Analysis of the cure-dependent dielectric relaxation behavior of an epoxy resin

Analysis of the Cure-Dependent Dielectric Relaxation Behavior of an Epoxy Resin

Y O N G D E N G and GEORGE C. MARTIN

Department of Chemical Engineering and Materials Science, Syracuse University, Syracuse, New York 13244

SYNOPSIS

The dielectric relaxation behavior of an epoxy-amine resin was investigated using the Wil- liams-Watts relaxation function. Phenomenologically, the dielectric features of the resin during cure are similar to those of stable materials. The distribution parameter of the dipole relaxation decreases from the onset of cure to a conversion near the gel point and then maintains a constant value. Based on the experimental observations and theoretical con- siderations, a single-frequency approach has been proposed for extracting the relaxation time of maximum loss. The relaxation data so obtained are independent of the measurement frequency and are in agreement with those acquired directly from the dipole loss peaks. 0 1994 John Wiley & Sons, Inc. Keywords: dielectric relaxation epoxies thermosets curing - dipole relaxation time

I NTRO DUCT10 N

Cure monitoring is of practical importance for the processing of a thermoset. Among the frequently used monitoring techniques are dynamic mechanical analysis, various spectroscopic methods, and dynamic dielectric analysis. Dielectric analysis offers the advantages of nondestructive inspection, in situ measurement, and high ~ensitivity.'-~ One of the properties that may be evaluated using this tech- nique is the dipole relaxation time, which is independent of measurement frequency and is very sensitive to changes in other properties of the resin, especially in the post-gel stage. The dipole relaxation time may be quantitatively utilized as a measure of such important parametersjproperties as the degree of conversion, the glass transition temperature, the v is~os i ty ,~ and the diffusion ~oefficient.~

One way to determine the relaxation time during isothermal cure is to collect a number of loss data using a fixed frequency and to then find the cure time corresponding to the maximum dipole loss. The inverse frequency is equivalent to the relaxation time at the particular cure time.4 At a high conversion, when the relaxation time is long, a low frequency is required and a long period of time is needed. Thus,

Journal of Polymer Science: Part B Polymer Physics, Vol. 32,2115-2125 (1994) 0 1994 John Wiley & Sons, Inc. CCC 0887-6266/94/122115-11

it is difficult to acquire a series of relaxation data using this method for fast curing processes or a t high conversions.

For stable materials, Cole-Cole diagrams, or complex-plane plots of permittivity, may be obtained either with a set of frequencies at a constant temperature or with a single frequency at a varying temperature. The diagrams appear to be skewed arcs, which may be described by the Williams-Watts relaxation function.6 For thermosetting resins, the Cole-Cole diagrams may be constructed only with complex permittivities obtained at the same degree of conversion. However, Mangion and Johari7 have recently found that the evolution of dielectric features of thermosetting resins during cure is phenomenologically similar to that of stable materials and the complex-plane plot of permittivity obtained with a single frequency during isothermal cure may also be well described by the Williams-Watts relaxation function. Note that such a plot is not a true Cole-Cole diagram since the complex permittivities were collected over a range of conversions. It was also found that the distribution parameter y of the Williams-Watts relaxation function decreases in the pre-gel stage and becomes a constant upon gelation. Consequently, the relaxation times of the Williams- Watts relaxation function may be extracted from the results of single-frequency measurement^.^,^ These findings suggest an efficient way to acquire

2115

21 16 DENG AND MARTIN

relaxation data during thermoset cure, especially for relatively fast curing processes or at high conversions. Therefore, the dipole relaxation time may become a very useful parameter for cure monitoring purposes.

The objective of the present work was to analyze the evolution of the relaxation behavior of thermosetting resins and to develop a convenient method to determine the relaxation time of maximum loss based on single-frequency measurements. The relaxation time of maximum loss, or the inverse frequency at maximum loss, is more commonly used than the Williams-Watts relaxation time and is simply referred to as the relaxation time in this ar- ticle. The evolution of the relaxation behavior of an epoxy-amine resin was studied following an approach different from the one used by Mangion and J ~ h a r i . ~ Three existing algorithms for evaluating the Williams-Watts relaxation function were compared based on their accuracy. A simple equation was de- rived to convert the Williams-Watts relaxation time into the relaxation time of maximum loss. Finally, the relaxation times extracted from single-frequency measurements were compared to those obtained directly from the experimental loss data.

THEORETICAL BACKGROUND

In the permittivity complex plane, an ideal dipole relaxation forms a semicircle, and the Debye model adequately describes this relaxation behavior. For polymeric materials, skewed arcs are typical and the relaxation behavior is often modeled with empirical relationships, such as the Williams-Watts relaxation function6 and the Davidson-Cole relaxation function.”

For a stable material, the complex permittivity may be written as:

&*( w ) = &, + (CO - &,)N*( 0 ) (1)

N*( w ) = e-j’”‘( - g) dt ( 2 )

where c* is the complex permittivity, E, is the unrelaxed (high-frequency limit) permittivity, EO is the relaxed (low-frequency limit) permittivity, 0 is the angular frequency, N* is the normalized complex permittivity, $ is a decay function characterizing the decay of polarization after removal of the electric field, and t is time. At isothermal conditions, the normalized complex permittivity changes due to a change in frequency.

The decay function of the Williams-Watts relaxation function is given by:

( 3 )

where 70 is the relaxation time of the Williams- Watts relaxation function and ,6 (between 0 and 1 ) is the distribution parameter, which characterizes the non-idealness of the relaxation. When ,6 = 1, the relaxation is ideal.

Mangion and J ~ h a r i ~ ’ ~ have found that the evolution of dielectric features of thermosetting resins during cure is phenomenologically similar to that of stable materials and the complex-plane plots of permittivity of the resins obtained with a single frequency during isothermal cure may be considered similar to the Cole-Cole arcs of stable materials. Be- cause the properties of a curing resin vary during the measurements, the data collected correspond to a range of conversions instead of a fixed conversion and E * , E,, to, and 7 0 are all functions of the cure time. To describe the dielectric data at various conversions, the original Williams-Watts relaxation function was modified. Specifically, N* was rede- fined as a function of w i o instead of w alone and ,6 was replaced with y to indicate a reactive resin or an unstable material:

Hence, the distribution parameter y may be considered as an average of ,6 values over the range of conversions covered by the measurements. Eq. ( 4 ) in- dicates that, as the resin cures isothermally, the normalized complex permittivity at a constant frequency changes due to the increasing 7 0 .

If the distribution parameter is a constant, the relaxation behavior of the resin is independent of conversion. Then, the Williams-Watts relaxation time may be extracted from the dielectric data obtained during isothermal cure.728

EXPERIMENTAL

The epoxy resin (X-22) used in this study was the diglycidyl ether of bisphenol A (DGEBA) obtained from the Shell Development Co. The epoxide equivalent weight was 172. The crosslinking agent was

DIELECTRIC RELAXATION OF EPOXY RESIN 2 11 7

4,4'-diaminodiphenol methane (DDM) from Aldrich with a 99% purity. The epoxy resin was first melted at 70°C and cooled to 55°C. Then, the stoichiomet- rically balanced amount of diamine was poured into the epoxy. The mixture was mechanically stirred at a high speed with a magnetic stirrer for 15 min, when it became clear and transparent. The sample was then cooled and stored at -21°C.

A Micromet Eumetric System I1 microdielectrometer was employed to perform the dielectric measurements. The system, when equipped with a low conductivity sensor, can generate measurement sig- nals ranging from 10k to 0.005 Hz. Two types of isothermal experiments-curing experiments and frequency-scan tests-were conducted. In a curing experiment, the sensor was inserted into an electri- cally heated oven and the temperature was allowed to equilibrate. Then, a small amount of sample was placed on the surface of the sensor and covered the entire sensing area. In each measurement cycle, the microdielectrometer scanned the sample from high to low frequencies over a range of 10k to 0.01 Hz.

The frequency-scan tests were conducted in order to study the relaxation behavior of the pre-gel samples. Because the pre-gel samples have relatively short relaxation times, low temperatures were used in the tests so that the relaxation peaks were located within the frequency window of the dielectrometer. As illustrated in Figure 1, to conduct a frequency- scan test, the sample was first cured at 75°C. After a certain period of time, the sample and sensor were removed from the oven and left at room temperature for 20 to 30 min to allow the oven to cool to room temperature. They were then reinserted into the sealed oven. Cooling of the system was accomplished by running liquid nitrogen through tubing inside the oven and electric heating was controlled by an

several scans

Time Figure 1. Schedules of the frequency-scan tests.

Omega controller to maintain the system at the de- sired temperature. In each test, the sample was scanned once from high to low frequencies.

Up to 32 frequencies ranging from 10k to 0.005 Hz were used in the frequency-scan tests. Depending on frequencies used, 5 to 30 min were needed for each scan, but most scans at relatively high temperatures were completed within 15 min. The test schedules may be classified into two groups. As indicated in Figure 1, Group a included those scans at temperatures below 20°C. Each sample was tested at several temperatures. Group b included those scans at temperatures above 2O"C, and each sample was tested several times at the same temperature after the sample was held at the temperature for various periods of time.

The epoxide conversion was determined with Fourier transform infrared spectroscopy (FTIR) experiments using an IBM Instruments IR/32S FTIR spectrometer. The sample holder was first heated to slightly higher than the cure temperature. Then, the liquid sample between NaCl cells was placed in the sample holder. The infrared scans were initiated when the cell temperature was 2°C below the cure temperature. The cell temperature reached the cure temperature in less than 2 min. Each IR spectrum consisted of 15 co-added interferograms at 2 cm-' resolution. The disappearance of the epoxide peak area at 916 cm-' was monitored, and the aromatic ring-carbon-aromatic ring stretch at 1184 cm-' was chosen as the reference peak. Thus, the epoxide conversion was calculated using:

where a is the epoxide conversion, Ag16 is the peak area at 916 cm-', and AIla4 is the peak area at 1184 cm-' .

The glass transition temperature of the resin was determined with a Mettler DSC-30 differential scanning calorimeter (DSC) . In each of the DSC tests, approximately 10 mg sample was first placed in an aluminum pan, cured for a period of time, cooled to -70°C and scanned at a heating rate of 10°C /min. The temperature corresponding to the onset of the endothermic deflection of the baseline was taken as the glass transition temperature.

RESULTS AND DISCUSSION

Figure 2 shows the results of the FTIR studies at isothermal conditions between 55 and 115°C. Those

DENG AND MARTIN 2118

1 .o

0.8 C 0 .- t? $ 0.6 C 0 0

-$! 0.4 X 0 Q

W

.-

0.2

0.0,

Figure 2 . arrows indicate vitrification times.

FTIR data for the epoxy-amine resin. The

data at low conversions may be well-fitted to an au- tocatalytic rate law. As indicated by the arrows in the figure, the glass transition temperature of the sample reached the corresponding cure temperature prior to the end of each experiment.

The frequency-scan tests were performed at temperatures ranging from -11.1 to 40°C. The DSC results show that the scan temperatures were typi- cally at least 10°C above the glass transition temperatures. For the tests below 20"C, the samples were cured at 75°C for no more than 100 min and scanned at 4 or 5 temperatures with 4.4"C incre- ments. Figure 3 shows the permittivity and loss factor measured at -4.4,0.0,4.4, and 89°C for a sample cured for 60 min (01 = 13% ) . As the temperature increases, the permittivity and loss curves shift to higher frequencies. The relaxed permittivity co may be found from the plateau of each of the permittivity curves. co decreases with increasing temperature. The unrelaxed permittivity em was not reached in some of the experiments (e.g., at 4.4 and 8.9"C). The dipole loss peaks are located at relatively high frequencies and the peak value decreases with increasing temperature. At lower frequencies, the ionic contribution to the loss factor becomes significant.

Figure 4 is a plot of the Cole-Cole arcs of the same sample. The skewed arc becomes smaller as the temperature increases. The solid curves in the figure are the best fits of the Williams-Watts relaxation function. All the Cole-Cole arcs may be well described by the relaxation function. The Davidson- Cole relaxation function" was also used to fit the

arcs and was found to be, generally, inferior to (or exhibiting larger deviations than) the Williams- Watts relaxation function. Thus, only results from the Williams-Watts relaxation function are reported.

The procedure for evaluating the Williams-Watts relaxation function is described below. Dishon et a1.l' have recently presented an algorithm to cal- culate the normalized complex permittivity N* of the Williams-Watts relaxation function and tabulated the results. The algorithm is exact; the com- putation involves the evaluation of the infinite series over a wide range of w r o and the evaluation of the numerical integrations for the rest of W T ~ . Both the series and the integrations are equivalent to N*. Thus, highly accurate results may be obtained by using many terms of the series and small intervals for the integrations. The present work adopted a similar approach and used the tabulated results in ref. 11 whenever the numerical integrations were needed. Specifically, for small values of W T O given in Tables 1 and 2 of ref. 11, N* was determined by interpolation using the data listed in Tables 3 and 4 of ref. 11. For larger values of W T ~ , the two convergent series, Eqs. (3 ) , ( 5 ) , and ( 7 ) in ref. 11, were used as suggested by Dishon et al. This approach saves computing time by eliminating the numerical integrations. The conjugate directions algorithm" was employed to fit the Williams-Watts relaxation function to the Cole-Cole arcs. Since the unrelaxed permittivity was not reached in some experiments, E, was also treated as an adjustable parameter. Thus, 3 parameters, p, ro and c , , were optimized for each Cole-Cole arc.

12'0 i 3 ' 0

Frequency (Hz)

Figure 3. sample cured at 75°C for 60 min. (a = 13% ) .

Permittivity (0) and loss factor (0) for a

DIELECTRIC RELAXATION OF EPOXY RESIN 2 119

5.0

4.0 0 0

2 3.0 Y

cn cn 0

2.0

-

-

-

-

0

0

1 .o

.o Permittivity

Figure 4. Cole-Cole diagrams at -4.4 (0) , 0.0 (0) ,4.4 ( A ) and 8.9"C (0) for a sample cured at 75°C for 60 min. The curves are the best fits of the Williams-Whtts relaxation function.

For the samples scanned below 20"C, p was found to range from 0.57 to 0.37 and the epoxide conversions were between 0 and 34%. A number of trial experiments were conducted by keeping the samples at the scan temperatures for long periods of time, but the dielectric data obtained were not significantly different from those of the samples scanned immediately after the temperature was equilibrated. The IR conversion corresponding to the end of the cure at 75°C was then taken as the conversion of the sample. Hence, any reaction at the scan temperatures was neglected.

Frequency-scan tests were also conducted at 22.2, 26.7, 31.1, 35.6, and 40.0"C. Because the scan temperatures were relatively high, the reaction pro- ceeded slowly during the experiment. Significant changes in the dielectric data were observed when the samples were kept at the scan temperatures for different periods of time. Thus, a sample was scanned several times at the same temperature to observe the variation of the relaxation behavior. Figure 5 shows the permittivity and loss factor measured at 31.1"C after curing at 75°C for 99 min. As the reaction proceeds slowly, the permittivity and loss curves shift to the low frequencies, the relaxed permittivity becomes smaller and the loss peak decreases. The decrease in the dipole loss peak at the isothermal conditions may be attributed to the drop in the concentration of epoxide rings in the system (see below). Figure 6 is a plot of the Cole-Cole arcs

of the same sample and the best fits of the Williams- Watts relaxation function. The arcs are also well- fitted to the relaxation function.

For the samples scanned above 20"C, the distribution parameter p was found to be between 0.40 and 0.28, and the epoxide conversions ranged from 31 to 48%. The reported conversion was the epoxide conversion at the beginning of each scan, which was obtained from the IR experiment that followed the same schedule as the one of the scanned sample. Because the scans were short and the reaction was slow, any reaction during the dielectric measurements was not considered. Frequency scans at even higher temperatures were not conducted because the reaction is too fast to allow measurements of reasonable accuracy.

Dynamic dielectric monitoring of the isothermal cure was performed at temperatures between 55 and 115°C. Figure 7a and b shows the permittivity at 60°C and the loss factor at 115°C. As the cure proceeds, the permittivity and the loss due to ionic conduction decrease. The loss factor exhibits a peak when the dipole loss is larger than the loss due to ionic conduction. Similar observations were also found for other epoxy-amine systems.'-3 The dipole loss peaks are mainly located in the post-gel region. For example, a t 115"C, the loss peaks are found at cure times between 20 and 40 min, but the resin gels (gel conversion = 58% according to recursive mod- eling) after curing for 22 min.

Figure 8 shows the complex-plane plots of per-

Figure 5. Permittivity (0) and loss factor ( 0 ) at 31.1"C. The sample was cured at 75OC for 99 min. and then kept at the test temperature for various periods of time.

2120

3.5

3.0

2.5 L

c 0

0 LL 0 2.0

$ 1.5 0 1

1 .o

0.5

0.4

DENG AND MARTIN

4.0 5.0 6.0 7.0 8 Permittivity

Figure 6. Cole-Cole diagrams for a sample cured at 75°C for 99 min. and then kept at 31.1"C for 66 ( A ) and 204 (0) min. The curves are the best fits of the Williams- Watts relaxation function.

mittivity for the resin cured at 115°C. The data form skewed arcs similar to the Cole-Cole arcs at fixed conversions. An examination of the time series of the experimental data shows that the high loss tails of the plots are due to the ionic conduction.

To fit the complex-plane plots to the Williams- Watts relaxation function, the relaxed and unrelaxed permittivities were first obtained by extrapolation. Next, the data close to the maximum loss were fitted to a third-order polynomial. The maximum of the polynomial was evaluated and was taken as the maximum loss c:. Then, the maximum reduced loss factor ek/ (eo - &,) was calculated and the value of y was obtained from a table listing both p and the corresponding E : / ( E , , - E , ) values.

The plots obtained using high frequencies at 115OC were fitted to the Williams-Watts relaxation function. Figure 9 shows that the experimental and calculated values are in agreement. The reduced permittivity is the real part of the N* and the reduced loss factor is the negative imaginary part of N*. For the data obtained using low frequencies, significant ionic conduction (close to the tails ) does not allow extrapolation of the relaxed permittivity with reasonable accuracy. Hence, curve fitting can- not be performed. This is not the case at lower temperatures. As shown in Figure 10a and b, the low frequency data obtained during cure at lower temperatures may also be well described by the relaxation function. Thus, phenomenologically, the di-

electric behavior of the epoxy resin during cure is similar to that of a stable material-both exhibit Williams-Watts arcs.

Because each arc or dipole peak was obtained while the resin continued to cure, the distribution parameter 0 may vary in its value in the process. Thus, each y value found from the single-frequency measurements represents an average of values over the range of conversions corresponding to the peak. Since the dipole peaks are mainly located in the post- gel region, the y values essentially reflect the /3 values of the post-gel resin.

9.0 I

8.0 h 7.0

x > Y .- .- =: 6.0 E a 5.0

.- L Q,

4.0

3'0 d $0 140 lA0 240 360 360 4;O 4hO 5 Time (min)

I I I I I 10 -4 10 20 30 40 50

Time (min)

Figure 7. Permittivity and loss factor measured during cure of the epoxy-amine cure. (a) permittivity at 60°C; (b) loss factor at 115°C.

DIELECTRIC RELAXATION OF EPOXY RESIN 2121

v) 0.2 v)

3 0.6

0.6 I I

. I

1 .

-

d" I 1 I

100 Hz 0.4 1

v, 0.2 v,

3 0.6

0.2

- a a .

. I I I

I 0.0 0 g 0.4 -w

L L

0.15

- . . . . A

..A I I I

-

. .a .. I

10k Hz 0.4 1 0.2 1

t 1 .

I I A ...A& I 4.5 5.0 5.5 6.0

Permittivity

0.6 I

0.1 Hz 0.4

I I

. a .a

1 I I 4.5 5.0 5.5 6.0

Permittivity

Figure 8. during cure at 115°C.

Complex-plane plots of permittivity measured

Figure 11 illustrates the cure dependence of the distribution parameter. For each y obtained, the epoxide conversion corresponding to the maximum loss of the peak was taken as the characteristic conversion of the distribution parameter. As shown in the figure, the distribution parameter decreases in the pre-gel stage. All the y data are located in the post- gel region. The range of y is between 0.26 and 0.30, but the variation of y for any one temperature is only kO.01. Since a new sensor was used in each curing experiment and the sensors were calibrated in air, the differences in the y values are considered insignificant. Thus, the distribution parameter may

be considered as a constant in the post-gel stage. This finding and the average y of 0.28 are in agreement with the results of Mangion and Johari7>' for similar resin systems. While Mangion and Johari 7,8 suggested that the distribution parameter decreases in the pre-gel stage and does not vary significantly in the post-gel stage, the range of the distribution parameter they obtained was rather narrow (y = 0.26-0.31 for the epoxy-DDM resin and p was not evaluated). The data reported here cover a much wider range and show that the distribution parameter decreases in the pre-gel stage and may be considered as a constant in the post-gel stage.

From the single-frequency measurements, only y, or some average of a range of p, may be obtained. When 0 is a constant, the y value gives immediately the p value, which makes it possible to accurately extract relaxation data from single-frequency measurements.

Figure 11 also shows that, at low conversions, the distribution parameter is linearly dependent on the conversion since the data do not exhibit much cur- vature up to a conversion of approximately 30%. At higher conversions, the distribution parameter con- tinues to drop rapidly. Since the y values mark the lower limit of the distribution parameter, there must exist a transition region in which the distribution parameter changes its trend from a rapid decrease to a constant value. The transition region is narrow in terms of conversion as may be seen from the figure. This phenomenon is interesting and must be related to the structural evolution of the thermo-

0.45 I I 100 Hz

0.30 1 y = 0.29

b0.15 - -w 0 2 0.00 I I

J

-0 e, 0.15 -

2 0.45

Reduced Permittivity

Figure 9. function to the complex permittivity data of 115OC.

Best fits of the Williams-Watts relaxation

DENG AND MARTIN 2122

0.45

0.30

0.15 -Id

0 9 0.00

0.30

0.15

0.45

g 0.30

0.15

0 -1

-0 a,

-0

0.00,

10 Hz y = 0.28

1 I

100 Hz y = 0.27 .

I

I k Hz y = 0.27

: i f 1 + 0.0 0.4 0.8 1.2 1 .


0.45 I I 1 Hz

o.30 y = 0.28 I I

1.6 .~


Figure 10. Best fits of the Williams-Watts relaxation function to the complex permittivity data of 90°C ( a ) and 55°C (b) .

setting resin. The dipole relaxation studied here is commonly referred to as the a-relaxation which, in the case of epoxy resins, is mainly due to the epoxy ring.13 Other groups that may contribute to the relaxation include the C - 0 bonds in the epoxy and the dipoles in the amine. As the resin cures, more and more epoxy rings are opened and the relaxation peak becomes smaller, which may have the effect of varying the distribution parameter. On the other hand, as the epoxides react with the amine hydro- gens, the molecular species become increasingly in- terconnected, causing structural changes in the system. At low conversions, the structural changes are

mainly due to chain growth or formation of short chains. Close to the gel point, chain branching is dominant and many large molecules are created. Then, in the post-gel stage, free chains are contin- uously being incorporated into the infinite network structure. Such structural changes have the effect of hindering the dipolar movement and, hence, may also affect the distribution parameter. While, for the same increment in conversion, a fixed amount of the epoxy rings disappears, the structural changes are not the same in different stages of cure. Thus, the distribution parameter may also be considered as an indicator of the structural changes in the system. The cure dependence of the distribution parameter is the result of the competition between the effects of the dipole concentration variations and the structural changes. The constant distribution parameter in the post-gel stage is probably related to the formation of an infinite network in the system.

Generally speaking, the separation between the a - and @-relaxations increases with increasing conver~ion.'~ The Cole-Cole arcs of the epoxy resin show that its a-relaxation is isolated and well separated from its @-relaxation even at low conversions. While this may be true for many resin systems, it has been found that the two relaxation arcs may, to some degree, overlap for certain ~ys t ems . '~ J~ In the later sit- uations, it is impractical to fit the arcs to the Wil- liams-Watts relaxation function.

Based on the y values, the dipole relaxation time may be extracted from single-frequency measurements. The relaxation time of maximum loss is a

0.6 0.7 ~

h0.5 h 0.3

3

Figure 11. sion.

Distribution parameter vs. epoxide conver-


more commonly used parameter than the relaxation time of the Williams-Watts relaxation function. If a set of frequencies is used to monitor the response of a stable material a t a constant temperature, the inverse frequency corresponding to the maximum loss is the relaxation time. If the dielectric data are found with a single frequency at a varying temperature, the inverse of the measurement frequency is the relaxation time at the temperature of the maximum loss. For thermoset cure, the inverse measurement frequency is the relaxation time at the cure time when the dipole loss at the measurement frequency reaches its peak value. In order to extract the relaxation time from single-frequency measurements, the maxima of the Williams-Watts arcs need to be determined. More precisely, ( w T ~ ) ~ , or W T O

corresponding to the maximum reduced loss factor, is needed. Moynihan et a1.16 used the sum of 14 ideal decay functions (0 = 1 ) for 0.5 I p I 1 and the sum of 17 ideal decay functions for 0.3 I p I 0.5 as approximations to the decay function of the Williams- Watts relaxation function. The maximum loss of the approximate decay function was evaluated and the values of ( W 7 0 ) m were obtained for 0.3 I /3 I 1.0. Since the Williams-Watts arcs are rather flat close to the maxima and using approximate forms of the decay function may introduce significant errors to ( W ~ O ) ~ , Lindsey and Patterson17 approximated the decay function by a cubic spline fit to evaluate ( W 7 0 ) m . In the present work, the approach described earlier for evaluating the Williams-Watts relaxation function was employed to determine the maxima. Since its basis is the exact algorithm of Dishon et al." and the interpolation was not used close to the maxima, this approach is exact for the evaluation of ( W T ~ ) ~ . Highly accurate results may be obtained by including many terms of the convergent series in the calculation. Table I shows the calculated ( ~ 7 0 ) ~

values over the range of /3 between 0.05 and 1.00 with 0.05 intervals. A comparison of the results from different algorithms is shown in Figure 12. The data provided by Moynihan et al. exhibit significant pos- itive deviations from the results obtained based on the exact algorithm of Dishon et al. The trend of the data of Moynihan et al. changes at ,6 = 0.5 due to the different approximations used for /3 < 0.5 and /3 > 0.5. The accuracy of the data provided by Lind- sey and Patterson depends on the value of p.

The ( W T ~ ) , , , data calculated based on the approach of Dishon et al. show that ( 6 x 0 ) ~ is almost linearly dependent on /3. Thus, linear interpolation using the data listed in Table 1 will maintain the accuracy of ( ~ 7 0 ) ~ needed for the extraction of the relaxation time.

Table I. Relaxation Function

( ~ 7 ~ ) ~ Values for N* of the Williams-Watts

1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05

1.0000 0.9712 0.9425 0.9142 0.8863 0.8592 0.8329 0.8075 0.7830 0.7595 0.7370 0.7154 0.6948 0.6751 0.6561 0.6380 0.6207 0.6041 0.5885 0.5741

0.000e0 -1.269e-2 -2.572e-2 -3.896e-2 -5.242e-2 -6.591e-2 -7.941e-2 -9.286e-2 -1.062e-1 -1.195e-1 -1.325e-1 -1.455e-1 -1.581e-1 -1.706e-1 - 1.830e- 1 - 1.952e- 1 -2.071e-1 -2.189e- 1 -2.303e- 1 -2.410e-1

By following the method proposed by Mangion and Johari,8 the Williams-Watts relaxation times under various curing conditions were determined for the epoxy resin. For each cure temperature, the dielectric data of a single frequency were fitted to the Williams-Watts relaxation function to obtain y. Then, complex permittivities corresponding to a series Of @To ranging from to lo5 were calculated and were compared to the experimental data in order to find the series of cure times corresponding to the series of W T ~ . The Williams-Watts relaxation times were then determined for the resin at different cure times.

To find the relationship between the relaxation time of maximum loss and the Williams-Watts relaxation time T ~ , suppose an infinitely fast frequency-scan test could be conducted for a resin with a distribution parameter y to obtain a Cole-Cole arc. The Williams-Watts relaxation time of the arc is T O

and the distribution parameter /3 is equal to y, which corresponds to a unique ( ~ 7 ~ ) ~ . Now, what would be the frequency at which the maximum loss was reached? The maximum loss occurs when W T O

= ( Thus, the frequency of maximum loss is equal to ( 0 T 0 ) m / ( 2 7 r T 0 ) . The relaxation time of maximum loss is simply:

2124 DENG AND MARTIN

0 0 P

0.9 i o.6 t ’

I I I I 0.3.L 0.5 0.4 0.6 0.8 1 .o P

Figure 12. ( W T ~ ) , , , values evaluated using the approach based on Dishon et al.” (. . . . . .) as compared with the results given in Refs. 16 (0) and 17 ( A ) .

where r is the relaxation time. In this way, the 7 0

values were converted into the relaxation times. For each cure temperature, the relaxation times

of the epoxy resin were also obtained directly from the dipole loss peaks. To do so, the loss data close to the maximum of each peak were fitted to a third- order polynomial and the position of the polynomial maximum was determined in the form of a cure time. The relaxation time is equal to the inverse frequency at the particular cure time.

Figure 13 shows the relaxation times for eight temperatures. The data obtained from the Williams- Watts relaxation function approximations or single- frequency measurements are in agreement with those obtained directly from the loss peaks. The relaxation times extracted from various frequencies form curves that almost overlap and the differences between the curves are not significant considering the accuracy of the method itself. Therefore, good approximations of the relaxation data may be found by following the single-frequency approach and the results are independent of the measurement frequency. Figure 13 also shows that the range of relaxation times obtained from the single-frequency approach is much wider than that of the data obtained from the loss peaks directly. In addition, the range of relaxation times extracted varies with the measurement frequency used. Dielectric data obtained with a high frequency offer mostly short relaxation times and those obtained with a low frequency offer mostly long relaxation times. Hence,

sometimes more than one frequency may be needed to acquire relaxation data over a very wide range of conversions.

Cure monitoring at high conversions has been conducted using different parameters including the glass transition temperature.’* Among the advantages for using relaxation time is that the relaxation time may be obtained through in-situ measurements and is highly sensitive to any changes in the resin system. As the single-frequency approach presents an effective way to extract a wide range of relaxation data, the dipole relaxation time may be conveniently utilized as a parameter for cure monitoring purposes.

CONCLUSIONS

For the epoxy-amine resin studied, the permittivity and loss data measured with a single frequency during isothermal cure form a skewed arc in the complex-plane of permittivity. The skewed arcs are similar to the Cole-Cole diagrams of stable materials and may be described by the Williams-Watts relaxation function. Thus, a phenomenological similarity was found between the dielectric features of the resin during cure and those of stable materials. The distribution parameter of the Williams-Watts relaxation function was determined for the epoxy resin over a wide range of conversions. This parameter

l o ’I peak 1 0 4 o 3 m ~ 10k HZ

mm l k b

1 0 3 - 100 00(900 10 b

W

0

I 1 I I 10 i’13 1.6 1.9 2.2 2.5 2

Log Time (min) 3

Figure 13. Dipole relaxation times extracted from the Williams- Watts relaxation function approximations as compared with those obtained directly from the relaxation peaks. From left to right, the cure temperatures were 115, 98,90,82, 75,68,60, and 55’C.


decreases in the pre-gel stage and may be considered as a constant in the post-gel stage. The transition from a rapid decline to a constant value occurs over a narrow range of conversions. The cure-dependence of the distribution parameter may be related to the dipole concentrations variations and the structural changes in the cure system. A nearly constant distribution parameter over certain range of conversions is essential for extracting relaxation time from the single-frequency measurements over the same range.

A single-frequency approach has been developed to determine the relaxation time of maximum loss, which includes the method of Mangion and Johari' for extracting the Williams-Watts relaxation time, an exact approach for evaluating the maxima of the Williams-Watts arcs, and an equation, Eq. ( 6 ) , for converting the Williams-Watts relaxation time into the corresponding relaxation time of maximum loss. The relaxation times of the epoxy resin were obtained from single-frequency measurements over a very wide range. The relaxation data are independent of the frequency and are in agreement with those obtained directly from the dipole loss peaks. This single-frequency approach may be useful for cure monitoring purposes, especially at high conversions.

The epoxy resin used in this study was supplied by the Shell Development Co.

REFERENCES A N D NOTES 1. S. D. Senturia and N. F. Sheppard, Jr., Adu. Polym.

Sci., 80, 1 (1986).

2. W. W. Bidstrup and S. D. Senturia, Polym. Eng. Sci.,

3. D. R. Day, ibid, 26, 362 (1986). 4. D. Kranbuehl, S. Delos, M. Hoff, L. Weller, P. Hav-

erty, and J. Seeley, ACS Symp. Ser., 367,100 ( 1988). 5. Y. Deng and G. C. Martin, Macromolecules, in press

(1994). 6. G. Williams and D. C. Watts, Trans. Faraday SOC.,

6 6 , 8 0 (1970). 7. M. B. M. Mangion and G. P. Johari, J. Polym. Sci.,

Polym. Phys., 28,1621 (1990). 8. M. B. M. Mangion and G. P. Johari, ibid, 29, 1127

(1991). 9. M. G. Parthun and G. P. Johari, Macromolecules, 25,

10. D. W. Davidson and R. H. Cole, J. Chem. Phys., 18,

11. M. Dishon, G. H. Weiss, and J. T. Bendler, J. Res.

12. M. J. D. Powell, Comp. J., 7, 155 (1964). 13. P. Hedvig, Dielectric Spectroscopy of Polymers, John

14. G. Levita, A. Livi, and P. A. Rolla, private commu-

15. M. A. Bachmann, J. W. Lane, and J. C. Seferis, Polym.

16. C. T. Moynihan, L. P. Boesch, and N. L. Laberge,

17. C. P. Lindsey and G. D. Patterson, J. Chem. Phys.,

18. G. Wisanrakkit and J. K. Gillham, J. Coatings. Tech.,

29, 290 (1989).

3254 (1992).

1417 (1950); 19, 1484 (1951).

Nut. Bur. Stand., 90, 27 (1985).

Wiley, New York, 1977.

nication, 1993.

Sci. Eng., 26, 346 (1986).

Phys. Chem. Glasses, 14, 122 (1973).

73, 3348 (1980).

62, 35 (1990).

Received December 29, 1993 Revised April 15, 1994 Accepted April 18, 1994

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Analysis of the cure-dependent dielectric relaxation behavior of an epoxy resin