time with

the equation fm ) 1/(2πτ), Mangion and Johari14 haverewritten eq 7 as

Good straight-line plots show that eq 8 gives a goodrepresentation of our data, where both fm(0) and kdepend upon the reaction temperature. Table 1 showsthe increase of k as the temperature is increased. Theplots of k as a function of reciprocal of temperature ofcure are drawn in Figure 4a and they are found to lieon a straight line with a good linear correlation coef-ficient r ) 0.99. Thus k may be considered to obey theArrhenius law as

where Qapp is an apparent activation energy, R is thegas constant, and k0 is the value of k for infinite

Table 1. Dielectric Parameters Observed for the Measurement Frequency f ) 105 Hz during the Cure of theEpoxide-Amine Thermoseta

T (K) ε0 ∆ε tm × 10-3 (s) ε′′max HW (s-1) area (s-1) k × 104 (s-1) fm(0) (Hz) r

290 8.30 4.60 13.90 0.809 17820 13560 3.90 0.26 × 108 0.997293 7.80 4.00 10.30 0.703 12670 7550 6.00 0.69 × 108 0.995303 7.70 3.90 8.35 0.644 7440 4330 8.90 8.70 × 108 0.992308 7.25 3.55 6.80 0.556 5040 2800 10.60 7.70 × 108 0.997313 7.05 3.35 5.10 0.515 3630 1830 16.60 86.0 × 108 0.997323 6.78 3.08 4.05 0.510 2170 925 22.80 126 × 108 0.998333 6.50 2.80 2.30 0.391 1280 468 72.30 135 × 1010 0.984353 6.00 2.40 1.50 0.221 470 142 220.2 990 × 1010 0.990a T is the temperature of the cure, ε0 is the relaxed permittivity, ∆ε is the total decrease of permittivity, which can also be written (ε0

- ε∞), tm is the time of maximum loss, and ε′′max is the height of the loss peak and HW its half-width, both of which have been measuredon plots ε′′ vs t not shown in this article. The seventh column displays the area of the loss peak, which was also determined from a plotsε′′ vs t (in a linear scale). k and fm(0) are the temperature-dependent constants included in eq 8: fm ) fm(0) exp(-ktm). The correlationcoefficient r, in the last column, corresponds to Figure 3, where tm is drawn as a function of log(fm).

τ ) τ0 exp(ktm) (7)

Figure 4. (a) ln(k) and (b) ln(tm/s) against reciprocal temper-ature. k is an empirical constant in eqs 7 and 8, and tm is thetime of maximum loss.

fm ) fm(0) exp[-ktm] (8)

k ) k0 exp(-Qapp

RT ) (9)

7100 Fournier et al. Macromolecules, Vol. 29, No. 22, 1996

temperature. We found ln(k0) ) 14.0, where k0 isexpressed in s-1 and Qapp ) 764 kJ/mol. Thus thefrequency of maximum loss is found to follow “first-orderkinetics” as indicated by eq 8 with a rate coefficient kobeying an Arrhenius law. In this case, the influenceof the temperature in the change of tm is implicit.Another approach is to consider a direct relationship

between the time of maximum loss and the temperatureof the cure, based on the kinetic theory16,18 of polyad-dition reactions of epoxide with amine. Mangion andJohari12,21 proposed that the time ∆t necessary for thethermosetting material to reach a predefined extent ofreaction or a particular physical state varies with thetemperature and can be given approximately by theequation

Mangion and Johari assumed that the time of maximumloss for a fixed measuring frequency may characterizea certain extent of reaction and they used ∆t ) tm in eq10, where E corresponds to an activation energy and∆t0 is the extrapolated value of tm when the curingtemperature becomes infinite. The values of tm corre-sponding to the present thermoset epoxide-amine reac-tion are plotted in Figure 4b against the reciprocal ofTcure for three fixed measurement frequencies. Thecalculation of E has been carried out for 12 measure-ment frequencies in the range 103-105 Hz and the meanvalue of E at 99% confidence level was found to be 38.2( 2.9 kJ mol-1. Mangion and Johari10 studied differentsystems and they obtained ∆t0 ) 668 µs and E ) 47.1kJ mol-1 for the DGEBA-DDM thermoset and ∆t0 )56.4 ms and E ) 44.5 kJ mol-1 for the DGEBA-DDSthermoset at f ) 103 Hz, where DDM is 4,4′-diamino-diphenylmethane and DDS is 4,4′-diaminodiphenyl sul-fone. Our data are very similar. They show that ∆t0increases significantly with the frequency of measure-ment and the apparent activation energy E may alsobe considered as a frequency-dependent variable. Thusthe significance of eq 10 can be queried insofar as themain parameters of this relation are obviously linkedwith the measurement frequency. The concept of ac-cumulated equivalent curing time yielding eq 10 isbased on the main assumption21 that the reaction pathis the same at all temperatures, and it tends to relatethe time ∆t required to reach a predefined extent or achemical and physical state with the temperature of thecure. The parameters ∆t0 and E must be independentof the spectroscopic technique of measurement used todetermine them. Since ∆t0 and E are found to beobviously dependent on the measurement frequency inDRS as seen in Figure 4b, eq 10 is only valid for a fixedfrequency of measurement. It is difficult to see aninterpretation from a chemical or kinetic point of view.We also consider that no dielectric event associated withthe extent of reaction has yet been established since tmis strictly defined as the time required for the loss factorto reach a maximum for the fixed frequency fm. Thenthe concept of accumulated equivalent curing time isnot appropriate to the analysis of dielectric quantitiessuch as the time of maximum loss, which does not definea chemical state independent of the measurementfrequency. So the dielectric parameter tm has to beconsidered as a two-variable function, taking in accountthe temperature and the frequency of measurement. Athree-dimensional plot in Figure 5 presents our experi-mental data obtained for nine temperatures in the range

290-353 K and for eleven frequencies of measurement.At a fixed frequency, tm is seen to increase exponentiallywith decreasing temperature. These results representa good approximation until the temperature is loweredto around 293 K. Afterward the experimental datadeviate strongly from the theoretical fitting curve, whoseequation can be written as

where tm0(fm) and δ(fm) are two coefficients dependingon the frequency of measurement. Measurements below293 K are extremely difficult to perform because of thecrystallization of the diamine (Tme1 ) 288 K) and thevery high viscosity of the epoxide at this temperature,which make complete mixing of the components bymanual means in a reasonable time almost impossible.Thus the reaction is diffusion controlled as soon as itstarts and the time of maximum loss increases stronglywith decreasing temperature for all the measurementfrequencies. At low temperatures, the progress of thecure, in practice, depends upon the duration and themethod of mixing the diepoxide with the diamine, andlarge variations of tm can be observed for the sameexperimental conditions. Above 293 K, tm obeys eq 11satisfactorily and the two parameters have been deter-mined using 11 frequencies of measurement:

The deviation between experimental data and thetheoretical 3D curves is shown in Figure 5 and demon-strates that eq 11 gives a good fit of the behavior of tmas a function of Tcure until a limit temperature around293 K.

4. Thermochemical DataThe thermochemical studies give the conversion R(tr)

as a function of tr at each reaction temperature. Ourresults are shown in Figure 6 and will be discussedfurther below.To model the cure kinetics of epoxy-amine thermo-

setting systems, it is necessary to derive an equationexpressing dR/dt, the rate of change of conversion R withtime, as a function of R and temperature. Sheppard andSenturia6 related different analysis using the followingdifferential equation to describe an isothermal cross-linking kinetics:

∆t(T) ) ∆t0 exp( ERT) (10)Figure 5. Time of maximum loss tm against the temperatureof cure and the frequency of measurement, on a logarithmicscale.

tm ) tm0(fm) exp[δ(fm)T] (11)

tm0(fm) ) 72670 - 9800 × log(fm) (r ) 0.997)

δ(fm) ) -0.051 + 0.0030 × log(fm) (r ) 0.998)

Macromolecules, Vol. 29, No. 22, 1996 Molecular Dynamics during Polymerization 7101

where k is the rate coefficient and m is the empiricalreaction order. However, one difficulty with suchempirical approaches is that they tend to oversimplifythe chemical kinetics. In 1970, Horie et al.22 proposedthe following equation to describe the kinetics of po-lymerization of a resin epoxy and an amine:

where k1 is a rate constant for the reaction catalyzedby groups initially present in the resin, k2 is the rateconstant for the reaction catalyzed by newly formedhydroxyl groups, and B is the ratio of primary amineN-H bonds to epoxide groups in the initial mixture.They22 concluded that the curing reaction of DGEBAwith aliphatic diamines, which proceeded through athird-order mechanism, was followed by the diffusion-controlled mechanism at the later stage of conversion,and the reaction ceased at the conversion where anysegmental diffusion of functional groups was suppressedbecause of the onset of the glass transition of the system.Similar equations11,23 were used for the thermosettingreactions of different systems DGEBA-diamine systemsbut they do not consider the reactions occurring in thelater stage of the cure, such as the ether reaction,18which is believed to become particularly significant athigh temperatures. Kamal24 proposed the followingsemiempirical equation, taking in account etherificationreactions:

Many workers19 have proposed different values of theempirical reaction orders m and n. Cole16 found thatthe values of n and m varied significantly with temper-ature. He developed recently a more rigorous model fordescribing the cure kinetics of general epoxy-aminesystems. He concluded that the reaction could bedivided into two stages. At the beginning of the cure,the amine-epoxide reaction dominates and the effectof the etherification reaction is insignificant, so theHorie22 equation (13) is valid for the initial reaction. Butin the later stages of curing, the isothermal cross-linkingreaction tends to become simply first order with respectto epoxide concentration and can be described in termsof the simple differential equation

Here the rate coefficient k(R) is dependent upon theextent of reaction and will fall substantially when thereaction becomes diffusion controlled toward the end ofthe cure. Cole expressed this “diffusion factor” as

In eq 16 the denominator is a cut-off function, whichwe note is formally the same as the Fermi-Diracfunction in the band theory of metals. It allows k(R) tobe constant for R < Rc and to decrease rapidly to zerofor R > Rc. Equation 16 gives a point of inflection fork(R) at R ) Rc. Physically, it is more acceptable to expectthat -dk(R)/dR should increase monotonically in thediffusion-controlled regime. Therefore we modifiedCole’s equation (16) as follows:

In this equation, k0 is the rate coefficient for non-diffusion-controlled kinetics, which is given by experi-ment for 0.4 < R < 0.55, where (dR/dt)/(1 - R) isapproximately constant. Equation 17 is different fromCole’s equation in the following ways:(i) Rf is the final degree of polymerization, which is

an experimental result (Rf ) Qi/(Qi + Qr)), whereas Rcis a certain critical value that is difficult to determinewith accuracy.(ii) Equation 17 allows us to fit the experimental data

to the final degree of conversion Rf as seen in Figure 6.The constant b is the only unknown in eq 17, and it

can be easily calculated in our case to give the bestcorrelation coefficient between the theoretical curve andthe experimental data that were obtained using DSCat four curing temperatures ranging from 213 to 253K. The results in Figures 7-9 were derived from theDSC data presented in Figure 6, where the values of R,plotted with respect to the time of cure, describe typicalcurves characterizing an autocatalytic polymerizationreaction. Figure 7 shows a normalized plot of the ratecoefficient in the later stages of the cure at 333 K. Wesee that k(R)/k0 decreases dramatically for R > 0.65 asthe network vitrifies and the reaction becomes diffusioncontrolled. This behavior is observed in Figures 8 and9 at respectively 323 and 353 K for high values of R.

Figure 6. Extent of reaction R against the time of reaction(in minutes) for the curing temperatures T ) 313 K (3), T )323 K (O), T ) 333 K (2), and T ) 353 K (4).

dRdt

) k(1 - R)m (12)

dRdt

) (k1 + k2R)(1 - R)(B - R) (13)

dRdt

) (k1 + k2Rm)(1 - R)n (14)

Figure 7. Normalized kinetic rate k(R)/k0 against the extentof reaction R during the later stages of the cure of the systemDGEBA-PACM at T ) 333 K.

dRdt

) k(R)(1 - R) (15)

k(R) )kc

1 + exp[C(R - Rc)](16)

fd(R) )k(R)k0

) 2[ 11 + exp[(R - Rf)/b]

- 12] (17)

7102 Fournier et al. Macromolecules, Vol. 29, No. 22, 1996

The kinetic parameters k0, Rf, and b, correspondingto eqs 15 and 17, together with the correlation coefficientr for the later stages of the present diepoxide-diaminethermoset reaction at 313, 323, 333, and 353 K aresummarized in Table 2. The rate coefficient k0 wasfound to obey the following Arrhenius relationship, anda good linear correlation was observed (r ) 0.999).

We calculated ln(A) ) 13.39 and the apparent activationenergy Eq ) 59 kJ/mol. This energy value is similar tothose obtained by Cole17 and Chiao18 for epoxide-aminereactions of different systems. In Table 2, Rf is seen toincrease as the temperature of cure is increased. Cole17proposed a linear relationship of Rf against the temper-ature of the cure. However, this relationship is not validoutside the experimental temperature range. A betterempirical equation to express Rf as a function of T was

found to be

Hence eq 19 gives a good approximation of the evolutionof Rf over a larger range of temperature than a linearequation. The empirical constant bwas expected to riseas the temperature of the cure was lowered, accordingto the theory that the reaction is slower at low temper-ature, and the network changes are also slower at thelater stage of the cure. But the b values obtained fromthe analysis that are summarized in Table 2 do notfollow such behavior. In further studies by us, it wasconcluded that no systematic trend with temperaturewas observed. The average value for b was 0.0232 witha standard variation s ) 1.16 × 10-2.It was interesting to incorporate our diffusion control

term, eq 17, in a theoretical equation which completelydescribes the kinetics of the cure for all the tempera-tures. For temperatures below 333 K, the best fit wasobtained by combining Horie’s equation (13) with thediffusion control factor eq 17. In eq 13, the constant Bis the ratio of primary amine N-H bonds to epoxidegroups in the initial mixture. The chosen ratio of(diepoxide/diamine) was (2/1) so the value of B was 1.In this case, Horie’s equation corresponds to Kamal’sequation (14), with m ) 1 and n ) 2. These values ofm and n were proposed by Riccardi19 to fit the kineticsof an amine adduct cured DGEBA. Thus, only thereactions between the epoxide group, the primaryamines, and the secondary amines have to be consideredfor these temperatures. In the first stage of curing, thekinetics are found to be second order with respect toepoxide groups, but in the later stage, first and secondorder can be assumed to be present as well. Theincorporation of eq 17 in Horie’s equation gives

Figure 8 shows the excellent correlation between ourexperimental data and the theoretical fit using eq 20for the cure of the DGEBA-PACM system at 323 K overthe entire range of R. The kinetic rates k0, k1, and k2,the constant b, and the correlation factor r are given inTable 3 for the epoxide-amine thermoset at threedifferent curing temperatures. The values of the con-stant b are slightly different from those summarized inTable 2 because of the second-order rate of eq 20.At 353 K, the experimental data were not fitted with

eq 20, but a good result was obtained by a combinationof Kamal’s equation (14) (with m ) 0.6 and n ) 2) andthe diffusion control factor (17), giving the followingrelation:

Figure 8. Normalized kinetic rate k(R)/k0 ) (dR/dt)[1/(k0(1 -R))] against the extent of reaction R for the cure of the systemDGEBA-PACM at T ) 323 K.

Figure 9. Normalized kinetic rate k(R)/k0 ) (dR/dt)[1/(k0(1 -R))] against the extent of reaction R for the cure of the systemDGEBA-PACM at T ) 353 K.

Table 2. Kinetic Parameters for the Diffusion FactorExpressed in Eq 17 Describing the Thermosetting of the

DGEBA-PACM Systema

T (K) k0 (min-1) Rf b r

313 1.27 × 10-4 0.659 0.017 0.990323 1.96 × 10-4 0.696 0.034 0.998333 3.77 × 10-4 0.747 0.032 0.991353 1.50 × 10-3 0.820 0.010 0.990a T is the temperature of the cure, k0 is the rate coefficient for

the non-diffusion-controlled kinetic regime, Rf is the final degreeof conversion of the epoxide-amine thermoset, b is an empiricalconstant, and r is the correlation coefficient between the experi-mental data and the theoretical curve corresponding to eq 17.

k0 ) A exp(-Eq/RT) (18)

Table 3. Evaluation of the Kinetic Parameters of Eqs 20and 21a for the Thermoset of the DGEBA-PACM System

T (K) k0 (min-1) k1 k2 b r

313 1.27 × 10-4 0.306 3.073 0.0127 0.998323 1.96 × 10-4 0.415 3.124 0.0264 0.999333 3.77 × 10-4 0.346 3.233 0.0203 0.991353 1.50 × 10-3 0.110 2.697 0.0100 0.999a Equation 20 was used to fit the data obtained during the cure

of the thermoset DGEBA-PACM at 313, 323, and 333 K while eq21 was used at 353 K.

Rf ) 1.354 ln(T) - 7.127 (r ) 0.998) (19)

dRdt

) g(R)fd(R) ) k0(k1 + k2R)(1 - R)2 ×

[ 2

1 + exp(R - Rf

b )- 1] (20)

Macromolecules, Vol. 29, No. 22, 1996 Molecular Dynamics during Polymerization 7103

The correlation coefficient between this equation andthe fit to the data obtained from the present experi-ments is r ) 0.999; the curve using eq 21 and theexperimental data for the cure at 353 K are drawn inFigure 9.

5. DiscussionDuring the epoxide-amine thermosetting reaction,

irreversible changes in the physical properties areobserved in both dielectric and DSC experiments as theextent of reaction R increases from zero to the finaldegree of conversion. We observe that an obviouscorrelation between DRS and DSC dynamics is possible,but is difficult to quantify during a reaction. Thus DRSmeasurements predict that if molecular motions, at agiven reaction temperature TR, have reached a timescale t > 102 s, then the material may be regardedoperationally as a “glass”. When the time scale formolecular motion reaches this region, then as thereaction proceeds, it is expected that the reaction wouldtend to be approaching the diffusion-controlled limitingrange and would effectively “stop”. Extrapolations ofdielectric data at medium frequencies to the low-frequency range should give indications of the vitrifica-tion region in which polymerization actually stops (i.e.the rate of reaction becomes effectively zero). We shallexamine, through comparisons of DRS and dynamicDSC data, the extent to which DRS can predict thekinetic behavior in the vitrification region and hencedraw up a set of guidelines based on general workingrules.The present attempt to establish qualitative and

quantitative relationships between the kinetic anddielectric behavior of an epoxide-amine thermosetcuring requires careful consideration and a precisedefinition of terms. The relaxed permittivity ε0, whichis the short-term value of ε′ for each plot in Figure 1a,was dependent on the concentration of the dipole chainsegments in the initial mixture. The average dielectricrelaxation time for molecular motions in the reactingmixture increases with the extent of reaction as theeffective local viscosity increases. In our experimentalrange, the plots of log fm vs tm were linear, in accordwith eq 8, at each reaction temperature. In thisanalysis, as a working rule, we shall assume that suchplots are linear within and beyond the ranges of fm andTR shown in Figure 10 and we shall use the valuesextrapolated to higher and lower frequencies as a partof a systematic method for comparison of DRS and DSCdata. Clearly, the extremely low values of log(fm) couldnot be observed experimentally but are predicted usingthe explicit assumption that we make in our systematicanalysis. That forms a basis for the changes in fm inthe later stages of the diffusion-controlled range. Therelationship between fm and tm shown in Figure 10 canbe expressed by eq 22 derived from eq 8:

where fm(0) is a temperature-dependent quantity andk is a rate coefficient summarized in Table 1. Thesemiempirical kinetics equations (20) and (21) were

found to fit the whole range of experimental data at 313,323, 333, and 353 K and can be rewritten as

where fd(R) is the diffusion control factor expressed ineq 17 and g(R) is derived from Horie’s equation (respec-tively Kamal’s equation) that fit the normal kineticregime data in eq 20 (respectively eq 21). The functionalform of eq 23 allows us to calculate the time t(R) neededto reach a given extent of reaction by a numericalintegration:

Thus according to eqs 22 and 24, the frequency ofmaximum loss fm at which the dielectric absorptionreaches its maximum at a fixed TR can be expressed asa function of the degree of conversion of the reaction asfollows:

A different way to plot log(fm) vs R is to use the DSCdata which gives for each experimental extent of reac-tion the corresponding measurement time. Thus thistime can be transformed with the aid of eq 24 in orderto obtain the values of log(fm) for all the values of Robtained through the DSC measurement. The plots oflog(fm) and (dR/dt)/(k0(1 - R)), which represents thenormalized kinetic rate coefficient k(R)/k0, are drawn asa function of R in Figure 11 for T ) 333 K. It appears

dRdt

) k0(k1 + k2R0.6)(1 - R)2[ 2

1 + exp(R - Rf

b )- 1](21)

log(fm) ) log(fm(0)) -ktmln(10)

(22)

Figure 10. Frequency of maximum loss, on a logarithmicscale, against the time of reaction for the curing temperaturesT ) 313 K (3), T ) 323 K (O), T ) 333 K (4), and T ) 353 K(0).

dRdt

) g(R)fd(R) (23)

t(R) ) ∫0R dRg(R)fd(R)

(24)

log(fm) ) log(fm(0)) - kln(10)∫0R dR

g(R)fd(R)(25)

7104 Fournier et al. Macromolecules, Vol. 29, No. 22, 1996

that the curves log(fm) vs R are somewhat similar forall the cure temperatures T e 333 K, as seen in Figure12. log(fm) decreases approximately linearly as R in-creases in the range 0-0.6; then it drops suddenly tolower values as the reaction becomes diffusion con-trolled. The kinetics curves indicate the advent of thediffusion control when they decrease quickly in thelatter stage of the reaction. Owing to this observation,we take as a reference that diffusion control becomesimportant at a fixed extent of reaction that we defineas R/Rf ) 85%. This numerical value was also proposedby Enns and Gillham25 as the characteristic extent ofreaction for the conversion from gel to glass. Thisproposition required a comment. Indeed the gel is anetwork of limited stiffness which can be quantified bya modulus as distinct from a liquid of high viscositywhose modulus is zero. After a long observation time,the viscous liquid becomes a gel and then a glass.However, no characteristic state of the thermoset willbe associated at the value of R ) 0.85Rf in this study,but for this extent of reaction, log(fm) is found to beapproximately equal to zero for the three curing tem-peratures considered. Reciprocally, for log(fm) ) 0 theextents of reaction are found to be 86, 88, and 86% of Rffor respectively T ) 313, 323, and 333 K. ConsideringFigure 12 and the observations above, the followingworking rules can therefore be established:(i) The diffusion control of the reaction is character-

ized by the declination of the kinetic curves, whichentails the sharp fall in the dielectric curves log(fm) vsR. Now it appears in Figure 12 that for log(fm) > 1 and

T ) 313, 323, and 333 K, the dielectric data points forour frequency range 10-105 lie approximately on straightlines, which corresponds to a normal kinetic regime.This behavior can be seen in Figure 13, where, for ourdata, log(fm) is drawn as a function of R for the normalkinetic range (R < 0.6). Thus the reaction is notdiffusion controlled at the time tm when the dielectricloss peak occurs for every measurement frequency inour range (101.2-105 Hz). At T ) 333 K for instance,when ε′′ reach its maximum at f ) 105 Hz, the value ofR represents only 50% of the final degree of conversionand is 67% of Rf for f ) 103 Hz. Hence at low temper-atures, the present definition of the time of maximumloss peak as an indicator of the onset of the vitrificationshould be reconsidered carefully.(ii) The time at which the diffusion control of the

thermoset occurs can be defined operationally (for thereasons seen above) by dielectric spectroscopy as thetime required for the loss factor to reach its maximumvalue for the measurement frequency f ) 1 Hz in ourexperimental temperature range (313-333 K). For T) 353 K, the curve log(fm) vs R is qualitatively differentfrom those at 313, 323, 333 K insofar as the markeddrop due to diffusion control occurs very close to Rf. Itwas determined that the extent of reaction is 84% forlog(fm) ) 5, 92% for log(fm) ) 3, and around 98% for log-(fm) ) 0.The curves log(fm) vs R can be partially seen in Figure

13 for the range 0 < R < 0.65. They were obtained fromthe extrapolated lines presented in Figure 10b whichfit the experimental data in the range 101.2 < fm < 105.Then in the curves log(fm) vs R drawn in the normalkinetic regime (R < 0.6), the values of log(fm) seem tobe linearly dependent on R. So we may write in thisregime

The values of f ′m(0), kp, and the correlation coefficient rare presented in Table 4 for different cure temperatures.A good agreement between eq 26 and the data is seenin Figure 13 for the lower temperatures. These param-eters displayed in Table 4 can be combined with (i) datafor Rf and (ii) the diffusion control function fd(R) topredict log(fm) vs R. Indeed eq 17, which describes thedrop of the kinetic curves, can be used as a cut-offfunction to fit the curves log(fm) vs R in the diffusioncontrol region, where the constant b is chosen to be 0.01for all the curves and Rf(T) is defined in eq 17. Thenwe can build a family of theoretical curves for log(fm)

Figure 11. Normalized kinetic rate k(R)/k0 ) (dR/dt)[1/(k0(1- R))] against the extent of reaction R for the cure of the systemDGEBA-PACM at T ) 333 K. Also shown are log(fm/Hz)against R, where fm is the frequency of maximum loss.

Figure 12. Normalized kinetic rate k(R)/k0 ) (dR/dt)[1/(k0(1- R))] against the extent of reaction R for the cure of the systemDGEBA-PACM at T ) 333 K (s), T ) 323 K (- - -), and T) 313 K (‚‚‚). Also shown are log(fm/Hz) against R forcomparison.

Figure 13. Frequency of maximum loss, on a logarithmicscale, against the extent of reaction R for the curing temper-atures T ) 313 K (4), T ) 323 K (O), T ) 333 K (0), and T )353 K (3).

fm(R) ) f ′m(0) exp[-(kpR)] (26)

Macromolecules, Vol. 29, No. 22, 1996 Molecular Dynamics during Polymerization 7105

vs R, as shown in Figure 14 and corresponding to eq27, for a fixed temperature of the cure:

In eq 27, the term A fits the data for log(fm) in the rangecorresponding to the real kinetic regime. This part ofthe equation is based on the assumed linear dependenceof log(fm) with respect to time in the high-frequencydomain. The term B is the cut-off function whose onlyvariable is Rf(T). a(T) and c(T) are two temperature-dependent quantities which were extrapolated from theparameters summarized in Table 4. Figure 14 is onlyan approximate representation of the evolution of log-(fm) vs R during the cure of an epoxide-amine thermosetat different temperatures and it is based on theworkingassumptionmade here that log(fm) is linearly dependenton tm over a large range of frequencies, as presented inFigure 10b.The curves presented in Figure 14 provide a general

trend of the variation of log(fm) as the extent of reactionincreases and give some important information:(i) The slopes of the straight lines log(fm) vs R which

form the theoretical model outside the diffusion-controlled regime are seen to decrease with increasingtemperature of cure. At higher Tcure, the loss peakremains in the high-frequency domain until the effectivevitrification process occurs, after which log fm decreasesrapidly.(ii) In the diffusion-controlled range for fixed curing

temperature, R is always very close to the value of Rf.We have taken as a reference log(fm) ) 0 to define the

onset of diffusion control at low cure temperature. Thusfor log(fm) < 0, the diffusion controlled region is achievedat all the experimental temperatures and, according toFigure 14, we can consider that Rf is reached in practice.We may make a quantitative correlation between the

time-dependent dielectric relaxation process and thechemical kinetics over the whole range studied, andespecially as the vitrification is approached and thereaction becomes diffusion controlled. The diffusioncontrol limit and its temperature dependence are seenin Figure 14. As TR increases, the pattern of the curveschanges and the diffusion control sets in at successivelyhigher frequencies. It was previously noticed that forTR < 333 K, our experimental range of fm (101.2-105 Hz)corresponded to the normal kinetic regime, but at 353K, the dielectric loss peaks, occurring at time tm for fm< 105 Hz characterize the diffusion control region andthen the final degree of conversion. Thus for highcuring temperatures (T > 353 K), which are widely usedin industry for epoxy-amine curing, it is possible tofollow the extent of reaction in the vitrification region,by following the shift of the dielectric loss peak fromhigh to low frequencies. The only data required to buildthe correlation curves in Figure 14 is the final degreeof conversion of the network at the chosen temperature,which can be determined by DSC or FTIR measure-ments or also calculated with eq 8 in our case, or usinganother relationship for the kinetics, for example, thatproposed by Cole.17 Then by using a similar relationas eq 18, a set of correlation curves as shown in Figure14 for our study, can be obtained for other systems.

6. Conclusions

We have made diverse comparisons between thedielectric data, for molecular dynamics, and the DSCdata, for chemical kinetics, in order to understand howdielectric data can give information on the changes inkinetic behavior that lead to glass formation. Thevariation of the molecular dynamics over a large rangeof frequencies is complementary to the variation of thenormalized kinetic rate coefficient k(R)/k0(T) and hasbeen studied here as a function of conversion R(t) for agiven temperature. As the temperature is lowered, theloss peak measured in our frequency range does notoccur in the diffusion-controlled kinetic regime, whichobviously represents the effective vitrification of thenetwork.For temperatures above 353 K, at which epoxy-

amine cures are usually performed, it is observed thatthe dielectric loss peak measured for any frequency inour experimental range coincides with the diffusion-controlled kinetic regime. In this case, the interpreta-tion of the dielectric loss peak as the onset of thevitrification process, which has been widely used in thedifferent studies of curing system, may be consideredas valid. Our studies suggest that for all the curingtemperatures studied, the reaction becomes diffusioncontrolled at times when the dielectric absorptiondescribed by ε′′(t,f,L,T) reaches its maximum value ε′′maxat measurement frequencies f e 1 Hz. This resultshould be considered as an quantitative indicator whichrelates the dielectric properties of the curing system tothe real chemical changes in the thermoset.

Acknowledgment. We are pleased to acknowledgean Erasmus Grant to J.F. and thank Dr. John S. Daviesfor advice concerning the materials used in this study.

Table 4. Calculation of Parameters Deriving from Eq 26for the Thermoset of the DGEBA-PACM Systema

T (K) f′m(0) Hz kp r

313 1.6 × 108 34.7 0.99323 2.1 × 108 35.7 0.99333 2.0 × 108 33.6 0.99353 4.9 × 108 16.9 0.98

a T is the temperature of the cure, f′m(0) is the frequency ofmaximum loss for the noncured DGEBA-PACM mixture, kp isthe rate coefficient, and r is the linear correlation coefficientbetween the experimental data and the theoretical straight linepresented in Figure 13.

Figure 14. Theoretical plots of log(fm/Hz) against the extentof reaction R.

log(fm(R)) ) a(T)[log(f ′m(0))A

- kRln(10)] ×

[ 2

1 + exp[R - Rf(T)0.01 ]

B

- 1] + c(T) (27)

7106 Fournier et al. Macromolecules, Vol. 29, No. 22, 1996

We thank EPSRC for their support of the dielectricsresearch group.

References and Notes

(1) McCrum, N. G.; Read, B. E.; Williams, G. Anelastic andDielectric Effects in Polymeric Solids; Dover: New York, 1991.

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(3) Hill, N.; Vaugan, W.; Price, A. H.; Davies, M. DielectricProperties and Molecular Behaviour; Van Nostrand: London,1969.

(4) Williams, G. In Comprehensive Polymer Science, Allen, G.,Bevington, J. C., Eds.; Pergamon: Oxford, 1989; Vol. 2,Chapter 7, p 601.

(5) Williams, G. In Dielectric and Related Molecular Processes;Specialist Periodical Reports; The Chemical Society: London,1975; p 151.

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