4848
J . Phys. Chem. 1993,97, 4848-4854
Relaxation and Conformational Studies on Polymers with Large Conductive Contributions to the Dielectric Loss Ricardo Dhz-Calleja,+Evaristo Riande,'?$and Julio San Roman$ Departamento de Termodinbmica Aplicada, E TSII, Universidad Polithica de Valencia, 46071 Valencia, Spain, and Instituto de Ciencia y Tecnologia de Polimeros (CSZC), 28006 Madrid, Spain Received: January 4, I993
This work focuses on the study of the dielectric behavior of asymmetric molecular chains with high electronic unsaturation on the side groups, taking as a model poly(4-benzoylphenyl acrylate) (PBA). This polymer presents a prominent glass-rubber relaxation, named a, followed by two secondary processes that under decreasing temperatures are called j3 and y absorptions. Similar relaxations are found in the mechanical spectrum. At low frequencies, free charges and interfacial contributions to the dielectric a relaxation are dominant and a method is proposed to determine each of them. The experimental intramolecular correlation coefficient gintra = 1 Ci+,(cos yij) for PBA chains, where yij is the angle between the dipoles associated with i a n d j repeating units, is 0.81 at 30 OC and its temperature coefficient d In gintra/dT has a value of 1.3 X le3 K-1 in the interval 30-60 OC. A 4 X 4 rotational states scheme gives a good account of these experimental results. The correlation coefficient g, which includes both the intermolecular and intramolecular interactions, is in very good agreement with the value of gintra,suggesting that intermolecular interactions are negligible in this polymer.
+
Introduction The dielectric relaxation strength of polymer chains is straightforwardly related to the correlation coefficient g that accounts for intermolecular and intramolecular interactions between relaxing dipoles. This parameter is expressed by'
earlier developed for acrylate based polymers.' Comparison of the correlation coefficient thus obtained with that determined from dielectric measurements carried out on the bulk will give information about the magnitude of the intermolecular interactions in this polymer.
Experimental Section where y is the angle between the elemental dipoles of the chains. For low polarity polymers, dipolar intermolecular interactions are negligible'.* in comparison with the intramolecular ones so that g can be written as
g E gintra= 1
+
(cos y i j P r a m i#j
However, significant dipolar intermolecularinteractions can occur in systemswith relaxing speciesof high polarity. Thusa systematic work carried out on the dielectric behavior of acrylate based polymers has shown that intermolecular and intramolecular interactions can be comparable in systems with halogen atoms located in the side groups.3~~ In order to get a deeper insight into the relationship between the chemical structure and the dipolar correlation coefficient, it is advisable to determine the value of g in very dilute solutions, where intermolecular interactions are negligible, and in the bulk, where these interactions should be important. Pursuant to the objective of correlating chemical structure and dipolar interactions, the study of the dielectric behavior of poly(4-benzoylphenyl acrylate) (PBA) was undertaken. This polymer was chosen because the side group has an exceptionally high polarity while, on the other hand, its high electronic insaturation presumably will produce significant conductivity losses that will mask the dipolar loss. It is the purpose of this study to separate bothcontributionsto thedielectricloss to express the dipolar contribution in terms of a nonexponential decay parameter, or Kohlrausch-Williams-Watts (KWW) function.5~6 In addition the dipolar correlation coefficient that accounts for the intramolecular interaction will be measured and the results obtained will be interpreted by a 4 X 4 rotational states scheme +
Universidad Politbnica de Valencia. de Ciencia y Tecnologia de Polimeros.
1 Instituto
0022-3654/93/2097-4848$04.00/0
Synthesis and Characterization of Poly(Qbenzoylpbenyl acrylate). 4-Benzoylphenyl acrylate was synthesizedfrom a solution of 4-benzoylphenol and freshly distilled acryloyl chloridefollowing the procedures described in detail elsewhere.s Poly(4-benzoylphenyl acrylate) was obtained at 50 OC by radical polymerization under vacuum of 4-benzoylphenyl acrylate in benzene solution, using AIBN as initiator and keeping the conversion below 12%. The polymer was precipitatedwith methanol,filtered, washed with methanol, and dried under vacuum. The number average molecular weight of the polymer was 80400. The stereoregularity of PBF was determined by I3C NMR spectroscopy, at room temperature, with a Varian XL-300 spectrometer at 75 MHz, using TMS as internal reference. The molar fractions of the iso-, syndio-, and heterotactic triads were 0.20,0.30, and 0.50, respectively. The glass-transition temperature Tg of the sample used in this study was measured with a Perkin-Elmer DSC-4 calorimeter at a heating rate of 10 'C/min; the value of Tgdetermined on the onset of the thermogram was found to be 75 OC. Dielectric Measurements. The dielectric permittivity of solutions of the polymer in dioxane was measured at 10 kHz with a capacitance bridge (General Radio type 1620 A) and a three terminalcell. Incrementsof the refractiveindexAn of the solutions with respect to that of the solvent were measured at 632.8 nm with a differential refractometer (Chromatix Inc.). The components of the complex dielectric permittivity e* of the polymer in the bulk were measured with a plane condenser and a Dupont EA 2970 dielectric analyzer, over the frequencyand temperature ranges of 0.001-30 kHz and -140 to +lo0 OC, respectively. Dynamic Mechanical Experiments. Values of the components of the complex relaxation modulus E* were obtained at a heating rate of 1 OC/min with a PL-DMTA Mark I1 apparatus in single cantilever bending. The experiments were conducted in the frequency and temperature ranges 0.3-30 Hz and -150 to +lo0 OC, respectively. 0 1993 American Chemical Society
Dielectric Behavior of Polymers
The Journal of Physical Chemistry, Vol. 97, No. 18, 1993 4849 3
31
2
2b 10
B 6
4
3 2
IO8
B 6
4
3
Figure 1. Dependence of the storage relaxation modulus E'and the loss modulus E"on temperature at several frequencies: (A)0.3; ( 0 )1; (0)
-100
60
-20
3 Hz.
60
loo
140
Figure 3. Variation of the dielectric loss tan 6 with temperature for PBA at several frequencies: ( 0 )0.1; (A) 1.0; (m) 10; (0) 100; (A) 1000; (0) 10 000 Hz. (Insert is the temperature dependence of e'.)
I._.
9.0
#)
T,'C
-
-8.5 & iu 4"&0 -
7.5
-
7.0
t
20
Y-V-1 40
*
T,'C
80
100
Figure 2. Details of the mechanical glass-rubber relaxation for PBA at several frequencies: (V)0.3; ( 0 )1; (m) 3; (A) 10; ( 0 )30 Hz.
Results A. Mechanical Relaxation Spectra. Storage E'and loss E" components of the complex relaxation modulus E* of PBA are plotted as a function of temperature in Figure 1. In the glassy region the mechanical loss exhibits a peak in the low temperature zone, centered at -85 OC at 0.3 Hz, followed by another peak of higher intensity, centered at -23 OC at the same frequency. These peaks, named 7 and 8, respectively, follow Arrhenius behavior with activation energies of 14 and 21 kcal mol-I, respectively. The spectrum also presents a prominent a absorption associated with the glass-rubber relaxation, centered at 78 O C at 0.3 Hz. Details of the a relaxation at several frequencies are given in Figure 2. B. Dielectric Relaxation Spectra. a. General Results. The real e' component of the complex dielectric permittivity and the loss tan 6 are plotted against temperature at several frequencies in Figure 3. As usual, the real permittivity slowly increases with temperature in the glassy region and then rapidly through the
glass-rubber zone. The loss presents at 0.1 Hz two secondary absorptions in the glassy zone centered at -107 and -18 OC, named y and 8 peaks, respectively. Both absorptions, however, overlap at frequencies higher than 1 Hz. In the glass-rubber transition the loss does not present a maximum at frequencies below 100 Hz presumably as a consequence of the fact that the contribution of the conductivity to the loss becomes dominant at low frequencies. In order to determine this contribution,it is advisableto express the dielectric results in terms of the complex electric modulusl~9M* = (e*)-'. The values of the components of the dielectric permittivity can be converted to the componentsof the electric modulus by means of the expressions
M* = (e*)-'
= M'+ iM"
(3)
M'= + M"= e"/(e'2 + e l f 2 ) (4) Plots showing at several frequencies the dependence of M" on temperature are represented in Figure 4. The curves present a relaxation in the high temperature zone that is associated with conductive processes followed by a peak in the low temperature region associated with the glass-rubber dipolar relaxation. Whereas the intensity of the maximum of the dipolar peaks decreasesas the frequencyincreases, this intensity remains nearly independent on frequency for the peaks associated with the conductivity. As the frequencydecreases,the overlapping of both peaks increasesso that below 0.3 Hz only a single peak is detected. The real part M'of the complex electric modulus M*is plotted against temperature in Figure 5. The curves present a plateau separating the first part of the curve, associated with the dipolar contribution, from the second one, associatedwith the conductive process. The plateau becomes more prominent as the frequency increases so that at extremely low frequenciesit is not detectable. The height of the plateau also increases with increasing temperature.
DEaz-Calleja et al.
4850 The Journal of Physical Chemistry, Vol. 97, No. 18, 1993
Figure 4. Dependence of the loss electric modulus on temperature for PBAat several frequencies: ( 0 )0.1; (A)0.3; (e) 1; ( 0 )3; (W) 10; (V) 30; (A) 100; (0) 3000; ( 7 ) 1000; (0) 3000; (0) 10 000 Hz.
The contribution of the conductivity to the electric loss modulus M”was obtained a t temperatures above 80 O C from the plots of Figure 4. The curves of M” versus frequency, represented in Figure 6, reveal the existence of a prominent and narrow peak, ostensibly associated with the conductive process, followed by a broad and low intensity peak in the high frequencies region produced by the dipolar relaxation. The conductivity peaks are symmetric and the half-width values are given in Table I. b. Dipolar and Conductive Contributions to the Glass-Rubber Relaxation. Although the half width of the conductive peaks decreases with increasing temperature as a consequence of the diminishing of the overlapping of the dipolar and conductive processes, the value found for this quantity at high temperatures (1.25) is significantly larger than that corresponding to a simple conductiveprocess (1.14). This seems to suggest that in addition to free charges, interfaceelectrode phenomena intervene in the process. Since these effects become important in the glass-rubber relaxation at low frequencies, the determination of the dipolar a peak requires one to substract from the total loss the contributions of free charges and/or interface-electrode phenomena.10~1I By assuming that additivity holds, the complex dielectric permittivity can be expressed by
= t*d
E*
+ e*, +
(5)
where the subindices d, c, and i refer, respectively, to the contributions to e* from dipoles, free charges, and interfaceelectrode phenomena. Therefore the real and loss components of e* can be written as e‘ 6‘‘
= e’,
+ e‘, + e’,
(6)
= e’’,
+ effc + pi
(7)
Separation of the different contributions to the total dielectric loss requires some mathematical handling that will be described in detail, taken as reference the loss isotherm of 95 O C . In fact both erfcand e”i curves nearly overlap each other so that for their separation it will be taken into account that for materials with no dipolar processes, the equivalent circuit is simply a parallelRC combination.’* Consequently e*,
= e,, [l
+ (ja7,)-’]
(8) In this equation the conductivity relaxation time T~ is defined by
= fgemc/Q (9) where eo is the permittivity of free space ( 4 . 8 5 1 4 pF/m) and u is the conductivity. The free charges contribution can be 7,
D
Figure 5. Change of the electric storage modulus with temperature for PBA at several frequencies: (0)0.1; ( 0 )1; (W) 10; (0) 100; (+) 1000; (A) 10000 Hz.
I
4.0~
f(Hz)
Figure 6. Plot showing the dependence of the electric loss modulus on frequency for PBA at several temperatures: ( 0 )80; ( 0 )85; (A)90; (0) 95; (+) 100; (0) 105; (0) 110; (V) 115; (A) 120; (W) 125 ‘c.
expressed in terms of the dielectric modulus by
M*, = Mmc( 1
1
)=
Mmc(1 -l+jw7c
)
+ (jarc)-‘
(10)
where Mi, = em,-!, whereas the effect due to partial blocking of charges in the electrodes can be expressed by
that can also be written as
M*i = Mmi where M,i = 6,i-l. These equations suggest that in the low frequency region ep -fI and ti’’ -f”,and therefore, the former contribution can be separated from the total dielectric loss by substracting from e” the values of trfcgiven in the straight line of slope-1 of the doublelogarithmic plot e” versus5 As indicated in Figure 7, the loss that only includes the dipolar and the interfacial contributions is obtained, whose slope in the low frequency region is -0.48; in other words, e’’i p 48. It should be pointed out that Cole and Tombari13J4have shown that a is equal to 0.5. By substracting this contribution from the loss, the dipolar peak shown in Figure 7 appears.
-
Dielectric Behavior of Polymers
The Journal of Physical Chemistry, Vol. 97, No. 18, 1993 4851
TABLE I: Influence of the Temperature on Both the Conductivity and the Half Width of the Relaxing Peak That Includes Free Charges and Electrode-Interface Contributions ~~
T,"C
half-width
85 90 95 100 105 110 115 120
1.5 1.45 1.40 1.35 1.30 1.30 1.25 1.25
lo%,
s
0.20 1.52
7.78 27.8
As the effects were assumed to be additive, separation of the real and imaginary components of 2* = ac* + ei* gives
d - atd = Zf = (emi a"
+ a,,) +
E , ~ ( w T ~ ) * COS
-x2 a
(13)
x
- affd = ; I J = c,~(wT~)-"sin a + E ~ , ( w T , ) - ' (14) 2
For simplification purposes, the real and imaginary parts of c* will be written as
2 = aw-" + d e
bw-"
(15)
+ cw-l
(16)
-
8-
where the expressions for the parameters a, b, c, and d can easily be obtained by comparing these equations with eqs 13 and 14. Therefore the real and imaginary componentsof M* = (a*)-' are
a' =
+d + 2bcw-('+") + c ~ w +- ~d2
aw-a (a2
+ b 2 ) i 2 "+ 2adw"
(17) &/f
=
bw"
(a2
+ cw-'
10.'
100
10'
loz
10'
10'
f,(Hz)
Figure 7. Dependence of the dielectric loss on frequency at 95 "C.Open circles represent the total loss whereas filled symbols only include the dipolar and electrodeinterfacecontributions. The peak corresponds to the a dipolar relaxation.
+ b2)w-2"+ 2adw-" + 2bcw-('+") + c ~ w +- ~d2 (18) a,
The conductivity c can be obtained by considering that
=we0 where eo is the permittivity of free space. From the values of a", correspondingto the straight line of slope -1 (see Figure 7), the results for the conductivity are obtained. Values of these quantities at different temperatures are given in Table I. From the '@i versus MJfplot shown in Figure 8, one obtains M, = 0.092 and hence e,, + ami = l/M, = 10.87. Both ~i and ami can be obtained by solving eqs 13 and 14 for f = 1 and W T ~= 1, using for a the valueof 0.48 indicated above; thevaluesof thesequantities amount to 0.16 s and 4.66, respectively. Finally, e,, = 10.87 - c,i = 4.66 and T~ = (amc/u)ao = 2.7 X 10-2 s. It can easily be seen that the parameters a, b, c, and d by which the values of M'and MJJ at 95 OC can be obtained are a = 10.91, b = 10.24, c = 171.3, and d = 10.87. The complex dielectric plot calculated by using these values and shown in Figure 8 is in good agreement with the experimental results, except in the high frequency region where the influence of the dipolar peak is important. C. Mpolar Correlation Coefficients. The strength of the relaxation processes is related to the mean square dipole moment (&) of the relaxingdipoles by means of Onsager type equations15 such as the Frahlich equation16
and unrelaxedglass-rubber and subglass relaxations, respectively, N pis the number of relaxing dipoles per unit of volume, Tis the absolute temperature, and K is the Boltzmann constant. The value of tar was taken to be 10.87, the reciprocal of M, at 95 OC;this parameter seems to be very little sensitive to the temperature for this polymer. The value of 68" was taken to be 2.56, the square of the refractive index of the polymer in the bulk. By using these parameters in eq 1 1 the correlation coefficient g is found to be 0.89 at 110 OC. The dipolar intramolecular correlation coefficient gint,, was determined from dielectric measurements carried out in solution by using the method of Guggenheim and Smith1' according to which
where the subindicesa r and @u refer, respectively, to the relaxed
where M Ois the molecular weight of the repeating unit of the
a",
= a,,(w7,)-'
Q
- - - - - - - -l- 0
0.02
0
-r- - - -l- - - - ~- - - - -j- - - - - ~
0.04 - 0.06
M'
0.08
0.10
Figure& Complexdielectricplot of &f"versusk'at95 "C. The symbols 0 and represent calculated and experimentalvalues, respectively.
+
a
4852 The Journal of Physical Chemistry, Vol. 97, No. 18, 1993
Diaz-Calleja et al.
TABLE 11: Summary of Dielectric Results and Values of the Intramolecular Correlation Coefficient at Different TemDeratures for Polv(44enzovl~henvlacrvlate) ~
T, O C 30 40 50 60
2nl dnldw 0.00 0.00 0.00 0.00
dc/dw 4.92 4.75 4.60 4.46
&?intra
0.808 0.819 0.832 0.840
TABLE III: Components of the Dipole (in D) in the Reference Frame of the Cn-CH2Bond of the Repeating Unit as a Function of the Rotational Angle about Cn-C* (x) and 0-C" (W,)Bonds x , deg $1 PX PV PZ 0 180 0 180 0 180 0 180
60 60 -60 -60 120 120 -120 -1 20
0.405 1.324 3.772 -2.043 -0.717 2.447 2.650 -0,921
-4.066 1.87 1 -1.593 -0.602 -2.478 0.283 -0.005 -2.190
0.396 3.406 0.292 3.510 1.824 1.978 1.720 2.082
solute, dc/dw and dnldw are, respectively, the derivatives of the dielectric constant and index of refraction of the solutions with respect to the weight fraction of solute, w, in the limit w 0; € 1 , n l , and p are, respectively, the dielectric permittivity, index of refraction, and density of the solvent. Experimental values of the derivatives indicated above, measured in dioxane at several temperatures, are given in Table 11. In the last column of the table the values of gin,,,,with an uncertainty of f5%, are also given. The intramolecularcorrelationcoefficient slightly increases with temperature obtaining ginIra c 0.88 at 95 'C by extrapolating the experimental results to this temperature. Theoretical Calculation of the Intramolecular Correlation Coefficient. The theoretical evaluation of the intramolecular correlation coefficient requires the determination of the dipoles associated with each of the conformations permitted to the side groups. Earlier studies carried out on 4-benzoylphenyl isobutyrate (BPI)l8 indicate that the experimental dipole moments of this molecular compound can be obtained by addition of three contributions shown in Figure 9: p1 = 1.75 D representing the dipole moment of a methyl ester that forms an angle T of 123' with the CH-C*O* bond,Ig1.12 = 0.3 D representing the effect of replacing the methyl group by another group having different polarity, and p3 corresponding to the contribution of diphenyl ketone. l e Semiempiricalconformationalcalculationssuggest that the potential minima about the Mar bonds, + I , are located at f60° and f120°, taking as = 0 for the conformation in which the phenyl group is coplanar with the carbonyl group. Reasons of symmetry indicate that the mean-square dipole moment of BPI does not depend on the rotational angle +2, because after averaging, cancels any modification produced by the former conformationalangle. The evaluationof the dependenceof (po2) on p3 indicates that in order to reproduce the experimental result the value of the latter contribution should be ca. 3.0 D, a value that coincides with the experimental result for diphenyl ketone.20 The components of the dipoles for each conformation of the side group in a reference frame associated with the Cm-CH2 bonds are given in Table 111. A two rotational states scheme describes the statistics of polyacrylates. Each skeletal bond is restricted t o t and g states, each of these states being split into two as a consequence of the fact that the rotational angles x about the Cu-C*O* bonds are 0, and T , where the reference angle is that one at which the carbonyl group is cis to the methine bond.' However, since the polarity of the side groups depends on the rotational angles about &Car bonds, 16 X 16 statistical weight matrices should be formulated in order to account for all the rotational states of these groups. This was avoided by using in parallel with the 4
-
Figure 9. Racemic diad for PBA in trans conformation. X 4 statistical weight matrices a Monte Carlo simulation of the state about the Mar bond of each repeating unit that conditions the componentsof the dipoles in the reference frame of the main chain. The statistical weight matrices in the order (t,X = 0; t,X = T ; g,x = 0; g,x = T ) are7J8
\o
p
0 o/
for the CH2-Ca-CH2 bond pair, and
for the meso and racemic configurationsof the C'-CH2-Ca bond pair. The skeletal bond angles were assumed to be 112' and the values used for the rotational angles about the skeletal bonds of the repeating unit were (41,42)t,t = 16, 16' and (41,42)~,~ = ( 4 1 , 4 2 ) ~=, ~3, 114' for the meso and (41,4~),,, = 3, 3' for the racemic. Monte Carlo methods were used to generate 20 chains of 100 repeating units each, for which the fraction of meso replacements was 0.45; for each of these chains 50 chains were also generated in which the probability of finding J.1 in each one of the four permitted rotational angles is 0.25 f 0.01, Values at 30 OC of gin,,,were calculated by matrix multiplicationmethods,described in detail elsewhere.21,22Preliminary calculations showed that the correlation coefficient is strongly dependent on 72, the statistical weight that governs the relative orientationof the dipoles of the side groups in racemic diads. Thereforeginlra was evaluated as a function of 7 2 using the set of statistical weights p = 1.1, fl = 5.0, and y = 4.0 and y1 = 1.4, already used in the critical interpretation of the dipole moment of 4-benzoylphenyl-2,4dimethylglutarate (BPDG).I8 The results obtained, shown in Figure 10, indicate that ginIra decreases rapidly as y2increases as a consequenceof the fact that this increases the fraction of racemic diads in which the dipoles of the ester groups are placed in a nearly antiparallel direction; once that 7 2 reaches the value of ca. 1.90, the decrease of the intramolecular correlation coefficient with this statistical weight parameter is smaller. Good coincidence between theory and experiment is found for y2 = 2.70, in agreement with the value found for this parameter in BPDG.18 The intramolecular correlation coefficient is also sensitive to y I , its value changing from 0.66 f 0.06 to 0.91 0.04 when y1 increases from 0.44 to 3.77. Other statistical weight parameters associated with the meso diads seem to have little influence on the value of ginIra. Thus in changing fl from 5.0 to 1.18, the value of gin,,, only increases in 9%. The sensitivity of the intramolecular correlation coefficient to y is even lower. When the temperature is increased, the population of tT, t r conformations in racemic diads that place the dipoles in antiparallel direction decreases and, as a consequence, gintra increases. Therefore the theoretical analysis suggests that the intramolecular correlation coefficient must show a positive
*
The Journal of Physical Chemistry, Vol. 97, No. 18, I993 4853
Dielectric Behavior of Polymers
It 't.. \I'
1.1L
i
........
+.\..
0.9
0.8
c
........................... 1
.b........ ...........*. 1
I
1 4 -
I\
I
I
I
I
1.o
2.0
3.0
v, , v,
I
Figure 10. Dependence of the intramolecular correlation coefficient of PBA on y I and
1
.cuI
0.8
.\ Y
0.6
72.
temperature coefficient. This is corroboratedby the experimental results according to which d In gin,, E = 1.32 X K-I, in fair agreement with the theoretical value that amounts to (1.26 f 0.1) x 10-3 K-1. PhenomenologicalDescription of the Glass-Rubber Relaxation. The phenomenological theory of linear processes states that a compliancefunction r,such as the complex dielectric permittivity or the complex mechanical compliance, is related to the decay function @p by the familiar expre~sion2+~3
r*(w) - ru =
rr - ru
J:[-2]
exp(-iot) dt
(24)
where Qr is the decay function associated with the process that is given by the Kohlrauch-Williams-Watts (KWW) equationss6
o
