Frequency-Dependent Dielectric Analysis - American Chemical Society

Chapter

8

Frequency-Dependent Dielectric Analysis

Downloaded by UNIV OF SOUTHERN CALIFORNIA on June 15, 2013 | http://pubs.acs.org Publication Date: April 18, 1988 | doi: 10.1021/bk-1988-0367.ch008

Monitoring the Chemistry and Rheology of Thermosets During Cure D. Kranbuehl, S. Delos, M . Hoff, L. Weller, P. Haverty, and J . Seeley Department of Chemistry, College of William and Mary, Williamsburg, V A 23185

Previous published reports have demonstrated how the frequency dependence of ε*(ω) can be used to qualita tively monitor the viscosity of a curing resin. The key is in using the frequency dependence in the Hz to MHz range to separate and determine parameters govern ing ionic and dipolar mobility. This paper reports on the use of the frequency dependence to determine ionic and dipolar diffusion processes. The quantitative relationship of the ionic and dipolar mobility parameters to the viscosity and degree of cure during the cure reaction of TGDDM epoxy is discussed. The temperature and degree of cure dependence of the ionic and dipolar mobility is analyzed in terms of an Arrhenius and WLF dependence. As an example of an application, the in-situ on-line measurement capability of the technique to measure the cure processing parameters in a thick laminate during cure in an autoclave is reported. The results support the use of the WLF equation for analyzing the advancement of a curing reaction in TGDDM epoxies. Frequency dependent complex impedance measurements made over many decades of frequency provide a s e n s i t i v e and convenient means f o r m o n i t o r i n g the cure process i n thermosets and thermoplastics [1-4]. They are of p a r t i c u l a r importance f o r q u a l i t y control m o n i t o r i n g o f cure i n complex r e s i n systems because the measurement of d i e l e c t r i c r e l a x a t i o n i s one of only a few i n s t r u m e n t a l techniques a v a i l a b l e f o r s t u d y i n g m o l e c u l a r p r o p e r t i e s i n both the l i q u i d and s o l i d states. Furthermore, i t i s one of the few experimental techniques a v a i l a b l e f o r s t u d y i n g the polymerization process of going from a monomeric l i q u i d of varying v i s c o s i t y to a c r o s s l i n k e d , i n s o l u b l e , high temperature s o l i d . In the past, impedance or d i e l e c t r i c studies have been examined as an experimental technique to monitor the flow properties, e f f e c t s of composition, and the advancement of a r e a c t i o n d u r i n g cure [1] . U n t i l a paper by Zukas et a l [2], l i t t l e emphasis had been placed on the frequency dependence except to note the s h i f t i n p o s i t i o n and m a g n i t u d e o f impedance maxima and minima. Furthermore, most measurements on c u r i n g systems r e p o r t e d r e s u l t s i n terms o f 0097-6156/88/0367-0100S06.00/0 © 1988 American Chemical Society In Cross-Linked Polymers; Dickie, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1988.


8. K R A N B U E H L ET A L .


e x t e n s i v e parameters, such as conductance G, capacitance C or t h e i r r a t i o d i s s i p a t i o n D [1-4]. Frequency dependent impedance measure ments have l o n g been used to p r o v i d e i n f o r m a t i o n on a molecular l e v e l about the mobility of charged s p e c i e s and d i p o l a r groups i n l i q u i d s and s o l i d s [5]. Over the recent years, our laboratory and others [3,4,6-11] have focused on using the frequency dependence o f the impedance to determine and separate the i o n i c and dipolar con t r i b u t i o n s to the impedance. F u r t h e r , measurements are made i n terms of the i n t e n s i v e geometry independent parameter, the complex p e r m i t t i v i t y e*. The angular frequency dependence, ω, and magnitude of ε*(ω) are determined by the time scale of the i o n i c groups' and the polar groups' mobility as w e l l as by the number and charge o f the i o n i c / p o l a r species. The emphasis of t h i s work i s on continuous measurement of the frequency dependence o f e* to measure both the ions and the dipolar groups' changes i n m o b i l i t y and then to r e l a t e t h i s changing m o b i l i t y to cure p r o c e s s i n g p r o p e r t i e s such as v i s c o s i t y and d e g r e e o f cure (see F i g u r e 1 ) . Thus, r a t h e r than focusing on extensive e l e c t r i c a l impedance properties such as C, G, or D, t h e i o n i c and d i p o l a r m o b i l i t y parameters are used as molecular probes of the cure reaction. These m o l e c u l a r probes are used to sense i n - s i t u the v i s c o s i t y and degree of cure as a function of time. A major long-range objective of t h i s research i s to develop on l i n e instrumentation using commercially available instruments, novel sensor t e c h n i q u e s , and a molecular understanding of the frequency dependence of the impedance f o r quantitative nondestructive material e v a l u a t i o n and c l o s e d loop "smart" cure cycle c o n t r o l . The key to achieving t h i s goal i s to r e l a t e the c h e m i s t r y o f the cure c y c l e process t o the d i e l e c t r i c properties of the polymer system by cor r e l a t i n g the time, temperature, and frequency dependent impedance m e a s u r e m e n t s w i t h c h e m i c a l and r h e o l o g i c a l measurements. Measurement of the wide v a r i a t i o n i n magnitude o f the complex per m i t t i v i t y with both frequency and state of cure, coupled with other c h a r a c t e r i z a t i o n work, have been shown t o have the p o t e n t i a l t o determine: r e s i n q u a l i t y , composition and age; cure cycle window boundaries; onset of flow and point of maximum f l o w ; e x t e n t o f and completion of reaction; evolution of v o l a t i l e s ; Τ ; c r o s s l i n k i n g and molecular weight buildup [3,4,6-11]. ^ In previous published reports we demonstrated how the frequency dependence of ε*(ω) can be used to q u a l i t a t i v e l y monitor the v i s c o s i t y o f a c u r i n g r e s i n [6,7]. The key i s i n using the frequency dependence i n the Hz t o MHz range t o s e p a r a t e and d e t e r m i n e parameters governing i o n i c and dipolar mobility. This paper focuses on the use o f the frequency dependence t o determine i o n i c and d i p o l a r d i f f u s i o n processes. The quantitative r e l a t i o n s h i p of the i o n i c and dipolar mobility parameters to the v i s c o s i t y and degree of cure d u r i n g the cure r e a c t i o n of a TGDDM epoxy i s discussed. The temperature and degree of cure dependence of the i o n i c and d i p o l a r m o b i l i t y i s a n a l y z e d i n terms of an Arrhenius and WLF dependence. As an example of an a p p l i c a t i o n , the i n - s i t u o n - l i n e measurement c a p a b i l i t y o f t h e t e c h n i q u e t o measure t h e cure p r o c e s s i n g parameters i n a t h i c k laminate d u r i n g cure i n an a u t o c l a v e i s reported.

In Cross-Linked Polymers; Dickie, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1988.

101


Heat Pressure Time

Processing Tool Sensor

Dipolar Mobility

σ,η

Molecular Parameters Ionic M o b i l i t y

Computer

Frequency/Time Molecular M o D l l l t y Models

Figure 1. Experimental approach.

τ

^ 7 .

Z»(oo), e:#(co)

Impedance Analyzer

Frequency Dependent Impedance Measurement


Processing Parameters n, degree of cure Tg, Tm volatization

Computer

Frequency/Time Molecular Models Data Base

W c/5

W

Ο

Ό

w


8. KRANBUEHL ET AL.


Experimental Measurements were made w i t h a now commercially a v a i l a b l e patented Dek Dyne p e r m i t t i v i t y s e n s o r . The s e n s o r , d e v e l o p e d i n our l a b o r a t o r y , consists of a fine array of two comb electrode^ mounted on a t h i n i n e r t substrate. The active surface area i s 2 cm and the thickness i s 3 mm. The surface area, thickness and substrate can be v a r i e d to withstand the resin's reaction conditions. In many cases, a p o l y i m i d e f i l m i s adequate f o r cure reactions below 200°C. At higher temperatures polyimides begin to l o o s e some s t r u c t u r a l i n t e g r i t y and a h i g h e r temperature s u b s t r a t e such as AÔ^ i s more r e l i a b l e . The sensor was designed f o r use w i t h c o n v e n t i o n a l , com m e r c i a l l y a v a i l a b l e b r i d g e s such as Hewlett Packard, GenRad or Tetrahedron. We used a Hewlett-Packard 4192A LF Impedance Analyzer with the sensor. Continuous measurements of both the r e a l and imaginary components o f the2Complegc p e r m i t t i v i t y , e* = e' - i e " , were made over a range of 10 to 10 i n magnitude. Since the sensor we used was i n e r t , c o n s t r u c t e d from noble metals and a h i g h temperature ceramic and does not c o n t a i n any s o l i d s t a t e c i r c u i t r y , i t i s capable of being used i n high temperature cure reactions as w e l l . The Impedance Analyzer was c o n t r o l l e d by a 9836 Hewlett-Packard computer which a l s o c o n t r o l l e d the time-temper-ature of the press. Measurements at f r e q u e n c i e s from 5 to 5 χ 10 Hz were taken at regular i n t e r v a l s during the cure cycle and converted to the complex p e r m i t t i v i t y . Further d e t a i l s of the e x p e r i m e n t a l procedure has been given elsewhere [10]. V i s c o s i t y measurements were made using a Rheometrics System IVdynamic mechanical spectrometer. Glass t r a n s i t i o n and degree of c o n v e r s i o n measurements were made using a Perkin-Elmer DSC-7 d i f f e r e n t i a l scanning calorimeter. The t e t r a g l y c i d y l 4, 4' diaminodiphenyl methane epoxy (TGDDM) r e s i n s u s e d were H e r c u l e s 3501-6 ( c a t a l y z e d r e s i n ) and 3502 (uncatalyzed resin). The c r o s s l i n k i n g r e a g e n t i s 4, 4' diaminodiphenyl sulfone and the c a t a l y s t i n 3501-6 i s BFîNH^R [12]. These were supplied by Hercules through NASA-Langley Research Center and stored i n a freezer u n t i l used.

Theory Measurements of capacitance C and conductance G were used to calcu l a t e the complex p e r m i t t i v i t y e* - e' - i e " , £

C material

- c (1)

„ _ G material " Cο2ïïf i s the e f f e c t i v e a i r replaceable capacitance and f i s the 6

where C

n


103

CROSS-LINKED POLYMERS

104

frequency. Both the r e a l and the imaginary p a r t s o f e* have a dipolar and an i o n i c component [5]. €'

+

d d

e'.

ι

(2)

+


The d i p o l a r component a r i s e s from d i f f u s i o n o f bound charge o r m o l e c u l a r d i p o l e moments. The frequency dependence of the polar component may be represented by the Cole-Davidson function:

d

"

(l+ior/

where e and are the l i m i t i n g low and high frequency values of e, r i s a c h a r a c t e r i s t i c r e l a x a t i o n time and β i s a parameter which measures the d i s t r i b u t i o n i n r e l a x a t i o n times. The d i p o l a r term i s g e n e r a l l y the major component o f the d i e l e c t r i c s i g n a l a t h i g h frequencies and i n highly viscous media. The i o n i c component, e.*, often dominates e* at low f r e quencies, low v i s c o s i t i e s and/or higher temperatures. The presence of mobile ions g i v e s r i s e to l o c a l i z e d layers of charge near the electrodes. Since these space charge layers are s e p a r a t e d by v e r y s m a l l m o l e c u l a r d i s t a n c e s on the order o f A , the corresponding space charge^capacitance can become extremely large, with e' on the order o f 10 . Johnson and Cole, while studying formic acid, derived empirical equations f o r the i o n i c contribution to e* [13]. I n t h e i r e q u a t i o n s , e' . i s frequency dependent due to these space charge i o n i c e f f e c t s and has the form Q

σ .2 - > 8.85x10" where Ζ* = Ζ (ίω) i s the electrode impedance induced by the i o n s and η i s an e m p i r i c a l parameter between 0 and 1 [5,13,14]. The imaginary part of the i o n i c component o f the p e r m i t t i v i t y has the form c

, ~ r, i - o o C

Z

S l n

(ηπ) -(n+1), 2 ω

N

(

( 4 )

14

1

14

2

14

8.85χ10· ω ° ° 8.85x10' where σ i s the conductivity (o^hm "^ cm " , an intensive v a r i a b l e , i n contrast to conductance G(ohm ) which i s dependent upon c e l l and sample s i z e . The f i r s t term i n Eq. 5 i s due to the conductance of ions t r a n s l a t i n g through the medium. The second term i s due t o e l e c t r o d e p o l a r i z a t i o n e f f e c t s . The second term, due to electrode p o l a r i z a t i o n , makes d i e l e c t r i c measurements i n c r e a s i n g l y d i f f i c u l t to i n t e r p r e t and use as the frequency of measurement becomes lower. E l e c t r o d e p o l a r i z a t i o n , r e p r e s e n t e d by the second term i n e q u a t i o n ( 5 ) , i n general i s a s i g n i f i c a n t and d i f f i c u l t to account for factor at frequencies below 10 Hz and/or f o r h i g h v a l u e s o f σ u s u a l l y a s s o c i a t e d w i t h a highly f l u i d r e s i n state. The frequency dependence e* due to dipolar mobility i s generally observed a t frequencies i n the KHz and MHz regions. For t h i s reason an analysis of the frequency dependence of e*, equations 3 and 5, i n the Hz t o


8.

K R A N B U E H L ET AL.


MHz range i s , i n g e n e r a l , optimum f o r measuring both the i o n i c m o b i l i t y parameter σ and the dipolar m o b i l i t y parameter r. The magnitude of the i o n i c m o b i l i t y σ and the r o t a t i o n a l m o b i l i t y of the dipole τ depends on the extent of the r e a c t i o n and the physical state of the material (5). As such, σ and τ determined from the frequency dependence of β*(ω), provide two molecular probes f o r m o n i t o r i n g the r e a c t i o n advancement and the v i s c o s i t y during cure.


Results and Discussion The v a r i a t i o n i n the magnitude of e" with frequency and with time i s w e l l r e p r e s e n t e d by F i g u r e 2 f o r the 165 °C isothermal run of^ the uncatalyzed resin's reaction. Note that c" changes by over 10 i n magnitude during the course of the cure r e a c t i o n and that the measurement s e n s i t i v i t y of 10"^ on the non-log p l o t Figure 3 can be used to monitor the long time, f i n a l stages of cure which continues for hours. As shown previously (6,8), a p l o t of e"* frequency or conductance, Figure 4, i s a p a r t i c u l a r l y informative representation of the cure because as seen from equations 1-5, the o v e r - l a p of e"(ω) f o r d i f f e r i n g frequencies indicates that i o n i c d i f f u s i o n i s the dominant physical process a f f e c t i n g the loss ( f i r s t term of eq. 5). S i m i l a r l y the peaks i n e"*w f o r i n d i v i d u a l frequencies indicate dipolar or bound charge d i f f u s i o n processes are c o n t r i b u t i n g to e" (Equation 3). The frequency dependence of the loss c" i s used f i r s t to deter mine σ by determining from a computer analysis or a p l o t of € *ω (Figure 4), the frequency r e g i o n where ε"*ω i s a constant. Over t h i s frequency r e g i o n the v a l u e of σ i s determined from the 1/ω dependence of c", eq. 5. The i o n i c c o n t r i b u t i o n e£ i s substrated from e" measured to determine the dipolar component c" The time at which a peak occurs i n the dipolar portion, e" , f o r a p a r t i c u l a r frequency, ω, i s used to determine the time of occurrence of the corresponding mean r e l a x a t i o n time r = 1/ω. In a p a r a l l e l experiment, the e x t e n t of the r e a c t i o n a i s measured using the p a r t i a l heat to a p a r t i c u l a r time d i v i d e d by the t o t a l heat of the isotherm p l u s the r e s i d u a l heat of a subsequent 10°/min ramp. Figures 5 and 6 show the observed r e l a t i o n s h i p of I n σ and In r to a. As expected f o r a s i m i l a r degree of advancement a, the i o n i c m o b i l i t y σ increases with temperature. S i m i l a r l y f o r the same v a l u e of a, the dipolar m o b i l i t y increases. An increase i n dipolar m o b i l i t y corresponds to a shorter r e l a x a t i o n time. Thus τ decreases as temperature increases. Somewhat unexpected, both In σ and In τ e x h i b i t a nearly l i n e a r dependence on a. Curvature i n the In σ and In r versus a p l o t i s most pronounced f o r small values of a and at the highest temperature. There i s no evidence of a break i n the In σ or In τ dependence on a which would indicate g e l . The rate of change of log σ with time and a approaches 0 as the advancement the r e a c t i o n approaches completion. The point at which — and — decreases s h a r p l y , appears to i n d i c a t e Τ . The values of a a ? t h i s point, .72 at 165°, .80 at 180°, and . 8 8 ^ t 195° are i n good agreement with our values determined by DSC scans. Η

d

0

The c o r r e l a t i o n d u r i n g cu§e of -log σ with l o g η ( v i s c o s i t y ) measured at 10 radians/sec and the a b i l i t y to use frequency depend e n t ε*(ω) measurements to determine σ, thereby accurately


105


106


Figure 3. e" vs. time f o r the uncatalyzed epoxy during a 165C isothermal cure showing the long time s e n s i t i v i t y a t 500 Hz and 5kHz.


KRANBUEHL ET AL.


9.0

250 H 200


150 ~ a> υ S

100 8.

I 50 80

200

120

Time (min)

Figure 4.

M

Log(€ *w) vs. time f o r the uncatalyzed epoxy during 165C isothermal cure.

-5

Temperalure • 165 °C

° ·°°

-6

c

Δ 180

Λ

# Δ

X

ο 195 °C

Ο

a 2 0 5 °C

-7 * -8 -9 -10 -II

0

.4

.6

10

a Figure 5.

Log σ ( i o n i c d i f f u s i o n ) v s . a (degree of cure) f o r the uncatalyzed epoxy during isothermal cure.



108


detecting points of maximum flow during cure, was f i r s t shown by us s e v e r a l years ago [6,7]. The q u a l i t a t i v e c o r r e l a t i o n of -log σ and log η i s c l e a r l y demonstrated i n Figure 7 · The q u a n t i t a t i v e r e l a t i o n ship of log σ and log η i s seen far two isothermal curves i n Figure 8 . To a f i r s t approximation one might look f o r a p l o t of l o g (σ) vs log (η) to be l i n e a r . Figure 8 shows there i s a breakîn the σ-η dependence as the r e s i n approaches a v i s c o s i t y of 10 poise, a value o f t e n a s s o c i a t e d w i t h g e l . The break i n the curve at t h i s point undoubtably i s due to the fact that the i o n i c and dipolar d i f f u s i o n processes r e f l e c t a molecular v i s c o s i t y [5] , while the v i s c o s i t y measured by the rheometer r e f l e c t s a macroscopic resistance to flow. At g e l the macroscopic v i s c o s i t y begins to r i s e r a p i d l y with small changes i n a. In contrast, the α dependence of the m o l e c u l a r v i s c o s i t y or the r a t e o f change i n the i o n i c d i f f u s i o n remains unchanged as seen i n Figure 5 · The r e l a t i o n o f l o g r to log η i s shown i n Figure 9· The r e l a t i o n s h i p i s indeed approximately l i n e a r at 165°. This apparent l i n e a r dependence i s due i n part to the smaller range of ryând the fact that the relaxation times a l l occur before η reaches 10 poise, t h a t i s before g e l , the region of curvature i n the log σ versus log η p l o t Figure 8. The temperature dependence at constant α of σ and r has been f i t to an Arrhenius dependence 6

In (a, r) - fè RT and a modified WLF dependence

Frequency-Dependent Dielectric Analysis - American Chemical Society

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