Thou, nature, art my goddess; to thy laws My services are bound. . . - Carl Friedrich Gauss DIELECTRIC LOSS AND RELAXATION-I T he dielectric constant and loss are important properties of interest to electrical engineers because these two parameters, among others, decide the suitability of a material for a given application. The relationship between the dielectric constant and the polarizability under dc fields have been discussed in sufficient detail in the previous chapter. In this chapter we examine the behavior of a polar material in an alternating field, and the discussion begins with the definition of complex permittivity and dielectric loss which are of particular importance in polar materials. Dielectric relaxation is studied to reduce energy losses in materials used in practically important areas of insulation and mechanical strength. An analysis of build up of polarization leads to the important Debye equations. The Debye relaxation phenomenon is compared with other relaxation functions due to Cole-Cole, Davidson-Cole and Havriliak-Negami relaxation theories. The behavior of a dielectric in alternating fields is examined by the approach of equivalent circuits which visualizes the lossy dielectric as equivalent to an ideal dielectric in series or in parallel with a resistance. Finally the behavior of a non-polar dielectric exhibiting electronic polarizability only is considered at optical frequencies for the case of no damping and then the theory improved by considering the damping of electron motion by the medium. Chapters 3 and 4 treat the topics in a continuing approach, the division being arbitrary for the purpose of limiting the number of equations and figures in each chapter. 3.1 COMPLEX PERMITTIVITY Consider a capacitor that consists of two plane parallel electrodes in a vacuum having an applied alternating voltage represented by the equation
Transcript
Thou, nature, art my goddess; to thy lawsMy services are bound. . .
- Carl Friedrich Gauss
DIELECTRIC LOSS AND RELAXATION-I
The dielectric constant and loss are important properties of interest to electricalengineers because these two parameters, among others, decide the suitability of amaterial for a given application. The relationship between the dielectric constant
and the polarizability under dc fields have been discussed in sufficient detail in theprevious chapter. In this chapter we examine the behavior of a polar material in analternating field, and the discussion begins with the definition of complex permittivityand dielectric loss which are of particular importance in polar materials.
Dielectric relaxation is studied to reduce energy losses in materials used in practicallyimportant areas of insulation and mechanical strength. An analysis of build up ofpolarization leads to the important Debye equations. The Debye relaxation phenomenonis compared with other relaxation functions due to Cole-Cole, Davidson-Cole andHavriliak-Negami relaxation theories. The behavior of a dielectric in alternating fields isexamined by the approach of equivalent circuits which visualizes the lossy dielectric asequivalent to an ideal dielectric in series or in parallel with a resistance. Finally thebehavior of a non-polar dielectric exhibiting electronic polarizability only is consideredat optical frequencies for the case of no damping and then the theory improved byconsidering the damping of electron motion by the medium. Chapters 3 and 4 treat thetopics in a continuing approach, the division being arbitrary for the purpose of limitingthe number of equations and figures in each chapter.
3.1 COMPLEX PERMITTIVITY
Consider a capacitor that consists of two plane parallel electrodes in a vacuum having anapplied alternating voltage represented by the equation
98 Chapter 3
where v is the instantaneous voltage, Fm the maximum value of v and co = 2nf is theangular frequency in radian per second. The current through the capacitor, ij is given by
~) (3-2)2where
m (3.3)z
In this equation C0 is the vacuum capacitance, some times referred to as geometriccapacitance.
In an ideal dielectric the current leads the voltage by 90° and there is no component ofthe current in phase with the voltage. If a material of dielectric constant 8 is now placedbetween the plates the capacitance increases to CQ£ and the current is given by
(3.4)
where
(3.5)
It is noted that the usual symbol for the dielectric constant is er, but we omit the subscriptfor the sake of clarity, noting that & is dimensionless. The current phasor will not now bein phase with the voltage but by an angle (90°-5) where 5 is called the loss angle. Thedielectric constant is a complex quantity represented by
E* = e'-je" (3.6)
The current can be resolved into two components; the component in phase with theapplied voltage is lx = vcos"c0 and the component leading the applied voltage by 90° isIy = vo>e'c0(fig. 3.1). This component is the charging current of the ideal capacitor.
Dielectric Relaxation-I 99
The component in phase with the applied voltage gives rise to dielectric loss. 5 is the lossangle and is given by
S = tan ' — (3.7)
s" is usually referred to as the loss factor and tan 8 the dissipation factor. To completethe definitions we note that
d= Aco 8s"E
The current density is given by
J = — = coss"E
Fig. 3.1 Real (s') and imaginary (s") parts of the complex dielectric constant (s*) in analternating electric field. The reference phasor is along Ic and s* = s' -je". The angle 8 isshown enlarged for clarity.
100 Chapters
The alternating current conductivity is given by
'+ &'-£„)] (3.8)
The total conductivity is given by
3.2 POLARIZATION BUILD UP
When a direct voltage applied to a dielectric for a sufficiently long duration is suddenlyremoved the decay of polarization to zero value is not instantaneous but takes a finitetime. This is the time required for the dipoles to revert to a random distribution, inequilibrium with the temperature of the medium, from a field oriented alignment.Similarly the build up of polarization following the sudden application of a direct voltagetakes a finite time interval before the polarization attains its maximum value. Thisphenomenon is described by the general term dielectric relaxation.
When a dc voltage is applied to a polar dielectric let us assume that the polarizationbuilds up from zero to a final value (fig. 3.2) according to an exponential law
JPao(l-*0 (3.9)
Where P(t) is the polarization at time t and T is called the relaxation time, i is a functionof temperature and it is independent of the time.
The rate of build up of polarization may be obtained, by differentiating equation (3.9) as
,at T T
Substituting equation (3.9) in (3.10) and assuming that the total polarization is due to thedipoles, we get1
Dielectric Relaxation-I 101
dt(3.11)
Neglecting atomic polarization the total polarization PT (t) can be expressed as the sum ofthe orientational polarization at that instant, P^ (t), and electronic polarization, Pe whichis assumed to attain its final value instantaneously because the time required for it toattain saturation value is in the optical frequency range. Further, we assume that theinstantaneous polarization of the material in an alternating voltage is equal to that underdc voltage that has the same magnitude as the peak of the alternating voltage at thatinstant.
Fig. 3.2 Polarization build up in a polar dielectric.
We can express the total polarization, PT (t), as
(3.12)
The final value attained by the total polarization is
(3.13)
We have already shown in the previous chapter that the following relationships holdunder steady voltages:
102 Chapters
Pe=e0(S<a-l)E (3.14)
where ss and c^ are the dielectric constants under direct voltage and at infinity frequencyrespectively.
We further note that Maxwell's relation
s^ = n2 (3.15)
holds true at optical frequencies. Substituting equations (3.13) and (3.14) in (3.12) weget
(3.16)
which simplifies to
j-fc / \ -j-^i f *\ -\ T\P^ = £0(ss - £X)E (3.17)
Representing the alternating electric field as
77 77 a^mt CZ 1 8 A
^ ~ ^max^ ^J.io;
and substituting equation (3.18) in (3.11) we get
-P(t}] (3.19)7 i_ - u \ - j - ou / m V / J V /
<^ r
The general solution of the first order differential equation is
-- (r -r }E ejcot
P(t) = Ce *+8Q l ^ ^ m (3.20)
1 + J(DT
where C is a constant. At time t, sufficiently large when compared with i, the first termon the right side of equation (3.20) becomes so small that it can be neglected and we getthe solution for P(t) as
Equation (3.23) shows that /Y() is a sinusoidal function with the same frequency as theapplied voltage. The instantaneous value of flux density D is given by
„ „ 77 l<Ot /"> O/l\: 6^0^ Eme (3.24)
But the flux density is also equal to
Equating expressions (3.24) and (3.25) we get
* J7* .jj^^ r. Z7 /yJ^t j J)( + \ C\ ^f\\
substituting equation (3.23) in (3.26), and simplifying we get
Equating the real and imaginary parts we readily obtain
£?' — C- I S CO /O 00\^-^00+^- T^ i3-28)
104 Chapters
+ CO T
It is easy to show that
- (3-30)£s
Equations (3.28) and (3.29) are known as Debye equations2 and they describe thebehavior of polar dielectrics at various frequencies. The temperature enters thediscussion by way of the parameter T as will be described in the following section. Theplot of e" - co is known as the relaxation curve and it is characterized by a peak at e'7s"max
= 0.5. It is easy to show COT = 3.46 for this ratio and one can use this as a guide todetermine whether Debye relaxation is a possible mechanism. The spectrum of theDebye relaxation curve is very broad as far as the whole gamut of physical phenomenaare concerned,3 though among the various relaxation formulas Debye relaxation is thenarrowest. The descriptions that follow in several sections will bring out this aspectclearly.
3.3 DEBYE EQUATIONS
An alternative and more concise way of expressing Debye equations is
8*=^+^^ (3.31)1 + JCOT
Equations (3.28)-(3.30) are shown in fig. 3.3. An examination of these equations showsthe following characteristics:
(1) For small values of COT, the real part s' « es because of the squared term in thedenominator of equation (3.28) and s" is also small for the same reason. Ofcourse, at COT = 0, we get e" = 0 as expected because this is dc voltage.
(2) For very large values of COT, e' = 800 and s" is small.(3) For intermediate values of frequencies s" is a maximum at some particular
value of COT.
The maximum value of s" is obtained at a frequency given by
Dielectric Relaxation-I 105
resulting in
(3.32)
where cop is the frequency at c"max.
Log m
Fig. 3.3 Schematic representation of Debye equations plotted as a function of logco. The peakof s" occurs at COT = 1. The peak of tan8 does not occur at the same frequency as the peak ofs".
The values of s' and s" at this value of COT are
106 Chapters
f' = ±^ (3-33)
~
The dissipation factor tan 5 also increases with frequency, reaches a maximum, and forfurther increase in frequency, it decreases. The frequency at which the loss angle is amaximum can also be found by differentiating tan 6 with respect to co and equating thedifferential to zero. This leads to
(3.35).8(<ot) "(e,+s^V)
Solving this equation it is easy to show that
®r = 4p- (3.36)
By substituting this value of COT in equation (3.30) we obtain
(3.37)
The corresponding values of s' and e" are
*' = -^- (3.38)
(3.39)
Fig. 3.3 also shows the plot of equation (3.30), that is, the variation of tan 8 as a functionof frequency for several values of T.
Dielectric Relaxation-I 107
Dividing equation (3.28) from (3.27) and rearranging terms we obtain the simplerelationship
s"(3.40)
s"According to equation (3.40) a plot of — against co results in a straight line
passing through the origin with a slope of T.
Fig. 3.3 shows that, at the relaxation frequency defined by equation (3.32) e' decreasessharply over a relatively small band width. This fact may be used to determine whetherrelaxation occurs in a material at a specified frequency. If we measure e' as a function oftemperature at constant frequency it will decrease rapidly with temperature at relaxationfrequency. Normally in the absence of relaxation s' should increase with decreasingtemperature according to equation (2.51).
Variation of s' as a function of frequency is referred to as dispersion in the literature ondielectrics. Variation of s" as a function of frequency is called absorption though the twoterms are often used interchangeably, possibly because dispersion and absorption areassociated phenomena. Fig. 3.4 shows a series of measured &' and e" in mixtures of waterand methanol4. The question of determining whether the measured data obey Debyeequation (3.31) will be considered later in this chapter.
3.4 Bi-STABLE MODEL OF A DIPOLE
In the molecular model of a dipole a particle of charge e may occupy one of two sites, 1or 2, that are situated apart by a distance b5. These sites correspond to the lowestpotential energy as shown in fig. 3.5. In the absence of an electric field the two sites areof equal energy with no difference between them and the particle may occupy any one ofthem. Between the two sites, therefore, there is a particle. An applied electric field causesa difference in the potential energy of the sites. The figure shows the conditions with noelectric field with full lines and the shift in the potential energy due to the electric fieldby the dotted line.
108 Chapter 3
"•
60
lOOX-90X -eox-60XSOX40X30X20X
40 H 10Xl
(• ox .Volume friction of w«ur
(b)
100
100
1000Fr*qu«ncr<UHz)
10000
1000Prequ«ney(tlHz)
10000
•: 00X w«Urb: SOX w«tere: 10X water
20 40 60 60 100
Fig. 3.4 Dielectric properties of water and methanol mixtures at 25°C. (a) Real part, s' (b)Imaginary part, s" (c) Complex plane plot of s* showing Debye relaxation (Bao et. al., 1996).(with permission of American Physical Society.)
The difference in the potential energy due to the electric field E is
i ~~02 =ebEcos0
Dielectric Relaxation-I 109
where 0 is the angle between the direction of the electric field and the line joining 1 and2. This model is equivalent to a dipole changing position by 180° when the charge movesfrom site 1 to 2 or from site 2 to 1. The moment of such a dipole is
f* = -eb
which may be thought of as having been hinged at the midpoint between sites 1 and 2.This model is referred to as the bistable model of the dipole. We also assume that 0 = 0for all dipoles and that the potential energy of sites 1 and 2 are equal in the absence of anexternal electric field.
Electric Field
a* 1
d I
position
Fig. 3.5 The potential well model for a dipole with two stable positions. In the absence of anelectric field (foil lines) the dipole spends equal time in each well; this indicates that there is nopolarization. In the presence of an electric field (broken lines) the wells are tilted with the'downside' of the field having a slightly lower energy than the 'up' side; this representspolarization.
I
HO Chapters
We assume that the material contains N number of bistable dipoles per unit volume andthe field due to interaction is negligible. A macroscopic consideration shows that thecharged particles would not have the energy to jump from one site to the other. However,on a microscopic scale the dipoles are in a heat reservoir exchanging energy with eachother and dipoles. A charge in well 1 occasionally acquires enough energy to climb thehill and moves to well 2. Upon arrival it returns energy to the reservoir and remain therefor some time. It will then acquire energy to jump to well 1 again.
The number of jumps per second from one well to the other is given in terms of thepotential energy difference between the two wells as
kT
where T is the absolute temperature, k the Boltzmann constant and A is a factor denotingthe number of attempts. Its value is typically of the order of 10~13 s"1 at room temperaturethough values differing by three or four orders of magnitude are not uncommon. It mayor may not depend on the temperature. If it does, it is expressed as B/T as found in somepolymers. If the destination well has a lower energy than the starting well then the minussign in the exponent is valid. The relaxation time is the reciprocal of Wi2 leading to
forw
The variation of T with T in liquids and in polymers near the glass transition temperatureis assumed to be according to this equation. Other functions of T have also beenproposed which we shall consider in chapter 5. The decrease of relaxation time withincreasing temperature is attributed to the fact that the frequency of jump increases withincreasing temperature.
3.5 COMPLEX PLANE DIAGRAM
Cole and Cole showed that, in a material exhibiting Debye relaxation a plot of e"against c', each point corresponding to a particular frequency yields a semi-circle. Thiscan easily be demonstrated by rearranging equations (3.28) and (3.29) to give
/~» \2 , f „! „ \1 _ (£S ~£ao)
Dielectric Relaxation-I 111
The right side of equation (3.41) may be simplified using equation (3.28) resulting in
(G"}2 + (8' - O2 = (*, - *.)(*' - O (3.42)
Further simplification yields
*'2-*'(*,+O + *A,+*'r2 = 0 (3-43)
Substituting the algebraic identity
£,£a,=-[(£3+£j2-(£s+£j2]
equation (3.43) may be rewritten as
(£' _ £t±f« )2 + (£»y = (£LZ .)2 (3 44)
G — G G ~\~ GThis is the equation of a circle with radius — — having its center at (— — ,0 ).
It can easily be shown that (SOD, 0) and (es, 0) are points on the circle. To put it anotherway, the circle intersects the horizontal axis (s') at Soo and ss as shown in fig. 3.6. Suchplots of s" versus e' are known as complex plane plots of s*.
At (Dpi = 1 the imaginary component s" has a maximum value of
The corresponding value of s' is
Of course these results are expected because the starting point for equation (3.44) is theoriginal Debye equations.
112 Chapter 3
001?= 1
increases
( e,+ e J/2
Fig. 3.6 Cole-Cole diagram displaying a semi-circle for Debye equations for s*.
In a given material the measured values of s" are plotted as a function of s' at variousfrequencies, usually from w = 0 to eo = 1010 rad/s. If the points fall on a semi-circle wecan conclude that the material exhibits Debye relaxation. A Cole-Cole diagram can thenbe used to obtain the complex dielectric constant at intermediate frequencies obviatingthe necessity for making measurements. In practice very few materials completely agreewith Debye equations, the discrepancy being attributed to what is generally referred to asdistribution of relaxation times.
The simple theory of Debye assumes that the molecules are spherical in shape andtherefore the axis of rotation of the molecule in an external field has no influence indeciding the value of e . This is more an exception than a rule because not only themolecules can have different shapes, they have, particularly in long chain polymers, alinear configuration. Further, in the solid phase the dipoles are more likely to beinteractive and not independent in their response to the alternating field7. The relaxationtimes in such materials have different values depending upon the axis of rotation and, asa result, the dispersion commonly occurs over a wider frequency range.
Dielectric Relaxation-I 113
3.6 COLE-COLE RELAXATION
Polar dielectrics that have more than one relaxation time do not satisfy Debye equations.They show a maximum value of e" that will be lower than that predicted by equation(3.34). The curve of tan 5 vs log COT also shows a broad maximum, the maximum valuebeing smaller than that given by equation (3.37). Under these conditions the plot e" vs. s'will be distorted and Cole-Cole showed that the plot will still be a semi-circle with itscenter displaced below the s' axis. They suggested an empirical equation for thecomplex dielectric constant as
~ , ; 0 < a < l ; (3.45)\l — c/ ~ ~ \. s]
a — 0 for Debye relaxation
where ic.c is the mean relaxation time and a is a constant for a given material, having avalue 0 < a < 1. A plot of equation (3.45) is shown in figs. (3.7) and (3.8) for variousvalues of a. Debye equations are also plotted for the purpose of comparison. Nearrelaxation frequencies Cole-Cole relaxation shows that s' decreases more slowly with cothan the Debye relaxation. With increasing a the loss factor e" is broader than the Debyerelaxation and the peak value, smax is smaller.
A dielectric that has a single relaxation time, a = 0 in this case, equation (3.45) becomesidentical with equation (3.29). The larger the value of a, the larger the distribution ofrelaxation times.
To determine the geometrical interpretation of equation (3.45) we substitute l-a = n andrewrite it as
(3-46)/ vi, • • ^(o)Tc_c) (cos— + 7 sin— )
Equating real and imaginary parts we get
114 Chapter 3
(3.47)
2(a)Tc_c)" cos( wr / 2) v2« (3.48)
Fig. 3.7 Real part of s* in a polar dielectric according to Cole-Cole relaxation, a =0 givesDebye relaxation.
Using the identity
ja =
o
equations (3.47) and (3 .48) may be expressed alternatively as
s -, sinhn-scosh «5 + si / 2)
(3.46a)
Dielectric Relaxation-I 115
ss -
s" _ 1 ( cos(cr;r/2)
2\^ cosh ns + sin(a;r / 2)(3.47a)
where
5 = Lncor
X-- 11=0.75--""- n=l
(Dtbve)
Jtl HF 10
Fig. 3.8 Imaginary part of s* in a polar dielectric according to Cole-Cole relaxation, a =0 givesDebye relaxation.
Eliminating OOTC.C from equations (3.47) and (3.48) Cole-Cole showed that
(3.49)
Equation (3.49) represents the equation of a circle with the center at
116 Chapter 3
and having a radius of
We note that the y coordinate of the center is negative, that is, the center lies below the s'axis (fig. 3.9).
Figs. 3.7 and 3.8 show the variation of s' and e" as a function of cox for several values ofa respectively. These are the plots of equations (3.47) and (3.48). At COT = 1 thefollowing relations hold:
„ £• - £• nn8 = — — tan —
/2, cot(n7T/2)x(- es+eJ/2
Fig. 3.9 Geometrical relationships in Cole-Cole equation (3.45).
Dielectric Relaxation-I 117
As stated above, the case of n = 0 corresponds to an infinitely large number ofdistributed relaxation times and the behavior of the material is identical to that under dcfields except that the dielectric constant is reduced to (ss - Soo)/2. The complex part of thedielectric constant is also equal to zero at this value of n, consistent with dc fields. As thevalue of n increases s' changes with increasing COT, the curves crossing over at COT = 1. Atn=l the change in s' with increasing COT is identical to the Debye relaxaton, the materialthen possessing a single relaxation time.
The variation of s" with COT is also dependent on the value of n. As the value of nincreases the curves become narrower and the peak value increases. This behavior isconsistent with that shown in fig. 3.8.
Let the lines joining any point on the Cole-Cole diagram to the points corresponding toSoo and ss be denoted by u and v respectively (Fig. 3.9). Then, at any frequency thefollowing relations hold:
oo ' / Mu = s -s- v = r^; — = (COT}-c / \ I— n ^ '(COT) u
By plotting log co against (log v-log u) the value of n may be determined. Withincreasing value of n, the number of degrees of freedom for rotation of the moleculesdecreases. Further decreasing the temperature of the material leads to an increase in thevalue of the parameter n.
The Cole-Cole diagrams for poly(vinyl chloride) at various temperatures are shown infig. 3.109. The Cole-Cole arc is symmetrical about a line through the center parallel tothe s" axis.
3.7 DIELECTRIC PROPERTIES OF WATER
Debye relaxation is generally limited to weak solutions of polar liquids in non-polarsolvents. Water in liquid state comes closest to exhibiting Debye relaxation and itsdielectric properties are interesting because it has a simple molecular structure. One isfascinated by the fact that it occurs naturally and without it life is not sustained. Hasted(1973) quotes over thirty determinations of static dielectric constant of water, alreadyreferred to in chapter 2.
118 Chapter 3
n
s'
Fig. 3.10 Cole-Cole diagram from measurements on poly (vinyl chloride) at various temperatures(Ishida, 1960). (With permission of Dr. Dietrich Steinkopff Verlag, Darmstadt, Germany).
The dielectric constant is not appreciably dependent on the frequency up to 100 MHz.The measurements are carried out in the microwave frequency range to determine therelaxation frequency, and a particular disadvantage of the microwave frequency is thatindividual observers are forced, due to cost, to limit their studies to a narrow frequencyrange. Table 3.1 summarizes the data due to Bottreau et. al. (1975).
Dielectric Relaxation-1 119
Table 3.1Selective Dielectric Properties of Water at 293 K (Bottreau, et. al., 1975).
800 = n2= 1.78, es = 80.4(With permission of Journal of Chemical Physics, USA)
a single relaxation of Debye type0.00260.00770.0188
0.00950.01460.0184
0.00210.00710.0179
0.00960.01670.0227
The following definitions apply for the quantities in shown Table 3.1.
'=^^; E* =£ ~ 8
120 Chapter 3
10and is reproduced from ref. . Fig. 3.11 shows the complex plane plots of e' - js" forwater (Hasted, 1973) and compared with analysis according to Debye equations andCole-Cole equations. The relaxation time obtained as a function of temperature fromCole-Cole analysis is shown in Table 3.2 along with £w used in the analysis.
Fig. 3.11 Complex plane plot of s* in water at 25°C in the microwave frequency range. Pointsin closed circles are experimental data, x, calculations from Cole-Cole plot, +, calculationsfrom Debye equation with optimized parameters [Hasted 1973]. (with permission of Chapman& Hall, London).
Dielectric Relaxation-I 121
Earlier literature on c* in water did not extend to as high frequencies as shown in Table3.1 and it was thought that £<» is much greater than the square of refractive index, n — 1.8(Hasted, 1973), and this was attributed to, possibly absorption and a second dipolardispersion of e" at higher frequencies. However more recent measurements up to 2530GHz and extrapolation to 13760 GHz shows that the equation 800 = n2 is valid, asdemonstrated in Table 3.1. The relaxation time increases with decreasing temperature inqualitative agreement with the Debye concept. The Cole-Cole parameter a is relativelysmall and independent of temperature. Recall that as a—» 0 the Cole-Cole distributionconverges to Debye relaxation.
At this point it is appropriate to introduce the concept of spectral decomposition of thecomplex plane plot of s*. If we suppose that there exist several relaxation processes,each with a characteristic relaxation time and dominant over a specific frequency range,then the Debye equation (3.31) may be expressed as
+ CO Tj
•an, (3.51)
where Acs and i\ are the individual amplitude of dispersion (siow frequency - Shigh frequency) andthe relaxation time, respectively. The assumption here is that each relaxation processfollows the Debye equation independent of other processes.
This kind of representation has been used to find the relaxation times in D2O ice11.Polycrystalline ice from water has been shown to have a single relaxation time of Debyetype at 270 K12 and the observed distribution of relaxation times at lower temperatures165-196 K is attributed to physical and chemical impurities13. However the D2O iceshows a more interesting behavior. Focusing our attention to the point under discussion,namely several relaxation times, fig. 3.12 shows the measured values of s' and s" in thecomplex plane as well as the three relaxation processes. The As and i are 88.1, 57.5 and1.4 (see inset) and, 20 ms, 60 ms and 100 us respectively.
A method of spectral analysis which is similar in principle to what was described above,but different in procedure, has been adopted by Bottreau et. al. (1975) who use a functionof the type
122 Chapter 3
(3.52)
where Q is the spectral contribution to ith region and <DI is its relaxation frequency. Thecondition EQ = 1 should be satisfied.
is 25 as 45
was
,..i -t-
\t
Mil. i ., i _ i J
fVSif
80
Fig. 3.12 The analysis of Cole-Cole plots into three Debye-type relaxation regions indicatedby semi-circles at 191.8 K. The numbers beside the filled data points are frequencies in kHz.Closed circles: low frequency bridge measurements; Open circles: high frequencymeasurements [11]. (with permission of the Royal Society, England).
This scheme was applied to H2O data shown in Table 3.1. The results obtained areshown in Table 3.3. Three Debye regions are identified with relaxation times as shown.Application of Cole-Cole relaxation (3.45) equation yields a value of cop = 107 x 109 rads"1 with a = 0.013 which agrees with the major region of relaxation in Table 3.3.
3.8 DAVIDSON - COLE EQUATION
Davidson - Cole14 have suggested the empirical equation
•J^d-cY(3.53)
Dielectric Relaxation-I 123
where 0 < P < 1 is a constant characteristic of the material.
Separating the real and imaginary parts of equation (3.53), the real and complex parts areexpressed as
'-ea>=(es-eao )(cos ( cos
s" - (ss - sx )(cos sn
(3.54)
(3.55)
where tan (j) = COTO.
Table 3.3Spectral contributions and relaxation frequencies of the three Debye constituents
of water at 20° C [Bottreau et. al. 1975].
RegionIIIIII
Ci0.05070.91360.0357
/ (GHz)5.57 ±0.5017.85 ±0.303440.3±8.0
(with permission of J. Chem. Phys.)
These equations are plotted in Figs. 3.13 and 3.14 and the Debye curves (P = 1) are alsoshown for comparison. The low frequency part of s' remains unchanged as the value ofP increases from 0 to 1. However the high frequency part of s' becomes lower as P isincreased, P = 1 (Debye) yielding the lowest values.
Similar observations hold gold for B" which increases with P in the low frequency partand decreasing with P in the high frequency part. The main point to note is that the curveof s" against COT loses symmetry on either side of the line that is parallel to the s" axisand that passes through its peak value.
Expressing equations (3.54) and (3.55) in polar co-ordinates
124 Chapter 3
Davidson-Cole show, from equations (3.54) and (3.55) that
tan 0 = tan
(3.56)
(3.57)
(3.58)
3,0
Fig. 3.13 Schematic variation of e' as afunction of COT for various values of p. Thelow frequency value of s' has beenarbitrarily chosen.
Fig. 3.14 Schematic variation of s" as afunction of COT for various values of p. Thevalue of T has been arbitrarily chosen.
The locus of equation (3.53) in the complex plane is an arc with intercepts on the s' axisat ss and £«> at the low frequency and high frequency ends respectively (fig. 3-15). Asoo—>0 the limiting curve is a semicircle with center on the s' axis and as co—>QO the
Dielectric Relaxation-I 125
limiting straight line makes an angle of $ii/2 with the e' axis. To explain it another way,at low frequencies the points lie on a circular arc and at high frequencies they lie on astraight line.
If Davidson-Cole equation holds then the values of ss , So, and (3 may be determineddirectly, noting that a plot of the right hand quantity of eq. (3.54) against co must yield astraight line. The frequency oop corresponding to tan (9/(3) =1 may be determined and Tmay also be determined from the relation cop T We quote two examples todemonstrate Davidson-Cole relaxation in simple systems. Fig. 3-16 shows the measuredloss factor in glycerol (b. p. 143-144°C at 300 Pa), over a wide range of temperature and
frequency . The asymmetry about the peak can clearly be seen and in the highfrequency range, to the right of the peak at each temperature, a power law, co"'3 ((3<1)holds true.
Fig. 3.15 Complex plane plot of s* according to Davidson-Cole relaxation. The loss peak isasymmetric and the low frequency branch is proportional to ro. The slope of the highfrequency part depends on p.
The second example of Davidson-Cole relaxation is in mixtures of water and ethanol[Bao et. al., 1996] at various fractional contents of each liquid, as shown in fig. 3-17. TheDavidson-Cole relaxation is found to hold true though the Debye relaxation may also beapplicable if great accuracy is not required. The methods of determining the type ofrelaxation is dealt with later, but, as noted earlier, the Davidson-Cole relaxation isbroader than the Debye relaxation depending upon the value of (3.
We need to deal with an additional aspect of the complex plane plot of s* which is due tothe fact that conductivity of the dielectric introduces anomalous increase of e" at both
126 Chapter 3
the high frequency (see the inset in fig. 3-12) and low frequency ends of the plots16 (fig.3.18). Equation (3.8) shows the contribution of ac conductivity to e" and thiscontribution should be subtracted before deciding upon the relaxation mechanisms.
1QQ
0.01
0.001
Fig. 3.16 s" as a function of ro in glycerol at various temperatures (75, 95, 115, 135, 175, 185,190, 196, 203, 213, 223, 241, 256, 273 and 296 K) [15]. (with permission of J. Chem. Phys.,USA).
3.9 MACROSCOPiC RELAXATION TIME
The relaxation time is a function of temperature according to a chemical rate processdefined by
T = r0 expkT
(3.59)
in which TO and b are constants. This is referred to as an Arrhenius equation in theliterature.
There is no theoretical basis for dependence of x on T and in some liquids such as thosestudied by Davidson and Cole (1951) the relaxation time is expressed as
Dielectric Relaxation-I 127
T = r0 expb
k(T-Tc)(3.60)
where Tc is a characteristic temperature for a particular liquid.
ao
60
40
20
- i-90% waterSOX water10X water
,1
40
30
20
10
0
100 1000Frequency(UHz)
10000
a: 90% waterb: SOX waterc: 10% water
too 1000Frequency(UHz)
10000
a: COX waterb: SOX watere: 10X water
100
Fig. 3.17 Dielectric properties of water-ethanol mixtures at 25°C. (a) Real part s' (b)Imaginary part, s" (c) Complex plane plot of s* exhibiting Davidson-Cole relaxation [Bao et.al. 1996] (with permission of American Inst. of Physics).
128 Chapter 3
In some liquids the viscosity and measured low field conductivity also follow a similarlaw, the former given by
k(T-T) (3.61)
At T - Tc the relaxation time is infinity according to equation (3.60) which must beinterpreted as meaning that the relaxation process becomes infinitely slow as weapproach the characteristic temperature. Fig. 3.19 (Johari and Whalley, 1981) shows theplots of T against the parameter 1000/T in ice. The slope of the line gives an activationenergy of 0.58 eV, a further discussion of which is beyond the scope of the book. Wewill make use of equation (3.60) in understanding the behavior of amorphous polymersnear the glass transition temperature TG in chapter 5.
75
80-
25
0SO e' 100
Fig. 3.18 Complex plane plot of s* in ice at 262.2 K. (a) sample with interface parallel to theelectrodes; Curve (b), true locus; curves (c) and (d), samples with electrode polarization arisingfrom dc conductance. Numbers beside points are frequencies in kHz (Auty and Cole, 1952). (withpermission from Am. Inst. Phys.).
The observed correspondence of T with viscosity is qualitatively in agreement with themolecular relaxation theory of Debye17 who obtained the equation
Dielectric Relaxation-I 129
T =3r/u
(3.62)
where Tm is called the molecular relaxation time (see next section), r| the viscosity, v themolecular volume (= 47ia3/3 where a is the molecular radius) , assumed spherical. Themolecular volume required to obtain agreement with the relaxtion times is too small forglycerol and propylene glycol (~ 10~31m3) and of reasonable size for w-propanol (60-900xlO-30m3).
101-1
COA
<U
10-2
I 10-3
10-4
10-5
*UT»
4.0 5.0 6.0
1000/T (K-1)
Fig. 3.19 Arrhenius plot of the relaxation time of ice I plotted as a function of 1/T (Johari andWhalley, 1981). (with permission of Am. Inst. Phys.).
The spread in these values arises due to the fact that the molecular relaxation time, TD,and the relaxation time, T, obtained from the dielectric studies may be related in severalways. The inference is that for the first two liquids the units involved are much smaller
130 Chapters
where as for the third named liquid the unit involved may be the entire molecule. Thesedetails are included here to demonstrate the method employed to obtain insight into therelaxation mechanism from measurements of dielectric properties.
3.10 MOLECULAR RELAXATION TIME
The relaxation time T obtained from dielectric studies that we have discussed so far is amacroscopic quantity and it is quite different from the original relaxation time rm used byDebye. im is a microscopic quantity and usually called the internal relaxation time or themolecular relaxation time. im may be expressed in terms of the viscosity of the liquid andthe temperature as
(3-63)
where a is the molecular radius. The molecular relaxation time is assumed to be due tothe inner friction of the medium that hinders the rotation of polar molecules. Hence rm isa function of viscosity. As an example of applicability of equation (3.63) we considerwater that has a viscosity of 0.01 Poise at room temperature and an effective molecularradius of 2.2 x 10"'° m leading to a relaxation time of 2.5 x 10"ns. At relaxation, thecondition COT = 1 is satisfied and therefore co = 4xl010 s"1 which is in reasonableagreement with relaxation time obtained from dielectric studies. Equation (3.63) ishowever expected to be valid only approximately because the internal friction hinderingthe rotation of the molecule, which is a molecular parameter, is equated to the viscosity,which is a macroscopic parameter.
It is well known that the viscosity of a liquid varies with the temperature according to anempirical law:
in which c is a constant for a given liquid. Therefore eq. (3.63) may now be expressed as
1 crmcc-exp— (3.65)T kT
Dielectric Relaxation-I 131
The relaxation time increases with decreasing temperature, as found in many substances,(see Table 3.2).
3.11 STRAIGHT LINE RELATIONSHIPS
There are many convenient methods for measuring the relaxation time experimentally"I ft
and the formulas for calculating T have been summarized by Hill et. al . Let COT = 1 andn2 = 800. Equation (3.28) and (3.29) may be expressed in several alternative ways:
s'-n2 11.
ss-n 1 + x
s"
s-n 1 +•S
Dividing the second equation from the first
x= S . (3.66)s' -n
2. It is easy to show that
Equation (3.66) shows that a graph of e"/x against e' will be a straight line. The graph ofs"x as a function of z' will also be linear, n2 and ss are obtained from the respectiveintercepts and T from the slope.
The equations may also be written as :
~ x x2 13. — = - +s" ss-n
2 ss-n2
4 _ = L_e'x (e,-n2
132 Chapter3
5 1 _ x2 Is' -n2 ss- n2 ss - n2
6.
e,-e'
Equations (3)-(6) above have the advantage that they each involve only one of theexperimentally measured values of e' and s", but the disadvantage is that the frequencyterm, x, enters through a squared term distorting the frequency scale. The advantage ofthese relationships lie in the fact that they may be used to check the standard deviationbetween computed values and measured values using easily available software.
The relation between the molecular relaxation time, im; and the macroscopic relaxationtime, i, is given by (Hill et. al.)
n +2
The macroscopic relaxation time is higher in all cases than the microscopic relaxationtime.
3.12 FROHLICH'S ANALYSIS
It is advantageous at this point to consider the dynamical treatment of Frohlich19 whovisualized a relaxation time that is dependent upon the temperature according to equation(3.63). To understand Frohlich's model let us take the electric field along the + axis andsuppose that the dipole can orient in only two directions; one parallel to the electric field,the other anti-parallel. Let us also assume that the dipole is rigidly attached to amolecule. The energy of the dipole is +w when it is parallel to the field, -w when it isanti-parallel, and zero when it is perpendicular. As the field alternates the dipole rotatesfrom a parallel to the anti-parallel position or vice-versa. Only two positions are allowed.A rotation through 180° is considered as a jump.
Frohlich generalized the model which is based on the concept that the activation energiesof dipoles vary between two constant values, W] and w2. Frohlich assumed that each
Dielectric Relaxation-I 133
process obeyed the Arrhenius relationship and the relaxation times corresponding to w}
and w2 are given by
r, = T0ekT
-.kT
The dipoles are distributed uniformly in the energy interval dw and make a contributionto the dielectric constant according to Jm and H^. The analysis leads to fairly lengthyexpressions in terms of the difference in the activation energy w2 - wj, and the finalequations are
J =(a
1—/•»25
rV
1= -(e2 tan x-e2 tan x)
s(3.68)
where
s =
We have used here the form of expressions given by Williams20 because they have theadvantage of using cop which is the radian frequency at which s" is a maximum. Since s isa function of temperature the shape of the Jffl and H^ curves will vary with temperature,tending towards a single relaxation time at higher temperatures. cop is related to TI and
1 ~I^ 1
m=—e 2kT =-—-
Fig. 3.20 shows the factor e"/e"maxas a function of co/cop calculated according to
134 Chapter 3
_, o) N r7 , , o)(tan ' —) -'--(tan1 —
cop \T! ^
tan -tan
(3.69)
The width of the loss curve increases with increasing (w2 -wj), and in the limit therelaxation time will vary from zero to infinity and s" will have a constant value over theentire frequency range.
Fig. 3.20 Dependence of dielectric loss s"(co) on co according to equation (3.69) for threevalues of the parameter V(t2/ TI) = 1, 5, 10. They correspond to a range of heights of thepotential barrier. cop is the frequency of £"max [1986]. (with permission of Clarendon Press,Oxford.)
The double potential well of Frohlich leads to a relaxation function that shows the peakof the e" - co plots are independent of the temperature. However, measurements of s" in awide range of materials show that the peak increases with increase in T. For examplemeasurement of dielectric loss in polyimide having adsorbed water shows such abehavior21
The temperature dependence of s"max in the context of Frohlich's theory is oftenexplained by assuming asymmetry in the energy level of the two positions of the dipole.At lower temperatures the lower well is occupied and the higher well remains empty.The number of dipoles jumping from the lower to higher well is zero. However, withincreasing temperature the number of dipoles in the higher well increases until both wells
Dielectric Relaxation-I 135
are occupied equally at kT» (w2-wi). Therefore there will be a temperature range,depending upon the energy difference, in which s" increases strongly. According to thismodel the loss factor is
1
3kT-cosh-2 W,
2kT
where (w2-Wi) is the difference in the asymmetry of the two positions. Plots of /(T)versus T for various values of V are shown in the range of V = 0 to 100 meV (fig. 3.21).By comparing the observed variation of s" vs T it is possible to estimate the averageenergy of asymmetry between the two wells.
T C K 3
Fig. 3.21 Temperature dependance of the relaxation strength calculated for various values of theasymmetry potential wi-W2. The upper curve corresponds to the symmetric case wi-wa = 0(Melcher et. al., 1989). (with permission of Trans. IEEE on Diel. and El. Insul.)
3.13 FUOS8-KIRKWOOD EQUATION
j22The Fuoss-Kirkwood dispersion equation is
136 Chapters
r (3-70)
Let us denote the frequency at which the loss factor is a maximum for Debye relaxationby CDP. Then the loss factor in terms of its maximum value, s"maX; is
2s"cn _ ^bmiCs
(Dp CO
which leads to
s" =
In the Fuoss-Kirkwood derivation this expression becomes
The empirical factor 8, between 0 and 1, is related to the Cole-Cole parameter aaccording to
\-a
cos
This expression shows that Fuoss-Kirkwood relation holds true for most materials inwhich the Cole-Cole relaxation is observed; only the value of the parameter will bedifferent.
When e" =1/2 e"max expression (3.70) leads to (coT)6= 2 ± V3. To apply Fuoss-Kirkwoodequation we plot arccosh(e'Yc') against (lna>) to get a straight line. This line intersects thefrequency axis at co = <DP with a slope of 8 which is related to the static permittivity inaccordance with23
Dielectric Relaxation-L 137
? _ 2s"
Fig. 3.22 shows such a plot for PMMA at 353K and the evaluated parameter (ss - 800)has value of =2.
Fig. 3.22 The dependence of arccosh(s"max/s") on log(o)) for PMMA at 353 K. T is the
relaxation time. Slope gives the value of (3. From the slope (SS-SQO) is evaluated as =2 [Mazur,19971. (with permission of Institute of Physics).
Kirkwood and Fuoss24 also derived the relaxation time distribution function for a freelyjointed polyvinyl chloride (PVC) chain, including the molecular weight distribution.Recall that the number of monomer units in a polymer molecule is not a constant andshows a distribution. The theory takes into account the hydrodynamic diffusion of chainsegments under an electric field (Williams, 1963). Polyvinyl chloride andpoly(acetaldehyde) show common characteristics of the dipole forming a rigid part of thechain backbone. The Kirkwood-Fuoss relation is
138 Chapters
, .„ '({I + u + u(u + 2}euEi(-u}][\ - jxu]duJa,-J
Ha>=\ : ~2 (-3-71)J 1 + x u
here u — n / <n>, n is the degree of polymerization, <n> its average value, Ei(u) theexponential integral, x = < n>a>T*, T* is the relaxation time of the monomer unit. Hro
reaches its peak value at x = 0.1 x 2ju and is symmetrical about this value. The mainfeature of Kirkwood-Fuoss distribution is that it is free of any empirical parameterbecause Jm and H^ are expressed in terms of log x.
The shape and height of no-parameter distribution of Kirkwood-Fuoss is independent oftemperature and this fact explains the reason for their theory not holding true for PVC.The distribution in PVC is markedly temperature dependent (Kirk and Fuoss, 1941).
Many polymers exhibit a temperature dependent distribution and at higher temperaturessome are noticeably temperature independent. At these higher temperatures the Cole-Cole parameter has an approximate value of 0.63 (Kirkwood and Fuoss, 1941). It seemslikely that the Kirkwood-Fuoss relation holds when the shape of the distribution isindependent of the temperature.
3.14 HAVRILIAK AND NEGAMI DISPERSION
We are now in a position to extend our treatment to more complicated molecularstructures, in particular polymer materials. The dispersion in small organic or inorganicmolecules is studied by measuring the complex dielectric constant of the material atconstant temperature over as wide a range of frequency as possible. The temperature isthen varied and the measurements repeated till the desired range of temperature iscovered. From each set of isothermal data the complex plane plots are obtained andanalyzed to check whether a semi-circular arc in accordance with Cole-Cole equation isobtained or whether a skewed arc in accordance with the Davidson-Cole equation isobtained.
The complex plane plots of polymers obtained by isothermal measurements do not lendthemselves to the simple treatment that is used in case of simple molecules. The mainreasons for this difficulty are: (1) The dispersion in polymers is generally very broad sothat data from a fixed temperature are not sufficient for analysis of the dispersion. Datafrom several temperatures have to be pooled to describe dispersions meaningfully. (2)The shapes of the plots in the complex plane are rarely as simple as that obtained with
Dielectric Relaxation-I 139
molecules of simpler structure rendering the determination of dispersion parameters veryuncertain.
In an attempt to study the a-dispersion in many polymers, Havriliak and Negami25 havemeasured the dielectric properties of several polymers, a-dispersion in a polymer is theprocess associated with the glass transition temperatures where many physical propertieschange in a significant way. In several polymers the complex plane plot is linear at highfrequencies and a circular arc at low frequencies. Attempts to fit a circular arc (Cole-Cole) is successful at lower frequencies but not at higher frequencies. Likewise, anattempted fit with a skewed circular arc (Davidson-Cole) is successful at higherfrequencies but not at lower frequencies.
The two dispersion equations, reproduced here for convenience, are represented by:
Combining the two equations, Havriliak and Negami proposed a function for thecomplex dielectric constant as
(3.72)
This function generates the previously discussed relaxations as special cases. When p= 1 the circular arc shown above is generated. When a = 0 the skewed semicircle isobtained. When a = 0 and (3 = 1 the Debye function is obtained. For convenience weomit the subscript H-N hereafter.
To test the relaxation function given by equation (3.72) we apply successively theDeMoivre's theorem and rationalize the denominator to obtain the expressions
(3.73)
140 Chapters
(3.74)
where
r = [1 + (fiw)1-" s i n ( ) ] 2 + [(ow)- c o s ( ) ] (3.75)
l-a cos(—)0 = arc tan [ - 2 - ] (3 .76)
1x \ 1 — /y • x
+ (&>r) sm(
Equation (3.72) may be examined for extreme values of co, namely co — »co and co — »026.
For the first case, as co — »oo, equation (3.72) becomes (for definition of /" see section3.15)
At very high frequencies c" oc (e'-Eoo) oc cop(1"a)
For the second case, as co—>>0,
= 1-P(CDTH}(\-c
(antan —
I 2 ,
Dielectric Relaxation-I 141
At very low frequencies, e" <x (SS-E') oc co(1~a). These results have led to the suggestionthat the susceptibility functions have slopes as shown in the two extreme cases.
From equations (3.73) and (3.74) we note the following with regard to the dispersionparameterI. As COT -> co, &' -> SQO and e"-> 0. Therefore 8* = 800
II. As COT —> 0, e' -» esand e" -> 0. Therefore s* —>> ss.
We can therefore evaluate es and Soo from the intercept of the curve with the real axis. Tofind the parameters a and (3 we note that equations (3.73) and (3.74) result in theexpression
s"= tan J30 = tang> (3.77)
where we have made the substitution (30 = (|). By applying the condition COT —> oo toequation (3.76) and denoting the corresponding value of (j) as (j)L (see fig. 3.23) we get
fa=(l-a)fr/2 (3.78)
which provides a relation between the graphical parameter <j)L and the dispersionparameters, a and [3. Again the relaxation time is given by the definition COT = 1, and letus denote all parameters at this frequency by the subscript p. Havriliak and Negami(1966) also prove that the bisector of angle <j)L intersects the complex plane plot at sp*.The point of intersection yields the value of a by
*— >-log[2 + 2sina(;r/2)] (3.79)
The analysis of experimental data to evaluate the dispersion parameters is carried out bythe following procedure from the complex plane plots:
1. The low frequency measurements are extrapolated to intersect the real axis fromwhich ss is obtained.
142 Chapter3
2. The high frequency measurements are extrapolated to intersect the real axis fromwhich 800 is obtained. If data on refractive index is available then the relation Soo = n2
may be employed in specific materials.3. The parameter q>L is measured using the measurements at high frequencies.4. The angle q>L is bisected and extended to intersect the measured curve. From the
intersection point the frequency is determined and the corresponding relaxation timeis calculated according to co = I/T. The parameters ep' - s^ and e" are also determined.
5. The parameter a is decided by equation (3.79)6. The parameter P is calculated by equation (3.78)
Havriliak and Negami analysed the data of several polymers and evaluated the fivedispersion parameters (es, £«, , a, (3, T) for each one of them (see chapter 5). A morerecent list of tabulated values is given in Table 3.327. The Havriliak and Negami functionis found to be very useful to describe the relaxation in amorphous polymers whichexhibit asymmetrical shape near the glass transition temperature, TG. In the vicinity of TG
the e"-logco curves become broader as T is lowered. It has been suggested that the a-parameter represents a quantity that denotes chain connectivity and P is related to thelocal density fluctuations. Chain connectivity in polymers should decrease as thetemperature is lowered. The a-parameter slowly increases above TG which may beconsidered as indicative of this28. A detailed description of these aspects are treated in ch.5.
Fig. 3.23 Complex plane plot of s* according to H-N function. At high frequencies the plot islinear. At low frequencies the plot is circular (Jonscher, 1999).
A final comment about the influence of conductivity on dielectric loss is appropriatehere. As mentioned earlier measurement of e" - co characteristics shows a sudden rise inthe loss factor towards the lower frequencies, (e. g., see fig. 3.12) and this increase is due
Dielectric Relaxation-I 143
to the dc conductivity of the material. The Havriliak-Negami function is then expressedas
where adc is the dc conductivity.
3.15 DIELECTRIC SUSCEPTIBILITY
In the published literature some authors29 use the dielectric susceptibility, %* = %' - j%"of the material instead of the dielectric constant s* and the following relationshipshold between the dielectric susceptibility and dielectric constant:
/=**-*. (3-80)
*' = *'-*„ (3-81)
Z" = e" (3.82)
The last two quantities are often expressed as normalized quantities,
s' — s s"?\s — • >y —
S\s ? As— c
°
Equations (3.81) and (3.82) may also be expressed in a concise form as
1X oc
•7
where cop is the peak at which %" is a maximum. Alternately we have
*x 1 = 11 + JCOT 1 + G)2T2 I + ,
(3.83)
144 Chapters
Equation (3.83) is known as the Debye Susceptibility function. Expressing thedielectric properties in terms of the susceptibility function has the advantage that theslopes of the plots of %'and %" against co provide a convenient parameter for discussingthe possible relaxation mechanisms. Fig. 3.24 (a) shows the variation of %' and %" as afunction of frequency30. As discussed, in connection with the Debye equations, thedecreasing part of %' <x co"2. The increasing part of %" at co « cop changes in proportionto co+1 and the decreasing part of %" at co » cop changes in proportion to co"1.
Table 3.4Selected Dispersion Parameters according to H-N Expression
(Havriliak and Watts, 1986: with permission of Polymer)
which is broader than the Debye function, though both are symmetrical about thefrquency at which maximum loss occurs. The Davidson-Cole has the form
Dielectric Relaxation-I 145
X**- - - (3.85)
These susceptibility functions are generalized and the relaxation mechanism is expressedin terms of two independent parameters, a and P, by Havriliak and Negami, theirsusceptibility function having the form:
Figs. 3.24 (b) and (c) show the variation of these functions as a function of frequency.The similarities and divergences of these functions may be summarized as follows:
1 . The Debye function is symmetrical about cop and narrower than the Cole-Cole andDavidson-Cole functions. The low frequency region of the dispersion has a slope thatis proportional to CD and the high frequency region has a slope that is proportional to1/co. In this context the low frequency and high frequency regions are defined as co «cop and co » cop In the high frequency region %' has a slope proportional to 1/co2 (Fig.3. 24 a).
2. The Cole-Cole function (Fig. 3.24b) shows that %" has a slope proportional to co1"" inthe low frequency region and a slope proportional to co™"1 in the high frequency region.It is also symmetrical about cop. The variation of %' in the high frequency region isalso proportional to coa " L
3. The Davidson-Cole susceptibility function (Fig. 3.24 c) is asymmetrical about thevertical line drawn at cop. In the low frequency region %" is proportional to co which isa behavior similar to that of Debye. In the high frequency region %" is proportional toco"13. The real part of the susceptibility function, %', in the high frequency region is alsoproportional to co"13. It should be borne in mind that the Cole-Cole and Davidson -Colesusceptibility functions have a single parameter, whereas the Debye function has none.
To demonstrate the applicability of Davidson-Cole function we refer to the measurementof dielectric properties of glycerol by Blochowicz et. al. (1999) over a temperature rangeof 75-296 K and a frequency range of 10'2 </< 3000 MHz. A plot e"(/) as a function oflog (/) is shown in fig. 3.16. At /</max, s" behaves similar to a Debye function but at/>fmax, there are two slopes, -P and -y appearing in that order for increasing frequency.
146 Chapters
Since Davidson-Cole function uses one parameter only a modification to equation (3.85)is proposed by Blochowicz et. al (1999).
Here TO - l/cop and C0 is a parameter that controls the frequency of transition from (3-power law to y-power law.
The condition (3 = y signifying a single slope for the log^e" yields Davidson-Colefunction because the two slopes merge into one. The mean relaxation time is obtained bythe relation
CDThe temperature dependence of P, y and C0 is shown in their fig. 4(b). (3 is weaklydependent on T whereas y increases rapidly to approach (3.
At the glass transition temperature the two power laws merge into one in accordancewith equation (3.85). The relaxation in the glass phase (T < Tg) is determined by a singlepower law over a wide frequency range of 10~2 < f < 105 Hz. Below Tg the co-efficient yis not temperature dependent and very similar for many systems exhibiting this type ofbehavior. For glycerol y = 0.07 ± 0.02. The behavior below T « Tg occurs according toX" = co y- The high frequency contribution of a-relaxation is frozen out at T « Tg. If thedata on %" is replotted in the T domain at/= 1 Hz an exponential relationship is obtainedaccording to
=e
where Tf is a constant that is dependent on the material. Their fig. (6a) shows thisrelationship for many substances, the departure from this equation being due to the onsetof the a-process.
4. The Havriliak-Negami function is more general because of the fact that it has twoparameters. The comments made with regard to equations(3.73) and (3.74) are equally
Dielectric Relaxation-I 1 47
applicable to their susceptibility functions; we get the three functions listed above asspecial cases:
(i) a = 0 and |3=1 gives the Debye equation(ii) 0 < a < 1 and (3=1 gives the Cole-Cole function(iii) a = 0 and 0 <P < 1 gives the Davidson-Cole function
Though this susceptibility function has two parameters which are independentlyadjustable, the low frequency behavior is not entirely independent of the high frequencybehavior. This is due to the fact that that the parameter a appears in both the expressionsfor %' and %".
The parameters a and P have no physical meaning though these models are successful infitting the measured data to one of these functions. It has been suggested that adistribution of relaxation times is usually associated with one of the Cole-Cole,Davidson-Cole, and Havriliak-Negami functions. In these models the observed dielectricresponse is considered as a summation, through a distribution function G(i), ofindividual Debye responses for each group.
3.16 DISTRIBUTION OF RELAXATION TIMES
We have already considered the situations in which there are more than a singlerelaxation time (section 3.6) and we intend to examine this topic more closely in thissection. Polymers generally have a number of relaxation times due to the fact that thelong chains may be twisted in a complex array of molecular segments or due to thecomplicated energy distribution in crystalline regions of the polymer. Since thedistribution function is denoted by G(i) the fraction of dipoles having relaxation timesbetween T and T + AT is given as G(x) di and hence for all dipoles
= 1 (3.87)
Assuming that each differential group of dipoles with different relaxation times followsDebye behavior non-interactively, the complex dielectric constant becomes:
r* - c- + (P f SfG^WT n oox£ -£* + (£s ~ O h - : — (3.88)
A+JCOT
148 Chapters
By separating the real and imaginary parts of equations (3.88), the dielectric constant andthe loss factor are expressed as
In some texts (Vera Daniel, 1967) the normalization of equation (3.87) is carried out byexpressing the right hand side of equation (3.87) as (cs - Soo). This results in the absenceof this factor in the last term in equations (3.88) - (3.90). The existence of a distributedrelaxation time explains the departure from the ideal Debye behavior in the susceptibilityfunctions. However it has been suggested that a distribution of relaxation times cannot becorrelated with the existence of relaxing entities in a solid, to justify physical reality.
Equations (3.89) and (3.90) are the basis for determining G(i) from s' and e" data thoughthe procedure, as mentioned earlier, is not straight forward. Simple functions of G(i)such as Gaussian distribution lead to complicated functions of s' and s". On the otherhand, simple functions of e' and s" also lead to complicated functions of G(i).
Fig. 3.25 helps to visualize the continuous distribution of relaxation times according toH-N expression for various values of the parameter |3 and a(3= I using expressions tofollow31. The corresponding complex plane plots are also shown.
It is helpful to express equations (3.89) and (3.90) as (Williams, 1963)
G(r)J =£-£°°= f-J°> r - r Jl (3.92)
0 l + fl»V(3.93)
150 Chapter 3
To demonstrate the usefulness of equations (3.92) and (3.93) the measured loss factor inamorphous polyacetaldehyde (Williams, 1963) over a temperature range of -9°C to+34.8°C and a frequency range of 25Hz-100kHz is shown in Fig. 3.26. Polyacetaldehydeis a polar polymer with its dielectric moment in the main chain, similar to PVC. Itsmonomer has a molecular weight of 44 and it belongs to the class of atactic polymers. Itsrefractive index is 1.437. A single broad peak was observed at all temperatures.
0.25
0.2
0.15
0.1
5 0.05
-5 5LOG(TIME)
10 15
0.3
0.25
(0(0Q 0.2
o
\j
20.15
0.1
0.05
0.2 0.4 0.6 0.8
DIELECTRIC CONSTANT
2 3 4 5 6" " 1
Fig. 3.25 Distribution of relaxation times for various values of (3 according to H-N dispersion. Thecorresponding complex plane plots of s* are also shown for ap=l [Runt and Fitzgerald, 1997].(with permission of Am. Chem. Soc.).
Dielectric Relaxation-I 151
Fig. 3.27 shows these data replotted as Jro and H^/H^ as a function of (oo/cOp). Theexperimental points all lie on a master curve indicating that the shape of the distributionof relaxation times is independent of the temperature.
Evaluation of the distribution function from such data is a formidable task requiring adetailed knowledge of Laplace transforms. The relaxation time distribution appropriate
"tj
to the Cole-Cole equation is
2ft cosh[(l - a) In r / r0 ] - cos an(3.94)
in which TO is the relaxation time at the center of the distribution.
1 0 2 3 4 5
Fig. 3.26 Plot of s" against log (w/2p) for 0.598 thick sample. 1-34.8°C, 2-30.5 °C, 3-25 °C,4-18.5 °C, 5-9.7 °C, 6-3.25 °C, 7—3.5 °C, 8--9 °C, 9—19.2 °C, 10—21.8 °C, 11—24.5 °C,12—26.4 °C, 13—28.7 °C [Williams, 19631 (with permission of Trans. Farad. Soc.).
As demonstrated earlier (fig. 3-10) the complex plane plot of the Cole-Cole distributionis symmetrical about the mid point and therefore the plot of G(i) against log T or log(t/^mean) will be symmetrical about the line. The graphical technique for the analysis ofdielectric data makes use of fig. 3.9. The quantity u/v is plotted against log v and the
152 Chapter 3
result will be a straight line of slope 1-a. Without this verification the Cole-Colerelationship cannot be established with certainty.
-3,0 -2.0 -1.0 0 i.o 2.0 3,0
Fig. 3.27 Master curves for Jw , eq. (3.92) and Hco/Homax, eq. (3.93) as a function of log(cfl/Omax)- 0.598 thick sample. Symbols are the same as in fig. 3-26 [Williams, 1963]. Adoptedwith permission of Trans. Farad. Soc.]
The distribution of relaxation time according to Davidson-Cole function is
Sin PTC \ r(3-95)
T >Tn (3.96)
The distribution of relaxation times for the Fuoss-Kirkwood function is a logarithmicfunction:
G(T) = -cos(-
(3.97)
Dielectric Relaxation-I 153
Where S is a constant defined in section (3.12) and s= log (co/Op). The distribution ofrelaxation times for H-N function is given by [Havriliak and Havriliak, 1997]
G(r) = | -}y a f i (sm/30)(y 2 a + 2yacosxa +\ n J
In this expression
(3.98)
y = (3.99)
9 = arctansnna }
+ cos/raj
.„ , „„.(3.100)
The distribution of relaxation times may also be represented according to an equation ofthe form, called Gaussian function (Hasted, 1973) given by
(3.101)
where a is known as the standard deviation and indicates the breadth of the dispersion.From the form of this function it can be recognized that the distribution, and hence thee"-co plot, will be symmetrical about the central or relaxation time (fig. 3.28). As thestandard deviation increases the log(s") - log(co) plots become narrower, and for the caseI/a = 0, the distribution reduces to a single relaxation time of Debye relaxation. In fig.3.28 the frequency is shown as the variable on the x-axis instead of the traditional T/Tmean;conversion to the latter variable is easy because of the relationship coi=l. In almostevery case the actual distribution is difficult to determine from the dielectric datawhereas its width and symmetry are easier to recognize.
A simple relationship between (ss-Soo) and s" may be derived33. The area under the s"-logco curve is
XJ LXJ UU
\e"d(Lnco)=(8s-8x} J J (3.102)r=0ffl=0
154 Chapter 3
-0.5
-1.0to
9 -1.5
(5O -2.0
-2.S
-3.0- 1 0 1
LOG(FREQUENCY)
0.2 0.3 0.5 0.9 2.5
Fig. 3.28 Log s" against log(frequency) for Gaussian distribution of relaxation times. Thenumbers show the standard deviation, a. Debye relaxation is obtained for l/s=0. Note that theslope at high frequency and low frequency tends to +1 and -1 as a increases. [Havriliak andHavriliak, 1997]. (Permission of Amer. Chem. Soc.)
Using the identity
ooCOT
11-0} T
expression (3.102) simplifies into, because of equation (3.87),
XI ff jt „ 7 , T , f£ aco n . ,\s'd(lM&) = J =-(*, - ^), J co 2
(3.103)
The inversion formula corresponding to equation (3.103) is
,,^_n_ de'c* —
2(3.104)
Dielectric Relaxation-I 155
which is useful to calculate e" approximately.
Equation (3.103) may be verified in materials that have Debye relaxation or materialsthat have a peak in the s" - log co characteristic though the peak may be broader than thatfor Debye relaxation. Such calculations have been employed by Reddish34 to obtain thedielectric constant of PVc and chlorinated PVc (see Chapter 5). For measurements thefrequency range can be extended by making measurements at different temperaturesbecause cop, x and T are related through equations
coT = l (3.105)
wr = r0exp— (3.106)
Fig. 3.29 (Hasted, 1963) summarizes the dielectric properties e'- s" in the complex plane,the shape of the distribution of relaxation time and the decay function which will bediscussed in chapter 6.
3.17 KRAMER-KRONIG RELATIONS
Expressions (3.89) and (3.90) use the same relaxation function G(t) and in principle wemust be able to calculate one function if the other function is known. This is true only ifs' are related z" and these relations are known as Kramer-Konig relations35:
2(3-107)
ft x -co
s"(co} = -— \8'~8\(h (3.108)n $x -co
Integration is carried out using an auxiliary variable x which is real. Equations (3.107)and (3.108) imply that at oo = oo , s' = Soo and e" = 0. Daniel (1967) lists the conditions tobe satisfied by a system so that these equation are generally applicable. These are:
156 Chapter 3
C<D
(a)
fro
(b)
fm
(c)
-1
In*
I -2
2 I/To
In*
I\
Time Variation of polarization
Fig. 3.29 Graphical depiction of dielectric parameters for the three relaxations shown at top.Theapplicable equations are also shown (Hasted, 1973). (with permission of Chapman andHall).Row and column designations are: al= eq. (3.31), bl = eq. (3.53), cl = eq. (3.45), a2 =single value, eq. (3.31), b2 = eq. (3.95), c2 = eq. (3.94), See chaapter 6 for row 3, a3 = eq.(6.46), b3 = eq. (6.49), c3 = eq. (6.47) and (6.48).
1. The system is linear.2. The constitution of the system does not change during the time interval under
consideration.3. The response of the system is always attributed to a stimulus.
There is no advantage in deriving these equations as we are mostly interested in theirapplication. Application of Kramer-Kronig relations is laborious if the known values arenot analytical functions of co. Jonscher36 lists a computer program for numericalcomputation. However the area of the curve s" - Ln o readily gives (ss - c^) according tothe equation (3.103). Kramer-Konig relations have been used to derive s' from s" - co
Dielectric Relaxation-I 157
data and compared with measured s' as a means of verifying assumed e" - GOrelationship37.
3.18 LOSS FACTOR AND CONDUCTIVITY
Many dielectrics possess a conductivity due to motion of charges and such conductivityis usually expressed by a volume conductivity. The motion of charges in the dielectricgives rise to the conduction current and additionally polarizes the dielectric. Theconductivity may therefore be visualized as contributing to the dielectric loss. Equation(3.8) gives the contribution of conductivity to the dielectric loss. The loss factor isexpressed as
e" = £*(&) + — (3.109)0)S0
Substituting, for s"((o), from equation (3.29) one gets
2fit T
If one substitutes
i~e^S° (3.1H)T
one obtains
(3.112)CO T
The conductivity increases from zero at co = 0 to infinity at co = oo in a manner similar tothe mirror image of the decrease of s' with increasing CD (fig. 3.3). If d.c. conductivityexists then the total conductivity is given by
(3.H3)
158 Chapter3
Jonscher has compiled the conductivity as a function of frequency in a large number of38materials and suggested a "Universal" power law according to
<r(o>) = 0^+00" (3.114)
where the exponent is observed to be within 0.6 < n < 1 for most materials. The exponenteither remains constant or decreases slightly with increasing temperature and the rangementioned is believed to suggest hopping of charge carriers between traps. The real partof the dielectric constant also increases due to conductivity. A relatively small increase ins' at low frequencies or high temperatures is possibly due to the hopping charge carriersand a much larger increase is attributed to the interfacial polarization due to space charge,as described in chapter 4.
Dielectric Relaxation-I 159
3.19 REFERENCES
1 J. F. Mano, J. Phys. D; Appl. Phys., 31 (1998) 2898-29072 Polar Molecules: P. Debye, New York, 1929.3 J. B. Hasted: Aqueous Dielectrics, Chapman and Hall, London, 1973, p. 19.4 J. Bao, M. L. Swicord and C. C. Davis, J. Chem. Phys., 104 (1996) 4441-4450.5 H. Frohlich, Theory of Dielectrics, Oxford University Press, London, 1958
Vera V. Daniel, Dielectric Relaxation, Academic Press, London, 1967, p. 206K.S. Cole andR. H. Cole, J. Chem. Phys., 9 (1941) 341-351.7 D. K. Das-Gupta and P. C. N. Scarpa, IEEE Electrical Insulation Magazine, 15 ( 1999)
23-32.o
V. V. Daniel, "Dielectric Relaxation", Academic press, London, 1967, p. 979 Y. Ishida, Kolloid-Zeitschrift, 168 (1960) 23-3610 A. M. Bottreau, J. M. Moreau, J. M. Laurent and C. Marzat, J. Chem. Phys., 62 (1975)
360-365.11 G. P. Johari and S. J. Jones, Proc. Roy. Soc. Lond., A 349 (1976) 467-49512 F. Bruni, G. Consolini and G. Careri, J. Chem. Phys., 99 (1993) 538-547.13 G. P. Johari and E. Whalley, J. Chem. Phys., 75 (1981) 1333-1340.14 D. W. Davidson and R. H. Cole, J. Chem. Phys., 19 (1951) 1484 - 1490.15 T. Blochowitz, A. Kudlik, S. Benkhof, J. Senker and E. Rossler, J. Chem. Phys., 110
(1999)12011-12021.16 R. P. Auty and R. H. Cole, Jour. Chem. Phys., 20 (1952) 1309-1314.17 P. Debye, Polar Molecules (Dover Publications, New York, 1929), p. 8418 Dielectric Properties and Molecular behavior, Nora Hill et. al, Van Nostrand, New
York, P. 4919 H. Frohlich, "Theory of Dielectrics", Oxford University Press, London, 1986.20 G. Williams, Trans. Farad. Soc., 59 (1963) 1397.21 J. Melcher, Y. Daben, G. Arlt, Trans, on Elec. Insu. 24 (1989) 31-38. Figure 5 is
misprinted as fig. 8.22 R. M. Fuoss and J. G. Kirkwood, J. Am. Chem. Soc., 63 (1941) 385.23 K. Mazur, J. Phys. d: Appl. Phys. 30 (1997) 1383-1398.24 J. G. Kirkwood and R. M. Fuoss, J. Chem. Phys., 9 (1941) 329.25 S. Havriliak and S. Negami, J. Polymer Sci., Part C, 14 (1966) 99-117.26 F. Alvarez, A. Alegria and J. Colmenco, Phys. Rev. B., 44 (1991) 7306.27 S. Havriliak and D. G. Watts, Polymer, 27 (1986) 1509-1512.28 R. Nozaki, J. Chem. Phys., 87(1987) 2271.29 A. K. Jonscher, J. Phys. D., Appl. Phys., 32 (1999) R57-R 70.30 D. K. Das Gupta & P. C. N. Scarpa, Electrical Insulation, 15, No. 2 (1999) 23-32
160 Chapters
T 1J S. Havriliak Jr. and S. J. Havriliak, ch. 6 in "Dielectric Spectroscopy of PolymericMaterials", Ed: J. P. Runt and J. J. Fitzgerald, American Chemical Soc., Washington,D. C., 1977
32 J. B. Hasted, "Aqueous Dielectrics", Chapman & Hall, London, 1973, p. 2433 Daniel (1967). Page 72. Daniel's normalization in equation (3.80) is es - Soo and not 1.34 W. Reddish, J. Poly. Sci., Part C, (1966) pp. 123-137.35 H. A. Kramers, Atti. Congr. Int. Fisici, Como, 2 (1927) 545.
R. Kronig, J. Opt. Soc. Amer., 12 (1926) 547.36 A. K. Jonscher, "Dielectric Relaxation in Solids", Chelsea Dielectric Press, London,
1983.37 R. M. Hill, Nature, 275(1978) 96.~»o
A. K. Jonscher, "Dielectric Relaxation in solids", Chelsea Dielectric Press,London(1983),p. 214.
