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Phenomenology of addition polymerization
]ourrral ofMolm~lur Ekvicr
Liqrrids, .S6 (1993) 15%174
Scicncc Pubkhcrs
B.V., Amsterdam
PHENOMENOLOGY
OF ADDITION
POLYMERIZATION
G.P. Johari, Department of Materials Science and Engineering, McMaster University, Hamilton, Ontario, Canada LSS 4L7
ABSTRACT The phenomenology of macromolecular growth by addition reactions, which is envisaged as a process of negative feedback between molecular diffusion and chemical reactions, has been described in terms of dielectric erXects, and a formaiism is given. According to this formalism the dielectric properties of a time-variant system irreversibly and spontaneously change in much the same manner as that of a time-invariant system when the ac-frequency used for the measurement is deliberately increased_ The dielectric spectroscopy shows an approach of dc conductivity towards a singularity for which equations are provided. The increase in the relaxation time of a liquid with the progress of reaction are shown to be interrelated. As the reaction occurs, a second, low-frequency, relaxation process separates and rapidIy shifts to lower frequzncies. Concomitantly, its strength decreases as does the strength of the remaining highfrequency process. The phenomenological consequences of this negative feedback between chemical reactions and molecular diffusion is remarkably similar to that observed on supercooling a molecular or polymeric liquid whcse structure remains time-invariant. GENERAL
ASPECTS
During the chemical
growth of a macromolecule,
the mclccular
diffusion
coef&ient irreversibly decreases with time until the chemical reactions that allow this growth come to a virtual halt. This :legative feedback tetween molecular diffusion ard chemical reactions, which cavuses both to become progressively
slower,
uItimately
converts a molecular liquid irreversibly to a linear chain or cross-linked solid polymer. Ii variety of thermodynamic changes occur during the course of this irreversible process. For example, as the reaction time approaches infinity, the extent of conversion of monomers to a macromolecuIe
approaches
-unity, the diffusion
approaches zero, the volume approaches its lower bound (limiting!
coefficient
value, and the
configurational contri-bution to entropy and enthalpy approach zei-o as the glass transition temperature of the molecular stmcture increases and enough time is available for
0167~7322/93/!%6.00
0 1993 - Elsevier Science Publishers
B.V.
AI1 rights reserved
its spontaneous relaxation towards the minimum
volume, entropy and enthalpy states
corresponding to vibrational frequencies at the temperature Equally
interesting
liquid as it irreversibly time in a sigmoidal correspondingly maximum
changes occur in the molecular
polymerizes.
reaches a maximum
zero value at infinite time Ill.
approaches
before decreasing
kinetic behaviour
of the
The velocity of ultrasonic waves increases with
manner and the attenuation
decreasing to a virtually
of the reaction.
a limiting
value
The mechanical
and the mechaniral
to a near zero vdue.
value before modulus
loss reaches
a
As the structure and chemical
composition of the liquid irreversibly and continuously
changes, its electric properties
change in at least seven ways, namely: (i)
a genera! decrease in the dc conductivity as the diffusion coefficient of impurity ions in the reacting mixture decreases with increase in its viscosity arId any proton transfer
along H-bonds in the mixture
is virtually
ehminated
by the
formation of cross-links; (ii)
an increase in the molecular difiusion or relaxation time as a resuIt of which the dielectric permittivity
measured at a fixed frequency monotonically
decreases
towrrd the value corresponding to the infrared region; (iii)
a change in the number
of dipoles
per unit
volume,
contribution to ezrmittivity from orientation pclarization,
and therefore
in the
L.e, as a result of both
the chemical reactions that alter the dipole moment and the volume contraction that raises the number density of dipoles; (iv)
a change in the dielectric relaxation function as the chemical structure of the liquid changes and its viscosity and density increase;
(VI
a change in Aeir, the contribution to permittivity the vibrational polymerization
(vi)
frequencies of the various modes in the structure change
a change in %&eoptical refractive in&_: or optical polarizg?‘;.l the splitting
.‘LS on
and densification;
occurs and the structure de&&s; (vii)
due TVinfrared polarization,
of a unimodal
as polymerization
and,
relaxation
function
into a bimodal
relaxation
function. Thus in addition to the frequency and temperature dependence. the complex dielectric permittivity, E*, for such systems becomes time-dependent and is expressed by the equation, c*lio. T. tc) = E’(o, T. tc) -
k-m, T. tc~
(11
155
where the dielectric permittivity, c’, and loss. E”~contain contributions from both the dipolar orientation polarization and dc or ionic conduction, w is ‘the angular frequency of the applied electric field and T and & refer to the temperature beginning
of the chemical
reaction.
(Terms
in parenthesis
and time since the
refer to independent
Here the time k includes the effects of irreversible chemical and physical cb.anges of the reacting system. These changes are related to k by a
variables in such studies.)
suitable functional form, which can be determined by experiments. In ion-containing
materials, the measured permittivity
and loss include contri-
butions from both the interfacial effects and dc conduction, and the dipolar orientation, such that E’(o.T. tc) = (i+n (ad2 10-“+a’~C’~w,T.t,)/Co
+ C&
h.T.te)
(21
and, L’*(w, T, tc) = (a,,(T. tYoed
G’? o. T. tcVCO+ E”
- (ZOms(ad210’aJ
&P
(3)
!a. ‘I’, tc)
where the first term on the right hand side of Eqn. (21 and the second in Eqn (3) refer to the cant-ibution
from a %onstant phase element” for an inkrfacial
in series with the bulk properties of a material. is tha characteristic conductivity
In these terms in Eqns. (2) and (3), Zo
impedance of the electrode-insulator
which
is equal
to the reciprocal
impedance which is
interface, G is &&emeasured
of resistivity,
Co is the geometric
capacitance of the sample, and a= 0.5, as Johnson and Cole I21 and McDonald 131 have shown.
In the !aet terms of Eqns (2) and (3) E’dIp and &‘*&pare the contributions to E’
and E” from dipolar orientation alone. orein Eqn. (3) is the dc conductivity and eo the permittivity ( = 8.8514 pFm-‘1 of free space. For relatively
iow ionic concentration or dc conductivity,
interfacial impedance are generaIly insignificant
contributions
from
and can be neglected. so that E’ and
en may be written as. (4)
and, E-(0,T. tc) = Cion tT. tcl
+ Cdi.
:o, T, tc)
Wbf. lX f”ion = to[r/weg) is the contribution to c” from dc co-rductivity. that the permittivity
(5, Here it is clear
during the occurrence of a chemical reaction is determined only
by the dipolar orientation polarization but the loss is determined by both the ionic or dc conduction and dipolar reorientation process. We first
consider the change in the dc or ionic conductivity contribution, e”ion, as
the chemical reactions occur.
A number of our earlier studies on the reactions of
l-56
diamine
with bisphenol-A-diepoxides
[4-6] have shown that the decrease
in the-dc
conductivity during its reaction is described by a power law, oO(T.tC)=
t&T) - t,
tC +
0)
t gel IT)
(6)
I
where oo(T,t;t--tO) is the dc conductivity at the beginni,lg
of the reaction at a tempera-
ture T, kc1 is the time to reach the gelatiDn point and x is the critical exponent of the scaling equation used as a generalized
property function by Stauffer,
Adam [7] and Djabourov ES].
equation
oo(T,tc)
ConIglio and
= ~Iexp-(Mt0-t)),
which implies a singularity at t = to. was used also by Mangion and Johari [4], Johati and Mangion [!5] and Parthun and Johari [S]. but was found to be a less satisfactory description of the measured conductivity than the power law when a network structure formed and gelation occurred 3s a result of chemica1 r eactionsf. The significance of x in Eqn. (6) is that its value determines the approach ‘1: the liquid towards gelation, the rate at which a0 decreases with time.
Le.
The higher the value of x, the more rapid is
the decrease of ergwith the reaction time and the more rapid the approach to gelation. The description of the data by Eqn. (6) allows extr;rpolation of the measured ductlvity so that the time for the gel&ion of a chemicaIly-reacting
de con-
system during both
its isothermal and temperature-ramp conditions can be estimated. By combining Eqns. (5) and (61, the dielectric loss during the macromolecule formation Is given by,
We now consider the dipolar contributions to r* and E”. According to the classical theory of dielectrics, the complex permittivity
due to orientation po!arization
alone in
a material is given ‘k-y,
where EOand ca: are the Iimiting low- and high-freque1ir.y permittivitics. and t refers to the time for the decay of polarization
for a chemical
respectively, structure
‘Aat
remains unchanged during the measuremen, + time, or when the chemical reaction In Eqn. (S:l, 9 occurs too slowly in comparison with the time for measurement. represents a relaxation function whose mrmalized given by,
form for a time-invariant
system is
where
p is an empirical
relaxation
time, p is equal
in the range (Km)-,or
parameter. tc
unity.
zero to unity and stretched
polarization For
during
a chemically
approximation
n,gligible also useful
here,
structural
represents
a one-sided
a change period
X*
of time.
therefore
an
of 1 kHz or more, t in
in both the chemical structure
can be justifiabIy
assumed to
be
the reaction is slow and the total reaction allows
us to write Eqn_ (9) in a form
iot
exp-
(10)
is itst:lf a func’jon
at any
Laplace
of
as follows,
I-
=
state of the liquid
this
when
system,
quantity
from
of the decay
of time and
For frequencies
approximation
0
the normalized
c91
prob:esses are considered
time. remains invariant
invariant
as follows:
during
This
for a time-varia3t
- VVilliams-Watts
stable materic.1 t and t in
the relaxation
relaxation
the effects
S*(iCat)
where
Uihen
I: is not
temperat*clres
hours.
to as Kohlrausch
it is the time for the observation
system,
state of the sys’tem several
of viscous liquida and solids. P is
and physiczl!y
‘I:- the average
needs to be made
at th e usual
time exceeds
meaning.
1 ms and
process, i.e.. with a single
parameter.
in series,
reacting
Eqn. (9) is less than and physical
than
which
it is referred
or a chemically
Eqns. (8) and (9) ha-,-e a strict rather
For the majority
here
exponential
For a time-invariant, l a be in parallel
For a 135ebye relaxation
given
instan:
trar.sfox-m
of the product of o and r cc
since the beginning
_
of the reaction,
and E’ and z” are written
in terms of
Ku
and IV by: A
c’CT.tc)
N’(wr!T,tc)I
= c, + -
T
Cl?!
and z”(T,tJ
where
(&,-,-a& = Am with A as
For most polymerization
A 7
a cons+&nt reactions
the end olr reaction is by about to Eqn. (8) is much less than
=
20%
Y"(wr(T.t
11
according
c z3, tc the Curie Law.
the totzl decrease
or !ess.
the effect
_
in ~0 from the beginning
The et’;ect of this &crease
to
on E*, according
from the sever al orders of tnagnit-dde increase
in
r, which is, typically, from 10-S
to 10 + 3s. As a first approximation,
we consider that
both co and h remain constant with & and Eqns. (8) and (10) may be combined to obtain, c+(io,T,tc)
= E=(T) + (c,jT)-
It is easy to see tl,at Eqn. (14) is phenomenologically cne’s choice of an experimental
variable.
-he relaxation spectrum or Cole-Cole Thus during the polymerization fdback
(IS)
SeIior(T,tc)I
c_(T))
invariant
of o and t as
This means that any of the several shapes of
plots can be obtained by varying either o or L
process,
which is envisaged
here as a negative
between molecular diffusion and chemical reactions, the shape of the ~10”s of
c’ and 1” against
the reaction
time resembles
the dielectric
dispersion
and loss
spectrum (i.e. plots of e’ and c” against the logarithmic frequency) of a chemically physically stable substance. can be mathematically
and
The detailed shape of the complex plane plots of E’ and E”,
described by a relaxation
function when measurements
are
made for a fued frequency. The form of Eqn. (9! is therefore written as: WC) = exp [- Ctlritc)‘l where r(k) is now a pseudo-equilibrium
,
(15)
average value of T at instant 4.
which is
datermined by using the limiting short-time and long-time values of e*(iw,&i with I: as a variable of k_
The new parameter,
y, has been named
the curing
or reaction
parameter [4-61. Before considering the application of this fo&malism, one r’urther aspect needs to be discussed here; that is, that the dc conductivity easily measurable because of the requirement
of a time-variant
that experiments
system
is not
be done by decreasing
th- li-equency till a plateau value at a low enough frequency is reached and that during the period of measurement the chemical structure of the substance remain unaItered. This requirement cannot be fulfilled for a time-variant system. Therefore, it is necessary to devise an alternative
procedure for determining
value of ac frequency used for the measurement to oo(T,t&oeo
within the experimental
whether or not a fixed
is low enough for s”(o,T,tJ
to be equa1
error, We describe this procedure as follows.
The measured values of K’ and E” may be transformed into the complex electrical modulus, M*, formalisms by the equations, M*(io,T.tc)
where,
= W(io.T,te)
I- ’ = M’iu.T.tJ
+ i.tf[o,T,tc).
(161
lS9
&I’ and M” are the real and imaginary When
components
of the
complex electrical
it is related
M* is entirely due to ionic conduction,
to Maxwell
in!XiUlUS.
or single.
conductivity relaxation time, ro, by the equation, / wSere ‘co = eoedoo, and Mo=m-1.
irdL#, cT.tcl (18)
In Eqn. (181, M* is invariant
equivalently 00) as one’s choice ofvariables
of o and Q, (or
anti 34’ and M” obey the expression,
Accordingly, a complex plane plot of M” against M’ would be a semicircle with a radius equal to -2 MO and centre. on the M’ axis, provided o.T
and t
were such that no If other contri-
processes other than the ionic conduction contributed to e*(io,T,tJ. butions were p.resent, the plot would deviate from the semicirrular plex plane plot can therefore be used to determine
shape.
the time during measured
Such a comthe chemical
reaction up to which the conductivity
of the substance
for a fixed ac
frequency is equal to its dc conductivity.
This is analogous to Debye single relaxation
time representation where a complex plane plot of 8” against E’ is a semicircle with a radius equal to 4 (co - -1
and centre on the E’ axis provided dc conduction
contribute to &*(io)_ Thus the Xiaxwell conductivity
relaxation
did not
time for conductors
becomes formally analogous to Debye dielectric relaxation time in dipole containing insulators. OBSERVATIONS
ON IRREVERSIBLE
MACROMOLECULB
FORMATION
Since the rate of chemical reaction*c _s sensitive to temperature.
changes in the
dielectric properties with both the temperature and time become important variables for a time-variant
system.
Therefore, measurements
are needed as a function of both
temperature and time. As a typical example of the behaviour observed, measurements of the changes in dielectric properties during the course of reactions in a stoichiometric mixture of diglycidyl ether of b&phenol-A IDGEBA) and propylene diamine (PDA) at six different temperatures
are shown iE
components of the dielectric permittivity
Figure
1,
where the
and e ktrical
modulus
real
and imaginary
are pIott&
against
the Iogarithmic reaction timeFor short periods of curing, B’ slightly
decreases first towards a plateau
and thereafter in a stepwise manner to e’ of about 4.4 or less approaches tinity. decreased.
me
as
the
value
reaction time, k,
This step z:hifts towards longer k as the reaction temperature
corresponding value of E” first decrees
is
fern a near plateau value to
Figure 1. The real and imaginary
components of dielectric permittivity
and eIectric;al
modulus of the DGEBA-propylene
diamine mixture measured for a fixed frequency of 1 kHz are plotted against the reaction time. The isothermal temperature for reactions ar.e: (1) 284.3K.
(2) 296.3K.
(3) 304.2K,
i-S 312.3K,
(5) 324.3K
and (6) 336SK
(after
ref. 6). reach a local minimum.
which is foliowed by a peak.
As &-s-;.
E” dci:: Cadet TVreach a
limiting low value of less than 0.02, which corresponds to that. ~,f 3 vitrified solid at a high temperature.
As the reaction temperature
is decreased. the minimum
shallovyer, the peak ‘becomes higher and both the minimum longer Limes.
becomes
and the peak shift towards
141
The corresponding
plots of AM’ in Figure 1 show an increase
in M’ with thz
reaction time, that occurs in two steps W,vards an ultimate value M,.
Both steps shift
to longer times with decrease in the reaction temperature,
and ‘he first step becomes
smaller in height than the second step, while the corresponding first piateau becomes Broader on a logarithmic
scale.
The M” plots in Figure 1 show two peaks whose
positions shift to longer & and the width of the first peak increases. The formalism
given in the preceeding
section
here is more appropriately
considered by examining the shape of the complex p1ar.e pIots of E*’and E’ and of M” and M’, which are shown in Figures 2 and 3. The shape of the c” against E’ plots resembles that of the Cole-Cole relatively
large
plots of chemically
ionic conductivity_
distinction between dielectric relaxation
and physically
Nevertheless,
stabie
materials
it is advisable
with a
to maintain
a
these plots for which changes in the e* and M* occur when the time irreversibly
(and spor;taneously)
increases
and each data
corresponds to a certain &, and the Cole-Cole plots, for which changes in the I* and M* are produced by varying (s’&4~-E*~t~~=J
the measurement
1) increases with &crease
contribution to I” from ionic conductivity,
frequer_cy.
The width
of the plot, i.e.
in the reaction temperature, which, although
as does the
large in the beginning of
the reaction, becomes too small to alter the shape of the compiex plane plot as the reaction proceeds. The ccrresponding
complex
plane plots of M’ and W* measured
for a fixed
frequency of 1 kHz in Figure 3 have the shape of a semicircle, with centre on the axis, which is followed by an apparently deviations
fiorn the shape of the skewed
temperature semicircle
skewed arc.
But, as &4,
arc appear.
progressively
A change
in the reaction
also affects the shape of these plots in that both :5;_ diameter and the width of the skewed
arc generally
increase
more of the
as the reaction
temperature is increased. The shape of the complex plane plots of E* and M” indicate that at the initial stages of the addition reaction the dc conductivity rattler than the dipolar reorientation dominates the dielectric behaviour-
As the reaction time increases, deviations
from a .s.emicircular shape of the M” against 3%’ plot occur and a new shape of a skewed arc emerges, which is a refIection of 2 dipolar relaxation process. Therefore, only part of the measured conductivity is due to ionic conduction and this part lies at times of reaction shorter than the time &in. conductivity.
at which a minimum
appears in Figure3.
o(h), is equal to the dc ccnductivity when t
The
but exceeds the true
a~(‘-) as & approaches t min. and this limitation should be considered in the analysis of the kinetic ef&cts during the addition reactions.
Figure 2.
The complex plane plots of e* for the DGEBA-propylene
measured for a fixed frequency of i M-k
d--&--e uA -b
_!Gr
l
diamlne
.rnirture
reaction at *he same temper&Lure
as in Fig. 1. The triangles are the calculated values from the parameters, Ae = 4.85 and y = 0.32 for (1); 4.35 and 0.32 for (2): 3.90 and 0.32 for (3); 3.48 and 0.31 for (4),3_Gi) 2nd 0.30 for (5) and 2.45 and 0.34’. for (5). The time of reaction increases from right tu left (after ref. 61. Figure 3.
The complex pIane plots of M* for the DGEBA-propylene
measured for a fixed frequency of IkHz during the isothermal perat.ure indicated.
diamine mixture
reactions at the tem-
The semicircle
represents the conductivity relaxation and the skewed arc due to dipoIar relax&or&. The time of reaction increases from let1 *a right (after ref. 6). Typical plots _ r the measured conductivity against the reaction time between DGEBA
and 4,4’-diemino
diphenyl
methane
calculated from Epn. (6) for value 3 of tio(+o),
are shown
in Figure 4.
The curves
x a2d tgel have been shown (by the
dashed line) to indicate the adequacy of Eqn, (6) for describing the reaction kinetics. Another description, which is an alternative also been used by Johari and coworkers [4-61.
to the power law of Eqn. (61, has
Its piausibility
lies in recognizing that
the increase in viscosity during the early stages of the polymerization
process deter-
mines the ionic diffusion according to the Stokes-Einstein
This is parti-
equation.
163
DGEBA-DDM
4.5
4. c
3, !i
5. II
log10 (curing time/s) Figure 4.
*
- ..
logarithmic plots or tne mcssure, d conductivity against the reaction time for th2 DGERA-4,4’ diaminodiphenyl methane mixture at the temperatures The
indicated. The dashed line was ca!cuIatted from Eqn. (6) or power law, and the f?rl! line from Eqn. (201, or singularity
equation. (After ref. 5).
cularly so in view of Pa’hmanathan
and -JoharPs !lO! argument that the r *uer law of
Eqn (6). when applied to temperature dependent3 the well-known ments.
VogeLFulcher-Tamman
of relaxation time, is equivalent
to
equation over a narrow range of measure-
This alternative equation impIies an approach of og(t) towards a singularity
during the curing process acctirding to, UJ!-.Zc! = A*.T) exp [ - BtTV(t,cT)-
tc)l
1201
whe;-e b is the point of singularity or the time taken to reach a value oo(T,&) = 0 and A and B are temperature-dependent
empirical parameters which determine the rate at
which conductivity approaches the singularity at to_ A typical example for the validity of Eqn. (20) is shown in Figurr 4, butMangion
and Johari [4j and Parthun and Johari
[6: have also noted that the value of b obtained from Eqn. (20) is close to the reaction time when the errpeak for 1 kHz frequency measurement
appears, and which in turn is closer to the vitrification time then to the gelation time. In addition, Parth.. I a?d
Johari ISI have found that for the reaction between DGEBA the value of +l
of Eqn. (6) is virtuaily
independent
and a variety of diamines,
of the frequency
measurement, whereas that of to of Eqn. (20) systematically
changes.
The changes in E* during the addition polymerization conductivity’s contribution to E**has become negligible,
used for the
reaction, but after the dc
are of interest here.
These
changes seen in Figure 2 show that the fixed frequency values of E’ and E” during the chemical reaction or of a time-variant
system resemble the E*and E” plots of chemically
and physically stable dipolar liquids and solids or a time-invariant a fixes temperature
but with varying frequency.
As
system measured at
mentioiled
earlier,
this is a
reflection of the fact that relaxation functions are invariant of w or r as one's choice of an independent variable.
Mangion and Johari 141 have used Eqns. (14) and (15i to
calculate y, the curing parameter,
by using a modification
Ikloynihan. Boesch and LabergeIllI
and Be;;dler
of procedure
given
by
and coworkers [12-141 and Parthun
and Johari [61 have calculated it by a procedure given by Muzeati, Perez and Johari [IS]. The data calculated by Parthun and Johari 161 have been shown along with the experimental vaIues in Figure 2. The agreement seen here demonstrates the adequacy of the formalism for time-invariant Given
that the measured
systems already
described here.
dielectric
arc
d?ta
Eqn. (15), it now becomes possible to calculate during an addition reaction of polymerization.
satisfactorily
the relaxation
de.3:ribed
by
time at ar.y instant
This is so because
in the plots cf
Figures 1 and 2, each data point corresponds to a unique value of or(&) in Eqn. (141, which in turn corresponds to a unique value of N*.
Thus both the time-dependence
E*, and of r(k) can be caiculated when AC and y are known. calculations [4, 51, has been confIrmed by the extensive
The adequacy
of
of such
work by Parthun and Johari
161,and Tombari and Johari 1161 who used a variety of reacting aminecc to demonstrate the variation of r(&: with the reaction time,
An example,
which will suffke
here, is
shown in Fig. 5 where the calculated values of dielectric relaxation time, permittivity and loss of the DGEBA
hexarnethylene diamine mixture have been plotted against the
reaction time in Figure 5. The range of teiaxation time, as determined by CXZ above-given
procedure, is of
course limited by the insensitivity of X*(or) to WY for a given value of y in Eqn. (15) But or can also be varied by independentiy varying o which then allows one to determine a wider range oft over 3 brokder time scale of a time-invariant is a remarkable feature in that it allows one to determine virtualiy in r from picoseconds to kseconds as a liquid converts
system.
This
the entire change
to a solid polymer,
By using
frequencies of 50 Hz, 1kHz and lOOkHz, Parthun and Johari [SJ and frequencies from
. D
IIGKIIA
IIMlb%
fj
-1 C-2 ” xl-
0
3 -4 -5
-6 -7
Figure
5.
The
relaxation
time.
the
permittivity
and
loss
of the
DGEBA-
hexamethylene against
diamine mixture -measured for a fixed frequency of 1 kHz is plotted (1) 294.3K. (2) 303.8K. (31 3155K, the reaction time at temperatures:
(4) 323.7K, (51 335.9K measured, data.
and (6) 346.OK.
Circles
are the calculated.
and dots the
10 kHz to 20 MHz, Tombari and Johari 1161 have been able to obtain the reaction time dependence of r of a number of poiymerizing systems, a typical example of which is shown in Figure 6, The data obtained from different frequencies of measurement overlap over a narrow region of curing time, but all lie on the same clurve which is now much broader, covering a range of 10 -9 to 10 seconds. The single plot of the relaxation time against
the curing
time
seems to be a satisfactory
correctness of the formalism given here.
demonstration
of the
DGEBA-PDA
-1
-9
3.0
3.25
3.75
3.50
-
log10 (curing timekj Figure 6.
The relaxation time of ‘LheDGEBA-propylene
diamine mixture at 32Q.OK is
plotted against the reaction time. The data correspond to measurements
made at 50Hz
for circles, 1 kHz for triangles and X00 kHz for squares.
TEMPERATURE AND TIME EVOLUTION PROCESSES DURING POLYMERLZATION
OF
The fact that a single frequency measurement
THE
during the isothermal
can provide information on both the gelation time and the relaxation in determining occurs.
how the various relaxation
In virtualIy all physical
processes
measurements
RELAXATION
evolve
at a certain
occurrence the measured values correspond! to the relaxation
when
reaction
time is valuable polymerization
instant
during
characteristics
this
of the
s-ubstance’s structure at that instant for the frequency used in the measurement. any instant,
this (fixed) structure
distribution of relaxation tims,
is expected
known as a- and k-relaxations
liquid is near-or below the reaction temperature, relaxation
to relax with at least
At
a bimodal
1173, when Tg of the
but only with one distribution
times when T, is much greater than this temperature,
of
i.e. the condition
when the a- and B-relaxation processes are merged. Thus as the reaction progresses or
167
& increases, the structural sbte
of the substance changes From that of a liquid, when
the reaction temperature is much greater than T,, to a rigid solid, when it is equal to or lesser than Tg- For a fixed frequency measurement. the reaction progresses, continuously changing
one would therefore observe, as
dielectric properties belonging
to each
of the continuously different structural states as these states traverse with time from the ones with a very short relaxation time of a fluid, say IO-9 s, to the ones with a very long relaxation time, say 104, of its vitrified state. This is illustrated in Figure 7 where the anticipated increase in the relaxation with decreasing
time for the a- and P-relaxation
temperature are illustrated
for measurements
made
processes
at different
+! 3 r
+:
F
Figure 7.
An illustration of the change in relaxation time for the a- and l3-relaxation
processes of the macromolecular
structure at a given instant during
the chemical
reactions for addition polymerization. For simplicity, the plots for the P-process are drawn to have the same slope and to merge with the a-process at a frequency of 107 Hz. The pre-exponent for all plots has been kept the same. instants processes
TV, tl+ t2, e+x_, of the reaction are drawn to have the same
measurements
For simplicity, the plots of p-reIaxation slopes and to merge at a frequency of
of about 107 Hz, and the pre-exponential
factor is necessarily
kept the
same for all plots. In Figure: 7, the measured 6’ and E” at 1 kHz frequency for a given value of y would initially correspond to a structure and relaxation at a point T,-l
on
curve t+ With the passage of time, and in a continuous manner, this would be followed
with those properties which correspond to the structure and relaxation times for points at Tc-’ on curves tl. t:! .._ etc., and ultimately t,it is reached, all contributions
on curve h-it. the vitrification
from the a-relaxation
time. As
process may not yet reach their
minimu.m value if the frequency of measurement corresponds to the P-relaxation rate of the network structure.
In the previous Section we surmised that the deviations of e”
and c’ from a stretched exponential
decay function in the plots of Figures 2 and 3 as
are due to the contribution from &relaxation, and earlier studies (181 have tc-= shown that the strength of p-relaxation initially increases with time during the polymerizing reactions at the expense of the height of another sub-Tg, y-relaxation, whose strength decreases.
Simultaneously,
the a-relaxation
process shifts to lower
frequencies and thus its contribution to e” decreases. In measurements
made at a fixed
instant after tvit, where reactions in a vitrified state cccurred sufficiently
slowly to
allow the measurement
of the spectrum. one finds, as shown in earlier studies [18], a peak in the frequency spectrum of E”, and a shift in the position of the &relaxation peak towards high frequencies with increase in the temperature. .UtemativeIy, if ‘;he reaction temperature
was the same but different
frequencies
were
used for the
measurement of the dielectric properties of the time-variant system, one would deduce from Figure 7 that at &>&it, +&e measured E” is frequency-dependent, showing a peak. The position of this peak would shift to a higher frequency if the temperature of the reaction was increased.
CHEMICAL
KINETICS
AND
We now consider whether
DIELECTRIC a useful
BEHAVIOUR
reiation
between
the chemical
kinetics
measured in terms of the extent of reaction and the dielectric properties can be found. Attempts for seeking such a relation require that each of the seven effects noted in the first Section here be considered because chemical reactions affect both the dynamic and static properties from near zero frequency principle, the magnitude experimentally,
to optica
frequencies.
of thes e effects can be determined
the eflort required for doing so is enormous.
that includes only the most pror;linent effects can provide
2
Although,
both theoretically Therefore,
in and
a procedure
useful relation between the
chemical kinetics and dielectric properties of a time-variant
system, and these effects
are: (1) changes in the conductivity, o. and (2) the average dipolar relaxation time, r, of the liquid. Mangion and Johari f41 have concluded that the dipolar related to the reaction kinetics.
relaxation
time is
Their studies have shown *Aat the dipolar relaxation
time increases with the progress of reaction and ‘hat the logarithmic
plot of I: against &
-, 6
2
.6
3.
4
Icrg(time/s) Figure 8.
The relaxation
time calculated
from the dielectric studies
of DGEBA-
ethyIene diamine mixture reacting at 296.2K is compared against the prediction from Eqn. i22). ? ull line is cakulated from Eqn. (22) but here, S = [r[T, m)- r(T, 011 with t(T,O) = 6ns, r(T,=) = 52s and n = 1.95. Also plotted are the extent of reaction, a, and the heat capacity, Cp. thermodynamics
These are useful for a comparative
analysis
of the chemistry,
and diffusion process [after refs. 16 and 201 as po!ymerization
is sigmcidal in its shape up to ‘I:= 103-
occurs.
lWs, beyond which r could not be determined as
the time required for such measurements
becomes prohibitively
long.
Parthun and
Johari (61 have analytically examined the dependence of K on the curing time and have shown that the shape of the plot of the extent of reaction against
the logarithmic
reaction time is related to that of the corresponding plot of the average dielectric relaxation time determined from a fixed frequency measurement.
This is given by an
empirical equation: r(T.t c) =
where S = ln(rIT,=)KT,O)) parameter.
r(T.0)explS a”tT.tC)?,
(221
is a constant for a given epoxy and n is an empirical
The factor S normalizes the plot of ln(r(T,t)I against &
This normalization
allows, for a direct comparison of the plot of In t
by allowing parameter
the values of IS-1 In (r(T, t+)/t(T,O))l n alters the shape
correspond with a(T,t,&). In(z(T,t& The dielectric
of the sigmoidal
When
diamine
at 296.2K,
measured
plot so that r(T,bl)
t(m) = 52 s and n = 1.95,
The to
n = 1, the extant of reaction is directly proportional
to
in the kHz and MHz frequency that For the reaction
Eqn. (22) provides o satisfactory of reaction
extent
zero and unity.
can be made
measurement
and Johari 1161 have confirmed
to vary between
and
the dipolar
between
range by Tombari
DGEBA
and ethylene
description of the calorimetrically
relaxation
but here, S = Cr(T, =+r;(T,
time
with
O)]. A typical
t(O) = 6.0 ns,
plot showing
this
agreement is given in Figure 8. There is need For Further measurements of dielectric and calorimetric behaviours of time-variant systems For demonstrating the usefulness of this relaticn.
EVOLUTION OF THE a-RELAXATION PROCESS Dielectric properties measured in the frequency range 6.01-26 and Johari il63 have led to observations and structurally particular
arrested states of a time-variant
of the DGEBA-ethylene
They demonstrate obtained
&amine
the fixed frequency
measurements
and Johari [6].
The normalized
process which comes in the strength
and as the viscosity the low-frequency
of relaxation
feature,
nameIy,
as the reaction
time
progresses,
io, shown
of a second
of 1_25h, i.e.,
of the reaction
These relaxations
appears
that
what
splits into two progresses
are referred
to as a- for
at First glance to be the new process in that it
with respect to the p-relaxation
from the P-process as it shifts towards
and a
processes.
and separates
that it is instead the a-process
in
stable liquids
more From each other as the reaction
of the Iiquid increases.
as do
I161 and earlier
the present?
aftsr a reaction
and p for the high-frequency
seems to be the one that emerges indicate
plots generally
with the spectra of structurally
peak at the beginning
progressively
In Figure 9, &relaxation normalized
to the corresponding
in
at 296.2K.
form of one such spectrum
into evidence
appears to be a single relaxation peaks which separate
time periods,
its reaction
with reaction time made by them
and solids, Figure 9 shows a remarkable decrease
at different
during
of chemically
stable liquids and solids and yield the same value oft
Figure 9. In addition to the similarity relaxation
loss spectrum
system
mixture
that these spectra are similar
for structurally
hy Parthun
of the dielectric
MHz by Tombari
From the a-process.
peak,
which
which emerges
is not shown
here,
and progressively
a lower frequency.
these features in Figure 10, where the variation
But a spectrum
For clarity,
of the relaxation
would
separates
we illustrate
rates for the CL-and fi-
Figure 9.
The normalized plots of dielectric loss against frequency for the DGEBAethykme diamine mixture at several fmed instants during its reaction at 296.3.K. The instant of measurement ia given in ks for each plot. J?ull lines are for the values calculated for a stretched exponentialrelaxation fun&on with 6 = 0.4 in Eqn. (9) [after ref. 163. processes is shown against the reaction time at a fixed temperature. The plot illustrates a pattern of the development of relaxation processes and their progremive separation on a time scale. As the reaction proceeds, the relaxation process with a unimodal distribution of times acquires a bimodal distribution of the a- and prelaxation processes, each showing a peak for a fixed frequency of measurement. This feature of splitting of the relaxation process, we note, bears a remarkable resemblance to that observed on supercooling a liquid whose a-relaxation process also progressively separates from the P-relaxation process in a time-temperature plane Cl71 as reversible molecular clustering raises the liquids viscosity. Recent studies by Cassettari, et al. 1191 reported in this volume show an additional remarkable feature of the evolution of a low frequency relaxation during the addition polymerization reactions, namely the development of the slower relaxation yrocess at the expense of a faster process. This new process rapidly shifts towards the lower frequency side of the spectrum,and its dielectric polarization strength decreases This occurs in a manner as does that of the remaining high-frequency process. remarkably similar to that observed on supercooling a liquid through its glasstransition temperature, as noted earlier by Jchari [17]. It seems as if in ita relaxation
172
log(time/s) Figure 10.
An illustration
of the change
in relaxation
time with increase
reaction time as the Q and p-processes evolve at a fixed temperature,
in the
and progressively
separate on a time scale. effects the irreversible increase in the moleculs r weight of a macromolecule tively related to the inverse of temperature
0’ a structuraliy
is qualita-
and chemically
stable
glass-forming liquid.
SUMMARY
AND CONCLUSIONS
The phenomenolbgy formulated
of the dielectric
for a time-variant
system
diffusion
a liquid under virtually
process, the dielectric properties irreversibly similar
to that when
the ac frequency
been
described
reactions
The process of this conversion
as a negative feedback between molecular vitrifies
has
in which irreversible
convert a monomer into a macromolecule. ultimately
processes
and chemical
isothermal
and spontaneously
used for the study
continuously is envisaged
reactions
conditions.
and
which
During
this
change in a manner of a chemically
and
structuraIly stable material is deliberately increased. Dielectric
spectroscopy
shows
conductivity approaches a singularity, useful in determining
the gelation
that
as polymerization
progresses,
the dc
for which equations are provided which can be time
develops_ The relaxation time irreversibly
when a cross-linked
network
structure
increases with the reacticn time according
173
to a sigmoidal-shape
curve. Its value is related to the extent of reaction. relaxation As polymerization progresses, a second, low-frequency,
evolves and progressively
separates from the high-frequency
process
relaxation process on a
time scale.
This is analogous to the evoiution of relaxation processes when a liquid is supercooled through its glass transition temperature. The irreversible increase in the molecular
weight with the reaction time affects
manner quaIitatively
the relaxation
phenomznon
in a
similar to that observed on supercooling a molecular liquid.
References 1.
I. Alig, D. L&inger
and G.P. Johari, J. Polym. Sci. Part B, Polym. Phys. 30,791
(19921. 2.
J.F_ Johnson and R.H. Cole, J. Amer. Chem. Sot. 73,45X
3.
JR. McDonald, ed., “Impedance Spectroscopy” (Wiley, N.Y. 1987).
4.
M.B.M.
(1951).
Mangion atid C.P. Johari, J. Polym. Sci. Part B, Polym. Phys. a
(1990); Macromol. 23,3687 B, Polym. Phys. 29,437
(1990); Polymer 32, 2747 (1991); J. Polym. Sci. Part
(1991), ibid. 29,1117
(1991).
5.
G.P. Johari and M.B.M. Mangion, J. Noncryst. Solids 131-133,92lIl991).
6.
M.G. Parthun
and G.P. Johari, J. Pclym.
(19921, Macromol. 25.3149;
Sci. Part B, Polym. Phys, 30, 655
3254 !1992).
7.
D. Stauffer, A. Coniglio and M. Adam, Adv. Polym. Sci. 44.105
8.
M. Djabourov, Contemp. Phys. 29,273
9.
G. Williams
10.
K. Pathmanathar,
II_
CT.
(1982).
(1988).
and D.C. Watts. Trans. Faraday Sot., 66.80
Moynihan,
1621
and G.P. Johari, Phil. Mag. B, 62,525
(1970). (1990).
L.P. Boesch and N-L_ Laberge, Phys. Chem. Glasses,
14. 122
(1973). 12.
E.W. Montrol’l and J.T. Bendler, J. Stat. Phys. 34,129
13.
M. Dishon, G.H. Weiss and J.T. Bendler, J. Res. Natl. Bur. Stand. 90,27
14.
G.H. Weiss, J-T. Bendler, and M. Dishon, J. Chem. Phys. 83.1424
15.
E. Muzeau, J. Perez and G-P. Johari, Macromol. 24.4713
16.
E. Tombari and G.P. Johari, J. Chem. Phys. 00,OOOO (1992)_
17.
G.P. Johari, J. Chem. Phys. 58. 1766 ;1976).
(1973);
Ann.
(1984). (1985).
(1985).
(1991).
N.Y.
Acad. Sci. 279,
117
174
18.
M.B.M.
Mangion and G-P. Johari, J. Polym.
Sci. Part B, Polym. Phys. 28, 71
(1990). 19.
M. Cassettari,
G. Salvetti,
S. Veronesi,
E. Tombari
and G.P. Johari,
(this
volume). 20.
M. Casettari,
G. Salvetti,
S. Veronesi,
E. Tombari
and G.P. Johari,
Cimento (in press); J. Polym. Sci. B., Polym. Phys. (in press).
Nuovo
Liqrrids, .S6 (1993) 15%174
Scicncc Pubkhcrs
B.V., Amsterdam
PHENOMENOLOGY
OF ADDITION
POLYMERIZATION
G.P. Johari, Department of Materials Science and Engineering, McMaster University, Hamilton, Ontario, Canada LSS 4L7
ABSTRACT The phenomenology of macromolecular growth by addition reactions, which is envisaged as a process of negative feedback between molecular diffusion and chemical reactions, has been described in terms of dielectric erXects, and a formaiism is given. According to this formalism the dielectric properties of a time-variant system irreversibly and spontaneously change in much the same manner as that of a time-invariant system when the ac-frequency used for the measurement is deliberately increased_ The dielectric spectroscopy shows an approach of dc conductivity towards a singularity for which equations are provided. The increase in the relaxation time of a liquid with the progress of reaction are shown to be interrelated. As the reaction occurs, a second, low-frequency, relaxation process separates and rapidIy shifts to lower frequzncies. Concomitantly, its strength decreases as does the strength of the remaining highfrequency process. The phenomenological consequences of this negative feedback between chemical reactions and molecular diffusion is remarkably similar to that observed on supercooling a molecular or polymeric liquid whcse structure remains time-invariant. GENERAL
ASPECTS
During the chemical
growth of a macromolecule,
the mclccular
diffusion
coef&ient irreversibly decreases with time until the chemical reactions that allow this growth come to a virtual halt. This :legative feedback tetween molecular diffusion ard chemical reactions, which cavuses both to become progressively
slower,
uItimately
converts a molecular liquid irreversibly to a linear chain or cross-linked solid polymer. Ii variety of thermodynamic changes occur during the course of this irreversible process. For example, as the reaction time approaches infinity, the extent of conversion of monomers to a macromolecuIe
approaches
-unity, the diffusion
approaches zero, the volume approaches its lower bound (limiting!
coefficient
value, and the
configurational contri-bution to entropy and enthalpy approach zei-o as the glass transition temperature of the molecular stmcture increases and enough time is available for
0167~7322/93/!%6.00
0 1993 - Elsevier Science Publishers
B.V.
AI1 rights reserved
its spontaneous relaxation towards the minimum
volume, entropy and enthalpy states
corresponding to vibrational frequencies at the temperature Equally
interesting
liquid as it irreversibly time in a sigmoidal correspondingly maximum
changes occur in the molecular
polymerizes.
reaches a maximum
zero value at infinite time Ill.
approaches
before decreasing
kinetic behaviour
of the
The velocity of ultrasonic waves increases with
manner and the attenuation
decreasing to a virtually
of the reaction.
a limiting
value
The mechanical
and the mechaniral
to a near zero vdue.
value before modulus
loss reaches
a
As the structure and chemical
composition of the liquid irreversibly and continuously
changes, its electric properties
change in at least seven ways, namely: (i)
a genera! decrease in the dc conductivity as the diffusion coefficient of impurity ions in the reacting mixture decreases with increase in its viscosity arId any proton transfer
along H-bonds in the mixture
is virtually
ehminated
by the
formation of cross-links; (ii)
an increase in the molecular difiusion or relaxation time as a resuIt of which the dielectric permittivity
measured at a fixed frequency monotonically
decreases
towrrd the value corresponding to the infrared region; (iii)
a change in the number
of dipoles
per unit
volume,
contribution to ezrmittivity from orientation pclarization,
and therefore
in the
L.e, as a result of both
the chemical reactions that alter the dipole moment and the volume contraction that raises the number density of dipoles; (iv)
a change in the dielectric relaxation function as the chemical structure of the liquid changes and its viscosity and density increase;
(VI
a change in Aeir, the contribution to permittivity the vibrational polymerization
(vi)
frequencies of the various modes in the structure change
a change in %&eoptical refractive in&_: or optical polarizg?‘;.l the splitting
.‘LS on
and densification;
occurs and the structure de&&s; (vii)
due TVinfrared polarization,
of a unimodal
as polymerization
and,
relaxation
function
into a bimodal
relaxation
function. Thus in addition to the frequency and temperature dependence. the complex dielectric permittivity, E*, for such systems becomes time-dependent and is expressed by the equation, c*lio. T. tc) = E’(o, T. tc) -
k-m, T. tc~
(11
155
where the dielectric permittivity, c’, and loss. E”~contain contributions from both the dipolar orientation polarization and dc or ionic conduction, w is ‘the angular frequency of the applied electric field and T and & refer to the temperature beginning
of the chemical
reaction.
(Terms
in parenthesis
and time since the
refer to independent
Here the time k includes the effects of irreversible chemical and physical cb.anges of the reacting system. These changes are related to k by a
variables in such studies.)
suitable functional form, which can be determined by experiments. In ion-containing
materials, the measured permittivity
and loss include contri-
butions from both the interfacial effects and dc conduction, and the dipolar orientation, such that E’(o.T. tc) = (i+n (ad2 10-“+a’~C’~w,T.t,)/Co
+ C&
h.T.te)
(21
and, L’*(w, T, tc) = (a,,(T. tYoed
G’? o. T. tcVCO+ E”
- (ZOms(ad210’aJ
&P
(3)
!a. ‘I’, tc)
where the first term on the right hand side of Eqn. (21 and the second in Eqn (3) refer to the cant-ibution
from a %onstant phase element” for an inkrfacial
in series with the bulk properties of a material. is tha characteristic conductivity
In these terms in Eqns. (2) and (3), Zo
impedance of the electrode-insulator
which
is equal
to the reciprocal
impedance which is
interface, G is &&emeasured
of resistivity,
Co is the geometric
capacitance of the sample, and a= 0.5, as Johnson and Cole I21 and McDonald 131 have shown.
In the !aet terms of Eqns (2) and (3) E’dIp and &‘*&pare the contributions to E’
and E” from dipolar orientation alone. orein Eqn. (3) is the dc conductivity and eo the permittivity ( = 8.8514 pFm-‘1 of free space. For relatively
iow ionic concentration or dc conductivity,
interfacial impedance are generaIly insignificant
contributions
from
and can be neglected. so that E’ and
en may be written as. (4)
and, E-(0,T. tc) = Cion tT. tcl
+ Cdi.
:o, T, tc)
Wbf. lX f”ion = to[r/weg) is the contribution to c” from dc co-rductivity. that the permittivity
(5, Here it is clear
during the occurrence of a chemical reaction is determined only
by the dipolar orientation polarization but the loss is determined by both the ionic or dc conduction and dipolar reorientation process. We first
consider the change in the dc or ionic conductivity contribution, e”ion, as
the chemical reactions occur.
A number of our earlier studies on the reactions of
l-56
diamine
with bisphenol-A-diepoxides
[4-6] have shown that the decrease
in the-dc
conductivity during its reaction is described by a power law, oO(T.tC)=
t&T) - t,
tC +
0)
t gel IT)
(6)
I
where oo(T,t;t--tO) is the dc conductivity at the beginni,lg
of the reaction at a tempera-
ture T, kc1 is the time to reach the gelatiDn point and x is the critical exponent of the scaling equation used as a generalized
property function by Stauffer,
Adam [7] and Djabourov ES].
equation
oo(T,tc)
ConIglio and
= ~Iexp-(Mt0-t)),
which implies a singularity at t = to. was used also by Mangion and Johari [4], Johati and Mangion [!5] and Parthun and Johari [S]. but was found to be a less satisfactory description of the measured conductivity than the power law when a network structure formed and gelation occurred 3s a result of chemica1 r eactionsf. The significance of x in Eqn. (6) is that its value determines the approach ‘1: the liquid towards gelation, the rate at which a0 decreases with time.
Le.
The higher the value of x, the more rapid is
the decrease of ergwith the reaction time and the more rapid the approach to gelation. The description of the data by Eqn. (6) allows extr;rpolation of the measured ductlvity so that the time for the gel&ion of a chemicaIly-reacting
de con-
system during both
its isothermal and temperature-ramp conditions can be estimated. By combining Eqns. (5) and (61, the dielectric loss during the macromolecule formation Is given by,
We now consider the dipolar contributions to r* and E”. According to the classical theory of dielectrics, the complex permittivity
due to orientation po!arization
alone in
a material is given ‘k-y,
where EOand ca: are the Iimiting low- and high-freque1ir.y permittivitics. and t refers to the time for the decay of polarization
for a chemical
respectively, structure
‘Aat
remains unchanged during the measuremen, + time, or when the chemical reaction In Eqn. (S:l, 9 occurs too slowly in comparison with the time for measurement. represents a relaxation function whose mrmalized given by,
form for a time-invariant
system is
where
p is an empirical
relaxation
time, p is equal
in the range (Km)-,or
parameter. tc
unity.
zero to unity and stretched
polarization For
during
a chemically
approximation
n,gligible also useful
here,
structural
represents
a one-sided
a change period
X*
of time.
therefore
an
of 1 kHz or more, t in
in both the chemical structure
can be justifiabIy
assumed to
be
the reaction is slow and the total reaction allows
us to write Eqn_ (9) in a form
iot
exp-
(10)
is itst:lf a func’jon
at any
Laplace
of
as follows,
I-
=
state of the liquid
this
when
system,
quantity
from
of the decay
of time and
For frequencies
approximation
0
the normalized
c91
prob:esses are considered
time. remains invariant
invariant
as follows:
during
This
for a time-varia3t
- VVilliams-Watts
stable materic.1 t and t in
the relaxation
relaxation
the effects
S*(iCat)
where
Uihen
I: is not
temperat*clres
hours.
to as Kohlrausch
it is the time for the observation
system,
state of the sys’tem several
of viscous liquida and solids. P is
and physiczl!y
‘I:- the average
needs to be made
at th e usual
time exceeds
meaning.
1 ms and
process, i.e.. with a single
parameter.
in series,
reacting
Eqn. (9) is less than and physical
than
which
it is referred
or a chemically
Eqns. (8) and (9) ha-,-e a strict rather
For the majority
here
exponential
For a time-invariant, l a be in parallel
For a 135ebye relaxation
given
instan:
trar.sfox-m
of the product of o and r cc
since the beginning
_
of the reaction,
and E’ and z” are written
in terms of
Ku
and IV by: A
c’CT.tc)
N’(wr!T,tc)I
= c, + -
T
Cl?!
and z”(T,tJ
where
(&,-,-a& = Am with A as
For most polymerization
A 7
a cons+&nt reactions
the end olr reaction is by about to Eqn. (8) is much less than
=
20%
Y"(wr(T.t
11
according
c z3, tc the Curie Law.
the totzl decrease
or !ess.
the effect
_
in ~0 from the beginning
The et’;ect of this &crease
to
on E*, according
from the sever al orders of tnagnit-dde increase
in
r, which is, typically, from 10-S
to 10 + 3s. As a first approximation,
we consider that
both co and h remain constant with & and Eqns. (8) and (10) may be combined to obtain, c+(io,T,tc)
= E=(T) + (c,jT)-
It is easy to see tl,at Eqn. (14) is phenomenologically cne’s choice of an experimental
variable.
-he relaxation spectrum or Cole-Cole Thus during the polymerization fdback
(IS)
SeIior(T,tc)I
c_(T))
invariant
of o and t as
This means that any of the several shapes of
plots can be obtained by varying either o or L
process,
which is envisaged
here as a negative
between molecular diffusion and chemical reactions, the shape of the ~10”s of
c’ and 1” against
the reaction
time resembles
the dielectric
dispersion
and loss
spectrum (i.e. plots of e’ and c” against the logarithmic frequency) of a chemically physically stable substance. can be mathematically
and
The detailed shape of the complex plane plots of E’ and E”,
described by a relaxation
function when measurements
are
made for a fued frequency. The form of Eqn. (9! is therefore written as: WC) = exp [- Ctlritc)‘l where r(k) is now a pseudo-equilibrium
,
(15)
average value of T at instant 4.
which is
datermined by using the limiting short-time and long-time values of e*(iw,&i with I: as a variable of k_
The new parameter,
y, has been named
the curing
or reaction
parameter [4-61. Before considering the application of this fo&malism, one r’urther aspect needs to be discussed here; that is, that the dc conductivity easily measurable because of the requirement
of a time-variant
that experiments
system
is not
be done by decreasing
th- li-equency till a plateau value at a low enough frequency is reached and that during the period of measurement the chemical structure of the substance remain unaItered. This requirement cannot be fulfilled for a time-variant system. Therefore, it is necessary to devise an alternative
procedure for determining
value of ac frequency used for the measurement to oo(T,t&oeo
within the experimental
whether or not a fixed
is low enough for s”(o,T,tJ
to be equa1
error, We describe this procedure as follows.
The measured values of K’ and E” may be transformed into the complex electrical modulus, M*, formalisms by the equations, M*(io,T.tc)
where,
= W(io.T,te)
I- ’ = M’iu.T.tJ
+ i.tf[o,T,tc).
(161
lS9
&I’ and M” are the real and imaginary When
components
of the
complex electrical
it is related
M* is entirely due to ionic conduction,
to Maxwell
in!XiUlUS.
or single.
conductivity relaxation time, ro, by the equation, / wSere ‘co = eoedoo, and Mo=m-1.
irdL#, cT.tcl (18)
In Eqn. (181, M* is invariant
equivalently 00) as one’s choice ofvariables
of o and Q, (or
anti 34’ and M” obey the expression,
Accordingly, a complex plane plot of M” against M’ would be a semicircle with a radius equal to -2 MO and centre. on the M’ axis, provided o.T
and t
were such that no If other contri-
processes other than the ionic conduction contributed to e*(io,T,tJ. butions were p.resent, the plot would deviate from the semicirrular plex plane plot can therefore be used to determine
shape.
the time during measured
Such a comthe chemical
reaction up to which the conductivity
of the substance
for a fixed ac
frequency is equal to its dc conductivity.
This is analogous to Debye single relaxation
time representation where a complex plane plot of 8” against E’ is a semicircle with a radius equal to 4 (co - -1
and centre on the E’ axis provided dc conduction
contribute to &*(io)_ Thus the Xiaxwell conductivity
relaxation
did not
time for conductors
becomes formally analogous to Debye dielectric relaxation time in dipole containing insulators. OBSERVATIONS
ON IRREVERSIBLE
MACROMOLECULB
FORMATION
Since the rate of chemical reaction*c _s sensitive to temperature.
changes in the
dielectric properties with both the temperature and time become important variables for a time-variant
system.
Therefore, measurements
are needed as a function of both
temperature and time. As a typical example of the behaviour observed, measurements of the changes in dielectric properties during the course of reactions in a stoichiometric mixture of diglycidyl ether of b&phenol-A IDGEBA) and propylene diamine (PDA) at six different temperatures
are shown iE
components of the dielectric permittivity
Figure
1,
where the
and e ktrical
modulus
real
and imaginary
are pIott&
against
the Iogarithmic reaction timeFor short periods of curing, B’ slightly
decreases first towards a plateau
and thereafter in a stepwise manner to e’ of about 4.4 or less approaches tinity. decreased.
me
as
the
value
reaction time, k,
This step z:hifts towards longer k as the reaction temperature
corresponding value of E” first decrees
is
fern a near plateau value to
Figure 1. The real and imaginary
components of dielectric permittivity
and eIectric;al
modulus of the DGEBA-propylene
diamine mixture measured for a fixed frequency of 1 kHz are plotted against the reaction time. The isothermal temperature for reactions ar.e: (1) 284.3K.
(2) 296.3K.
(3) 304.2K,
i-S 312.3K,
(5) 324.3K
and (6) 336SK
(after
ref. 6). reach a local minimum.
which is foliowed by a peak.
As &-s-;.
E” dci:: Cadet TVreach a
limiting low value of less than 0.02, which corresponds to that. ~,f 3 vitrified solid at a high temperature.
As the reaction temperature
is decreased. the minimum
shallovyer, the peak ‘becomes higher and both the minimum longer Limes.
becomes
and the peak shift towards
141
The corresponding
plots of AM’ in Figure 1 show an increase
in M’ with thz
reaction time, that occurs in two steps W,vards an ultimate value M,.
Both steps shift
to longer times with decrease in the reaction temperature,
and ‘he first step becomes
smaller in height than the second step, while the corresponding first piateau becomes Broader on a logarithmic
scale.
The M” plots in Figure 1 show two peaks whose
positions shift to longer & and the width of the first peak increases. The formalism
given in the preceeding
section
here is more appropriately
considered by examining the shape of the complex p1ar.e pIots of E*’and E’ and of M” and M’, which are shown in Figures 2 and 3. The shape of the c” against E’ plots resembles that of the Cole-Cole relatively
large
plots of chemically
ionic conductivity_
distinction between dielectric relaxation
and physically
Nevertheless,
stabie
materials
it is advisable
with a
to maintain
a
these plots for which changes in the e* and M* occur when the time irreversibly
(and spor;taneously)
increases
and each data
corresponds to a certain &, and the Cole-Cole plots, for which changes in the I* and M* are produced by varying (s’&4~-E*~t~~=J
the measurement
1) increases with &crease
contribution to I” from ionic conductivity,
frequer_cy.
The width
of the plot, i.e.
in the reaction temperature, which, although
as does the
large in the beginning of
the reaction, becomes too small to alter the shape of the compiex plane plot as the reaction proceeds. The ccrresponding
complex
plane plots of M’ and W* measured
for a fixed
frequency of 1 kHz in Figure 3 have the shape of a semicircle, with centre on the axis, which is followed by an apparently deviations
fiorn the shape of the skewed
temperature semicircle
skewed arc.
But, as &4,
arc appear.
progressively
A change
in the reaction
also affects the shape of these plots in that both :5;_ diameter and the width of the skewed
arc generally
increase
more of the
as the reaction
temperature is increased. The shape of the complex plane plots of E* and M” indicate that at the initial stages of the addition reaction the dc conductivity rattler than the dipolar reorientation dominates the dielectric behaviour-
As the reaction time increases, deviations
from a .s.emicircular shape of the M” against 3%’ plot occur and a new shape of a skewed arc emerges, which is a refIection of 2 dipolar relaxation process. Therefore, only part of the measured conductivity is due to ionic conduction and this part lies at times of reaction shorter than the time &in. conductivity.
at which a minimum
appears in Figure3.
o(h), is equal to the dc ccnductivity when t
The
but exceeds the true
a~(‘-) as & approaches t min. and this limitation should be considered in the analysis of the kinetic ef&cts during the addition reactions.
Figure 2.
The complex plane plots of e* for the DGEBA-propylene
measured for a fixed frequency of i M-k
d--&--e uA -b
_!Gr
l
diamlne
.rnirture
reaction at *he same temper&Lure
as in Fig. 1. The triangles are the calculated values from the parameters, Ae = 4.85 and y = 0.32 for (1); 4.35 and 0.32 for (2): 3.90 and 0.32 for (3); 3.48 and 0.31 for (4),3_Gi) 2nd 0.30 for (5) and 2.45 and 0.34’. for (5). The time of reaction increases from right tu left (after ref. 61. Figure 3.
The complex pIane plots of M* for the DGEBA-propylene
measured for a fixed frequency of IkHz during the isothermal perat.ure indicated.
diamine mixture
reactions at the tem-
The semicircle
represents the conductivity relaxation and the skewed arc due to dipoIar relax&or&. The time of reaction increases from let1 *a right (after ref. 6). Typical plots _ r the measured conductivity against the reaction time between DGEBA
and 4,4’-diemino
diphenyl
methane
calculated from Epn. (6) for value 3 of tio(+o),
are shown
in Figure 4.
The curves
x a2d tgel have been shown (by the
dashed line) to indicate the adequacy of Eqn, (6) for describing the reaction kinetics. Another description, which is an alternative also been used by Johari and coworkers [4-61.
to the power law of Eqn. (61, has
Its piausibility
lies in recognizing that
the increase in viscosity during the early stages of the polymerization
process deter-
mines the ionic diffusion according to the Stokes-Einstein
This is parti-
equation.
163
DGEBA-DDM
4.5
4. c
3, !i
5. II
log10 (curing time/s) Figure 4.
*
- ..
logarithmic plots or tne mcssure, d conductivity against the reaction time for th2 DGERA-4,4’ diaminodiphenyl methane mixture at the temperatures The
indicated. The dashed line was ca!cuIatted from Eqn. (6) or power law, and the f?rl! line from Eqn. (201, or singularity
equation. (After ref. 5).
cularly so in view of Pa’hmanathan
and -JoharPs !lO! argument that the r *uer law of
Eqn (6). when applied to temperature dependent3 the well-known ments.
VogeLFulcher-Tamman
of relaxation time, is equivalent
to
equation over a narrow range of measure-
This alternative equation impIies an approach of og(t) towards a singularity
during the curing process acctirding to, UJ!-.Zc! = A*.T) exp [ - BtTV(t,cT)-
tc)l
1201
whe;-e b is the point of singularity or the time taken to reach a value oo(T,&) = 0 and A and B are temperature-dependent
empirical parameters which determine the rate at
which conductivity approaches the singularity at to_ A typical example for the validity of Eqn. (20) is shown in Figurr 4, butMangion
and Johari [4j and Parthun and Johari
[6: have also noted that the value of b obtained from Eqn. (20) is close to the reaction time when the errpeak for 1 kHz frequency measurement
appears, and which in turn is closer to the vitrification time then to the gelation time. In addition, Parth.. I a?d
Johari ISI have found that for the reaction between DGEBA the value of +l
of Eqn. (6) is virtuaily
independent
and a variety of diamines,
of the frequency
measurement, whereas that of to of Eqn. (20) systematically
changes.
The changes in E* during the addition polymerization conductivity’s contribution to E**has become negligible,
used for the
reaction, but after the dc
are of interest here.
These
changes seen in Figure 2 show that the fixed frequency values of E’ and E” during the chemical reaction or of a time-variant
system resemble the E*and E” plots of chemically
and physically stable dipolar liquids and solids or a time-invariant a fixes temperature
but with varying frequency.
As
system measured at
mentioiled
earlier,
this is a
reflection of the fact that relaxation functions are invariant of w or r as one's choice of an independent variable.
Mangion and Johari 141 have used Eqns. (14) and (15i to
calculate y, the curing parameter,
by using a modification
Ikloynihan. Boesch and LabergeIllI
and Be;;dler
of procedure
given
by
and coworkers [12-141 and Parthun
and Johari [61 have calculated it by a procedure given by Muzeati, Perez and Johari [IS]. The data calculated by Parthun and Johari 161 have been shown along with the experimental vaIues in Figure 2. The agreement seen here demonstrates the adequacy of the formalism for time-invariant Given
that the measured
systems already
described here.
dielectric
arc
d?ta
Eqn. (15), it now becomes possible to calculate during an addition reaction of polymerization.
satisfactorily
the relaxation
de.3:ribed
by
time at ar.y instant
This is so because
in the plots cf
Figures 1 and 2, each data point corresponds to a unique value of or(&) in Eqn. (141, which in turn corresponds to a unique value of N*.
Thus both the time-dependence
E*, and of r(k) can be caiculated when AC and y are known. calculations [4, 51, has been confIrmed by the extensive
The adequacy
of
of such
work by Parthun and Johari
161,and Tombari and Johari 1161 who used a variety of reacting aminecc to demonstrate the variation of r(&: with the reaction time,
An example,
which will suffke
here, is
shown in Fig. 5 where the calculated values of dielectric relaxation time, permittivity and loss of the DGEBA
hexarnethylene diamine mixture have been plotted against the
reaction time in Figure 5. The range of teiaxation time, as determined by CXZ above-given
procedure, is of
course limited by the insensitivity of X*(or) to WY for a given value of y in Eqn. (15) But or can also be varied by independentiy varying o which then allows one to determine a wider range oft over 3 brokder time scale of a time-invariant is a remarkable feature in that it allows one to determine virtualiy in r from picoseconds to kseconds as a liquid converts
system.
This
the entire change
to a solid polymer,
By using
frequencies of 50 Hz, 1kHz and lOOkHz, Parthun and Johari [SJ and frequencies from
. D
IIGKIIA
IIMlb%
fj
-1 C-2 ” xl-
0
3 -4 -5
-6 -7
Figure
5.
The
relaxation
time.
the
permittivity
and
loss
of the
DGEBA-
hexamethylene against
diamine mixture -measured for a fixed frequency of 1 kHz is plotted (1) 294.3K. (2) 303.8K. (31 3155K, the reaction time at temperatures:
(4) 323.7K, (51 335.9K measured, data.
and (6) 346.OK.
Circles
are the calculated.
and dots the
10 kHz to 20 MHz, Tombari and Johari 1161 have been able to obtain the reaction time dependence of r of a number of poiymerizing systems, a typical example of which is shown in Figure 6, The data obtained from different frequencies of measurement overlap over a narrow region of curing time, but all lie on the same clurve which is now much broader, covering a range of 10 -9 to 10 seconds. The single plot of the relaxation time against
the curing
time
seems to be a satisfactory
correctness of the formalism given here.
demonstration
of the
DGEBA-PDA
-1
-9
3.0
3.25
3.75
3.50
-
log10 (curing timekj Figure 6.
The relaxation time of ‘LheDGEBA-propylene
diamine mixture at 32Q.OK is
plotted against the reaction time. The data correspond to measurements
made at 50Hz
for circles, 1 kHz for triangles and X00 kHz for squares.
TEMPERATURE AND TIME EVOLUTION PROCESSES DURING POLYMERLZATION
OF
The fact that a single frequency measurement
THE
during the isothermal
can provide information on both the gelation time and the relaxation in determining occurs.
how the various relaxation
In virtualIy all physical
processes
measurements
RELAXATION
evolve
at a certain
occurrence the measured values correspond! to the relaxation
when
reaction
time is valuable polymerization
instant
during
characteristics
this
of the
s-ubstance’s structure at that instant for the frequency used in the measurement. any instant,
this (fixed) structure
distribution of relaxation tims,
is expected
known as a- and k-relaxations
liquid is near-or below the reaction temperature, relaxation
to relax with at least
At
a bimodal
1173, when Tg of the
but only with one distribution
times when T, is much greater than this temperature,
of
i.e. the condition
when the a- and B-relaxation processes are merged. Thus as the reaction progresses or
167
& increases, the structural sbte
of the substance changes From that of a liquid, when
the reaction temperature is much greater than T,, to a rigid solid, when it is equal to or lesser than Tg- For a fixed frequency measurement. the reaction progresses, continuously changing
one would therefore observe, as
dielectric properties belonging
to each
of the continuously different structural states as these states traverse with time from the ones with a very short relaxation time of a fluid, say IO-9 s, to the ones with a very long relaxation time, say 104, of its vitrified state. This is illustrated in Figure 7 where the anticipated increase in the relaxation with decreasing
time for the a- and P-relaxation
temperature are illustrated
for measurements
made
processes
at different
+! 3 r
+:
F
Figure 7.
An illustration of the change in relaxation time for the a- and l3-relaxation
processes of the macromolecular
structure at a given instant during
the chemical
reactions for addition polymerization. For simplicity, the plots for the P-process are drawn to have the same slope and to merge with the a-process at a frequency of 107 Hz. The pre-exponent for all plots has been kept the same. instants processes
TV, tl+ t2, e+x_, of the reaction are drawn to have the same
measurements
For simplicity, the plots of p-reIaxation slopes and to merge at a frequency of
of about 107 Hz, and the pre-exponential
factor is necessarily
kept the
same for all plots. In Figure: 7, the measured 6’ and E” at 1 kHz frequency for a given value of y would initially correspond to a structure and relaxation at a point T,-l
on
curve t+ With the passage of time, and in a continuous manner, this would be followed
with those properties which correspond to the structure and relaxation times for points at Tc-’ on curves tl. t:! .._ etc., and ultimately t,it is reached, all contributions
on curve h-it. the vitrification
from the a-relaxation
time. As
process may not yet reach their
minimu.m value if the frequency of measurement corresponds to the P-relaxation rate of the network structure.
In the previous Section we surmised that the deviations of e”
and c’ from a stretched exponential
decay function in the plots of Figures 2 and 3 as
are due to the contribution from &relaxation, and earlier studies (181 have tc-= shown that the strength of p-relaxation initially increases with time during the polymerizing reactions at the expense of the height of another sub-Tg, y-relaxation, whose strength decreases.
Simultaneously,
the a-relaxation
process shifts to lower
frequencies and thus its contribution to e” decreases. In measurements
made at a fixed
instant after tvit, where reactions in a vitrified state cccurred sufficiently
slowly to
allow the measurement
of the spectrum. one finds, as shown in earlier studies [18], a peak in the frequency spectrum of E”, and a shift in the position of the &relaxation peak towards high frequencies with increase in the temperature. .UtemativeIy, if ‘;he reaction temperature
was the same but different
frequencies
were
used for the
measurement of the dielectric properties of the time-variant system, one would deduce from Figure 7 that at &>&it, +&e measured E” is frequency-dependent, showing a peak. The position of this peak would shift to a higher frequency if the temperature of the reaction was increased.
CHEMICAL
KINETICS
AND
We now consider whether
DIELECTRIC a useful
BEHAVIOUR
reiation
between
the chemical
kinetics
measured in terms of the extent of reaction and the dielectric properties can be found. Attempts for seeking such a relation require that each of the seven effects noted in the first Section here be considered because chemical reactions affect both the dynamic and static properties from near zero frequency principle, the magnitude experimentally,
to optica
frequencies.
of thes e effects can be determined
the eflort required for doing so is enormous.
that includes only the most pror;linent effects can provide
2
Although,
both theoretically Therefore,
in and
a procedure
useful relation between the
chemical kinetics and dielectric properties of a time-variant
system, and these effects
are: (1) changes in the conductivity, o. and (2) the average dipolar relaxation time, r, of the liquid. Mangion and Johari f41 have concluded that the dipolar related to the reaction kinetics.
relaxation
time is
Their studies have shown *Aat the dipolar relaxation
time increases with the progress of reaction and ‘hat the logarithmic
plot of I: against &
-, 6
2
.6
3.
4
Icrg(time/s) Figure 8.
The relaxation
time calculated
from the dielectric studies
of DGEBA-
ethyIene diamine mixture reacting at 296.2K is compared against the prediction from Eqn. i22). ? ull line is cakulated from Eqn. (22) but here, S = [r[T, m)- r(T, 011 with t(T,O) = 6ns, r(T,=) = 52s and n = 1.95. Also plotted are the extent of reaction, a, and the heat capacity, Cp. thermodynamics
These are useful for a comparative
analysis
of the chemistry,
and diffusion process [after refs. 16 and 201 as po!ymerization
is sigmcidal in its shape up to ‘I:= 103-
occurs.
lWs, beyond which r could not be determined as
the time required for such measurements
becomes prohibitively
long.
Parthun and
Johari (61 have analytically examined the dependence of K on the curing time and have shown that the shape of the plot of the extent of reaction against
the logarithmic
reaction time is related to that of the corresponding plot of the average dielectric relaxation time determined from a fixed frequency measurement.
This is given by an
empirical equation: r(T.t c) =
where S = ln(rIT,=)KT,O)) parameter.
r(T.0)explS a”tT.tC)?,
(221
is a constant for a given epoxy and n is an empirical
The factor S normalizes the plot of ln(r(T,t)I against &
This normalization
allows, for a direct comparison of the plot of In t
by allowing parameter
the values of IS-1 In (r(T, t+)/t(T,O))l n alters the shape
correspond with a(T,t,&). In(z(T,t& The dielectric
of the sigmoidal
When
diamine
at 296.2K,
measured
plot so that r(T,bl)
t(m) = 52 s and n = 1.95,
The to
n = 1, the extant of reaction is directly proportional
to
in the kHz and MHz frequency that For the reaction
Eqn. (22) provides o satisfactory of reaction
extent
zero and unity.
can be made
measurement
and Johari 1161 have confirmed
to vary between
and
the dipolar
between
range by Tombari
DGEBA
and ethylene
description of the calorimetrically
relaxation
but here, S = Cr(T, =+r;(T,
time
with
O)]. A typical
t(O) = 6.0 ns,
plot showing
this
agreement is given in Figure 8. There is need For Further measurements of dielectric and calorimetric behaviours of time-variant systems For demonstrating the usefulness of this relaticn.
EVOLUTION OF THE a-RELAXATION PROCESS Dielectric properties measured in the frequency range 6.01-26 and Johari il63 have led to observations and structurally particular
arrested states of a time-variant
of the DGEBA-ethylene
They demonstrate obtained
&amine
the fixed frequency
measurements
and Johari [6].
The normalized
process which comes in the strength
and as the viscosity the low-frequency
of relaxation
feature,
nameIy,
as the reaction
time
progresses,
io, shown
of a second
of 1_25h, i.e.,
of the reaction
These relaxations
appears
that
what
splits into two progresses
are referred
to as a- for
at First glance to be the new process in that it
with respect to the p-relaxation
from the P-process as it shifts towards
and a
processes.
and separates
that it is instead the a-process
in
stable liquids
more From each other as the reaction
of the Iiquid increases.
as do
I161 and earlier
the present?
aftsr a reaction
and p for the high-frequency
seems to be the one that emerges indicate
plots generally
with the spectra of structurally
peak at the beginning
progressively
In Figure 9, &relaxation normalized
to the corresponding
in
at 296.2K.
form of one such spectrum
into evidence
appears to be a single relaxation peaks which separate
time periods,
its reaction
with reaction time made by them
and solids, Figure 9 shows a remarkable decrease
at different
during
of chemically
stable liquids and solids and yield the same value oft
Figure 9. In addition to the similarity relaxation
loss spectrum
system
mixture
that these spectra are similar
for structurally
hy Parthun
of the dielectric
MHz by Tombari
From the a-process.
peak,
which
which emerges
is not shown
here,
and progressively
a lower frequency.
these features in Figure 10, where the variation
But a spectrum
For clarity,
of the relaxation
would
separates
we illustrate
rates for the CL-and fi-
Figure 9.
The normalized plots of dielectric loss against frequency for the DGEBAethykme diamine mixture at several fmed instants during its reaction at 296.3.K. The instant of measurement ia given in ks for each plot. J?ull lines are for the values calculated for a stretched exponentialrelaxation fun&on with 6 = 0.4 in Eqn. (9) [after ref. 163. processes is shown against the reaction time at a fixed temperature. The plot illustrates a pattern of the development of relaxation processes and their progremive separation on a time scale. As the reaction proceeds, the relaxation process with a unimodal distribution of times acquires a bimodal distribution of the a- and prelaxation processes, each showing a peak for a fixed frequency of measurement. This feature of splitting of the relaxation process, we note, bears a remarkable resemblance to that observed on supercooling a liquid whose a-relaxation process also progressively separates from the P-relaxation process in a time-temperature plane Cl71 as reversible molecular clustering raises the liquids viscosity. Recent studies by Cassettari, et al. 1191 reported in this volume show an additional remarkable feature of the evolution of a low frequency relaxation during the addition polymerization reactions, namely the development of the slower relaxation yrocess at the expense of a faster process. This new process rapidly shifts towards the lower frequency side of the spectrum,and its dielectric polarization strength decreases This occurs in a manner as does that of the remaining high-frequency process. remarkably similar to that observed on supercooling a liquid through its glasstransition temperature, as noted earlier by Jchari [17]. It seems as if in ita relaxation
172
log(time/s) Figure 10.
An illustration
of the change
in relaxation
time with increase
reaction time as the Q and p-processes evolve at a fixed temperature,
in the
and progressively
separate on a time scale. effects the irreversible increase in the moleculs r weight of a macromolecule tively related to the inverse of temperature
0’ a structuraliy
is qualita-
and chemically
stable
glass-forming liquid.
SUMMARY
AND CONCLUSIONS
The phenomenolbgy formulated
of the dielectric
for a time-variant
system
diffusion
a liquid under virtually
process, the dielectric properties irreversibly similar
to that when
the ac frequency
been
described
reactions
The process of this conversion
as a negative feedback between molecular vitrifies
has
in which irreversible
convert a monomer into a macromolecule. ultimately
processes
and chemical
isothermal
and spontaneously
used for the study
continuously is envisaged
reactions
conditions.
and
which
During
this
change in a manner of a chemically
and
structuraIly stable material is deliberately increased. Dielectric
spectroscopy
shows
conductivity approaches a singularity, useful in determining
the gelation
that
as polymerization
progresses,
the dc
for which equations are provided which can be time
develops_ The relaxation time irreversibly
when a cross-linked
network
structure
increases with the reacticn time according
173
to a sigmoidal-shape
curve. Its value is related to the extent of reaction. relaxation As polymerization progresses, a second, low-frequency,
evolves and progressively
separates from the high-frequency
process
relaxation process on a
time scale.
This is analogous to the evoiution of relaxation processes when a liquid is supercooled through its glass transition temperature. The irreversible increase in the molecular
weight with the reaction time affects
manner quaIitatively
the relaxation
phenomznon
in a
similar to that observed on supercooling a molecular liquid.
References 1.
I. Alig, D. L&inger
and G.P. Johari, J. Polym. Sci. Part B, Polym. Phys. 30,791
(19921. 2.
J.F_ Johnson and R.H. Cole, J. Amer. Chem. Sot. 73,45X
3.
JR. McDonald, ed., “Impedance Spectroscopy” (Wiley, N.Y. 1987).
4.
M.B.M.
(1951).
Mangion atid C.P. Johari, J. Polym. Sci. Part B, Polym. Phys. a
(1990); Macromol. 23,3687 B, Polym. Phys. 29,437
(1990); Polymer 32, 2747 (1991); J. Polym. Sci. Part
(1991), ibid. 29,1117
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