Contributions from interfacial polarization, conductivity and polymer relaxations to the complex permittivity of a film of poly[(5-ethyl-1,3-dioxan-5-yl)methyl acrylate] containing ionic impurities

Article abstract of DOI:10.1039/a701239j

The complex permittivity of a film of the polymer poly[(5-ethyl-1,3-dioxan-5-yl)methyl acrylate] of width 0.4 mm and containing very small amounts of ionic impurities has been studied at a range of frequencies from 0.01 Hz to 100 kHz and at a range of temperatures from -135 to +140°C. Some mechanical determinations of the complex Young modulus have also been performed for the same polymer. To separate the surface polarisation effects and the conductivity effects from the dielectric relaxations of the polymer chains and side-chains we have used the same theoretical methods as earlier described for a film of a copolymer of vinylidene cyanide and vinyl acetate and for a film of poly[4-(acryloxy)phenyl-(4-chlorophenyl)methanone]. The diffusion coefficient of the most rapidly diffusing ion is studied as a function of temperature. The diffusion coefficient follows a non-Arrhenius Vogel relation with the same Vogel temperature as the α-relaxation (glass-rubber). Both phenomena may be interpreted using the Cohen-Turnbull theory of free volume as has previously been done for the diffusion of oxygen through poly(cyclohexyl acrylate). The fractional free volume at the glass transition temperature (ca. 36°C) is found to be 0.031, close to the range normally found (0.025 ± 0.005). A β-relaxation is also found at higher frequencies and lower temperatures. This relaxation shows Arrhenius behaviour with an activation energy E?/R = 5780 K. The α- and β-relaxations seem to merge at ca. 100°C and in addition a relaxation more slow than the α-relaxation is found at even higher temperatures. This relaxation can only be seen after correction of the dielectric loss for conductivity. The mean activation energy of this relaxation in the temperature range 90-140°C is practically identical with the mean activation energy of the α-relaxation in the same range of temperatures (E?/R ≈ 15 000 K). The slow relaxation is probably connected with the motion of the polymer molecule as a whole in the 'virtual tubes' of long-range, topological entanglements, for example by 'reptation'. At very high frequencies (50-100 kHz), isochronous graphs of dielectric loss vs. temperature exhibit a splitting of the α-peak into two peaks.

Full text of DOI:10.1039/a701239j

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Contributions from interfacial polarization, conductivity and polymer
relaxations to the complex permittivity of a Ðlm of
poly[(5-ethyl-1,3-dioxan-5-yl)methyl acrylate] containing ionic
impurities
Torben Smith SÊrensen,a* Ricardo Diaz-Calleja,b Evaristo Riande,c Julio Guzmanc and
Andreu Andriod
a Physical Chemistry, Modelling & T hermodynamics, DT H, NÏrager Plads 3, DK2720 V anlÏse,
Denmark
b Departamento de T ermodinamica Aplicada, Escuela T ecnica Superior de Ingenieros
Industriales, Universidad Politecnica de V alencia, E46020, V alencia, Spain
c Instituto de Pol•meros, Consejo Superior de Investigaciones Cient•Ðcas, E28006, Madrid, Spain
d Departament de Ciencies Experimentals, Universitat Jaume I, E12080 Castello, Spain
The complex permittivity of a Ðlm of the polymer poly[(5-ethyl-1,3-dioxan-5-yl)methyl acrylate] of width 0.4 mm and containing
very small amounts of ionic impurities has been studied at a range of frequencies from 0.01 Hz to 100 kHz and at a range of
temperatures from [135 to ]140 ¡C. Some mechanical determinations of the complex Young modulus have also been performed
for the same polymer. To separate the surface polarisation e†ects and the conductivity e†ects from the dielectric relaxations of the
polymer chains and side-chains we have used the same theoretical methods as earlier described for a Ðlm of a copolymer of
vinylidene cyanide and vinyl acetate and for a Ðlm of poly[4-(acryloxy)phenyl-(4-chlorophenyl)methanone]. The di†usion coeffi-
cient of the most rapidly di†using ion is studied as a function of temperature. The di†usion coefficient follows a non-Arrhenius
Vogel relation with the same Vogel temperature as the a-relaxation (glassÈrubber). Both phenomena may be interpreted using the
CohenÈTurnbull theory of free volume as has previously been done for the di†usion of oxygen through poly(cyclohexyl acrylate).
The fractional free volume at the glass transition temperature (ca. 36 ¡C) is found to be 0.031, close to the range normally found
(0.025 ^ 0.005). A b-relaxation is also found at higher frequencies and lower temperatures. This relaxation shows Arrhenius
behaviour with an activation energy Et/R \ 5780 K. The a- and b-relaxations seem to merge at ca. 100 ¡C and in addition a
relaxation more slow than the a-relaxation is found at even higher temperatures. This relaxation can only be seen after correction
of the dielectric loss for conductivity. The mean activation energy of this relaxation in the temperature range 90È140 ¡C is
practically identical with the mean activation energy of the a-relaxation in the same range of temperatures (Et/R B 15 000 K). The
slow relaxation is probably connected with the motion of the polymer molecule as a whole in the “virtual tubesÏ of long-range,
topological entanglements, for example by “reptationÏ. At very high frequencies (50È100 kHz), isochronous graphs of dielectric
loss vs. temperature exhibit a splitting of the a-peak into two peaks.
From the point of view of polymer science it might be of
value to study the dielectric relaxation of an amorphous
Introduction
In some previous studies, it has been demonstrated that the
complex permittivity (or the impedance) measured over
alveolar membranes1h4 or polymer Ðlms5h8 is, at medium and
low frequencies, dominated by conductance and by interfacial
polarisation phenomena. These e†ects are quite well described
by linearized theories based on the NernstÈPlanck equations
for electrodi†usion.9h11 In optimal cases, these “electro-
chemical e†ectsÏ may be subtracted from the measured dielec-
tric dispersion to reveal the “true dispersionÏ due to the
dielectric relaxations of the polymer backbone or of the side-
chains. Furthermore, the di†usion coefficient of the most
rapidly di†using of the ionic impurities may be estimated as a
function of temperature as well as the concentration of ion
impurities.8 The thermal behaviour of the di†usion process
may thus be studied and compared to that of the relaxations
of the polymer itself.
In the present paper we shall study another example of a
polymer Ðlm with marked “electrochemicalÏ e†ects. The
polymer studied is poly[(5-ethyl-1,3-dioxan-5-yl)methyl
acrylate] which, at higher temperatures and lower frequencies,
exhibits a distinct maximum in the loss tangent at low fre-
quencies which should be ideal for the determination of the
di†usion coefficients and the ion concentrations.7,8
acrylic polymer with a bulky side-chain, since there has
recently been much discussion about the seemingly quite
complex, dynamic glass transition in such polymers, its “Ðne
structureÏ and the precise manner in which the a- and the b-
relaxations merge at high temperatures.12,13 Furthermore, it
has earlier been shown, that the temperature dependence of
the di†usion coefficient of oxygen through a quite similar
polymer, poly(cyclohexyl acrylate), parallels the temperature
dependence of the glassÈrubber relaxation, and that both of
these are well described in the framework of a simple free
volume model.14 Thus, it is of interest to see if the same is true
for the di†usion of charged species (ions).
In more general terms, the study and elucidation of the
mechanisms involved in the relaxation processes of glass-
forming materials are very important from a practical point of
view.15 In general, the relaxation spectra of these systems
present an ostensible absorption associated with the glassÈ
liquid transition followed in decreasing order of temperature
by less pronounced “subglass absorptionsÏ. It is noteworthy
that while the glassÈliquid transition seems to be governed
mainly by the free volume, the subglass relaxations show Arr-
henius behaviour.
Among the most important glass formers, polymers stand
J. Chem. Soc., Faraday T rans., 1997, 93(14), 2399È2411
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out. In spite of the large number of internal degrees of
freedom, both the glassÈliquid (glassÈrubber) and the subglass
relaxations in polymers are almost independent of molecular
weight when this is above a quite small critical value (around
104 u, see for example ref. 16b. This behaviour suggests that
cooperativity plays a pivotal role in the local motions that
produce the subglass relaxations.17 Cooperativity must also
be involved in the “generalized microbrownian motionsÏ of
larger segments of the backbone with their side-groups that
give rise to the glassÈrubber relaxation.18
no requirement from the second law to have a maximum in
the loss functions JA or DA, since the dissipation per cycle may
well tend to inÐnity in a “cycle of inÐnite durationÏ in visco-
elastic polymers, since these do have not an equilibrium
modulus. For example, the low-frequency contribution to
JA(u) with the frequency dependence u~1 is proportional the
reciprocal viscosity at zero frequency (the analogue of conduc-
tivity for electrical phenomena). This di†erence at low fre-
quencies between dielectric and mechanical relaxations should
be borne in mind when these are compared which is often
done in the case of polymers.16
In the present paper we shall attempt to subtract the com-
bined e†ects of conductivity and surface polarisation from
dielectric measurements at low frequencies in order to reveal
the detailed nature of the glassÈrubber relaxation and perhaps
even a “normal-mode processÏ from the masking of the com-
bined e†ects of conductivity and surface polarisation.
For uncrosslinked, amorphous polymers of high molecular
weight, there is much evidence that topological
“entanglementsÏ, where the polymer chains loop around each
other in their long-range contour, restrict the cooperative
microbrownian motions of the glassÈrubber relaxation.19a
This explains the M-independence of the glassÈrubber tran-
sition in such polymers, since the microbrownian motions
only involve chain segments conÐned between two entangle-
ments. The entanglement concept also explains that at still
lower frequencies than the glassÈrubber relaxation (a-relax-
ation), the mechanical relaxation spectra of polymers, complex
Young modulus E*(u) or shear modulus G*(u), exhibit
another relaxation which is strongly dependent on molecular
weight (M). The relaxation time of this absorption approx-
imately scales as M3.4 for chains in which M is larger than a
Experimental
Materials
2-Ethyl-2-hydroxymethylpropane-1,3-diol (Fluka), toluene,
chloroform, para-formaldehyde, toluene-p-sulfonic acid, tri-
ethylamine and acryloyl chloride (Fluka) were used as
received.
critical value M .19a This relaxation involves motions of the
cr
chain as a whole and various theories have been proposed to
Synthesis of 5-hydroxymethyl-5-ethyl-1,3-dioxane (HED)
explain this relaxation. For example the entanglement theory
of Bueche20,21 predicting a power 3.5 and an M ca. twice the
This intermediary compound was prepared by equimolar
reaction of 2-ethyl-2-hydroxymethylpropane-1,3-diol with
para-formaldehyde, reÑuxing toluene and using toluene-p-sul-
fonic acid as catalyst. The reaction was carried out under a
nitrogen atmosphere for 4 h with azeotropic removal of water.
The product was distilled and collected under vacuum [bp
75 ¡C (0.11 mbar, yield 95%)].
cr
mean molecular weight between two entanglements, or the
“tube modelÏ of de Gennes,22,23 Doi and Edwards24 and
Graessley25 where the entanglements are described as a
“virtual tubeÏ, from which the easiest escape of the polymer
molecule in focus is by snakelike “reptationÏ along its own
contour. Clearly, this reptation gives rise to very long relax-
ation times.
In the dielectric spectra, the slow relaxation connected with
the motion of the whole chain relative to the conÐnements of
the entanglements should also be visible in some cases. The
theory for polymer melts (or rubbers) has been reviewed in ref.
26 and 27. However, the peak in the dielectric loss eA (negative
imaginary part of the complex permittivity) corresponding to
this “normal-mode processÏ is quite often completely masked
by conductivity and surface polarisation phenomena. These
phenomena are introduced by minute traces of ions left over
from the preparation of the polymer. MaxwellÈWagnerÈSillars
(MWS) calculations28h31 predict that the dielectric loss is pro-
portional to (frequency)~1, so that the dissipation per cycle
tends to inÐnity when the frequency tends to zero.
This situation is unrealistic, however, since the sample
polymer Ðlm has a Ðnite thickness (L ). Because of the polarisa-
tion at the two surfaces of the Ðlm, the system has to tend
towards electrochemical quasi-equilibrium (determined by the
instantaneous electric Ðeld) at very low frequencies. Therefore,
the dissipation per cycle and the dielectric loss also tend to
zero at very low frequencies as it should according to the irre-
versible thermodynamics version of the second law.8,32 The
position of the maximum in eA (or in tan d) is determined by
the di†usion coefficients of the ions and by iL , where i is the
inverse Debye length.5h9 Usually, it is very difficult to
measure down to frequencies where the maximum in eA can be
seen, but the maximum in the loss tangent tan d ( \ eA/e@) is
positioned at somewhat higher frequencies and can often be
observed at high temperatures.6,8
Synthesis of (5-ethyl-1,3-dioxan-5-yl)methyl acrylate
(monomer, EDMA)
The monomer was prepared by equimolar reaction between
acryloyl chloride and HED in chloroform solution under a
nitrogen atmosphere. Triethylamine was used to neutralize
hydrogen chloride evolved during the reaction. Acryloyl chlo-
ride was added dropwise into the solution of alcohol during a
period of 30 min, and the reaction medium was kept cold in
an iceÈwater bath. The triethylammonium chloride was
removed by Ðltration and the reaction mixture was washed
several times with distilled water and extracted with benzene.
Finally, the crude product was distilled under vacuum to give
a liquid [bp 78 ¡C (0.35 mbar)].
Polymerisation
Poly[(5-ethyl-1,3-dioxan-5-yl)methyl acrylate] was made by
radical polymerisation of EDMA in benzene solution at 50 ¡C
using AIBN (2,2@-azo-bisisobutyronitrile) as initiator. The
polymer was isolated from the reaction medium by several
precipitations in n-hexane and was Ðnally puriÐed by freeze
drying from benzene solutions. Impurities in the solvents and
precipitants are usually metallic salts of Al, Pb, Sn, Zn, etc.
The proportion of each of these impurities is in most cases
\0.000 01 wt.%. The polymerisation and the above men-
tioned preparations were all performed in the Instituto de
Ciencia y Tecnologia de Polimeros (Consejo Superior de
Investigaciones CientiÐcas), Madrid. The repetition unit of the
polymer is shown in Fig. 1.
It should be noted that the negative imaginary parts (JA or
DA) of the complex compliance functions J*(u) \ 1/G* or
D*(u) \ 1/E*, which are mechanical analogues to e*, are simi-
larly found to scale as (frequency)~1 at low frequencies in
uncrosslinked, amorphous polymers above the glass transition
temperature.19b In the case of torsion or elongation, we have
Dielectric, mechanical and DSC measurements
The dielectric measurements were performed at the Poly-
technic University of Valencia using a ceramic parallel-plate
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Fig. 1 The chemical structure of the repetition unit
condenser with a gap width L \ 0.4 mm and a surface area
A \ 380 mm2. The polymer was moulded as a circular disc
and placed between the electrodes. After this, the assembly
was heated to a temperature 20 ¡C above the glass transition
temperature (T B 36 ¡C) and was pressed using a force equal
g
to 250 N in order to avoid irregular air gaps and voids of air
between the electrodes and in order to maintain a uniform
thickness of the sample. Experiments were carried out for each
frequency (isochronous measurements) from low ([135 ¡C) to
high temperature at 1 ¡C min~1 until room temperature.
From room temperature to temperatures well above the glass
transition temperature, measurements were carried out in the
isothermal mode at 5 ¡C steps equilibrating the sample about
20 min at each temperature. Successive runs of isothermal
measurements at the same frequencies were reproducible
(within ca. 4% relative uncertainty) which reveals the stability
of the properties of the sample. The measurements were per-
formed using a capacitance measuring apparatus TA DEA
2970. The frequencies used ranged from 0.01 Hz to 100 kHz.
(In the apparatus there is a built-in automatic device that dis-
charges the condenser for protection when the condenser
charge is extremely high. This occasionally a†ects the mea-
sured values of the real part of the permittivity at very low
frequencies and high temperatures. However, these discharges
are easily discernible as jumps in the measurements, and when
these were seen, the measurements were discarded.)
Mechanical measurements of the complex Young modulus
(for simple extension with lateral contraction, E*) were carried
out at the Polytechnic University of Valencia by means of a
DMTA-Mark II apparatus in the same range of temperatures
as the dielectric experiments. Measurements were made from
low- to room-temperature in the multiplexing mode using 0.3,
1, 3, 10 and 30 Hz at 1 ¡C min~1 and from room- to higher
temperatures in steps of 5 ¡C with ca. 15 min between two
successive measurements.
Fig. 2 Isochronous maps of the components of the complex YoungÏs
modulus (E) as a function of temperature at 0.3 Hz (rectangles) and 30
Hz (diamonds). The a- and b-relaxations are indicated.
ments at 5, 20 and 100 Hz, respectively. The maxima of the
dielectric losses are positioned approximately at the tem-
peratures where the given frequency is the frequency of
maximum loss (in an isothermal lossÈfrequency diagram) for
the relaxation in question. Thus, the frequency of maximum
loss ( f ) at a temperature corresponding to the maximum in
max
the lossÈtemperature diagram may be taken as the frequency
corresponding to the isochronous curve in question.
Following tradition, we label the relaxation peaks as a, b
and so on, starting from the high temperatures in the mecha-
nical or dielectric lossÈtemperature diagrams or from the low
frequencies in the isothermal lossÈfrequency diagram.16 For
amorphous polymers, as here, a-relaxation is invariably con-
nected with the glassÈrubber transition (motion of entire seg-
ments of the polymer backbone) whereas the subglass
transitions are connected with more rapid motions of limited
A glassÈrubber “transition temperatureÏ of the amorphous
polymer was determined as 36 ¡C by di†erential scanning
calorimetry (DSC) at a heating rate of 20 ¡C min~1. A peak in
the mechanical loss modulus of the polymer (EA) was also seen
at 35 ¡C at 3 Hz using the above apparatus. A subglass peak
in EA (b-relaxation) was observed at [70 ¡C, similarly at 3 Hz
mechanical
vibration
frequency.
Such
“transition
temperaturesÏ are kinetic properties (and not thermodynamic)
and depend on the frequency or heating rate. This is clearly
seen in Fig. 2(b) where the subglass peak in EA moves from
[80 to [50 ¡C when the frequency is altered from 0.3 to 30
Hz, and the glassÈrubber peak similarly moves from 30 to
40 ¡C. The downward steps in the real part of YoungÏs
modulus (E@) move correspondingly, see Fig. 2(a). Also the
glass transition temperature found by DSC depends on the
heating rate. For example for poly(n-butyl methacrylate), T
was found to vary from 16.6 to 22.3 ¡C when the heating rate
g
was varied from 0.5 to 20 ¡C min~1, ref. 33, Table 1.
Dielectric a- and b-relaxations of the polymer
Fig. 3 Isochronous maps of the dielectric loss (negative imaginary
part of the complex relative permittivity) as a function of temperature
at 100 Hz (rectangles), 20 Hz (diamonds) and 5 Hz (crosses). The a-
and b-relaxations are indicated together with the contribution from
conductivity at high temperature.
Fig. 3 shows the dielectric loss, the negative imaginary part of
the relative complex permittivity eA, as a function of tem-
r
perature (from [135 to ]80 ¡C) for isochronous measure-
J. Chem. Soc., Faraday T rans., 1997, V ol. 93
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spatial extension (rotations and librations) in the side-chains
and the local part of the backbone.16
Determination of the conductivity from medium- and
low-frequency data
In Fig. 3 we observe a glassÈrubber relaxation and a sub-
glass b-relaxation. When the frequency is lowered, the tail of a
marked rise due to conductivity (and to interfacial
polarisation) appears in the high-temperature end of the
diagram. Apart from this, the a-relaxation peak is clearly
dominant over the b-relaxation peak. All peaks move towards
lower temperatures when the frequencies corresponding to the
isochronous curves are lowered.
As in previous publications6h8 it is possible to obtain the spe-
ciÐc conductivity (p) of the polymer Ðlm at a given tem-
perature from a plot of the logarithm of the dielectric loss (eA)
r
vs. the logarithm of the frequency ( f ) at the temperature in
question. Thus, the data have to be rearranged from isochro-
nous data to isothermal data. In a certain range of low and
medium frequencies this plot is linear with a slope of [1. This
means
that
the
MaxwellÈWagnerÈSillars
(MWS)
Fig. 4 shows an Arrhenius plot of ln f
dielectric b-relaxation (data from run no. 1). With all the 20
measuring points included we have the regression line
vs. 1/T for the
max
approximation28h31
eA \ p/(e 2nf )
(4)
r
0
ln f \ 27.28 [ 5155/T
with
correlation
coefficient
is approximately valid at frequencies below the isothermal loss
peaks of the internal relaxation of the polymer. This a good
approximation even at quite low frequencies where the real
part of the relative complex permittivity rises to high values
because of the surface polarisation in the Ðlm in contrast to
max
r \ [0.9918. The regression in Fig. 4, however, is based on
the 16 middle points (since there are deviations in the two
ends) and is given by
ln( f
/Hz) \ 29.93 [ 5778/T ,
max, b
r
the MWS behaviour (where e@ should remain constant). This
r
is so because the peak maximum in eA due to
\ [0.9986 ([90 to 20 ¡C)
(1)
r
conductivity ] surface polarisation is situated at frequencies
far below the ones measured (in contrast to the case with the
The value of the activation energy (Et/R B 5800 K) is typical
for hindered rotations.
The a-relaxations in amorphous polymers are usually found
to have a non-Arrhenius dependence on temperature (in con-
trast to the subglass relaxations).34 Usually the so-called
Vogel equations apply, see ref. 34, eqn. (11)È(13) where A, B
loss tangent tan d \ eA/e@).6,8
r
r
In Fig. 5 we show typical examples of such doubly logarith-
mic plots of slope [ 1. The speciÐc conductivities are found
from the intercept. We measured speciÐc conductivities
between 50 and 140 ¡C. (At 40 and 45 ¡C the frequency inter-
val of linearity is too narrow.) The values of p and the fre-
quency ranges are listed in Table 1 together with information
about the maximum in the loss tangent.
and T are material constants.
0
ln( f
/Hz) \ ln A [ [B/(T [ T )]
T [ T
(2)
max, a
0
0
Although there is a slight indication of a break point at ca.
100 ¡C in a plot of ln p vs. 1/T , a straight line is a quite good
approximation in the given range of temperatures:
a-Relaxation is not activated below the Vogel temperature
T
. Plotting ln f
vs. 1/(T [ T ) for estimated values of T ,
0
max, a
0
0
we obtain a maximum in the absolute value of the correlation
coefficient o r o for T ca. 228 K ([45 ¡C):
ln(p/S m~1) \ 17.93 [ 13650/T ,
r \ [0.9979 (50 to 140 ¡C) (5)
0
ln( f
/Hz) \ 31.55 [ 2608/(T [ 228), r \ [0.990 03 (3)
max, a
In this regression 39 data points have been included from two
independent experimental runs.
Data treatment for nine temperatures showing a low-frequency
maximum in tan d
At 3 Hz we obtain from eqn. (3) a temperature of 40.5 ¡C.
This value is not far from the glassÈrubber peak seen at ca.
35 ¡C in isochronous measurements of the mechanical loss
modulus (EA) at 3 Hz, see Experimental section. On the other
hand from eqn. (1) we obtain at 3 Hz a temperature of
[72.7 ¡C close to the peak at ca. [70 ¡C found for EA at 3
Hz. Thus, the same two relaxations are seen in the mechanical
as well as in the dielectric relaxations for the polymer.
At the nine temperatures 75, 80, 85, 90, 95, 100, 105, 110 and
115 ¡C a maximum in the loss tangent tan d \ eA/e@ is clearly
r
r
Fig. 5 Examples of the Ðtting of conductivity lines to the low-
frequency end of the dielectric loss spectra at 100 ¡C (rectangles),
105 ¡C (diamonds) and 110 ¡C (]]]). The double logarithmic plots
have slopes of [1 and the conductivities are found from the intercept
with the vertical corresponding to 1 Hz.
Fig. 4 Arrhenius plot for the frequencies of the peak maxima corre-
sponding to the b-relaxation. The middle 16 points are used for the
regression.
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J. Chem. Soc., Faraday T rans., 1997, V ol. 93

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Table 1 Values of the speciÐc conductivity (p) and range of fre-
quencies for the validity of eqn. (4) and values of tan d
and f
in
max
max
the low-frequency region
frequency range
(Hz) of linear
ln p vs. ln f plot
T /¡C
p/10~9 S m~1
tan d
f
/Hz
max
max
50
55
60
65
70
75
80
85
90
0.025
0.058
0.125
0.21
0.36
0.61
1.00
1.60
2.6
3.8
6.1
9.2
16
30
50
80
150
240
350
0.01È0.13
0.02È0.2
0.01È0.2
0.01È0.2
0.01È0.5
0.01È3
È
È
È
È
È
26
26
24
23
23
25.5
26.5
26
29
È
È
È
È
È
È
0.0178
0.0224
0.0501
0.0631
0.1584
0.224
0.282
0.316
0.631
È
0.01È3
0.01È3
0.01È10
0.01È10
0.02È10
0.02È1
95
100
105
110
115
120
125
130
135
140
0.02È20
0.1È80
Fig. 6 Example of the Ðtting of a least-squares third-degree poly-
nomial to seven points in order to locate the maximum of the loss
tangent at low frequencies. Experimental points at 95 ¡C (rectangles).
0.01È100
0.01È80
0.02È200
0.02È2
È
È
È
È
È
È
È
È
0.02È1000
the two cases and when iL A 1, see Fig. 6È8 in ref. 8. The
dimensionless angular frequency is (in both cases) deÐned as
X 4 i~2(D D )~1@2u
(6)
seen at very low frequencies in isothermal plots of the data.
Below 75 ¡C, the maximum occurs at lower frequencies than
measured. At temperatures above 115 ¡C irregularities are
1 2
where i is the inverse Debye length, D and D are the two
1
2
di†usion coefficients and u is the angular frequency. In case
(a) D \ D \ D and in case (b) D [ 10D .¤
often seen in e@ at very low frequencies (especially due to dis-
r
1
2
1
2
charge of the “interfacial condenserÏ, see Experimental
In case (a), for iL A 1 we have:
section). Therefore, the values of tan d are unreliable here even
tan d B (iL /8)1@2
(7)
(8)
if the values of eA seem correct.
max
r
It should be realized that the presence of a maximum in tan
X
\ i~2D~1u B (2/iL )1@2
max
max
d at low frequencies is a speciÐc e†ect of the interfacial polari-
sation. In a usual MaxwellÈWagnerÈSillars treatment of com-
In case (b) we just have to multiply by the appropriate factors:
bined permittivity and speciÐc conductivity, the value of eA
1.13 tan d B (iL /8)1@2
(9)
r
varies as f ~1 and the value of e@ does not vary with frequency.
max
r
Such a MWS model cannot have a maximum in either eA or in
r
1.75X (D /D )1@2 \ 1.75i~2D~1 u B (2/iL )1@2
max fast slow
fast max
tan d. However, in previous papers6,8 we have shown that the
(10)
“dynamical electric double layersÏ, described by means of
NernstÈPlanckÈMaxwell electrodynamics, create an excess
impedance in addition to the MWS impedance, and the com-
bined e†ect of these impedances creates the maximum in tan d
Thus, from the low-frequency maximum of the loss tangent we
may estimate iL from eqn. (7) or (9). Since L is known
(L \ 4.0 ] 10~4 m) we can calculate the value of the inverse
Debye length i. Then, from the experimental value of the
angular frequency of maximum loss tangent we may calculate
the di†usion coefficient from eqn. (8) or (10).
since e@ rises with increasing frequency as a reÑection of the
r
capacitance of the macropolarisation. The maximum in eA
r
occurs at much lower frequencies that the maximum in tan d,
and in the cases we have analysed up to now, eA hardly devi-
In Tables 2 and 3, the calculated values of iL , the di†usion
r
ates from the MWS behaviour except perhaps for a very small
coefficient and the relative static permittivity (e /e ) are listed
s 0
downward deÑection at the one or two lowest values of the
frequency.
In order to obtain a better determination of the maxima,
tan d vs. log f for seven experimental points around the
maximum were Ðtted by a least-squares third-degree poly-
at temperatures from 75 to 115 ¡C for the two cases (a) and (b),
respectively. The latter quantity is calculated by isolating e in
s
Table 2 Values of iL , D and e /e in the case D \ D \ D
s
0
1
2
nomial. The values of tan d
and f
obtained from these
max
max
T /¡C
iL
D/10~13 m2 s~1
e /e
polynomials are listed in Table 1. A typical example of a
s
0
maximum in tan d is shown in Fig. 6.
75
80
85
90
95
100
105
110
115
5408
5408
4608
4232
4232
5202
5618
5408
6728
0.318
0.400
1.14
1.63
4.09
4.24
4.76
5.65
8.13
11.9
15.4
12.0
16.1
9.4
From the values of tan d
and f
we may estimate the
max
max
di†usion coefficient of the impurity ions in the following two
cases: (a) where the di†usion coefficients of the two dominat-
ing impurity ions are equal and (b) where one di†usion coeffi-
cient is much greater than the other. In case (b), the di†usion
coefficient of the most rapidly di†using ion is found, see ref. 8.
The basis is a theory for the dielectric dispersion due to a
combination of MWS e†ects and NernstÈPlanckian electro-
dynamics for the interfacial polarisation in the Ðlm.5h11 A plot
of tan d vs. the dimensionless angular frequency X in case (a)
9.6
11.1
17.5
14.7
coincides with a plot of 1.13 tan d vs. 1.75 X(D /D )1@2 \
¤ Unfortunately, there was a misprint in eqn. (6) of ref. 8, where
fast slow
1.75i~2D~1u in case (b). This is correct when the Ðlm thick-
fast
D /D was written instead of D D in the deÐnition of X. This mis-
1
2
1 2
ness measured in terms of the Debye length (iL ) is the same in
print has no consequences in ref. 8.
J. Chem. Soc., Faraday T rans., 1997, V ol. 93
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Table 3 Values of iL , D and e /e in the case D [ 10D
110 and 115 ¡C, respectively, we have:
fast
S
0
fast
slow
T /¡C
iL
D
/10~13 m2 s~1
e /e
ln(D /m2 s~1) \ ]16.78 [ 16 660/T ,
fast
S
0
fast
r \ [0.981 (75È95 ¡C) (13a)
75
80
85
90
95
100
105
110
115
6905
6905
5884
5404
5404
6642
7174
6905
8591
0.386
0.486
1.38
1.98
4.96
5.15
5.77
6.85
9.86
12.0
15.6
12.1
16.3
9.5
ln(D /m2 s~1) \ [15.62 [ 4715/T ,
fast
r \ [0.932 (95È115 ¡C) (13b)
9.7
From the intersection between the two regression lines we
calculate a break-point temperature of 95.4 ¡C. The high acti-
vation energy at low temperatures (ca. 16 600 K) probably
arises from motions of whole segments of the backbone chain
with their side-chains in order to create new holes and chan-
nels for the di†usion of trapped ions. The low activation
energy at high temperatures (4715 K) is 82% of the activation
energy found for the b-relaxation in the temperature range
[90 to ]20 ¡C.
11.2
17.7
14.9
the formulae (valid for ideal 1 : 1 electrolytes):
p \ (e /L2)(iL )2D
(D \ D \ D)
(11a)
(11b)
s
1
2
p B (e /L2)(iL )2D /2
(D A D
)
s
fast
fast
slow
Temperature dependence of the a- and b-relaxations
In Table 4, the calculated values of the electrolyte concen-
trations at the di†erent temperatures are shown for the two
cases (a) and (b). Assuming the electrolyte to be uniÈunivalent,
we calculate the electrolyte concentration (c ) from the
inverse Debye length i and e by isolating c in the formula
As mentioned previously, a-relaxation is usually of a non-
Arrhenius type. Fig. 8 shows ln f
calculated from the
max, a
Vogel plot (3) and ln f
calculated from the Arrhenius plot
salt
max, b
(1). The abscissa is 1/T in both cases and experimental points
are also shown. It is seen that the “localÏ activation energy for
the a-relaxation is high at low temperatures but diminishes at
higher temperatures. It is also seen that the a- and the b-
relaxation approach each other at higher temperatures. The
two relaxations practically merge at ca. 100 ¡C corresponding
to 1/T ca. 0.0027 K~1.
s
salt
i2 \ (2F2/RT e )c
(12)
s salt
Discussion
Broken Arrhenius temperature dependence of the di†usion
coefficient
The extrapolation shown in Fig. 8 is somewhat uncertain,
however, because of lack of data in the high temperature
region. The points of highest temperature shown for the b-
relaxation probably deviate from the Arrhenius line because
the maxima are distorted before their complete absorption in
the ostensible a-peak, and with a-relaxation there is a con-
siderable covariance between the choice of the B and T
parameters in the Vogel expression, and the speciÐc choice
a†ects the extrapolation.
In Fig. 7, the logarithm of the di†usion coefficient of the faster
ion (case b) is plotted vs. 1/T and there seems to be a break
point in the Ðgure. Indeed, Ðtting regression lines to the points
at 75, 80, 85, 90 and 95 ¡C and to the points at 95, 100, 105,
0
Table 4 Values of c
in the case (a) D \ D and in the case (b)
salt
1
2
D
[ 10D
fast
slow
It has been observed in many other cases and stressed espe-
cially by Williams34h39 that the dielectric a-relaxation and b-
relaxation seem to merge to an ab-relaxation above a certain
T /¡C
c
(a)/10~6 mol l~1
c
(b)/10~6 mol l~1
salt
salt
75
80
85
90
95
100
105
110
115
2.98
3.94
2.25
2.59
1.53
2.39
3.26
4.85
6.40
4.91
6.49
3.70
4.27
2.52
3.95
5.38
7.99
10.55
Fig. 8 Common plot for the a- and b-relaxations of the logarithm of
the maximum frequency vs. 1/T . The former has a non-Arrhenius tem-
perature dependence. Experimental points are shown. The line for the
b-relaxation is the regression line eqn. (1). The curve for the a-
relaxation is the Vogel curve given by eqn. (3). The extrapolations of
the two curves seem to merge at ca. 100 ¡C.
Fig. 7 Broken Arrhenius plot for the temperature dependence of the
di†usion coefficient of the faster ion. The break point is at ca. 95 ¡C.
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temperature and that the a- and b-processes cannot cross in
the frequency domain. According to Williams these processes
are coupled at high temperatures since they both involve
motions of the same dipolar group.35,37 However, mechanical
measurements of, for example, the shear loss module GA(u)
exhibit a similar merging of the a- and b-relaxation, see Fig. 2
in ref. 33. One possible explanation of this phenomenon might
be that the motions involved in the b-relaxation are the
“minimum motionsÏ involved also in long-range chain confor-
mational changes corresponding to the a-relaxation. If a-
relaxation corresponds a kind of “broken zip fastener e†ectÏ
involving shorter or longer sequences of the chains interacting
cooperatively with neighbouring chains, then at high tem-
peratures, when there are many unlocked loops in the “broken
zipperÏ the restrictions of the motions of the b-relaxation are
the only remaining “bottlenecksÏ.
unclear state of the art. Bartenev et al. seem to be at variance
with previous authors in their observation of two b-
relaxations for PMMA. In addition, in Table 8.2 of ref. 16, the
activation energy of the (unique) b-relaxation is listed from
various sources. In dielectric studies it varies from 79 to 96 kJ
mol~1, whereas mechanical studies values are grouped
between 71È75 and 121È125 kJ mol~1. The latter two groups
of values might be an indication of the existence of two di†er-
ent relaxations, but all the activation energies listed are far
higher than the 30È32 kJ mol~1 which Bartenev et al. state as
the “universalÏ activation energy for the b(CH )-relaxation
2
“observed in all linear polymers containing CH groups in the
2
chainÏ.40b In ref. 16 for other polymers containing CH groups
2
in the chain, no indication of any “universalÏ activation energy
is found.
The activation energy of 48 kJ mol~1 found for the b-
relaxation of PEDMA (an alkyl acrylate polymer) is lower
than the values found for PMMA and various alkyl meth-
acrylate polymers (Table 8.2 of ref. 16). For poly(methyl
acrylate) PMA, however, some authors have found a dielectric
b-relaxation with activation energy 31È40 kJ mol~1, and
others a dielectric b-relaxation with activation energy of
58È63 kJ mol.16c The activation energy of 48 kJ mol~1 found
here lies well inside this range.
It should be mentioned that the phenomenological situation
is not yet completely clariÐed. The group of Donth, for
example, maintain that there must be
a
“minimal
cooperativityÏ in such a way that there cannot be a smooth
transition from the a-relaxation to the ab-relaxation.12,13,33
At least a small number of monomer units must be involved
to have an a-relaxation. Therefore, a-relaxation should vanish
shortly before it would have merged with the b-relaxation at
high temperatures. Bartenev et al.40 have very recently
stressed that there are two mechanical b-relaxations in
poly(methyl methacrylate), PMMA, corresponding to the
Parallelism between the temperature dependence of the
di†usion coefficient and the a-relaxation
rotation of the CH group and the rotation of the more bulky
2
C(CH )CO CH group in the polymer backbone. The former
Whatever the true nature and coupling between the a- and the
b-relaxation for the present acrylic polymer, it seems quite
natural that the process of di†usion of ions through the
polymer network is related to the glassÈrubber relaxation
responsible for the creation of temporary holes and channels
for the di†usion path.
3
2
3
seems to approach the non-Arrhenius a-relaxation at high
temperatures whereas plots of ln f for the latter crosses the
a-relaxation in a plot vs. 1/T (at much lower frequencies), see
Fig. 1 in ref. 40. This seems to be in favour of the “broken
max
zipper modelÏ since the b(CH ) relaxation is likely to be the
2
high temperature “bottleneckÏ mode for the zipper.
Fig. 9 shows that the same “Vogel temperatureÏ T \ 228 K
0
From the data and extrapolations made in Fig. 8, it is
unfortunately not possible to clarify if (a) the a- and the b-
relaxations merge to a single ab-relaxation (as described by
Williams), if (b) the a-relaxation vanishes shortly before
merging with the b-relaxation because of the concept of
“minimal cooperativityÏ advocated by the group of Donth, or
(c) if the a-relaxation and the b-relaxation crosses each other
found for the a-relaxation serves quite well to straighten out a
ln D vs. 1/(T [ T ) plot. We obtain:
fast
0
ln(D /m2 s~1) \ [17.36 [ 1616[1/(T [ 228)],
fast
r \ [0.971 (75 to 115 ¡C) (13c)
Even if the correlation coefficient is quite good ([0.971) it
should be noticed that the data above 95 ¡C exhibits a system-
atic curvature. This curvature is also visible in the Arrhenius
plot of Fig. 8 above the “break pointÏ. The data here are too
scarce to determine whether this curvature is fortuituous or
covers a more complicated “singularityÏ. [For the sake of com-
pleteness it may be mentioned that instead of the Vogel repre-
sentation of the a-relaxation as in eqn. (3) it is also possible to
in the ln
f vs. 1/T plot as the case is with the
b[C(CH )CO CH ]-relaxation in the mechanical study of
3
2
3
PMMA of Bartenev et al.40
The activation energy found for the b-relaxation in the
present study is Et/R \ 5780 K (48.0 kJ mol~1). For PMMA,
Bartenev et al. found values of 30 kJ mol~1 for the b(CH )-
2
relaxation and 69 kJ mol~1 for the b[C(CH )CO CH ]-
3
2
3
relaxation while the value found for PEDMA is midway
between these values. Another clue is the limiting frequency at
high temperatures. For the b-relaxation of PEDMA this is
exp(29.93) \ 9.96 ] 1012 Hz. This value is about 10 times
higher than the limiting value of 1 ] 1012 Hz stated by Barte-
nev et al. for the b(CH )-relaxation in PMMA. It is usual to
have considerable shifts in frequencies between dielectric and
2
mechanical
measurements,
however.
For
the
b[C(CH )CO CH ]-relaxation, Bartenev et al.40 found a
3
2
3
much lower limiting frequency of 2 ] 109 Hz. The large mag-
nitude of the limiting frequency for the b-relaxation found for
PEDMA makes it likely that the relaxation observed is indeed
a b(CH )-relaxation and not, for example, a b(CHCO CH ,
2
2
2
C H , dioxane)-relaxation. If this is true, it is more likely that
2
5
the a- and b-relaxation for PEDMA merges in Fig. 8 without
crossing. However, if the b-relaxation found here corresponds
to hindered rotations in the side-chain (for example around
the OwCH bond) without any relation to the chain back-
2
bone, the b-relaxation might be crossing rather than merging
Fig. 9 Vogel plot for the logarithm of the faster di†usion coefficient.
with the a-relaxation.
Final conclusions are also made difficult by the present
The same Vogel temperature (T \ 228 K) is used as for the a-
0
relaxation, see eqn. (3) and Fig. 8.
J. Chem. Soc., Faraday T rans., 1997, V ol. 93
2405

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represent it as a broken Arrhenius dependence with a break-
point in the neighbourhood of 95 ¡C.]
wherein the directly calculated (and very scattered) values of
e /e were used instead of the mean value of 9.8.
s 0
Static relative permittivity, impurity electrolyte concentrations
and conductivity
Interpretation of Vogel relations in terms of free volume
theories
Fig. 10 shows the best values estimated of the static relative
permittivity of the polymer between 45 and 140 ¡C (rectangles,
see later for a discussion of the method of estimation). Up to
130 ¡C the value is 9.8 ^ 0.5 and is independent of tem-
perature. The crosses are values calculated directly from eqn.
(11a, b) for the two cases considered in Tables 2 and 3.
Assumption (a) D \ D and assumption (b) D [ 10D
In ref. 14, the permeability of oxygen through poly(cyclohexyl
acrylate), PCHA, was calculated as a function of temperature
from measurements of the stationary reduction current in an
electrochemical permeometer with an oxygen electrode. The
di†usion coefficient was also calculated at each temperature
using the transients and the time-lag method. Fig. 4 in ref. 14
shows that [ln(DO2) and the logarithms of the relaxation
times associated with the mechanical as well as the dielectric
relaxation times for the glassÈrubber relaxation are all straight
lines vs. 1(T [ T ) with a “Vogel temperatureÏ T \ 248 K
1
2
fast
slow
lead to essentially the same values of e /e . In these latter
s 0
values there is a considerable scatter due to the difficulties of
measuring at low frequencies and high temperatures.
However, six values are within 9.8 ^ 1.5 and only three values
seem too high.
0
0
([25 ¡C). The slopes found for the linear plots were B ca.
a
1300 K for the mechanical and dielectric a-relaxations com-
pared to a slope of B \ 392 K for the di†usion coefficient of
diff
oxygen. For the polymer here we have found a slope B \
a
2608 K for the dielectric a-relaxation and B \ 1616 K for
diff
the di†usion coefficient of the faster ion, at
a “Vogel
temperatureÏ T \ 228 K ([45 ¡C).
0
The observed parallelism between the temperature response
of the glassÈrubber relaxation and that of di†usion in amor-
phous polymers may be interpreted in terms of simple “free
volumeÏ theories. Free volume is a useful semiquantitative
concept which has been much employed in statistical thermo-
dynamical theories of the liquid state. According to Glasstone,
Laidler and Eyring,41 the free volume may be regarded as the
volume in which each molecule of a liquid moves in an
average potential Ðeld due to its neighbours. This volume is
more or less easily redistributed during time and is determin-
ing for transport processes like viscosity, self-di†usion and dif-
fusion in liquids (or rubbery polymers).
Fig. 10 Estimated, best values of the static relative permittivity for
various temperatures (broken curve with rectangles). Estimated values
are given in the Discussion. The crosses are values calculated from
eqn. (11a, b). (Practically no di†erence can be seen between the values
calculated by either of these formulae.)
A simpliÐed, but useful, deÐnition was given in the theory of
Cohen and Turnbull.42,43 They stated that the free volume (v )
f
is that part of the excess volume (total volume minus volume
occupied by molecules) which can be redistributed without
change in energy. The main points of their theory may be for-
mulated as follows: the translational friction coefficient of the
liquid is to some approximation only dependent on the tem-
perature through the so-called fractional free volume, and
since the relaxation times for viscosity, for self-di†usion and
for di†usion of foreign molecules in the liquid are all pro-
portional to the translational friction coefficient, the same is
true for these coefficients. More speciÐcally, we write for the
logarithm of the relaxation time relative to the relaxation time
of a reference state:
Fig. 11 shows the electrolyte concentration as a function of
the temperature assuming (a) or (b) but Ðxing the value e /e to
s 0
9.8 in eqn. (12). The impurity salt concentration is quite con-
stant (ca. 2È4 lmol l~1) perhaps with a slight increase at the
high temperatures (liberation of more free charge carriers?).
The concentrations shown in Fig. 11 are inside a much more
narrow range than the concentrations stated in Table 4
ln(q/q ) B cv*[(1/v ) [ (1/vref)]
ref fm fm
(14)
m
In this expression, subscript m stands for either the liquid
molecule or (in polymers) the chain segment involved in the
relaxation process, c is a constant between 0.5 and 1 account-
ing for the overlap of free volume, v* is the minimum hole
m
volume to be created by redistribution of free volume for the
molecule or segment to change place in the relaxation at
hand, and v is the average free volume per molecule or
fm
segment. We introduce the fractional free volume
U \ v /v
fm m
(15)
where v is the (total) average volume per molecule or
m
segment and write:
Fig. 11 The concentrations of the impurity electrolyte as a function
of temperature for the case of equal di†usion coefficients and the case
of one di†usion coefficient being 10 times (or more) greater than the
other. Eqn. (12) has been used with a static permittivity of 9.8 for all
the temperatures.
ln(q/q ) B c(v*/v )[(1/U) [ (1/U )]
ref m m ref
(16)
We consider a temperature T , lower than T as well as T
,
0
ref
and having the property that at T the free volume would be
0
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reduced to zero were the glassy state not formed. We have
then approximately (at constant pressure) that
di†using in PEDMA could be smaller than the O molecule.
According to the preparation of the polymer described in the
2
Experimental section, the main ionic impurity is likely to be
triethylammonium chloride. However, during the Ðltration
and washing process of the EDMA monomer with distilled
water in ambient air, the chloride might have been replaced
U \ a(T [ T )
T P T
(17)
0
0
where a would be the true thermal expansion coefficient if all
expansion were in the form of CohenÈTurnbull free volume.
Introducing eqn. (17) in (16) we obtain for the temperature
dependence of any relaxation time which is dependent on free
volume only:
by HCO ~ ions which are quite bulky but still probably
3
faster di†using than the triethylammonium ions which entan-
gle with the polymer chains. There might also be di†erent
overlap factors for the di†usion process and for the glassÈ
rubber relaxation in the same polymer.
At the glass temperature (ca. 36 ¡C) we have for the polymer
in this work:
ln q B constant ] B/(T [ T )
(T [ T )
(18)
(19)
0
0
B \ cv*/av
m
m
Using the latter equation we can also reformulate eqn. (16):
(1/q) B A exp([Ba/U) (20)
A \ (1/q )exp(]Ba/U (for any T [ T ) (21)
U(glass)/B a \ (T [ T )/B \ 81/2608 B 0.031 (PEDMA)
a
g
0
(29)
)
ref ref
ref
0
This value is almost in the range 0.025 ^ 0.005 obtained for
most amorphous polymers.19c This is often called the
Clearly, the Vogel expressions found here and in ref. 14 for
the a-relaxation and for the di†usion of foreign molecules or
ions conform to eqn. (18)È(21), and both the Vogel tem-
perature and the B parameter may now be interpreted physi-
cally. The Vogel temperature is the temperature where the free
volume (which can be redistributed without use of energy) is
zero in a hypothetical state where the glass transition does not
take place. Furthermore, the relaxation times tend towards
inÐnity when T ] T from above. Below T there cannot be
“fractional free volumeÏ at the glass temperature, since B a B
a
1. (The constant which is called B a in the present paper was
a
called B in ref. 14 and 19.) In ref. 14, the extraordinary high
value for U(glass)/B a \ 0.038 was found for PCHA.
a
Demasking a very low-frequency relaxation
In ref. 8 the dielectric properties of another polymer, poly[4-
(acryloxy)phenyl(4-chlorophenyl)methanone], was studied. In
that paper a low-frequency relaxation was observed in the iso-
thermal plots at temperatures from 120 to 140 ¡C. This relax-
ation seemed interdispersed in between what appeared to be
an a-relaxation and the relaxation due to the combined e†ects
of the conductivity and the surface polarisation, Fig. 12È14 of
ref. 8. Our Ðnal experiment is to look for a slow relaxation in
the dielectric data.
Plots are made of the dielectric loss corrected for conduc-
tivity and of the uncorrected real part of the permittivity
down to frequencies where the uncertainties of the corrected
dielectric losses are still acceptable. The corrected relative
dielectric losses are calculated as:
0
0
any relaxation. The coefficient B is the activation energy (in
K) of the relaxation at very high temperatures (T A T ), and it
0
is also proportional to the critical volume v* for accommodat-
m
ing the molecule or segment relaxing.
The Doolittle expression44 for the viscosity in terms of frac-
tional free volume, found to represent with high accuracy vis-
cosities of ordinary liquids of low molecular weight, is also of
the same type. Cohen and Turnbull used their theory for self-
di†usion in hard-sphere Ñuids, but the same relation has been
used for real di†usion of foreign molecules in polymers.14,45
Indeed, the concept of fractional free volume has been found
to be very useful in the study of di†usive transport pro-
cesses.46,47
eA(corr.) 4 eA(obs.) [ p/(e u)
(30)
r
r
0
In accordance with eqn. (20) we write for the maximum fre-
quency of the glassÈrubber relaxation and for the di†usion
coefficient:
Note that the values of the dielectric loss are corrected for
conductivity only and not for the surface polarisation. We
have seen here and in previous work that the surface polarisa-
tion a†ects the real part of the permittivity at much higher
frequencies than the imaginary part. In most cases we have
f B A exp([B a/U)
(22)
(23)
(24)
(25)
a
a
a
D \ A exp([B a/U)
diff
diff
studied, the MWS result for the dielectric loss eA (MWS) \
r
B \ c v* /av
p/(e u) Ðts well at least down to 0.01 Hz, whereas a substan-
a
a m, a
m
0
tial rise in the values of e@ is seen at low frequencies due to the
capacitance e†ects of the separation of charges in the Ðlm.
B
\ c v*
/av
r
diff
diff m, diff
m
Assuming that the overlap factor c is the same in the two
Thus, we assume that eA (corr.) is an approximation of the
r
cases we have
dielectric loss due to polymer relaxations down to frequencies
where eA (corr.) becomes a very small di†erence between two
B /B B v* /v*
(26)
r
a
diff
m, a m, diff
very large numbers, viz. eA (obs.) and p/(e u). On the other
r
0
since the total volume for the polymer segment active in the
hand, the uncorrected values of e@ are good approximations to
r
redistribution of free volume (v ) is the same in the two pro-
the real part of the permittivity for the polymer itself down to
m
cesses. The ratio between the two B-coefficients in the Vogel
frequencies where the observed values rise sharply due to
interfacial polarisations.
equation should therefore be equal to the ratio between the
critical void volume for the two processes. For the polymer
studied here (PEDMA) we have:
Fig. 12 shows isothermal plots of eA (corr.) and e@ vs. the
r
r
frequency at temperatures of 45, 60 and 75 ¡C. At 45 ¡C
(rectangles), two relaxations are clearly seen (two downward
v* /v*
B 2608/1616 \ 1.61 (PEDMA)
(27)
m, a m, diff
steps in e@ accompanied by two loss peaks). One loss peak has
r
For the PCHA studied in ref. 14 we have:
a frequency maximum at ca. 20 Hz. This is within the uncer-
tainty of the value of 14 Hz calculated from the Vogel expres-
sion (3) for the a-relaxation. The other peak has a maximum
at frequencies [100 kHz. Therefore the peak maximum
cannot be observed, but the left-hand side of the peak is
clearly visible at high frequencies. According to the regression
(1) for the b-relaxation, the frequency maximum should be ca.
129 kHz. Thus, the high-frequency relaxation in Fig. 14 is
v* /v*
B 1300/392 \ 3.32 (PCHA)
(28)
m, a m, diff
In both cases, the minimum hole volume is greater for the
glassÈrubber relaxation than for the di†usion of either the
fastest ion or the oxygen. This is as should be expected.
However, we would have expected the ratio for PEDMA to be
even higher than the ratio for PCHA, since the side-group in
PEDMA is more bulky than in PCHA and the fastest ion
surely the b-relaxation. The rise in e@ seen below 0.6 Hz should
r
J. Chem. Soc., Faraday T rans., 1997, V ol. 93
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Fig. 14 The maximum loss frequencies of the “X-peakÏ (slow
relaxation) show almost perfect Arrhenius dependence with tem-
perature between 90 and 140 ¡C. The activation energy is 14 620 K.
be ascribed to surface polarisation. In that case, the static
value (for u ] 0) of the relative permittivity for the polymer
would be 11 ^ 1 at 45 ¡C.
At 60 ¡C (diamonds in Fig. 12) the Vogel value of the
maximum frequency for the a-relaxation is found to be ca. 850
Hz which corresponds well with the ca. 1000 Hz observed in
the Ðgure. The maximum frequency for the b-relaxation is
found from the (extrapolated) regression (1) to be ca. 293 kHz,
far above the maximum value of 100 kHz observed. However,
the left-side tail of the b-relaxation is clearly seen at high fre-
Fig. 12 Isothermal plots of (a) the dielectric loss (corrected for
conductivity) and (b) the real part of the relative permittivity as a
function of frequency. Temperatures: 45 ¡C (rectangles), 60 ¡C
(diamonds) and 75 ¡C (crosses). The a-relaxation is seen and part of
the b-relaxation (at high frequencies). At very low frequencies some
noise is seen in the corrected dielectric loss due to the subtraction of
large and almost equal quantities.
quencies. There is a plateau in the values of e@ between 1 and
r
10 Hz. The rise in e@ below 1 Hz is due to surface polarisation
r
and the rise in eA (corr.) due to the e†ect of “small di†erence
r
between large numbersÏ (called the SDLN e†ect for brevity).
The static value of the relative permittivity seems to be 12 ^ 1
at 60 ¡C.
At 75 ¡C (Fig. 12, crosses) the a-relaxation has moved to still
higher frequencies with a maximum at ca. 20 kHz which is
well in accordance with the value of ca. 18.9 kHz calculated
from the Vogel eqn. (3). The b-relaxation has only moved
slightly and is overlapping to a considerable degree with the
a-relaxation at the highest frequencies. This is because the a-
and b-relaxation are merging together, but they have not yet
uniÐed as an ab-relaxation. There is evidence of a low-
frequency relaxation with a loss peak around 10 Hz, but it
cannot be completely excluded that this peak is a spurious
SDLN e†ect. The static value of the relative permittivity
seems to be 11 ^ 1 at 75 ¡C.
At 90 ¡C (Fig. 13, rectangles) the a-relaxation according to
the Vogel expression should have a maximum of ca. 210 kHz
which cannot be observed. However, the left-hand tail of the
loss peak is clearly seen at the high frequencies. There is surely
a low-frequency relaxation around 10 Hz, visible as a peak in
the loss as well as in the step downwards in e@ . The static
r
value of the relative permittivity seems to be 11 ^ 1 at 90 ¡C.
At 100 ¡C (Fig. 13, diamonds) the left-hand tail of the a-
relaxation is just visible at high frequencies and there also
seems to be a slow relaxation around 50 Hz, visible as a
shoulder in the SDLN loss rise as well as in the step in e@ . The
r
static value of the relative permittivity seems to be 11.5 ^ 1 at
100 ¡C.
At 110 ¡C (Fig. 13, crosses) only the leftmost part of the tail
of the ab-relaxation is visible from 10 to 100 kHz. The slow
Fig. 13 Isothermal plots of (a) the dielectric loss (corrected for
conductivity) and (b) the real part of the relative permittivity as a
function of frequency. Temperatures: 90 ¡C (rectangles), 100 ¡C
(diamonds) and 110 ¡C (crosses). The a-relaxation is disappearing to
the right and at low to intermediate frequencies another (slow) relax-
ation is seen. The dielectric loss peaks of this relaxation rapidly grow
with increasing temperature. They are interrupted at very low fre-
quencies by noise.
relaxation has moved to ca. 100 Hz. The static value of e@ is
r
12 ^ 1.
At 115 ¡C (Ðgure not shown) the low-frequency relaxation is
situated at ca. 200 Hz and the static value of e@ is 13 ^ 1. At
r
130 ¡C (Ðgure not shown) the “low-frequencyÏ relaxation is
seen from the turning tangent of e@ at ca. 700 Hz correspond-
r
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ing to a shoulder in the corrected loss. The static value of e@ is
r
12 ^ 1. Finally, at 140 ¡C (Ðgure not shown) the turning
tangent of e@ has moved to ca. 1.5 kHz, but the shoulder in the
r
corrected loss seems to be “swallowedÏ by the rapid rise at
“lowÏ frequencies (below 500 Hz) due to the SDLN e†ect. The
static value of e@ has dropped to 9.2 ^ 0.5.
r
The reason for the large SDLN e†ects at 140 ¡C may be
illustrated by the value of eA at 10 Hz of ca. 663. The value of
r
p/(2nfe ) is ca. 629. With an experimental uncertainty of ca.
0
4% we have an uncertainty of ^27 on the Ðrst Ðgure. Fur-
thermore, the second Ðgure also carries some uncertainty since
the value of p can only be stated approximately over a given
range of frequencies. Thus, the di†erence 663 [ 629 \ 34
simply represents the uncertainty. At 0.1 Hz, we measured eA
r
of ca. 66 600 which should be compared to p/(2nfe ) ca. 62 900.
0
The di†erence is 3700 which is 5.5% of the value measured.
The slow relaxation peak observed (denoted “XÏ in the
following) exhibits a maximum dielectric loss. These maxima
are approximately positioned at 10, 50, 100, 200, 700 and 1500
Hz at 90, 100, 110, 115, 130 and 140 ¡C, respectively. The tem-
perature dependence from 90È140 ¡C is Arrhenius-like, see
Fig. 14. We Ðnd:
Fig. 15 Isochronous plots of the dielectric loss corrected for conduc-
tivity. Frequencies: 1000 Hz (rectangles), 500 Hz (diamonds) and 200
Hz (crosses) with a-peaks indicated. At high temperatures the domi-
nant loss peaks of the X-relaxation are seen to be partially interrupted
by SDLN noise.
ln f
max, X
\ 42.82 [ 14 620/T ,
r \ [0.9945 (90È140 ¡C)
(31)
uncertainty, but the erratic variations (twin peaks) may well
be produced by SDLN e†ects.
The mean activation energy is 14 620 K. We can compare
this with the mean activation energy for the a-relaxation
extrapolated to the same range of temperatures. In general,
the “local activation energyÏ at the temperature T is given by
Fig. 16 shows isochronous plots at 2, 5 and 10 kHz. The
a-peaks are situated in the region 60È70 ¡C and there is a
(twin ?) X-peakÈstill situated in the region 125È135 ¡C. The
X-peaks are now smaller than the a-peaks, and they are very
small at 10 kHz. In the “worst caseÏ we have at 135 ¡C and 2
kHz eA of ca. 3.00 and p/(2nfe ) \ 2.16. The di†erence, 0.84, is
Et(T )/R \ [d ln f /d(1/T ). From the Vogel relation we
max
obtain:
r
0
now 28% of the observed value, so that the high-temperature
values are quite reliable in this case.
Et(T )/R \ [T /(T [ T )]2B
(32)
0
The mean temperature in the range 90È140 ¡C is 115 ¡C or 388
K. With a Vogel temperature equal to 228 K, the factor at the
right-hand side becomes 5.88 and we have (since B \ 2608 K
for the a-relaxation):
At 20, 50 and 100 kHz (Ðgure not shown), the a-peak seems
to split into two peaks (a and a ). The splitting is clear at 100
1
2
kHz with one minor peak (a ) situated at ca. 60 ¡C and
1
another major peak (a ) situated at ca. 85 ¡C. [According to
2
eqn. (1) the b-relaxation has a maximum at 60 ¡C at 290 kHz,
Et(115 ¡C)/R \ 15 340 K
(33)
so the a -peak cannot be the b-peak.] At 50 kHz the a -peak
a
1
1
is seen as a relatively sharp “shoulderÏ at ca. 60 ¡C at the left-
This mean activation energy is very close to the mean activa-
tion energy 14 620 K found for the slow relaxation. This might
be an indication that the same inter- or intra-molecular inter-
actions are involved in the a-relaxation and the slow relax-
ation.
hand side of the a -peak with a maximum at ca. 80 ¡C. At 20
2
kHz there is only a “soft shoulderÏ at ca. 60 ¡C at the left-hand
side of the a -peak with maximum at ca. 75 ¡C. In retrospect
2
we observe in Fig. 16 that the a-peak for 10 kHz is broader
than at 5 and 2 kHz. Thus, the a - and a -relaxations have
The X-peak does not seem to be seen in isochronous maps
of dielectric loss vs. temperature but is in fact observed in
these maps if the dielectric loss is Ðrst corrected for conduc-
tivity. This is demonstrated in Fig. 15 and 16. In the isother-
mal plots the X-peak is positioned at lower frequencies than
the a-peak, but in the isochronous plots the order of suc-
cession is reversed, and the X-peak should be found at higher
temperatures than the a-peak.
1
2
still not merged completely at 10 kHz.
In isochronous plots at 5, 20 and 100 Hz (Ðgure not shown),
the SDLN region seems to start above 120 ¡C at 100 Hz. At
20 and 5 Hz, the SDLN region instead begins above 100 ¡C.
At lower temperatures than the beginning of the SDLN
region, the left wing of a broad absorption peak is clearly
seen. At 100 Hz, the maximum is located at ca. 115 ¡C. These
broad peaks are clearly separated from the a-peaks with
maxima at 40 ¡C (100 Hz) and below.
This is more clearly seen in Fig. 15 which shows isochro-
nous plots at 200, 500 and 1000 Hz. The a-peaks are situated
in the region 55È60 ¡C and there is clearly an “X-peakÏ at high
temperatures with maxima in the region around 130 ¡C. The
absorption peak becomes more and more pronounced relative
to the a-peak with decreasing frequency. With respect to the
uncertainty, we have in the “worst caseÏ at 135 ¡C and 200 Hz
an eA of ca. 24.5 and p/(2nfe ) \ 21.6. The di†erence is 2.9
Fig. 16 Isochronous plots of the dielectric loss corrected for conduc-
tivity. Frequencies: 10 kHz (rectangles), 5 kHz (diamonds) and 2 kHz
(crosses) with a-peaks indicated. At high temperatures the loss peaks
of the X-relaxation are seen and the a-peaks dominate.
r
0
which is 12% of the observed value. This is well above the 4%
J. Chem. Soc., Faraday T rans., 1997, V ol. 93
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Fig. 16 also shows a very small (twin ?) X-peak in the
Education and Science of Spain) under project PB92-0773-
C03-03 and by BANCAJA under project P1B95-004. T.S.S. is
indebted to DGICYT for a half year prolongation of his sab-
batical year guest professorate enabling him to continue his
research at the Universitat Jaume I, Castellon, Spain.
region 125 to 135 ¡C at 20 kHz. At 50 kHz the a -peak is
2
overlapping to a considerable degree and the X-peak is only
visible as a “double shoulderÏ at the right-hand side of the
a -peak. Finally, at 100 kHz the X-peak is completely
2
“swallowedÏ by the dominating a -peak.
2
Thus, there are many indications of a quite complex dielec-
tric relaxation pattern of the polymer studied here. The cor-
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the generalized microbrownian motions of the polymer16d,19d)
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e
the normal mode relaxation times. This factor represents the
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0
14 V. Compan, E. Riande, J. San Roman and R. Diaz-Calleja,
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15 C. A. Angell, Science, 1995, 267, 1924.
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However, this model does not give us the correct frequency
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the limit of low frequencies.19e
However, the “tube evaporation by reptationÏ model of Doi
and Edwards24 also produce a sum of relaxation terms with
the longest relaxation time corresponding to the “reptationÏ
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to f .19f This model is valid for mechanical relaxations and
0
probably dielectric relaxations can be treated similarly.
A future project might be to measure the slow relaxation
found here with [(5-ethyl-1,3-dioxan-5-yl)methyl acrylate]
polymers of varying and well deÐned molecular weights to see
if the relaxation time scales as M3.4 as has been found experi-
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a mechanical relaxation study of a number of poly(alkyl
methacrylates)48 that keeping the polymers 10 min above the
glass transition temperatures is enough for a complete elimi-
nation of the “memoryÏ of the thermal history of the polymer.
The polymer investigated here seems to be an interesting
target for more detailed dielectric and mechanical investiga-
tions in the future because of the many complex features
exhibited. For example, it would be interesting to study
samples with di†erent and well deÐned mean molar masses
(M) of the polymer in order to investigate the M-dependence
of the di†erent relaxation peaks observed, especially the slow
“X-relaxationÏ.
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Paper 7/01239J; Received 24th February, 1997
J. Chem. Soc., Faraday T rans., 1997, V ol. 93
2411