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Dielectrics in Electric Fields
Gorur Govinda Raju
Dielectric Loss and Relaxation—I
Publication details https://www.routledgehandbooks.com/doi/10.1201/b20223-4 Gorur Govinda Raju Published online on: 13 May 2016
How to cite :- Gorur Govinda Raju. 13 May 2016, Dielectric Loss and Relaxation—I from: Dielectrics in Electric Fields CRC Press Accessed on: 02 Oct 2021 https://www.routledgehandbooks.com/doi/10.1201/b20223-4
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wher voltage by equation the represented alternating a vacuum applied in Consider having electrodes an consists a capacitor of that two plane-parallel 3.1 In thi In © 2017 Dielectric relaxation is studied to reduce energy losses in materials that are used in practically practically in used are that losses materials energy in reduce to relaxation is studied Dielectric about, norabout, what whether saying we is true. are the subjectin as which maywe never defined be Mathematics knowtalking are what we
e s equation, C s equation, 3 OPE PERMITTIVITY COMPLEX is the instantaneous voltage, V instantaneous v is the by Taylor and Relaxation—I Loss Dielectric & Francis 0 is the vacuum capacitance, sometimes referred to as geometric capacitance. geometric as to referred sometimes vacuum capacitance, is the Group, LLC m iI the maximum value of v maximum the 1 I =+ m v =V == m V z m co m cos ω cos s ω ω t CV (British philosopher and mathematician) and philosopher (British o t
π 2 m 1
is given, is by
, and ω=2 , and π f the angular frequency frequency angular the Bertrand Russell Bertrand (3.2) (3.3) (3. 3/31/2016 1:20:32 PM 83 1) - - - Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 84 © 2017 age is I
is givenand by capacitor. ideal of the current componentcharging is the This ε″ voltage (90° angle −δ but atan be now the with phase in will not phasor be of current clarity,sake εis dimensionless. The that noting C to increases capacitance the εis now plates, constant the of between voltage. placed dielectric the with phase rent in amaterial If 84 tric field. The reference phasor is along I is referencephasor The field. tric 3.1FIGURE for Testing (ASTM)Standard provides following the Materials definitions [1]. dissipationthe factor. complex by represented quantity wher is usua The current can be resolved be can volt applied two into components; the with current component phase in the The It is noted that the usual symbol for the dielectric constant is ε constant symbol for dielectric usual the the It that is noted The component in phase with the applied voltage applied the with component phase loss. in The loss gives dielectric to δis the angle rise In an ideal dielectric, the current leads the voltage the leads by current 90°, the dielectric, is no component ideal cur there of an and the In Loss index is defined as the complex part of the dielectric constant, ε″ constant, dielectric the complexthe of as index isLoss defined part of study. discipline upon the depending American The Some terminology exists confusion in by e Taylor x = V lly referred to as the loss the index as to (thelly referred factor loss previous term
ωε″ & Real Real Francis C o , and the component leading the applied voltage the component the leading , and by is 90° ( ε′ ) and imaginary ( imaginary ) and Group, LLC 0 ε , and the current is given current by the , and I c loss angle loss ), the where δis called δ ε″ iI c 2 and ε and ) parts of the complex dielectric constant ( constant complex of the dielectric ) parts =+ I x m * =ε′ I ε co δ m * =ε′ =ω ε″ = s I tan =ε′ ω −j C t −j − ε″ 0 1 D ε I is shown enlarged for clarity. shown δis enlarged angle . The ε y ε V ε″
′′ ′ π m 2
− δ
V r , but we for subscript the omit the . The dielectric constant is a constant dielectric . The Dielectrics in Electric Fields in Electric Dielectrics is now obsolete) δ tan and . I y ε = V *) in an alternating elec alternating *) an in ωε′ C o (Figure 3.1). (Figure (3.5 (3. (3.4) (3.7) 3/31/2016 1:20:33 PM 6) - ) - - K22709_C003.indd 85 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4
we d can Dielectric Loss and Relaxation—I Loss Dielectric dielectric relaxation dielectric term by general phenomenon the value. is described This maximum its attains before polarization the interval time voltagesudden applicationafinite of adirect takes followingthe Similarly,buildup the alignment. polarization of afield-oriented from medium, the of temperature the with equilibrium for in dipoles distribution, to revert the arandom to required time the is This time. a finite but takes value zero to is not instantaneous decay ofthe polarization voltage for adirect adielectric to applied asufficientlyWhen long is suddenlyremoved, duration 3.2 6. 5and Chapters
© 2017 Powe Conductivity is further considered at the end chapter. of considered atthe the Conductivity is further Dissipation factor ( Compari cur The To com The alternating current (AC) current conductivity alternating to expressedacomplex is often as The according quantity its discussion textpostpone till and the throughout We several equation times this encounter The total conductivity is given total The by
OAIAIN BUILDUP POLARIZATION by r factor is defined as r factor is defined educe that educe Taylor plete the definitions, plete definitions, that the we note rent density is given by ng equation with & Francis D Group, is defined as tan ( tan as δis defined ) or loss or tan tangent σ T LLC =σ I PF x D σ = ac ac = == =σ′ vC +σ ωε si no 1 δ +j − dc o PF J =σ () x ′′ PF σ″ == = σ =ωε = tan D =tan 1 dc =ωε I A J =σ v 2 + D +ωε x ωε D d 2 ωε o o E ε o [ o ε″ ε″ ′′ δ ε″ o A ε
+j +j ′′ rc = E ( onve AE ωε ε′
ωε −ε o ( rs ε′ o ∞ ε ely, −ε )] ′′ ε″
/ ∞ ε′
).
) . (3. 3/31/2016 1:20:33 PM 85 8) Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 86 © 2017 of the alternating voltage instant. atthat alternating of the peak the as magnitude voltage DC same voltage under the has that to that is alternating equal an in material of the polarization instantaneous the we that range. assume Further, frequency optical the in value is saturation for to attain it required time the because instantaneously value its final attain dipoles, we get [2] it time. and of is independent the ture, (Figure an 3.2) lawvalue to exponential afinal 0to from according 86 FIGURE 3.2 FIGURE where
orientational polarization at that instant, P instant, atthat polarization orientational
We can express the total polarization, P We polarization, express total can the Neglecting atomic polarization, the total polarization P polarization total the polarization, Neglecting atomic The rate of buildup of polarization may be obtained, by differentiating Equation 3.9 Equation by differentiating as may obtained, be of rate buildup of polarization The up builds polarization voltage the aDC that letWhen dielectric, usassume apolar to is applied Subs The final v final The by P Taylor tituting Equation 3.9 in Equation 3.10 and assuming that the total polarization is due to the is due to the 3.9 Equation polarization 3.10tituting Equation in total the that assuming and ( t ) - of τ is tempera a function relaxation time. the τ is called t and at time polarization is the
& Pola alue attained by the total polarization is polarization by total the alue attained Francis rization buildup in a polar dielectric. apolar in buildup rization Group, LLC Pt dP Pt dP P P dt P dt T 0 () )() () =− P = P P () μ T T tP T PP ( ( ( Pe t =ε t ∞ t ), and electronic polarization, P polarization, ), electronic and ) =P ∞ P(t) ), as ), =− − ττ 0 − ∞ ( ε ττ () μ 1 t s 1 ( − 1) − t ) +P ≅ e − τ − PP t E τ t µ = e
−
Pe t ( ∞ t ) can be expressed as the sum of the of sum the expressedthe be as ) can () t − τ t
Dielectrics in Electric Fields in Electric Dielectrics e , which is assumed to to , which is assumed (3.12) (3.10) (3.13 (3.1 (3.9) 3/31/2016 1:20:33 PM 1) ) K22709_C003.indd 87 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 to simplifies which where
3.18 substituting Equation and 3.11, Equation in we get
side of Equation 3.20 becomes so small that it can be neglected, and we and solution get neglected, the be itside that can for of3.20 so small Equation P becomes where voltages: Dielectric Loss and Relaxation—I Loss Dielectric hold © 2017 Representing the alternating electric field electric as alternating the Representing We further note that Maxwell’s that note We further relation Simp Subs gen The We following previous have shown the the chapterthat in relationships already hold steady under s true at optical frequencies. Substituting Equations 3.13 Substituting frequencies. Equations atoptical s true 3.14 and 3.12, Equation in we get ε by C s tituting Equation 3.21 Equation tituting 3.12, Equation in we get lification yields lification is a constant. At time t At time is aconstant. and and Taylor eral solution of the first-order differential equation is solutiondifferential eral first-order of the ε ∞ are the dielectric constants under direct voltage and at infinity frequency, voltagerespectively. atinfinity direct and under constants dielectric the are & Francis Group, Pt LLC () , sufficiently large when compared with τ , sufficiently large whencompared Pt dP Pt =− Pt dt () εε P () () 0 μ =− () =ε Pt =− =+ () ∞ ε Ce τ 1 0 P ∞ P ( = [( 1 ε μ εε e − =ε E = s =ε ε τ 0 t − 1) − Ee 1 0 ε m + () ∞ sm 0 εε ε 0 ( () ( =n jt sm 1 0 E −ε ε ωω εε ε max − s ∞ + () s ε 1 −ε εε + − 1) − − ∞ + e sm 2 j )( ∞ ωτ
εε − j Ee 1 ω j 0 ∞ 0 ∞ ωτ 1 ( t Ee +
() ) ε E E + ∞ ∞
s j
jt − 1) − ω ωτ ε j Ee − ωτ jt 0 ω Ee − ε ∞ m
Pt E jt ω
jt Ee ω )]
m
jt
, the first term on the right the on term first , the (3.20) (3.22) (3.23) (3.15) ( (3. (3. (3.21) (3.19) (3.18) (3. t ) as 16) 17) 14) 3/31/2016 1:20:33 PM 87 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 88 © 2017 is the narrowest. The descriptions that follow in several sections will bring out this aspect clearly. aspect follow out this that bring descriptions several The will narrowest. in sections is the [4,phenomena is concerned p. 19] Debye relaxation formulas, various the among relaxation though whole the Debye of as far the gamut of as broad is very physical relaxation curve spectrum The Debye whether relaxation is apossible determine to mechanism. aguide as this use one can and more concise way and alternative An of expressing Debye is equations 3.3 following characteristics: 3.30shown 3.28 in Equations are through at by a peak it and is characterized curve, eter discussion the enters by way param temperature of the The frequencies. atvarious dielectrics polar 88
voltage. The instantaneous value ofvoltage.density flux given Dis by instantaneous The
It t is easy Equatin Equation 3.23Equation shows P that Equatio But flux d the Equating Equations 3.24 3.25, and Equations weEquating get Substituting Equation 3.23 in Equation 3.26 and simplifying, we 3.23Substituting 3.26 Equation get simplifying, Equation in and
by as will be described in the following the in plot section.The of described ε″ be will τ as EY EQUATIONS DEBYE Taylor ns 3.28ns Debye 3.29 as known and [3], equations are behavior the of describe they and g the real and imaginary parts, we readily obtain we readily parts, imaginary and real g the & o show that Francis ensity is also equal to equal ensity is also Group, LLC ( t ) is a sinusoidal function with the same frequency as the applied applied the as frequency same the with ) is a sinusoidal function () εε ′ − ε tan 0 ε j D * δ E εε ′′ εε D ( ε ′ t * m = ′′ ) =ε F ( =+ =+ εε e t =+ igure 3.3igure ) =ε = j ′′′ ′′ ε ω 11 ε t / ′′ ′ ∞ =ε () ∞ 0 εε ma 1 = E s 0 ε + x m ε − 1 εε 0 () ∞ e 1 * E εε E εε εε ωτ = s + j s + s ω s −+ m + . ∞ 22 05 ωτ − t − − e +P An examination of these equations shows equations of these examination the An m . j j ωτ ωτ 22 ω e ∞ . It show to is easy ωτ t ∞ j ∞ ∞ ωτ +P ω 1 εε ( t
t ωτ 22
+ s )
− ( j t ωτ )
∞
– is known as the relaxation the as ω is known Dielectrics in Electric Fields in Electric Dielectrics =3.46 ratio, for this (3.26 (3.25 (3.27) (3.24) (3.30) (3.28) (3.29) (3.31) 3/31/2016 1:20:33 PM ) - ) K22709_C003.indd 89 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 where ω and equating the differential to 0. This leads to leads 0. to This differential the equating ωand to respect δwith found tan by differentiating be also can loss atwhich is the angle amaximum frequency frequency, in The increase decreases.
resu ωτ at occurs 3.3 FIGURE Dielectric Loss and Relaxation—I Loss Dielectric
© 2017 The dis The The values ofThe ε′ The maximum value of ε″ maximum The 2. 3. 1. lting in
For small valuesFor of small ωτ tor of3.28, ε″ Equation and For in get get For ve by p is the frequency at frequency is the Taylor ε″ also increases with frequency; reaches a maximum; and with further further with and amaximum; frequency; reaches with increases sipation δalso factor tan does not occur at the same frequency as the peak of ε″ peak the as frequency same the at notoccur δdoes of tan =1. peak The
= 0 as expected because this is DC voltage. is DC this because expected =0as termediate values of ε″ frequencies, termediate ry large values large of ωτ ry Schem & Francis and ε″ and atic representation of Debye equations plotted as a function of log ω afunction as plotted of Debye equations representation atic Group, (ε (ε at this value of ωτ atthis s s – ε + ε ε , the re , the is obtained at a frequency given atafrequency by is obtained LLC ma ′′ ∞ ∞ ∂∂ )/2 )/2 x ε ∂ ε (t . ∞ , ε′ s ∂ () an ω ∂ () is also small for the same reason. Of course, atωτ course, Of reason. for same the small is also ωτ ε al part ε′ part al t =ε ′′ ) =− =− ∞ , and ε″ , and () Tan δ () εε ε εε s ε ma ′′ s ′ are ≈ε x ω = = p ∞ is a maximum at some particular value of ωτ atsome particular is amaximum τ =1 εε ∞ s is small. is Log ω εε s - denomina the in term squared of the because () () s + () 2 εω εε () 1 − 2 ωτ s ∞ +
+ ∞ 22 ∞ ωτ 22
τε ∞ 22
− ωτ − 1 22 2 s = = 0 ε ε ´´
0 ´
. . The peak of ε″ peak . The =0, we (3.35) (3.34) (3.3 (3.33) . 3/31/2016 1:20:33 PM 89 2)
Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 90 © 2017 later in this chapter. this in later Debye obey3.31) the data (Equation equation measured the whether considered be will determining 3.4 interchangeably,used phenomena. associated possibly are absorption dispersion and because Variationtrics. of ε″ of ε′ relaxation, atrelaxation frequency.quency, rapidly temperature Normally, with absence the decrease in it will frequency. ataspecified ε′ If we measure a material in relaxation occurs whether over arelatively fact determine to may used be This bandwidth. small aslope with of τ origin the frequency forfrequency several values of τ 90 potential energy,potential shown as in b[6] by adistance (see Ref. apart also situated [9,are p. 20]). lowest the to sites correspond These model of molecular of adipole, the emay charge occupy aparticle In one of two sites, that 1 or 2, 3.4 Accord the shift in the potential energy due to the electric field by the dotted line. field electric due energy the to the dotted by potential the in shift the and line shows sites. of figure full energy the conditionsthe The withfield the tial no electric with poten the fieldin electric twoapplied difference sites,the a causes An is therefore,aparticle. there may occupy either particle one Between of the them. and them, between no with difference energy
Variation of ε′ 3.3Equation Figure defined by shows relaxation frequency 3.32, at the ε′ that, Dividing Equation 3.29 by Equation 3.28 and rearranging terms, we obtain the simple the relationship we obtain terms, 3.29 Equation Dividing by 3.28 rearranging Equation and By subst it equation, show to Solving is easy this that Fi The corre The
shows ε′ of measured aseries by gure 3 gure BISTABLE MODEL OFADIPOLE Taylor ing to Equation 3.40, Equation to ing aplot of .3 ituting this value of ωτ this ituting & sponding valuessponding of ε′ also shows of also afunction δas plot of the tan of3.30, variation Equation is, the that Francis should increase with decreasing temperature according to Equation 2.51. Equation to according temperature decreasing with should increase as a function of frequency is referred to as dispersion in the literature on dielec- literature the dispersion in to as of is referred a frequency function as as a function of frequency is called absorption, though the two terms are often often are two terms the though absorption, of is called frequency afunction as Group, . LLC F igure 3.5 igure . and ε″ and and ε″ and in Equation 3.30, Equation in we obtain εε . ε (t In the absence of an electric field, the two sites are of equal two field,the equal electric are absenceof of sites an the In ′ are ′′ an ε in mixtures of water and methanol [5]. of methanol water and question The of mixtures in − = ′′ εε ε δ ωτ ′ ′ ∞ εε εε ) ε − = ma s s ′′ results in a straight line passing through passing through line astraight in ωresults against + − = x εε ∞ 2 s = εε ∞ ∞ + = s ε εε ε ωτ ∞ s s ∞ as a function of temperature at constant fre atconstant of temperature afunction as εε ∞ εε − s
s
∞ ∞ ∞
Dielectrics in Electric Fields in Electric Dielectrics decreases sharply sharply decreases F (3.36) (3.37) (3.40) (3.38) (3.39) igure igure 3/31/2016 1:20:33 PM - - K22709_C003.indd 91 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 part, ε″ part, FIGURE 3.5 FIGURE 3.4 FIGURE Dielectric Loss and Relaxation—I Loss Dielectric lower energy than the “up side.” This represents polarization. side.” “up represents the lower This than energy the “down with the side”slightlyfield a of having tilted, line), are field(broken the wells electric of an presence the In polarization. is no there that indicates this each in well; time equal line),fielddipole spends the (full © 2017 J. Chem. Phys. Bao J. al., et Chem. M. L. . (Adaptedfrom by Taylor
Diel The po The & Francis ectric properties of water and methanol mixtures at 25°C. at ε′ (a) part, mixtures Real methanol of and water properties ectric tential well model for a dipole with two stable positions. In the absence of an electric electric of an absence the well twopositions.In for stable model with adipole tential (a) (b Group, ) Loss index (ε˝) Dielectric constant (ε´) 10 20 30 40 20 40 60 80 0 1 LLC 100%
Energy 20% 40% 60% 80% 0% Volume fractionofwater 10 2 10 2 Ele Fr Fr Po ctric fie equency (MHz) equency (MHz) equency sition 10 , Vol. 104, 1996.) 4441–4450, 3 ld 10 b 3 Volume fractionofwater 10 4 10 4 100% 20% 40% 80% 0% 60% ; (b) imaginary ; (b) imaginary 3/31/2016 1:20:33 PM 91 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 92 © 2017 of W of reciprocal is the exponent relaxation time signthe in The is valid. minus well, the then starting the for [6] liquids polar (see2000 Ref. also [9, p. 20]). well destination alower has the If than energy of attempts. Its value is typically of the order of order of Its value10of the attempts. is typically ture. If it it If does, is expressedB as ture. It not- may uncommon. or may tempera are not on the depend or four of orders magnitude by three 92 3.28 3.29 Equations and give to by rearranging strated demon easily be can yields frequency asemicircle. This aparticular to corresponding point each Cole exhibiting Debye Cole and [7]a plot showed in relaxation, amaterial of in that 3.5 temperature. increasing with of frequency increases jump the fact that the to uted is attrib temperature increasing with of relaxation time 5. consider decrease Chapter in The shall of which we Thave functions proposed, been Other equation. also this to according be to assumed where whic from sitefrom 1. 2to moment The of such adipole is model is equivalent position adipole to changing by 180° moves charge when the site from 2or 1to where
energy difference between the two wells the between as difference energy well to jump to energy 1again. acquire for then there It some will time. remains and reservoir the to energy moves and it returns well to hill Upon the 2. enough climb to arrival, energy acquires ally dipoles. well and exchanging in A charge other reservoir each aheat 1occasion with energy in are not have one site other. from jump to energy the to However, the dipoles the scale, on a microscopic is negligible.due interaction to would consideration Amacroscopic shows particles charged the that field. electric external absenceof an the in of of equal energy site site that 1and potential 2are the bistable the model as to dipole. ofis referred the We that dipoles θ=0for and that all assume also The difference in the potential energy due to the electric field electric due energy the to Eis potential the in difference The The var The The number of jumps per second from one well to the other is given in terms of the potential potential of the is given one well other from of second terms number the to per jumps in The of Nnumber bistable volume dipoles field unit contains per the and We material the that assume
by 12 h may be thought of as having been hinged at the midpoint between sites 1 and 2. This model This sites 2. between midpoint 1and atthe h may hinged thoughtof having be as been , leading to , leading COMPLEX PLANE DIAGRAM is the angle between the direction of the electric field and the line joining sites 1 and 2. This This 2. and sites 1 joining line fieldthe electric and of the direction the between angle θ is the T Taylor a factor denoting the number number Aafactor the denoting and constant, Boltzmann kthe absolute is the temperature, in liquids and in polymers near the glass transition temperature is temperature transition glass the polymers near in and liquids Tin iation of τwith & Francis Group, LLC () / εε has a value between 500 and found someT as and polymers. avalue in Bhas 500 between ττ ′′ = 22 ϕ + WA 1 0 −ϕ () 12 ex ′ = p µ −= 2 kT w =ebE ε = ∞ ex 1 2 13 fo eb p /s − 1 rw cos θ cos at room temperature, though values differing values though differing temperature, atroom wE
() 1 − kT εε µ + s − ωτ
µ 22 ∞ E
2
Dielectrics in Electric Fields in Electric Dielectrics ε″ against ε′ against (3.41) 3/31/2016 1:20:33 PM - - - , K22709_C003.indd 93 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4
FIGURE 3.6 FIGURE Debye equations. cou Of complex plane plots of ε of plots plane complex Dielectric Loss and Relaxation—I Loss Dielectric the horizontal axis ( axis horizontal the be shown ( be that © 2017 Substituting the algebraic identity algebraic Substituting the Further simplification yields simplification Further This i This side right of3.41The Equation Equation in using may simplified resulting 3.28, be The cor The Equat At ω At by rse, these results are expected because the starting point for Equation 3.44 is the original for point 3.44 original Equation is the starting the because expected are results these rse, p = 1, the imaginary component ε″ τ =1, imaginary the s the equation of equation acircle radius with s the ion 3.43 as may rewritten be Taylor responding valueresponding of ε′
Cole & ε ∞ Francis ,0) ( and –Cole diagram displaying a semicircle for Debye ε asemicircle equations displaying diagram –Cole ε′ ) atε Group, ε *. ´´ ε ∞ s ,0) are points on the circle. on,0) the To points are putway, it another circle intersects the and and LLC ( ε ε″ ε ε ∞ is εε s ε′ ′ , as shown in Figure 3.6. shown, as Figure in Such plots of ε″ ) ss − 2 2 (ε +( −ε′ ∞∞ s εε + ε ss =+ ε′ + 22 ( ∞ 4 1 ε −ε )/2 s [( ∞∞ +ε ε has a maximum value of amaximum has ε εεε εε ′′ ′ ∞ 2 εε = = ) ∞ 2 + s =( ) +ε ωτ =1 εε εε − 2 () s s ε + )( − 2 ′′ ε 2 ∞ 22 (ε s s −ε 2 −+ ε having its at center s ∞ ∞ ∞ = – ε +ε″
∞ ε s ∞ ´ )( εε )/2 ε′ 2 − = 0 = −ε ∞ )] ∞
ω incre
) 2
ε
s ases εε versus ε′ s *. + 2 ∞ 0, are known as as known are . It can easily . It can (3.44) (3. (3.43 42) 3/31/2016 1:20:33 PM 93 ) Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 94 © 2017 for equation complex as the constant empirical suggested dielectric an below its with displaced center asemicircle be ε′ ation the showed still plot the will that frequency range. frequency over occurs dispersion commonly awider the aresult, as of and rotation, axis uponing the fieldhave different alternating in such[8].materials - the depend to values times relaxationThe response their not interactivein moreindependent likely be to and dipolessolid are the phase, the in Further, configuration. shapes; long-chain polymers, in have, they alinear particularly of thevalue fielddeciding in influence external ofno has moleculeof rotation an the in axis the distribution as to what to referred is generally of relaxationattributed times completely being few very Debye with practice, discrepancy In the materials agree equations, measurements. for necessity obviating the frequencies, making atintermediate constant dielectric complex the obtain to used be then can exhibits Debye diagram relaxation. ACole–Cole material ation. α=0givesation. Debye relaxation. = 10 ω= ω=0to from cies, usually 94 FIGURE 3.7 FIGURE where 3.37.Equation Under plot conditions, the these ε″ versus log ωτ value of ε″ a maximum Debye do not have satisfy one relaxation that time show They equations. Polar dielectrics more than 3.6 A plot of3.45 Equation 3.7 is shown Figures in and are also plotted for the purpose of comparison. Near relaxation frequencies, Cole–Cole relaxation Cole–Cole relaxation frequencies, Near of comparison. for plotted purpose also the are ε The simple theory of Debye assumes that the molecules are spherical in shape and therefore, and shape in spherical of molecules Debye the are simple that theory The assumes In a given material, the measured values of ε″ measured the a givenIn material,
*. This is more an exception than a rule because not only can the molecules the have not only can *.because exception different a rule is more an than This by τ OECL RELAXATION COLE–COLE Taylor c–c is the mean relaxation time and α and relaxation time mean is the
& also shows a broad maximum, the maximum value being smaller than that given that by than value smaller being maximum the shows also maximum, abroad Schema Francis in a polar dielectric according to Cole–Cole relax Cole–Cole to according dielectric a polar of εin part of real representation tic Group, that will be lower than that predicted by Equation 3.34. The curve of tan δ by 3.34. of Equation curve tan lower predicted be The that will that than LLC
10 Real part (ε´) –3 εε * 10 10 10 =+ n =0.25 rad/s. If the points fall on asemicircle, we conclude fall points can the the that If rad/s. –2 α =0for D ∞ 10 1 –1 + is a constant for having is aconstant value agiven 0≤α material, () εε 10 j s ωτ 0 − ebye relaxation. cc are plotted as a function of a function as plotted are − versus ε′ ωτ 10 ∞ 1 1 α =1–n n =0.5 n =0.75 n =1(Debye) − α 3 .8 10 ;; 2 for values various of α.Debye equations n =0 01 will be distorted, and Cole–Cole relax Cole–Cole and distorted, be will ≤≤ 10 α 3
10 4
10 Dielectrics in Electric Fields in Electric Dielectrics 5 . ε′ - frequen at various axis. They They axis. (3.45) ≤ 1. ≤ 3/31/2016 1:20:33 PM - - K22709_C003.indd 95 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 index ε″ index Dielectric Loss and Relaxation—I Loss Dielectric as it times. 3.29. Equation identical with value the of larger α The shows ε′ that α=0gives Debyerelaxation. relaxation. 3.8 FIGURE © 2017 To determine the geometrical interpretation of3.45, Equation interpretation we geometrical To substitute the 1−α=n determine For a dielectric that has a single relaxation time, α asingle has relaxation time, that For adielectric Usin Equat by g the identity g the is broader than the Debye relaxation, and the peak value, ε Debye peak the the and relaxation, than is broader ing real and imaginary parts, we get parts, imaginary and real ing Taylor
decreases more slowly Debye decreases loss the α, the relaxation. With increasing ωthan with Schem & Francis in a polar dielectric according to Cole–Cole Cole–Cole to according dielectric apolar of εin part of imaginary representation atic εε Group, ′ εε =+ εε ′′ ′ =− − ∞∞ LLC
() ε j ´´ 0.0 0.2 0.4 0.6 0.8 1.0 s () ′′ εε j =+ α s ε ≡e − ε ∞ ∞ 12 j απ + 1 12 /2 ++ cos απ ≡cos + () ωτ () ωτ ωτ () 1 ωτ () cc 1 ωτ − cc − + n =1(Debye) cc n =0.75 − n cc α =1–n () n − ωτ co n =0.5 /2 + sin απ +sin /2 n εε co s( n c s cc − oos si − n
ss n =0.25 , the larger the distribution of relaxation distribution the larger , the = 0 in this case, Equation 3.45 Equation case, this becomes = 0in () n( π n n 22 ∞ / ππ n co n 2 2 π π ))( s / + 2 /2 j + n ωτ 2 in π () ωτ max cc n − cc , is smaller. ) − 2 n
2
n
and rewrite rewrite and (3.46) (3.47) (3.48) 3/31/2016 1:20:34 PM 95 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 96 © 2017
(Figure 3.9). (Figure 96 FIGURE 3.9FIGURE haviand where respectively. plots the 3.47of Equations are These 3.48. and At ωτ Equations 3.47Equations 3.48 and may expressed be alternatively [9, as p. 97] Figures 3.7Figures 3 and is negative, center of the belowWe lies center is, ycoordinate the ε′ that the the that note Equatio Eliminat by Taylor ng a radius ofng aradius n 3.49 at center of equation acircle the with the represents
ing ωτ ing & Geomet Francis ε ′ c–c − .8 from Equations 3.47 Equations from 3.48, relaxation and showed Cole–Cole that εε rical relationships in Cole–Cole equation (Equation 3.45). (Equation equation Cole–Cole in relationships rical Group, ss show the variation ofshow ε′ variation the + 22 ε ∞∞ ´´ LLC 22 + εε εε s ′ εε − − ε s ε ′′ − (ε ′′ εε ∞ + ∞ s ss ∞ + ε εε =− +− 22 εε πh/2 = ∞ s 1 2 − )/2, cot( ∞∞ − 2 1 2 ,/ 1 α =0(Debye) s =Ln ∞ εε Cole–Cole co cot( co π(1-h) and ε″ and co hsi sh nπ/2) ×(–ε u s( hsi sh n co ec ns π ωτ ns s( cot( + si n 2
απ as a function of ωτ afunction as π )/ + nh () n( n (1-h)π/2 2 πh/2 / s ns π ()/ n( = 2 )/ απ + ε ) v
απ 2 ∞ ε ) / )/2 2 s = 1, 3.9 relations Figure the in hold. −
2 2 εε
∞
co Dielectrics in Electric Fields in Electric Dielectrics ε s( ´ ec for several values of α n π / 2 ) 2
(3.49) axis axis 3/31/2016 1:20:34 PM , K22709_C003.indd 97 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 relaxation, the material then possessing asingle then relaxation time. material the relaxation, in in behavior is consistent shown This that value with increases. peak narrower, the become and curves curves crossing overcurves at ωτ value of n Dielectric Loss and Relaxation—I Loss Dielectric v by denoted uand be ( to is reduced constant fields, DC under dielectric is identicalthat the to except that behavior the material of and the times, Kolloid Z. Y.(Adapted from Kolloid Ishida, 3.10FIGURE By plott The Cole–Cole arc is symmetrical about a line through the center parallel to the ε″ the to parallel center the through about aline is symmetrical arc Cole–Cole The 3.10Figure (see Ref. [10]; includes also on polyvinyl paper studies the isobutyl nylon-16). and ether n. value parameter the of in the increase an to leads material of the temperature n © , the number of degrees of freedom for rotation of the molecules decreases. Further decreasing the the decreasing for Further moleculesof of ofrotation number the freedom decreases. degrees , the 2017 Fi Let the lines joining any point on the Cole–Cole diagram to the points corresponding to ε to corresponding points the to diagram Cole–Cole anyon point the joining lines the Let The variation of ε″ variation The = 0 corresponds to an infinitely large number of distributed relaxation distributed large of number infinitely to an of case n = 0 corresponds the stated, As The Cole–Cole diagrams for poly(vinyl diagrams Cole–Cole The shown chloride) in are (PVC) temperatures atvarious gure 3.8 gure by ing loging (log v−log ωagainst u Taylor increases, ε′ , consistent fields. DC with the As value of nincreases, .
& Cole–C Francis , respectively ( with ωτ with ole diagram from measurements on poly(vinyl chloride) at various temperatures. onpoly(vinyl temperatures. measurements chloride) various at from ole diagram ε Group, s − = 1. At n = 1, ε′ change in the
ε´´ ε ∞ )/2. The complex part of the dielectric constant is also equal to 0 at this 0atthis to equal is also constant dielectric of complex the The )/2. part 1 2 1 2 1 2 LLC uv is also dependent on the value dependent on the of is also n. =− ε , Vol. 168, of Springer-Verlag.) 1960. 23–36, permission With εε ′ 101.5°C 108°C 114.5°C 121.5°C 128°C = *; Figure 3.9Figure εε may be determined. With increasing value of With increasing ), value the of nmay determined. be ss ∞ + 22 ∞∞ ;t = ). ε Then, atany frequency,Then, following the relations hold: εε () ωτ ′′ * = − ε ´ 1 εε − ∞ n − ;( with increasing increasing with v u = an ωτ n 2 π ) changes with increasing ωτ increasing with changes increases, the the value the of As nincreases, 1
− n
111098765432 ωτ 1 2 1 2 is identical to the Debye is identical to the axis. ∞ and and , the , the 3/31/2016 1:20:34 PM 97 ε s
Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 98 © 2017 Bottreau et al. [12]. et al. Bottreau range. Table frequency from anarrow to data studies the their 3.1due limit to cost, to summarizes forced, microwave of disadvantage the are observers individual is that frequency aparticular and relaxation frequency, the microwave outthe in to determine range frequency carried are surements 1930–1995. years the of water constant during dielectric [11] et al. Fernández without it, isand not life sustained. of quote over static 36 determinations naturally it by fact that occurs the is fascinated One it asimple structure. has because ing molecular interest comes closest are liquidstate in exhibiting to Debye properties its dielectric and relaxation, solutions weak to solvents. nonpolar Debye in liquids of polar limited relaxation is generally Water 3.7 98 2530.00 f (GHz) of Water Properties Dielectric atSelective 293 K 3.1 TABLE 890.00 300.00 35.25 34.88 24.19 23.81 23.77 23.68 23.62 15.413 9.455 9.390 9.375 9.368 9.346 9.346 9.141 4.630 3.624 3.254 1.744 1.200 0.577 13,760 6880 3440 Note: Source: The followingThe apply definitions is shownthe quantities for Table 3.1. - mea isThe not upappreciably to 100 constant frequency dependent on MHz. the dielectric The
by
ε
DIELECTRIC PROPERTIES OFWATER Taylor ∞ =n A. M.Bottreauetal.,J. Chem.Phys., 2 Measured Complex =1.78,ε & Permittivity Francis 20.30 19.20 29.64 30.50 31.00 31.00 30.88 46.00 63.00 61.50 62.00 62.80 61.41 62.26 63.00 74.00 77.60 77.80 79.20 3.65 4.30 5.48 80.4 80.3 1.98 2.37 3.25 ε′ s =80.4. Group, Extrapolated Values from aSingleRelaxationofDebye Type LLC Measured Reduced Permittivity Vol. 62,360–365, 1975. 29.30 30.30 35.18 35.00 35.70 35.00 35.75 36.60 31.90 31.60 32.00 31.50 31.83 32.56 31.50 18.80 16.30 13.90 1.35 2.28 4.40 7.90 7.00 2.75 0.75 1.15 1.45 ε″ .28007 .200.0234 0.0230 0.0172 0.0238 .31009 .350.0272 0.0335 0.0290 0.0321 .41006 .340.0582 0.0384 0.0560 0.0471 .36032 .200.3764 0.2230 0.3727 0.2356 .26035 .220.3787 0.2262 0.3854 0.2216 .54047 .600.4478 0.3600 0.4475 0.3544 .63045 .670.4500 0.3667 0.4452 0.3653 .77044 .640.4502 0.3674 0.4541 0.3717 .77045 .600.4507 0.3690 0.4452 0.3717 .71044 .710.4510 0.3701 0.4547 0.3701 .65045 .650.4683 0.5645 0.4655 0.5625 .77045 .680.4003 0.7618 0.4057 0.7787 .56041 .610.3989 0.7641 0.4019 0.7596 .60047 .660.3986 0.7646 0.4070 0.7660 .71040 .690.3984 0.7649 0.4007 0.7761 .55044 .660.3980 0.7656 0.4049 0.7585 .63044 .660.3980 0.7656 0.4141 0.7693 .77040 .780.3934 0.7728 0.4007 0.7787 .16029 .250.2472 0.9215 0.2391 0.9186 .64027 .460.2016 0.9486 0.2073 0.9644 .69016 .560.1835 0.9576 0.1768 0.9669 .87010 .880.1030 0.9868 0.1005 0.9847 .97005 .950.0348 0.9985 0.0350 0.9987 .06009 .010.0096 0.0021 0.0095 0.0026 .07004 .010.0167 0.0071 0.0146 0.0077 .18008 .190.0227 0.0179 0.0184 0.0188 .0 .80093 0.0717 0.9936 0.0890 1.000 E m ′′ Calculated ReducedPermittivity Dielectrics in Electric Fields in Electric Dielectrics E m ′′′′ E c ′′ E c ′′′′ 3/31/2016 1:20:34 PM - K22709_C003.indd 99 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 Table 3.2 along ε with analysis is shown Cole–Cole in from of temperature afunction as obtained relaxation time The water [4, p. 19], equations. Debye to Cole–Cole analysis with and according equations compared it ε was thoughtthat
with parameters shown in Table 3.2. Ordinates are shown on elongated scale for clarity sake. for shown clarity onelongated scale are Table shown in 3.2. Ordinates parameters with plot; — Cole–Cole from –,calculations data. experimental are circles 3.11FIGURE plane plot of ε* of plot plane tion converges Debye to relaxation. as that α is relatively Recall of independent temperature. and small parameter Debye the Cole–Cole with agreement The qualitative concept. in temperature decreasing Dielectric Loss and Relaxation—I Loss Dielectric that the equation ε equation the that cies. However, up 2530 to extrapolation measurements more recent and 13760 to GHz shows GHz this was attributed to possibly absorption and a second dipolar dispersion of possibly to dipolar ε″ asecond and was absorption attributed this © 2017 Earlier literature on ε literature Earlier These are reproduced from Ref. from reproduced [12]. are These 3.11 Figure shows complex the plots plane of ε′ At this point, it is appropriate to introduce the concept of spectral decomposition complex of the concept of the spectral introduce to it point, is appropriate At this by εε *; Taylor ′ −
jj Complex . If we suppose that there exist several relaxation processes, each with a characteristic exist acharacteristic we severalwith each there . If relaxation processes, suppose that & εε ′′ Francis ∞ *; = n ∞ is much greater than the square of the refractive of index, the n square the than is much greater = ∞ plane plot of ε of plot plane used in the analysis. the in used T (°C) in WaterRelaxation Time 3.2 TABLE Source: 75 60 50 40 30 20 10 0 εε Group, 2 ′ * in water did not extend to as high frequencies as shown water as not did * in frequencies extend in high as to is valid, as demonstrated in Table in demonstrated as is valid, 3.1. with increases relaxation time The − Loss index (ε ) 01 ´´ ′′
LLC London, 1973. J. B.Hasted,AqueousDielectrics,Chapman&Hall, 860 GHz EE mm * 0203 * in water at 25°C in the microwave frequency range. Points in closed in Points range. microwave 25°C at water the * in in frequency = 4.49 ±0.17 4.21 ±0.16 4.13 ±0.15 4.16 ±0.15 4.20 ±0.16 4.23 ±0.16 4.10 ±0.15 4.46 ±0.17 50 ′ ε Diele ∞ − 30 jE 0 ctric c m ′′ Me Cole–Cole De = 05 07 80 70 60 50 40 20 asuremen bye εε εε onstant ( s * − − τ (10 ∞ ∞ ts .90.011 0.013 0.009 0.32 0.012 0.39 0.013 0.48 0.014 0.58 0.014 0.72 0.93 1.26 1.79 ; ε ´ 10 EEC −11 ) * ) (s) =+ 1 GHz ∑ 5 i 4 = 3 3 2 , calculations from Debye equation Debye equation from , calculations 1 α 0, the Cole–Cole distribu Cole–Cole → 0, the i 1 α – j f f i 2 =1.8 [4, p. 19], and at higher frequen athigher = Ej c ′ T − able 3.1 E −j c ′′
ε″ , and and for for 3/31/2016 1:20:34 PM 99 - - Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 100 © 2017 equation independent of other processes. of independent equation other respectively.ation time, follows relaxation process each is that here assumption The Debye the tion Σ tion where J. Phys. Chem. Ref.Data Chem. J. Phys. D. (From P. al., et measurements. Fernández high-frequency circles, open measurements; bridge frequency low-circles, Closed kilohertz. in frequencies are points data 191.8 at filled circles the beside numbers The K. where expressedbe as Debyethe overthen Equationfrequencyrange, dominant 3.31 aspecific and relaxation time may 100 FIGURE 3.12FIGURE T processes. The The processes. procedure, has been adapted by Bottreau et al. [12], et al. by Bottreau adapted been has type procedure, of who the afunction use Figure 3.12Figure values shows of ε′ measured the behavior. discussion,ing point under namely, to the Focusing attention our several relaxation times,
attributed to physical and chemical impurities [15]. physical to impurities chemical and attributed However, D the 270 K[14], atlower 165–196 of relaxation times distribution observed the temperatures and Kis shown water been ice to have has from Polycrystalline of a single Debye at relaxation time type able 3.3 This kind of representation has been used to find the relaxation times in D times relaxationthe find to used been of has representation kind This A method of spectral analysis that is similar in principle to what was described, but different in but in different to whatprinciple wasin described, is similar analysis that of spectral A method This scheme was H to applied This by C ∆ε C Taylor i =1should satisfied. be i is the spectral contribution to the i the contributionto spectral is the . i and τ and Three Debye regions are identified with relaxation times as shown. Debyetimes relaxationwith Application identified the regionsof are Three
& Francis The anal The ∆ε i are the individual amplitude of dispersion ( amplitude individual the are and τ and Group, ysis of Cole–Cole plots into three Debye-type relaxation regions indicated by semi indicated regions relaxation Debye-type three plots into ysis of Cole–Cole , Vol. 1995.) 34–69, 24, are 88.1, are 57.5, 1.4 and (see inset),100 respectively. and μs, ms, 60 ms, 20 and ε´´ 100 20 40 60 80 08 LLC 0.3 0.15 1.0 0.08 2 0.06 O data, shown in O data, ε εε 40 ε * 0.04 ′ ′′ =+ =+ = and ε″ and ∑ i ∑ = n th region ω th and ∞ in i = 1 = 1 Cj 1 ∑ i i i 0.025 = + = in the complex plane as well as the three relaxation relaxation complex the in three well as plane the as 1 n ∆ 1 ωτ ε ´ ε 0 1 22 i + Table 3.1 ∆ i ωτ ω ω ε ωτ 22 i 0.016 i i i − i 1 is its relaxation frequency. condi The ε
low frequency
120 The results obtained are shown in are obtained results . The 0.01 . . . 4.5 3.5 2.5 1.5 7.0 2.0 1.0 2 0.5 0.005 O ice shows amore interest Dielectrics in Electric Fields in Electric Dielectrics −ε 0.2 0.002 160 high frequency high ) and the relax the ) and 2 O ice [13]. (3.50) (3.52) (3.51) 3/31/2016 1:20:34 PM - - - - K22709_C003.indd 101 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4
lowestthe values. 1.to However, of ε′ part high-frequency the low-frequency of The for ε′ comparison. part with with metry on either side of the line that is parallel to the ε″ the to is parallel that on either side line ofmetry the which agrees with the major the region with of Tablewhich relaxation in agrees 3.3.
Dielectric Loss and Relaxation—I Loss Dielectric where tan ϕ=ωτ where tan
as expressed Davidson Cole and [16] have equation empirical suggested the 3.8 3.45) (Equation relaxation equation Cole–Cole yields avalue of ω where 0≤β1i
© 2017 Similar observations hold goldSimilar for ε″ in plotted are equations These Expressing Equations 3.54 3.55Expressing Equations and coordinates polar in Davids Separating the real and imaginary parts of Equation 3.53, the real and complex parts are of 3.53, are Equation complex and parts real parts the imaginary and real the Separating
β in the high-frequency part. The main point to note is that the curve of curve ε″ the is note to that point main The part. high-frequency the in AISNCL EQUATION DAVIDSON–COLE by Taylor on and Coleon and show, 3.54 3.55, Equations and from that & s a constant characteristic of the material. of the characteristic s aconstant Francis 0 . Region of Water Debye Three of Constituents the at 20°C Relaxation Frequencies and Contributions Spectral 3.3 TABLE I II III Source: Group,
A. M.Bottreauetal.,J. Chem.Phys., Vol. 62,360–365,1975. LLC ε′ Fi −ε gures 3.13gures 3.14 and ε″ , which increases with β with , which increases =( εε r =( ∞ * = [( r = =( =+ ε θ s tan tan ε −ε becomes lowerβ=1(Debye) becomes βis increased, as yielding = ε s ∞ −ε s ε′ increases from 0 from value the of as βincreases unchanged remains −ε tan 0.0507 0.9136 0.0357 θ =ta −ε ∞ C () )(cos ϕ )(cos ∞ − 1 ∞ i 1 )[(cos( + )(cos ϕ )(cos ∞ εε εε ) ′ n βϕ s j 2 ε ωτ − +ε″ − axis and that passes through its peak value. its peak through passes that and axis ′′ , ) and the Debye the ( and curves θ β dc ∞ ∞ sin ϕβ sin
/ − ) β 2 β
] )] cos ϕβ cos
β β in the low-frequency part and decreases low-frequency decreases the and in part
p
=107 ×10 3440.3 ±8.0 17.85 ±0.30 f 5.57 ±0.50 i (GHz) against ωτ against 9 β =1) shown also are = 0.013, α= with rad/s loses sym loses (3.56) (3.57 (3.54) (3.55) (3.53) 101 3/31/2016 1:20:34 PM ) - Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 102 © 2017 an angle of angle βπ an ε′ on center the the with is asemicircle curve from the relation the ω from simple systems. 3.16 Figure of ε′ of 102 3.14FIGURE 3.13FIGURE The frequency ω frequency The aplot of3.54ωmust that Equation yield against noting right-hand of quantity line. the astraight ε circular arc, and at high frequencies, they lie on a straight line. on astraight lie they frequencies, athigh and arc, circular arbitrarily chosen. arbitrarily ∞ at the low-frequencyat the ends respectively high-frequency and The locus of Equation 3.53 in the complex plane is an arc with intercepts on the ε′ locus on of3.53 the The Equation intercepts complex with the arc in is an plane If the Davidson–Cole equation holds, then the values holds, equation the of Davidson–Cole ε then the If has been arbitrarily chosen. arbitrarily been has by Taylor
& /2 with the ε′ the with /2 Francis Schemat Schemat p corresponding to tan ( tan to corresponding p τ =1. We quote two examples relaxation in Davidson–Cole demonstrate to Group, ic variation of ε″ ic variation ic variation of ε′ ic variation ε´´ 0.0 0.2 0.4 0.6 0.8 1.0 10 LLC axis. To axis. way, explain it another on atlow a lie points the frequencies, shows the measured lossshows 143–144°C glycerol point, index in measured the (boiling –3 ε´ 1.0 2.0 3.0 10 –3 10 as function of ωτ function as as a function of ωτ afunction as –1 θ 10 ωτ / β –1 may also be determined determined be τmay also and ) =1may determined, be 10 dc − 0 axis, and as ω as and axis, = 10 10 tan 1 1 ωτ ωτ for various values of values β for various θ β for various values of values β for various 10
3 10 β =1(D β =0.75 β =0.25 β =0 β =0.5 s , 3 ∞, the limiting straight line makes makes line straight → ∞, limiting the ε ( Fi ∞ β =1(D β =0.75 β =0.5 β =0.25 β =0 may be determined directly, βmay determined be , and gure 3.15gure 10 ebye 5 ) ebye 10 Dielectrics in Electric Fields in Electric Dielectrics 5 ) ). ω As . The low-frequency value. The has been been value of τhas . The → 0, the limiting limiting 0, the axis atε axis (3.58) s and and 3/31/2016 1:20:34 PM K22709_C003.indd 103 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 Figure 3.12 Figure (see in inset the of ε″ increase anomalous an introduces conductivity dielectric that of the β of Debye value upon the the depending relaxation, lier, than relaxation is broader Davidson–Cole the later, with ear dealt noted but, of as relaxation are type the of determining methods The required. relaxation analyses relate a geometrical representation of representation ε′ relate ageometrical relaxation analyses relaxation mechanisms. upon the deciding shows contributionof AC the conductivity ε″ to i 3.17 shown contents of as Figure liquid, each in fractional various Dielectric Loss and Relaxation—I Loss Dielectric power law, ω a temperature, ateach peak of right the range, the to high-frequency the in clearly and seen, can be [17]. frequency over peak and aboutat 300 Pa), the awide of range temperature asymmetry The 1981.) 1333–1340, 196, 203, 213, 223, 241, Phys. 273, 256, G. P. 296 K). Whalley, J. Chem. (From E. and and Johari 3.16 FIGURE ω to proportional is low-frequency the branch and 3.15FIGURE approximates a straight line. The hypothesis proposed to explain this phenomenon explain to is to this proposed hypothesis The line. astraight approximates have frequencies, yields athigh seen, askewed and, which, atlow arc, is semicircular frequencies, aphysical be to claimed phenomenon be it [19].and cannot we as representation, geometrical The temporal and spatial fluctuations of the dipole in the alternating field. the turbulence Due to severe alternating dipolethe the fluctuations in of spatial and temporal © s found to hold true, though the Debye iss foundnot the though to hold accuracy applicable relaxation may be great also true, if 2017 We need to deal with an additional aspect of the complex of the aspect additional plot plane We an with of ε* deal to need Before concluding this section, it is noted that the Davidson–Cole equation and other similar similar other and equation Davidson–Cole Before the it section, that concluding is noted this The second example of Davidson–Cole relaxation is in mixtures of water and ethanol [5] of ethanol water and example at second ofmixtures The relaxation is in Davidson–Cole . by Taylor
− β ( Complex & ε″ β <1) holds true. as a func as Francis according to Davidson–Cole relaxation. The loss peak is asymmetric, asymmetric, is loss peak The relaxation. Davidson–Cole to εaccording of plot plane Group, ε (ω) 10 10 10 ´´ 10 10 10 ε ´´ –3 –2 –1 tion oftion 0 1 3 2–0 –2 and low-frequency) and plots ends of the [18] ( LLC 196 K in glycerol at various temperatures (75, 95, 115, temperatures glycerolω in various at 135, 175, 185, 190, β /2 24 . The slope of the high-frequency part depends onβ depends part high-frequency slope of the . The 223 K , and this contribution should be subtracted before contributionshould subtracted be this , and log ( ε ´ f, Hz) 75 K 68 175 K and ε″ and Glycerol The Davidson–Cole relaxation Davidson–Cole . The 296 K with an analytical equation, equation, analytical an with at both the high-frequency high-frequency the atboth Figure 3.18Figure , which fact is due the to 10 ). Equation 3.8Equation consid , Vol. 75, . er the the er 103 3/31/2016 1:20:34 PM - Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 104 © 2017 Vol. 104, 1996. 4441–4450, © ε″ 104 1309–1314, 1952.) 1309–1314, P. R. (From Phys. Auty, Cole, H. R. kilohertz. J. Chem. in and frequencies are points beside Numbers conductance. DC from arising polarization electrode locus; with (c of samples (b) d) of curves true and curve 3.18FIGURE 3.17FIGURE J. Chem. Phys. Bao J. al., ; (c)et Chem. M. L. (From relaxation. complex Davidson–Cole plot plane of εexhibiting by Taylor
& Francis Diel Compl ectric properties of water–ethanol mixtures at 25°C. at ε′ (a) part, mixtures Real of water–ethanol properties ectric in ice at 262.2 K. (a) Sample with interface parallel to the electrodes; electrodes; the to (a) parallel 262.2 at K. ice εin of plot interface with Sample plane ex Group, (c) (b (a)
ε˝ ) ε˝ ε´ 10 20 30 40 ″ 10 20 30 40 20 40 60 80
25 ε 50 75 LLC
0 0 0 American Institute of Physics.) Institute American 0 02 70 15 0 c: 10%wate b: 50%water a 7 : 90%water 7 10 10 7 2 2 5 20 3 r Fr Fr 3 1.5 equency (MHz) equency (MHz) equency 075 50 0.5 c 3 10 10 40 ε (a) ε ´ 3 3 ′ 1.5 ba b 1.5 a c 60 0.5 c: 10%wate b: 50%water a 1 (b : 90%water 100 10 10 c: 10%wate b: 50%water a 0.3 0.5 : 90%water ) b a c 4 4 0.5 80 (d ) Dielectrics in Electric Fields in Electric Dielectrics (c) r 0.1 100 r ; (b) imaginary part, part, ; (b) imaginary , Vol. 20, 3/31/2016 1:20:35 PM , K22709_C003.indd 105 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 former givenformer by
by Davidson Cole and (1951), is expressed as relaxation time the in wh in where defined by process rate achemical to according of temperature is afunction relaxation time The 3.9 model is able yield to This expressions plots. Davidson–Cole explain the that amplitude. higher with of number dipoles oscillate asmall amplitude, of dipoles small with vibrate number alarge While amplitude greater with is, vibrate to motion, that rotary exhibit oscillatory to level, others molecular more able be at the some dipoles than prevailing will at any instant, FIGURE 3.19FIGURE Dielectric Loss and Relaxation—I Loss Dielectric temperature. Figure 3.19 Figure temperature. [13] shows plots the of slow infinitely becomes as relaxation process the characteristic the we approach that meaning as Johari and E. Whalley, J. E. and Johari © 2017 In some liquids, the viscosity and measured low field measured viscosity and some the liquids, conductivity also In law, follow a the similar There is no theoretical basis for of dependence T, is τon no theoretical There At At
ich τ ich T = MACROSCOPIC RELAXATION TIME by T c is a characteristic temperature for a particular liquid. for aparticular temperature is acharacteristic Taylor 0 are constants. This is referred to as an Arrhenius equation in the literature. the in equation Arrhenius an as to is referred This constants. bare and c , the relaxation time is infinity according to Equation 3.60 which must be interpreted Equation to 3.60 interpreted according which be must is infinity relaxation time , the
Arrh & Francis plotted as a function of 1/ afunction as of Iplotted ice time plot relaxation enius of the Chem. Phys. Chem. Group, 10 10 10 10 10 LLC –5 –4 –3 –2 –1 5 , Vol. 75, 1333–1340, 1981.) Auty andCole ηη ττ = = ττ o 45 = 0 ex ex (1000/T )K p 0 against the parameter 1000/ parameter the τ against p ex kT kT () p () kT b Da and Gough b b − − η –1 vidson T T
and in some liquids, such as those studied some studied such liquids, in those as and c c
6 in ice. The slope ice.T in The of T . (Adapted from G. P.. (Adaptedfrom (3.60) (3.59) (3.61) 105 3/31/2016 1:20:35 PM Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 106 © 2017 (~10 for glycerol small is too propylene relaxation times and the with agreement glycol obtain to required lar volumelar (= 4π tion of the medium that hinders the rotation ofmolecules. rotation polar the Hence, τ hinders that tion medium of the T temperature transition glass the near behavior polymers the of amorphous Weof3.60 use book. Equation understanding the make in will gives activation line an of energy the 0.58 eV, discussion of which is beyond of scope afurther the 106 Ta (see found substances many as in temperature, decreasing with increases relaxation time The whic in where τ relaxation time quantity, original the it from and is quite different we that have studies dielectric is a macroscopic so from far discussed τ obtained relaxation time The 3.10 where
cal law:cal viscosity, the to is equated which parameter. ecule, whichparameter, is amacroscopic is amolecular mol of rotation the the hindering friction internal the only approximately valid be to because expected 3.63 Equation studies. dielectric from is, however, obtained relaxation time with reasonable agreement
τ relaxation time. molecular or the relaxation time internal the called usually and scopic quantity at room temperature and an effective an of ×10 radius and 2.2 molecular temperature at room example an As of of3.63, applicability Equation we consider aviscosity of has water 0.01 that poise employed to obtain insight into the relaxation mechanism from measurements of dielectric properties. of dielectric measurements from employed relaxation mechanism insight the into obtain to method the included involved to demonstrate here unit are molecule. details entire may These the be the involvedliquid, units twothe liquids, named for first are much the third smaller, the whereas for τ ation time, 2.5 ×102.5 of Debyerelaxation theory [3, p. equation 84], the who obtained m may be expressed in terms of the viscosity of the liquid and the temperature as temperature the viscosity liquidand of of the the may terms expressed be in ble 3.2 The spread in these values arises due to the fact that the molecular relaxation time, τ relaxation time, molecular the fact due that the to values arises these in spread The It is well known that the viscosity of a liquid varies with the temperature according to an empiri an to according temperature the with viscosityIt of is well the aliquidvaries that known with viscosity is qualitatively in agreement with the molecular molecular of viscosity the τwith is qualitatively with correspondence agreement observed in The −31 by
a τ m Taylor is a constant for cis aconstant agivenh Therefore, 3.63 liquid. Equation may now expressed be as MOLECULAR RELAXATION TIME m is the mo is the −11 −11 is called the molecular relaxation time (see relaxation time molecular next the section), is called viscosity, ηthe molecu νthe and ). 3 ) and of reasonable size for n-propanol (60–900 ×10 of reasonable) and for size n-propanol (60–900 s. condition At ωτ the relaxation, , obtained from the dielectric studies may be related in several mayin related ways. studies be dielectric is that the inference from The , obtained & Francis a - fric inner due the to be to is assumed relaxation time molecular The radius. lecular 3 /3, radius),volume molecular molecular where a is the The spherical. assumed Group, LLC G in Chapter 5. Chapter in τ m τ = 1 is satisfied, and thereforeand ω =1is satisfied, η τ m ∝ m ∝ = T = 1 ex 4 ex 3 πη p kT kT η p kT υ c a kT 3 c
−30 −10 −10 m m, leading to a relaxation time of arelaxation to time leading m, m 3 ). used byused Debye. τ Dielectrics in Electric Fields in Electric Dielectrics m
is a function ofis a viscosity.function =4×10 D 10 , and the relax the , and /s, which is in m is amicro (3.65) (3.64) (3.63) (3.62) 3/31/2016 1:20:35 PM - - - - - K22709_C003.indd 107 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 is given by [20] slope. the have been summarized by Hill et al. [20]. et al. by Hill ωτ Let τhave formulas calculating summarized been for- experimentally, the and relaxation time convenient the many for are methods measuring There 3.11 the standard deviation between computed values and measured values using easily available deviation values computed measured between and software. standard the check to may they used fact be that advantagethe relationships of in scale. lies The these frequency the of of ε′ function The mac The Dielectric Loss and Relaxation—I Loss Dielectric 3.28 3.29 and may several expressed ways: be in alternative
© 2017 ε′ The equations may also be written as written be may equations also The The relation between the molecular relaxation time, τ relaxation time, relationmolecular the The between have involve they each advantage that values the Items 3–6 measured only one experimentally of the Equat 2. 6. 4. 3. 5. 1. and ε″ and
It is ea εε εε ε ε ε by ε ε xx TAGTLN RELATIONSHIPS STRAIGHT-LINE s s 11 ′ ′′ ′′ ′ s roscopic relaxation time is higher in all cases than the microscopic relaxation time. microscopic the than cases all in is higher roscopic relaxation time ion 3.66 shows of ε″ agraph that ε Divi − − x Taylor − − 11 − 11 = ′′ , but the disadvantage is that the frequency term, x, term, frequency the is that disadvantage , but the n n nx n = εε 2 ′ 2 22 22 sy show to that ss () ding the second equation from the first, the from equation second the ding will also be linear. n linear. be also will = = εε = = − & ss 2 () 1 1 nn − εε Francis ss + 22 + ss x 1 nx − x − + x 2 22 nn nx 22 22
Group, + − 1 + ε () + − () LLC − 1 n − 1 2 n 2 2 and ε and /x /x s τ from τfrom and respective intercepts the from obtained are x x against against = = = n ε ε εε s 2 ′ s + + ε ε − − ′′ ′′ 2 2 ε′ n τ 2 ′ will be a straight line. The graph of ε″ graph The line. astraight be will m
m ,
enters through a squared term, distorting distorting term, asquared through enters and the macroscopic relaxation time, relaxation time, macroscopic the and = 1 and n =1and 2 =ε ∞ . Equations . Equations (3.66) (3.67) x as a as 107 3/31/2016 1:20:35 PM τ , Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 108 © 2017
temperature, the shape of the J of shape the the temperature,
of the difference in the activation the w energy in difference of the wher
tage oftage using ω J to according constant dielectric dipoles vary between two constant values, two between w constant dipoles vary 180° through rotation is considered ajump. position versa. ortwo vice allowed. Only positions antiparallel A the to are aparallel from rotates field,the − amolecule. to dipole of energy is the + dipole The attached is the rigidly that assume also Let us antiparallel. field,the other electric the to one only parallel twoin directions: Fr 3.63. Equation to according To temperature is dependent upon the that a relaxation time understand of [21], Fröhlich treatment consider to point dynamical the It is advantageous atthis who visualized 3.12 108 the Arrhenius relationship, and the relaxation times corresponding to w to corresponding relationship, relaxation times the and Arrhenius the relaxation time at higher temperatures. ω temperatures. at higher relaxation time
ö We hav dip The Fr hlich’sdipole orient the that can and suppose fieldaxis electric along the + model, let us take by
e ö hlich generalized the model, which is based on the concept that the activation the energies concept that of model, on the which the is based generalized hlich Taylor FR e used here the formof expressions the here e used [22] given by have they Williams because advan the when it is antiparallel, and 0 when it is perpendicular. As the field alternates, the dipolethe field the alternates, As 0when it and w when is perpendicular. it is antiparallel, Ö oles are distributed uniformly in the energy interval dw interval energy the in uniformly distributed oles are & HLI Francis p ,
CH which is the radian frequency atwhich ε″ frequency radian which is the ’ Group, S ANALYSIS S LLC H ω ω J and H and ω = ω
and and = εε s ω εε εε ε − s ′ s ω ′′ p H − − curves will vary with temperature, tending toward tending asingle temperature, with vary will curves = == ∞ ω p . The analysis to fairly lengthy leads expressions. The terms in is related to τ to is related ww τ ∞ ∞ 11 =− 2 1 −w 21 ττ ττ kT =− e 1 s 1 − 20 10 and w and −− 1 () = = () ex ww 1 2 2 21 ss , and the final equations are equations final the , and kT ; 2 ta 1 e e s nt kT w w kT 2 Ln x − 1 . Fr . 2 1
=
1 and τ and ω 1 ö ττ ω 1 hlich assumed that each process obeyed process each that assumed hlich + 12 ex p + is a maximum. Since s is amaximum. − xe 2
xe 2 2
2 an , − s s − 1 and make a contribution to the the acontributionto make and
Dielectrics in Electric Fields in Electric Dielectrics 1 and w and when it is parallel to to w when it is parallel 2 are given are by is a function of is afunction (3.68) 3/31/2016 1:20:35 PM - K22709_C003.indd 109 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 for of range values shown V=0–100 meV the various of in Vare the observed variation of ε″ variation observed the Dielectric Loss and Relaxation—I Loss Dielectric of of time will vary from 0 to infinity, and ε″ infinity, 0to from vary will time
FIGURE 3.20 FIGURE where ( parameter √( parameter between the two wells. the between loss model, the index is this to According which ε″ in difference, energy upon the range, depending a temperature be will kT atkT occupiedequally wells both are until well higher increases the lower the from well higher to is 0. However, of number dipoles the in temperature, increasing with lowerthe empty. well higher of number remains well dipoles the The jumping and is occupied, level energy the in two of positions the dipole. of asymmetry the At lower assuming temperatures, as Figure 8). Figure as water shows havingloss polyimide adsorbed in such behavior (see Ref. [23], 5 is misprinted Figure T in increase with increases show peak the that materials the © 2017 ε The double wellThe of potential Fr The width o width The Figure 3.20Figure shows factor the The temperature dependence of dependence temperature The ma ′′ ε″ x – . (1986). . by plots are independent of the temperature. However, temperature. of independent ω plots the are of ε″ measurements w Taylor 2 − τ 2
w / τ 1 1 & Depend f the loss curve increases with increasing w increasing with loss increases f the curve ) is the difference in the asymmetry of the two of positions. the Plots of f asymmetry the in difference ) is the ) = 1, 5, 10. They correspond to a range of heights of the potential barrier. ω barrier. potential ) =1,of of the heights arange to 5, 10. correspond They Francis ence of dielectric loss ε″ of dielectric ence Group, 0.01 versus T, ε LLC ε m ′′ ′′ εε ε = ′′′ ′′ ′′ / ö ∝ hlich leads to a relaxation function that shows that the peak of shows that peak to a relaxation function leads the hlich that ε ma ta 0.1 ma ′′ 32 nt x it is possible to estimate the average energy of asymmetry it average is possible the of estimate to energy asymmetry as a function of ω afunction as kT 1 ε x −− will have value will range. over aconstant frequency entire the ´´ in the context the of in Fr 1 /ε ta ω ω co ´´ max nnt pp −− sh ( 1 ω according to Equation 3.69 for three values of the of values the 3.69 for Equation to three ) onωaccording − ω/ω 2 τ τ τ τ 11 1 1 2 2 1.0 ww 0.25 0.5 0.75 p − − 12 − kT an an / 1 . For example, of measurement dielectric ω 1 p 0 = ω 2 calculated according to according calculated τ τ ω
ö − w 1 2 fT hlich’s by explained is often theory () 1 and in the limit, the relaxation the limit, the in , and τ τ
1 2 100 ≫ ( (
F igure 3.21igure w 2 −
w increases strongly. increases 1 in awide of range in ). Therefore, there ). p is the frequency frequency the is By comparing By comparing ( T ) versus T (3.69) 109 3/31/2016 1:20:35 PM Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 110 © 2017
et al., Trans.et al., EIectr. Insul. metry potential w potential metry 110 which value, of its maximum loss terms the index in Then [24] is dispersion equation Fuoss–Kirkwood The 3.13 3.21 FIGURE In the Fu the In Let us denote the frequency at which the loss index is a maximum for Debye loss atwhich the index frequency is amaximum relaxation by the usdenote Let ω by
leads to leads Taylor US–IKOD EQUATION FUOSS–KIRKWOOD oss–Kirkwood derivation, this expression derivation, this becomes oss–Kirkwood &
Francis Temp 1 −w erature dependence of the relaxation strength calculated for various values of the asym of values the for various calculated strength relaxation of the dependence erature 2 Group, . The upper curve corresponds to the symmetric case w case symmetric the to corresponds curve upper . The , Vol. (1), 24 31–38, 1989.) f (T ) LLC 30 10 20 0 80 εε ′′ εε =+ 20 meV 10 meV εε ′′ ′′ = 30 meV ∞∞ = 43 meV V ε ma ′′
= ′′ 170 60 meV ma ′′ ()
0 meV x εε = x se s se ω − ω c c 2 hL p T (K) hL ε + ma ′′ ε δ δ ω ma ′′ x ω 1 n p x n + , is , ω () ω
ωτ ω 260 () ω ωτ p p δ
2
δ
Dielectrics in Electric Fields in Electric Dielectrics 350 1 −w 2 = 0. (From J. =0. (From Melcher (3.70 3/31/2016 1:20:35 PM p ) - . K22709_C003.indd 111 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 integral, integral, axis atω= axis relation is backbone. Kirkwood–Fuoss The poly(acetaldehyde) chain of the part arigid dipole of forming the show characteristics common Slope value gives of β the parameter ( parameter different. be will value relaxation is only observed; parameter the of the Cole–Cole This e This FIGURE 3.22 FIGURE where Vol. 30, 1383–1398, 1997.)
into account the hydrodynamic diffusion of chain segments under an electric field electric an segments under of chain diffusion hydrodynamic the account into takes apolymer shows molecule in and theory is not units mer aconstant The adistribution. of number mono the PVCjointed that weight molecular including the Recall distribution. chain, and Relaxation—I Loss Dielectric tion, we plot ( arccosh © 2017 When When The empirical factor δ empirical The Figu Kirkwood and Fuoss [26] also derived the relaxation time distribution function for function afreely distribution Fuoss [26] and relaxation time derived the also Kirkwood xpression shows that Fuoss–Kirkwood relation holds true for most materials in which the which the in xpression relationfor holds most shows materials true Fuoss–Kirkwood that by u =n re 3.22re shows such aplot for at353 (PMMA) polymethylmethacrylate evaluated the and K, x =
p < −ε with aslope with of δ n> & The de The ∞ Francis , ) has avalue) has of ≅ ωτ ε is the degree of polymerization,
xu Ei 22 ω (radHz) ≅ φ its average value, Ei(u) J. Phys. D: Appl. Phys. J.D: Appl. Phys. Mazur, K. (From 2.
uj )][ 1 − xu ] du 10 5
ω reaches its peak value its reaches peak the exponential the
[22]. PVC and (3.71) 111 3/31/2016 1:20:36 PM - , Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 112 © 2017 the shape of the distribution is independent of the temperature. of is independent the distribution of shape the the value of relation approximate 0.63 holds an has when [26]. Kirkwood–Fuoss the likely that It seems parameter Cole–Cole the temperatures, higher At independent. these noticeably temperature are dependent [26]. PVCin temperature is markedly for PVC. not holding true distribution theory for The factreason explains their the this and perature, tion is that it is free of any empirical parameter because J because parameter of itany empirical tion is that is free at 112 = 0, the skewed α=0, the When is semicircle is generated. dispersion shown equation the in arc circular as constant tric but frequencies not at lower athigher successful frequencies. (Davidson–Cole) arc fit with a is skewed attempted Likewise, frequencies. circular an not athigher arc at lower(Cole–Cole) loware successful fit to frequencies circular a Attempts but frequencies. way. at arc acircular several and complex frequencies polymers, In the athigh plot plane is linear where a significant changephysical many in properties temperatures, transition glass the with ated of several polymers. properties α dielectric the sured uncertain. very of dispersion parameters determination the rendering molecules with of simpler structure, obtained simple as those rarely as are plane (2)have dispersions meaningfully. complex plots describe to pooled of the shapes be to the in The are not sufficientfrom analysis dispersion. temperatures for several the Data of temperature fixed as are follows: difficulty from a (1)data that polymers dispersionso broad in is The generally very for of reasons case simple this the selves molecules.in main is used that The simple to the treatment is obtained. equation Davidson–Cole the with askewed or whether accordance in tion arc is obtained - equa Cole–Cole the with accordance in arc check to asemicircular whether analyzed and obtained complex the plots plane data, is are covered. From set of each of range isothermal temperature desired the till repeated measurements the and varied is then possible. of as range frequency temperature The over wide a as temperature atconstant complex material of the the constant dielectric measuring molecules by oris inorganic studied organic small dispersion in The ticular, polymer materials. par in We now structures, aposition molecular more to complicated are in extend to treatment our 3.14 x Many polymers exhibit a temperature-dependent distribution, and at higher temperatures, some temperatures, athigher and distribution, Many polymers exhibit a temperature-dependent of is independent tem of Kirkwood–Fuoss distribution height and shape of no-parameter The Comb for here two convenience, reproduced by dispersion equations, The represented are α study to the attempt an In do not lend measurements them complexThe by plots plane isothermal of polymers obtained This f This and is symmetrical about this value. The main feature of Kirkwood–Fuoss distribu of Kirkwood–Fuoss feature value. main about The this = 0.1 is symmetrical ×2πand by
Taylor HAVRILIAK AND NEGAMI DISPERSION ining the two equations, Havriliak and Negami proposed a function for complex a function the proposed Negami and dielec- two Havriliak equations, the ining unction generates the previously discussed relaxations as special cases. When β=1, When cases. previously the unction the generates special as relaxations discussed & Francis εε εε s Group, * − − εε εε ∞ ∞ s * − LLC − =+ () ∞ 1 -dispersion in many polymers, Havriliak and Negami [27] Negami and have- mea polymers, many Havriliak in -dispersion ∞ =+ j [( ωτ 1 εε εε s * 0 − − j − ωτ β ∞ ∞ : =+ )] 11 Sk −− [( α 1 ewed : j ωτ circ se -dispersion in a polymer is the process associ process apolymer in is the -dispersion HN micircle( ul − rar ar ω )] 1 and H and −− αβ c( Da Co
ω vidsson–Col are expressed in terms of log x terms expressed in are le–Cool Dielectrics in Electric Fields in Electric Dielectrics e)
e)
(3.72) 3/31/2016 1:20:36 PM . - - - - - K22709_C003.indd 113 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 Dielectric Loss and Relaxation—I Loss Dielectric expressions the obtain to denominator the rationalize rem and H subscript For β=1, convenience, α=0and Debye the When is obtained. we function the omit obtained. At ver Equat wher susceptibility functions have functions susceptibility slopes, two shown extreme as the cases. in © 2017 To given relaxation function the test by 3.72, Equation we apply successively Moivre’s De theo At ver ω→0, as case, second For the as ω→∞, case, 3.72 first For Equation the (for becomes definition of χ″ e ion 3.72 for extreme values may examined of be ω y high frequencies, ε″ frequencies, y high by Taylor y low ε″ frequencies, – N hereafter. & Francis εε εε εε εε * s * s − − − − Group, r ∞ ∞ ∞ ∞ =+ =− ∝( =− = 1 11 1 LLC () ()
− ωτ ∝ ( j ε′ ωτ () θ βω βω ωτ −ε () () tan[ εε ε = βα j s ′ () βα −ε′ τα εε arc −= () 1 ∞ − −− τ ′′ H βα tan 1 ) ∝ω αα − 1 =− () ∞∞ () {} si () tan 12 1 ) 1 − co − n( r −= α ∝ α απ β 1 s[ r β 2 {c / ω 2 (1− απ 1 β 1 βα 22 () π / (1− os + () 2 α () ωτ 12 ) / () s () = α [( ωτ εε ) . These results have results . These suggestion the to led the that ] s εε 2 1 εβ − − − s + α ∞ ε 1 − − π εε () ′′ co α ω () ′ ))] si / ω → πα si ωτ ε () ′ , namely, [28]. ω→0 ω→∞and s →∞ n( ω co 0 () ′′ ] n // ω → 21 −−− →∞ )c απ 0 s( 1 2 θ απ − +− j βθ 2
si ) j
n[ ∞ os si )
βα
n[ ()
() 12 απ π 2 π , see Section 3.15), see
/ 2 ] ]} ,
(3.76) (3.75) (3.73) (3.74) 113 3/31/2016 1:20:36 PM - Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 114 © 2017 w
this frequency by the subscript p. Havriliak and Negami [29] Negami and prove of bisector p. also by subscript ϕ angle frequency the the Havriliak that this β
denoting the corresponding value of ϕ ϕas corresponding the denoting where we have substitution the βθ made 114 J. Phys. D., Appl. Phys. Jonscher, K. Appl. A. D., J. Phys. (From plot circular. is the low frequencies, 3.23 FIGURE complex the from lowing plots: plane procedure complex the plot plane intersects at
β αand parameters . Again, the relaxation time is given relaxation time the by definition . the Again, hich provides a relation between the graphical parameter ϕ parameter provideshich graphical arelation the between From Equations 3.73From Equations 3.74 and we following the note dispersion parameter. the to regard with The analys The We therefore evaluate can ε 2. 2. 6. 4. 3. 5. 1. 1. by
As As employedmaterials. specific in The parame The = 1/ ω= to according lated is calcu relaxation time corresponding the and is determined, frequency the tion point, As As The parame The parame The ε high-f The obtained. is low-fThe The angle angle The Taylor ∞ is obtained. If data on available, refractive relation index data are the ε If then is obtained. ωτ ωτ 0, →0, ∞, →∞, &
is of experimental data to evaluate the dispersion parameters is carried out by fol the is carried evaluate to dispersion parameters the data is of experimental Francis Complex requency measurements are extrapolated to intersect the real axis from which ε from axis real the intersect to extrapolated are requency measurements ϕ requency measurements are extrapolated to intersect the real axis from which from axis real the intersect to extrapolated are requency measurements ter ϕ ter is calculated by 3.78. Equation βis calculated ter by 3.79. αis decided Equation ter ε′ L ε′ , we note that Equations 3.73, we Equations that note 3.74 and expression the result in - intersec curve. From the measured the extended intersect to and is bisected →ε →ε Group, L plane plot of of plot plane is measured using the measurements at high frequencies. athigh measurements using the is measured s ∞ φ and ε″ and 11 and ε″→ and L
LLC ε (ω)
ε ´´ ∞ lo g s τ and ε and . The parameters parameters . The →0. Therefore ε εε εε * p s 0. Therefore, ε − − according to H-N function. At high frequencies, the plot is linear. At plot linear. is the frequencies, At high H-N function. to ε according ε p * ∞ . The point of intersection yields value of point intersection the of. The αby εε from the intercept of the curve with the real axis. To axis. real the the with find curve of the intercept the from ∞ φ ∞ =ϕ ′ L ε − =− ϕ ′′ L . By applying condition ωτ the L ∞ =(1 −α φ (Figure 3.23, (Figure πα L == /2 () 1 ta − εε nt ε * →ε ´ * =ε p (ω) ′ βθ − ) βπ log[ ∞ ∞ s ωτ /2 and ε″ and . . an 22
Table 3.4 1, and let us denote all parameters at ≡ 1, parameters let all and usdenote + φ L
and αand dispersion parameters, the and si are also determined. also are ωτ =1 n( απ ), we get / Dielectrics in Electric Fields in Electric Dielectrics 2 )] ε s
∞ to Equation 3.76 Equation → ∞to and
, Vol. 32, R57–R70, 1999.) ∞ = n 2 may be (3.77) (3.78) (3.79) - s 3/31/2016 1:20:36 PM L -
K22709_C003.indd 115 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 glass transition temperature, T temperature, transition glass the near shape exhibit polymers that asymmetrical amorphous relaxation in the to describe ful values is given Table in 3.4 is found use Negami very to be and [30]. of Havriliak function The should decrease as the temperature is lowered. α The temperature the as should decrease β and connectivity is lowered. α the suggested that It been has
In the published literature, some [32] authors published susceptibility literature, the dielectric In the use 3.15 where expressed as is then H-N function The material. lowerthe (e.g., frequencies, 5.1), Figure see conductivity DC is due of the to the increase this and ofmentioned earlier, measurement ε″ 5. Chapter in which may considered indicative[31]. be of this is treated aspects of these description Adetailed ( parameters
Dielectric Loss and Relaxation—I Loss Dielectric constant: dielectric and susceptibility dielectric ε constant dielectric of the instead material © 2017 A final comment about the influencehere. the As of conductivityabout comment lossdielectric isappropriate on A final of five several evaluated data the polymers and dispersion the analyzed Negami and Havriliak
by σ ILCRC SUSCEPTIBILITY DIELECTRIC dc Taylor is the DC conductivity. DC is the Poly(carbonate) Polymer and TemperaturePolymer According Parameters Expression to Dispersion H-N Selected 3.4 TABLE Polychloroprene (−26°C) Poly(cyclohexyl methacrylate)121°C Poly(iso-butyl methacrylate)102.8°C oynbtlmtarlt)5° .924 .606 0.60 0.62 7.06 2.44 4.29 Poly(n-butyl methacrylate)59°C oynhxlmtarlt)4° .624 .007 0.66 0.74 4.40 2.48 3.96 Poly(n-hexyl methacrylate)48°C oynnlmtarlt)4.° .124 .807 0.65 0.73 8.18 2.44 3.51 Poly(nonyl methacrylate)42.8°C oynotlmtarlt)2.° .826 .007 0.66 0.73 9.60 2.61 3.88 Poly(n-octyl methacrylate)21.5°C Poly(vinyl acetal)90°C Poly(vinyl acetate)66°C ydoa oymty ehcyae .225 .605 0.55 0.53 7.96 2.52 4.32 Syndiotac poly(methyl methacrylate) Poly(vinyl formal) ε s , & ε Francis ∞ , is related to the local density fluctuations. Chain connectivity in polymersin connectivity density fluctuations.Chain local the to is related α , β Group, , τ ) for one each of (see them 5). Chapter of list A more recent tabulated LLC G εε εε . In the vicinity of T vicinity the . In s * − − – ∞ ∞ characteristics shows loss the index toward in ω characteristics asudden rise =+ [( 1 χ χ′ * =ε -parameter represents a quantity that denotes chain chain denotes that aquantity represents -parameter *, following the and relationships hold the between =ε′ χ″ .431 .507 0.29 0.77 6.85 3.12 3.64 .526 .705 0.51 0.57 7.37 2.63 5.85 .324 .307 0.33 0.71 5.33 2.45 4.33 .223 .807 0.50 0.71 8.28 2.36 4.02 .130 .109 0.51 0.90 7.11 3.02 8.61 .526 .705 0.51 0.56 7.37 2.62 5.85 jj . . .308 0.30 0.89 6.43 2.5 6.7 ωτ ε =ε″ s * −ε −ε 0 )] ()
1 − ∞ G ∞ αβ
, the ε″ , the
ε ∞ -parameter slowly-parameter above increases T − ωε ω –log σ log f dc o
max curves become broader as T as broader become curves β α , χ * =χ′ −j χ , of the , of the (3.80) (3. (3.81) 115 82) 3/31/2016 1:20:36 PM G - ,
Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 116 © 2017 nection with the Debye equations, the decreasing part of χ′ part Debye decreasing the with nection the equations, in proportion to ω to proportion in
Figure 3.24a showsFigure of χ′ variation the χ″ slopes the plots of advantage the that of the χ′ has function susceptibility of the terms in ties 116 (c) Davidson–Cole. 3.24 FIGURE where part of χ″ part against ω against Equat quantities, expressednormalized often as last two are quantities The Equat by ω Taylor p ions 3.81 3.82 and aconcise expressed formas be in may also Debye susceptibility function ion susceptibility 3.83 Debye the as is known is the peak atwhich χ″ peak is the atω & provide for aconvenient possible discussing the relaxation mechanisms. parameter
Francis ≪ Schem
ω −1 p . changes in proportion to ω to proportion changes in Group, atic of frequency dependence of susceptibility functions. (a) functions. Debye; of susceptibility (b) dependence Cole–Cole; of frequency atic LLC χ is a maximum. Alternately, we is amaximum. have * (c) (b (a) ∝ χ )
χ , χ χ , χ χ , χ ′ ´ ´´ ´ ´´ ´ ´´ 1 = and χ″ and + χ ´´ χ εε 1 εε χ χ ´´ j ´ ´ ωτ ss ′ χ ω − − ω χ 1–α * ´´ ω +1 χ = +1 as a function of [8]. frequency afunction as con in discussed, As ∝ ∞ ´ ∞∞ Fr Fr Fr 11 +1 ; 1 equency equency equency + , and the decreasing part of χ″ part decreasing the , and + ωτ χ 1 1 j ′′ 22 ω ω = p εε ω −
ω –1 ω ε − –2 ω α– ′′ + is proportional to ω to is proportional –β j 1 ωτ ωτ 22
. Expressing the dielectric proper dielectric . Expressing the
Dielectrics in Electric Fields in Electric Dielectrics atω −2 . The increasing increasing . The
≫
ω p changes changes (3.83) and and 3/31/2016 1:20:36 PM - - K22709_C003.indd 117 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4
Dielectric Loss and Relaxation—I Loss Dielectric
which is b tion form having the β αand of two parameters, independent terms form the has function Davidson–Cole The loss occurs. which maximum
ties and divergences of these functions may be summarized as follows: as divergences and ties may summarized be functions of these
© 2017
These sus These The Cole–Cole function has the form, in susceptibility terms, susceptibility in form, the has function Cole–Cole The Fi 2. 3. 1. gure gure
The Deby The The Cole– The f > ω ior similar to that of Debye. that to region, high-frequency χ″ the ior In similar portional to ω to portional functions have Debyefunctions none. the has whereas asingle function parameter, range ofrange 75–296 of range a frequency 10 K and of glycerol properties surement of by dielectric Blochowitz [17] et al. over atemperature is also symmetrical about ω symmetrical is also to to χ′ function, susceptibility of the part real of log ( In this context,low-frequencyas ω the defined this regionsIn high-frequency are and ω to proportional low-frequency The functions. Davidson–Cole region aslope is dispersion of has that the cal line drawn atω drawn line cal The DavidThe the low-frequencythe ω to region aslope and proportional 3.85 by is proposed Blochowitz [17]. et al. by
≫ ω Since the D Since the To we function, Davidson–Cole refer of- mea applicability the the to the demonstrate Taylor max 3.24b −
roader than the Debye function, though both are symmetrical about the frequency at frequency about the symmetrical are Debye both the though function, than roader ω β . It should be borne in mind that the Cole–Cole and Davidson–Cole susceptibility susceptibility Davidson–Cole and Cole–Cole the that mind in . It should borne be , p ceptibility functions are generalized, and the relaxation mechanism is expressed in relaxation mechanism the and generalized, are functions ceptibility f
. In the high-frequency region, high-frequency χ′ the . In there are two slopes, are − there ) is shown in & - of a frequency.function as functions c showof similari and these variation The the e function is symmetrical about ω is symmetrical e function son–Cole susceptibility function (Figure 3.24c) is asymmetrical about the verti about the 3.24c) (Figure function is asymmetrical susceptibility son–Cole Cole function (Figure 3.24b)Cole (Figure function shows χ″ that Francis α avidson–Cole function uses one parameter only,Equation one uses to parameter amodification function avidson–Cole −1 . Group, , and the high-frequency region has a slope that is proportional to 1/ to region high-frequency aslope is proportional has the that , and p . In the low-frequency the . In region, χ″ Fi gure 3.16gure LLC p . The variation of χ′ variation . The and − β and . χ χ At f < f * χ χ * = * * = ∝ γ ∝ [( , appearing in that order for increasing frequency. for order that increasing in , appearing 1 1 - func susceptibility their Negami, and , by Havriliak 1 [( , in the high-frequency region is also proportional region high-frequency proportional is also the , in () max + + 1 1 has a slope proportional to 1/ to aslope has proportional + + + j () , j ωτ ωτ C ε″ j j ωτ 1 ωτ j 1 1 oo ωτ o −2 behaves similar to a Debye but behaves at function, similar p )] ≤ 3000 MHz. A plot MHz. ≤ f ≤ 3000 1 o in the high-frequency region pro high-frequency is also the in − and narrower than the Cole–Cole and and Cole–Cole the than narrower and 1 )] − αβ β βγ α β − is proportional to ω to is proportional
has a slope proportional to ω to aslope has proportional
α −1 in the high-frequency region. high-frequency It the in is proportional to ω to is proportional , which is abehav ω ε″ 2 ( (Figure 3.24a). (Figure f ) as a function a function ) as
≪
− ω β . The . The 1− p and and α (3.86) (3.85) (3.84) in in ω - - - . 117 3/31/2016 1:20:36 PM Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 118 © 2017
appears in both the expressions the both for in χ′ αappears parameter the that fact that the low-frequency behavior is not behavior. entirely high-frequency of independent is due the to This G function adistribution through is considered response asummation, dielectric observed models, the these In H-N functions. Davidson–Cole, and one Cole–Cole, with of the associated is usually times of relaxation adistribution suggested that It been one functions. to has of these data measured the
and τ+∆τ τand between ation times by polymer. is denoted G the function distribution Since the regions of crystalline in distribution energy complicated of segments or due the to molecular array acomplex in may longhave twisted be the fact chains due that the to of anumber relaxation times (Section topic 3.6), more closely section.Polymers this we generally and this examine to intend in asingleWe relaxation time is more than which situations there in have considered the already 3.16 118 ( τ
Though this susceptibility function has two parameters that are independently adjustable, are the that two has parameters function susceptibility this Though have no physical meaning, though these models are successful in fitting fitting in β have successful αand models are no physical these though meaning, parameters The
4. ), Debye of individual for responses group. each by
with Equation 3.85. ( Equation phase with glass the relaxation in The The H-N f The onsetα ofdue the to being equation this from showset al. relationship departure for the this substances, many tibility functions; wetibility 1–3 get functions cases: special as 3.73 Equations to regard with ments made 3.74 and applicable suscep equally their to are Blochowitz dependent on T βis weakly et al. relation by the is obtained two slopes the relaxation time tion merge because mean one. into The law γ to replotted in the T the in replotted wher contributionof high-frequency α The Here, Here, power law over of range awide 10 frequency ior. For glycerol, γ of behav for type systems many exhibiting this similar is very dependent and temperature b. a. c. Taylor DISTRIBUTION OFRELAXATION TIMES
At the gl At the The The c The tem The = 0 and 0< α =0and 0 <α1an α =0and e τ T & o f = 1/ = -power law. -power is a constant that is dependent on the material. Figure 6a in the work the of in 6a Blochowitz Figure material. is dependent on the that is aconstant ondition Francis unction is more general because of the fact that it has two parameters. The com The it fact two of that has parameters. the because unction is more general perature dependence of dependence β perature ass transition temperature, the two power the laws temperature, transition ass merge accordance one into in ω p β =1gives Debye the equation , and C , and Group, = 1 gives the Cole–Cole function d β=1gives Cole–Cole the domain atf domain β signifying asingle slope forβ =γsignifying log f =0.07 behavior ±0.02. The below T <1gives function Davidson–Cole the LLC o is a parameter that controls the frequency of transition from β of from frequency transition controls the that is aparameter -process. τ is given G as = = 1 Hz, an exponential relationship is obtained according to according exponential relationship an is obtained = 1Hz, ω lim → 0 ε ω ′′ -relaxation is frozen out-relaxation atT is frozen =− , χ βτ γ ′′ , and C , and ( o τ = ) d −2 e increases rapidly approach to β γincreases , whereas () τ TT is shown in Figure 4b in the work the 4b is shown of in Figure in C τ 5 0 ( 0 Hz. Below T Below Hz. τ ), the fraction of), dipoles fraction having the relax ≪ – T T g - yields func aDavidson–Cole 0 occurs according to χ″ to according occurs g ) is determined by asingle ) is determined Dielectrics in Electric Fields in Electric Dielectrics ≪ and χ″ and g , the coefficient, the γ T g . If the data on χ″ data the . If . -power -power =ω is not not is are are . − γ - - - . 3/31/2016 1:20:36 PM - K22709_C003.indd 119 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 distribution lead to complicated functions of ε′ functions complicated to lead distribution cedure, as mentioned earlier, is not straight forward. Simple mentioned forward. of earlier, as is not G straight functions cedure, ε″ complex shown. also corresponding plots plane are The αβ H-N expression βand for values parameter various of the physical justify to reality. entitiesasolid, existence of in the relaxing with correlated be cannot of relaxation times distribution Debye However, ideal functions. the susceptibility behaviorfrom the in a suggested it that been has departure explains the 3.90.3.88 relaxation time Equations existence of through a The distributed hand sidehand of3.87 Equation ( as Dielectric Loss and Relaxation—I Loss Dielectric behavior noninteractively, complex becomes the constant dielectric index are expressed as index are © also lead to complicated functions of G functions complicated to lead also 2017 By separ Assumi It is helpful express to 3.89 Equations 3.90 and [22] as Fi G 3.89Equations basis 3.90 for the and determining are The suscep The In some out texts by [9,In expressing of3.87 right- Equation the is carried p. normalization 97], the gure 3.25 gure by Taylor ng that each differential group of dipoles with different relaxation times follows of group relaxation dipolestimes different with differential each ng Debye that ating the real and imaginary parts of Equation 3.88, the dielectric constant and the loss the and of3.88, constant Equation dielectric the parts imaginary and real the ating tibility function is expressed as function tibility & helps to visualize the continuous distribution of relaxation times according to the the to according of continuous relaxation times distribution the helps visualize to Francis Group, LLC ε s −ε εε εε εε ′ *( J ∞ ′′ =+ ω ). This results in the absence of this factor in the last term in in last term the factor in absenceof this the in results ). This =+ =− = χ ∞∞ () ∞∞ εε * εε ( τ s ∫ ∞ s 0 ′ = ). − () − Gd εε εε ∫ ∞ ε 0 and ε″ and s () s ∞ ττ ∞ ∞ − 1 − G + = ∫ ∞ 0 () ω j ωτ ∫ ∞ ωτ 0 ) 1 . On the other hand, simple of ε′ functions hand, other the . On ∫ = ∞ 0 1 ∫ ∞ 0 + Gd 1 1 + Gd G d 1 Gd ωτ + () ωτ τ + () () ττ () ττ τ 22 ωτ ττ 22 j ( = 1 using Equations 3.92 =1using Equations 3.93 and [29]. ωτ τ 22 ) from ε′ ) from and ε″ and data, though the pro the though data, ( τ ) such Gaussian as (3.87) (3.90) (3.88) (3.89) (3.92) (3.91) and and 119 3/31/2016 1:20:37 PM - Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 120 © 2017 temperatures. class polymers. ofto the Its refractive atactic index is 1.437. at all is observed A single peak broad 120 m 3.73. Equation in β values shown for also αβ complex plots of εare plane responding 3.25 FIGURE of 25 Hz–100 kHz is shownof Hz–100 25 in kHz phous polyacetaldehyde [22] of range +34.8°C overto −9°C atemperature range afrequency and equation is [4,equation p. 24] Cole–Cole to the appropriate distribution knowledge relaxation time The of transforms. Laplace temperature. ofindependent the is of relaxation times distribution of shape the the that indicating curve, on amaster lie all points oment in the main chain, similar to PVC. to weight Its monomer similar amolecular has it chain, of and belongs 44, main oment the in Figure 3.27Figure To 3.92 of Equations usefulness 3.93, the and loss demonstrate amor index in measured the Evaluation of the distribution function from such data is a formidable task requiring a detailed adetailed requiring task is aformidable suchEvaluation from data function distribution of the by Taylor & Francis shows these data replotted as J as shows replotted data these Distribu Group, according to H-N dispersion. The cor The H-N dispersion. to of values βaccording for various times oftion relaxation Dielectric loss Distribution function 0.25 0.05 0.15 0.25 0.05 0.15 0.1 0.2 0.3 0.1 0.2 LLC 0 0 50 –5 0 F igure 3.26igure 23456 H ω 0.2 = 23456 εε 0 . Polyacetaldehyde polymer its with is dielectric apolar ε − ′′ Diele ω and H and ∞ . 0.6 0.4 Log (time) = ctric c ∫ ∞ 0 5 ω 1 = 1 [29]. The numbers in the legend correspond to to legend correspond the =1[29]. in numbers The G /H + onstant () τω ωτ ω p 22 as a function of ( afunction as τ d 10 τ 0.8 Dielectrics in Electric Fields in Electric Dielectrics 1 15 ω / ω p ). The experimental ). experimental The (3.93) 3/31/2016 1:20:37 PM - - K22709_C003.indd 121 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 Dielectric Loss and Relaxation—I Loss Dielectric s Trans. Farad. Soc. Trans. Farad. for G. Williams, clarity. (From delineated vidually –28.7°C, 34.8°C notindi through range the in temperatures various denote Symbols sample. thick 0.598 mm 3.27 FIGURE Soc. Trans. Farad. (From G. Williams, 5, 9.7 C; 3.25 6, C; 7, –3.5 C; C; –9 8, 9, –19.2 C; 10, –21.8 C; 11, C; –24.5 12, –26.4 C; 13, –28.7 C. 3.26 FIGURE in which τ which in Figure 3.9of Figure use makes for data analysis of the dielectric technique graphical The line. about the symmetrical 1 − © ymmetrical about the mid point, and therefore, plot and the point, of mid G about the ymmetrical 2017 As demonstrated earlier earlier demonstrated As α . Without this verification, the Cole–Cole relationship cannot be established with certainty. with established be relationship cannot the Cole–Cole verification, . Without this by Taylor 0 is the relaxation time at the center of the distribution. of center the atthe relaxation time is the . The quantity u quantity The Master c Master & Plot of Francis ε″ urves for J urves Group, ag –3.0 ainst log (/2)ainst 1, sample. for thick 34.8 0.598 C; C; 30.5 3, mm 2, C; 4, 18.5 C; 25 ( / ε F ˝ is plotted against log against v is plotted v 0 1 2 3 4 5 6 7 8 LLC G – igure 3.10igure 4– () , Equation 3.92, and H 3.92, and , Equation 20–1.0 –2.0 τ = J 9 , Vol. 59, 1397–1413, 1963.) 3– ω 21 πα 10 2– ), complex the is plot distribution plane Cole–Cole of the co 13 12 sh 11 10 0.3 0.6 0.9 [( log (ω/ω . . 3.0 2.0 1.0 0 log (ω/2π) −− si )l n ω max 1 απ / n] , and the result will be a straight line of slope line astraight be result will the , and H ττ ) Hω/Hω ω / 2345 max 0 , Equation 3.93, of log ( , Equation afunction as 8 ( max τ co 7 ) against log) against τor log ( , Vol. 59, 1397–1413, 1963.) s απ 6 5 4 1 2 3 τ / τ mean ) will be be ) will ω (3.94) / ω max 121 3/31/2016 1:20:37 PM ). ). - Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 122 © 2017 lowing form, called Gaussian function [4, function Gaussian lowing p. called form, 24], given by for is given H-N function the by [29] malization in Equation 3.80 Equation in is ε malization where where 122 able x on the of asingleto Debye is 3.28,vari relaxationshown time frequency Figure the as the relaxation. In from the dielectric data, whereas the width and symmetry are easier recognize. to are symmetry and width the whereas data, dielectric the from relationshipof the ωτ will be symmetrical about the central or relaxation time or relaxation time central about the symmetrical be will ε″ hencethe and distribution, the that recognized be it can function, formof this the increases, the log the increases, T The distr The The distribution of relaxation time according to the Davidson–Cole function is function Davidson–Cole the to according of relaxation time distribution The In this ex this In A simple relationship ( between by he distribution of relaxation times may also be represented according to an equation of the fol of equation the an to according represented be may of also relaxation times he distribution δ Taylor σ is ac is known as the standard deviation and indicates the breadth of the dispersion. of From the breadth the deviation indicates and standard the as is known ibution of relaxation times for the Fuoss–Kirkwood function is a logarithmic function: is alogarithmic function ibution for Fuoss–Kirkwood of the relaxation times pression, -axis instead of the traditional τ traditional of the instead -axis & onstant defined in Section 3.12, defined onstant and s=log ( Francis ε″ ω –log Group, = 1. In almost every case, the actual distribution is difficult to determine determine to is difficult distribution =1.actual every the case, almost In ∫ Gy ∞ plots narrower, for 1/ become case and the 00 εω () LLC τ ′′ dL = () G s −ε G n () G τ π 1 ε () () τ τ s ∞ −ε = θ and not 1). and ε″ the under area The =− = = αβ = () σπ εε ∞ Sin π arctan (s ∂ ) and ε″ ) and s in 1 π co = 0 2 βπ / βθ co τ ss ∞ y mean 22 sc )( = ex y ττ ∞ = τ > ∂ yy may derived (see be Ref. [9, p. 72]; Daniel’s nor α ; conversion because is easy variable latter the to τ 0 ∂ 2 τ π 22 p ωτ 2 τ si 0 π αα + − ∫∫ ∞ = n 00 ++ co πα + 1 0 G − 21 s + s() osh( β () πα in 1 2 ω τω ωτ ( / h( F co ω 22 ττ ∂ igure 3.28igure p τ s s) σ < ). The distribution of relaxation times of). relaxation times distribution The y ∂∂ πα dd s ) τω 0 2 () Ln σ − = 0, the distribution reduces reduces distribution = 0, the Dielectrics in Electric Fields in Electric Dielectrics ). curve is ωcurve –log β / As the standard deviation standard the As – ω (3.100) (3.102) (3.101) (3.97) (3.96) (3.95) (3.99) (3.98) plot, plot, 3/31/2016 1:20:37 PM - - - K22709_C003.indd 123 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 of the distribution of relaxation time, and the decay function, which will be discussed in Chapter 6. Chapter in discussed be which will decay function, the and of relaxation distribution time, of the which is us Dielectric Loss and Relaxation—I Loss Dielectric making measurements at different temperatures because ω because temperatures atdifferent measurements making PVC extended chlorinated (seeby be and can range 5). frequency Chapter the For measurements, Such have calculations employed been by of [33] Reddish PVC constant dielectric the obtain to the in peak 3.28 FIGURE Society, Washington, DC, 1997.)Society, DC, Washington, in 6 Chapter Jr. S. J. and Havriliak, S. Havriliak +1 to (From tends low −1and and frequency σincreases. as σ deviation, show standard the © 2017 Dielectric Spectroscopy of Polymeric Materials Polymeric of Spectroscopy Dielectric Fi Equation 3.103Equation havethat havethat Debye materials in a relaxationmaymaterials verified or be inverThe Equatio Using identity the gure 3 gure by Taylor n 3.102Equationinto, because of simplifies 3.87, eful to calculate ε″ calculate to eful .29 ε″ sion formula corresponding to Equation 3.103 Equation to sion formulacorresponding is –log –log & [4, p. 19] summarizes the dielectric properties ε′ [4, properties dielectric the p. 19] summarizes Log Log Francis characteristic, though the peak may be broader than that for that Debye than may broader relaxation. be peak the though ω characteristic, ε″ agai Group, log (ε˝) –3.0 –2.5 –2.0 –1.5 –1.0 –0.5 nst log(frequency) for Gaussian distribution of relaxation times. The numbers numbers The times. of log(frequency)nst relaxation distribution for Gaussian . Debye relaxation is obtained for 1/s = 0. Note that the slope at high frequency frequency forslope high at 1/s the =0. that Note obtained is . Debye relaxation 3– 10 –1 –2 –3 LLC approximately. ∞ 0 εω ′′ dL () ∫ ∞ 0 n 1 ε − ′′ ωτ ττ ωτ = ≅− = 22 ω ∫∫ , J. P. Runt and J. J. Fitzgerald, eds., American Chemical , J. P. Chemical J. Runtand J. American eds., Fitzgerald, p 0 πε εω τ =1 2( dL log ( log σ =0.2 ex ′′ ω () d ∂ p 2.5 0.9 0.5 0.3 ∂ ln n f, Hz) kT ω w ω =− ′ ) π = 2 () p 123 π 2 , τ εε s are related through equations through related Tare , and – ε″ ∞ in the complex the in shape plane, the σ =0.2 2.5 (3.106) (3.105 (3.104) (3.103) 123 3/31/2016 1:20:37 PM ) Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 124 © 2017 calculate one function if the other function is known. This is true only if ε′ only if is true This is known. function other the if one function calculate 124 relations [36,37]: Kramers–Konig as known relations are these 3.89Equations G 3.90 relaxation function and same the use 3.17 relaxation law.two-component Watts Jonscher’s and equation, of cases the relaxation shownlaw universal particular to be are ation law, Cole–DavidsonH-N Kohlrausch–Williams– relaxation, equation, Cole–Cole Debye previously such the relaxation processes as different relax response. Seemingly described, hierarchy, slowly whichcollective the adynamic the in by tute constrained response is fluctuating notbut, ofrather, independent consti other each are responses two dielectric phase.nematic These ordered or apartially precursors structure high-temperature strongly correlated the to corresponds slow or molecules atoms of bonds. The the held chemical component together response by normal composed structures noncrystalline or the cooperative crystalline of response the to the responds components, acollective aslowly and response collectiveThe response. cor response fluctuating field systemconsistto electric applied two the presumed to is matter of condensed arbitrary of an 6for row 6.46; 6.49; 3:Chapter b3, 6.47 Equation 6.48. a3, c3, Equation and Equations b1, 3.53; Equation c1, 3.45; 3.31; Equation value, single Equation a2, 3.95; b2, Equation 3.94. c2, Equation See shown [4, follows: also as p. are are 24]. designations a1, equations applicable Row 3.31; column and Equation 3.29 FIGURE Fu [35] has expounded on a new dielectric relaxation theory in which the dielectric response response dielectric Fu which [35] the in expounded on anewrelaxation theory has dielectric for is given relaxation equations formulas of the useful by Bello [34]. et al. A summary by Taylor RMR–RNG RELATIONS KRAMERS–KRONIG & 3 2 1 Francis Grap In φ ε ε ∞ –2 –1 ˝ hical depiction of dielectric parameters for the three relaxations shown at the top. The top. The shown the at relaxations three for the parameters of depiction dielectric hical 0 Group, 12 τ (a) 0 LLC 0 –1 In τ/τ τ/τ f (τ) 0 0 ω εω εω ε ′ s ε ′′ () Time variationofp ´ In φ () ε ε ∞ –2 –1 ˝ 0 −= =− ε ∞ 2 12 π ω π 2 (b ∫ ∞ 0 ) ∫ ∞ 0 τ 00 0 x –1 εε x τ/τ In τ/τ xx olariz 22 ′ 22 ε − − 0 f − ′′ (τ) ω ω () ε ω 0 ε ∞ s ation ´ In φ ( ε τ dx dx ∞ ε –2 –1 ), and in principle,), we in and must able be to ˝ 0 12 τ (c) 0 Dielectrics in Electric Fields in Electric Dielectrics –1 τ/τ f and ε″ and (τ) ω 0 ε ε s In τ/τ ´ are related, and and related, are 0 (3.10 (3.108) 3/31/2016 1:20:38 PM 7) - - - K22709_C003.indd 125 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 sured ε′ sured transform are discussed in Chapter 6. Chapter in discussed are transform ε′ since is asimple matter This ated. negative the periodic, be to of ε′ transformed be to signal the Weerasundara and Raju and [38].Weerasundara computation of by direct the domain frequency the may implemented in be transform Hilbert The (~10 a form. form. consuming since it requires n ples, 3.111 Equation multiplication. formulaistime very Computation of this amatrix becomes sy where the [40]. convolution Then forms. domain other multiplication the becomes in one domain in - inverse trans Laplace by taking domain afrequency into data by transforming reduced greatly Kramers–Konig relations have derive to used ε′ been Kramers–Konig and Relaxation—I Loss Dielectric may expressed be as pair transform Hilbert The transforms. Hilbert the as known theorem follows: as applicable. generally are are equation These these so that f imply that atω imply= ∞, that © ∞ nd (Hz); the latter is deemed to have attained ε have to is deemed latter attained the (Hz); 2017 However, ε″ curve of the area the polyamide for Nomex) (trade results name aromatic applicationExperimental and of Hilbert x, x, variable out auxiliary using an is carried Integration In a practical situation, ε′ a practical In - trans using Fourier domain frequency original back the to transformed are Finally, data these Weeras These transformed data are then multiplied by then are data transformed These Kramers–Kronig relations, 3.107 Equations 3.108, example and Kramers–Kronig of a general a particular are 2. 4. 3. 1. −4 −4 The imped The respo The const The systeThe the impedance must be a continuous and finite valued function. valued finite must acontinuous and be impedance the ation (stability). F Hz is assumed to yield to is assumed ε Hz by igure 3.30igure as a means of verifying an assumed ε″ assumed an of verifying ameans as Taylor undara [39]undara given sam computation. has for For discrete a computer numerical program mbols & m is linear (linearity). m is linear nse of the system ofnse the (causality). is always astimulus to attributed itution of the system does not change during the time interval under consider under interval time itution the system of the not does change during ance must be a finite mustvalue atω afinite be ance Francis shows for steps the computation. ε′ εω ˆ ′ = () Group, ε ∞ and and and and is measured at evenly spaced intervals, ∆ at evenly intervals, is measured spaced LLC 2 s ). Further, since the Fast Fourier Transform (FFT) technique requires requires technique Fast Transform the since Fourier ). (FFT) Further, multiplications and additions. Such a computational burden can be be multiplications additions. Suchcan and burden acomputational ε″ εω ˆ ′′ = 0. [9, Daniel conditions satisfied p. be to the by a lists system 97] () – εε ( are the Hilbert transforms of the corresponding quantities. quantities. corresponding of the transforms Hilbert the are js ′′ readily gives ωreadily ( – ln f ) is an even) is by an easily ofgenerated software. fand function = ig εω ˆ n( ′′ ˆ εω () ′ ˆ () ′ t) ω () =− – ωrelationship [41]. = 0 and ω → 0and ∞ = εε −= jt . Also, the initial frequency is sufficiently small is sufficiently frequency small initial . Also,the ∞ εω =< πω jt 1 ′′ fo () fo from ε″ from r ′ () . *( r ∞ . εω ′ which is real. Equations 3.107 Equations which is real. 3.108 and ∞. At all intermediate frequencies, frequencies, → ∞. intermediate At all > 0 = 0 to −∞ Hz needs to be gener be to needs Hz −∞ f=0to from ε 0 s ) −ε – - mea with compare to and ω data ∞ ) according to Equation 3.103. Equation to ) according f (Hz) and up to 10and (3.109) (3.112) (3.110) (3.111) 11 Hz, Hz, - 125 3/31/2016 1:20:38 PM - - Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 126 © 2017 [42,43] to component according imaginary and a real component (j imaginary fields,therefore, componentconsists alternating twounder ( real of the components, conductivity The density, charge of the to rate change of current. constitutes charge temporal and where 126 law, Ohm’s aAC component DC is given componentan component.ponents, aDC The of and by variation the Basically, all. them by is served listing conductivity the consists of purpose two no com useful that loss. dielectric loss. 3.8 dielectric Equation the to gives contributing contribution of as conductivity the the to ized conductivity may The dielectric. therefore visual the be polarizes additionally and duction current expressed by a volume conductivity. con- gives to the dielectric motion of rise the The in charges conductivity possess Many dielectrics due motion to such of and charges, conductivity is usually 3.18 Toulouse, July pp. 5–9, Solid Dielectrics, 558–561, France, 2004.) on Conference Relation,” International Kronig Kramers Techniques Through FFT and Transform Hilbert Using Permittivity Complex the Dielectric of Computation for Numerical Algorithm Efficient Raju, An 3.30 FIGURE The real part of3.113 Equation part real The is given by From consideration of an equivalent circuit, the conductivity of a dielectric can be shownFrom be conductivity consideration have to can equivalent the of of adielectric an circuit, Equat Recal express that conductivity of equations it anumber different of and is felt adielectric, are There by is the current density. current whichJ is the is related AC The polarization, component from is deciphered Taylor LOSS INDEX AND CONDUCTIVITY AND LOSS INDEX ling ion 3.114 expression of image the mirror the be to may seen be & Francis Impl ementation of Hilbert transform for dielectric data. (From R. Weerasundara and G. G. G. G. and Weerasundara R. (From data. for dielectric transform of Hilbert ementation ε ´ Group, (F ) ), as expressed in Equation 3.113.σ″), Equation expressed as in LLC σω ′ IF () εω FT ωω lim ′ () →→ == 0 σω Re == ′ () σ [( Re σω ( ω [* ) =σ′ = εω σσ )] J =σ () dc ε ; ( ´ σ ω (T ] ) +j dc dc lim ) E + ε ∞ ∞ σ″ () σσ + –j sign (t) sign –j ′ ( ∞ () ω ωσ 1 1 () ) εε − + + s = ωτ ωτ − dc 22 22 ∞ ∞ ωτ 22 Dielectrics in Electric Fields in Electric Dielectrics ε ε ˆ ˆ ´ ´ (T (F FF ) ) T ) and the the σ′) and (3.115) (3.116) (3. (3.114) 113) 3/31/2016 1:20:38 PM - - K22709_C003.indd 127 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 a factor ofa factor 2×10 ing to Equations 3.116 Equations to ing 3.118. Equation in 3.29 and data and which 1h, isas hexagonal yields designated and valuesperature, of ε′ decrease of image ror the and conductivity at infinity frequency ( frequency infinity at conductivity and 3.31 FIGURE left-handthe side of3.119, Equation one gets right-hand side on the of3.119 Equation term 3.29 substituting Equation and may on neglected, be conductivity, DC contributionfrom contributionis negligible the to first compared polarization the University Press, Oxford, 2002.) Oxford, University Press, Dielectric Loss and Relaxation—I Loss Dielectric © 2017 The DC conductivity is visualized as making a contribution to the loss index according to loss the acontributionto index according making as conductivity DC is visualized The Figu Appl way expressing alternate 3.114An Equation is mir the to similar amanner atω=∞in infinity atω=0to zero from conductivity increases The There i There by - 3.31 accord re of ice at263 Kconstructed shows conducting parameters and dielectric the ication of3.114 Equation conductivity tem the to ofand ice atambient common pressure Taylor s an alternate way of obtaining Equation 3.117. Equation way of alternate obtaining s an which the in of case dielectrics the In εε Freq & ωτ −7 ∞ Francis =× == Ω uency dependence of dielectric constant ( constant of dielectric uency dependence −1 31 21 .; m 100 6 −1 Group, 0 ε , ε , σ , σ∞ 100 120 . (Redrawn from data of V. data from Ice of Physics F. W.. (Redrawn and Petrenko Whitworth, ´ ´´ ´ 20 40 60 80 45 0 ra 0 ss d/ LLC s, (Figure 3.31). ω(Figure increasing with =× σσ .; σ 83 12345 .610 5.06 ∞ To σε ) on frequency for 1h σ 263 at ice K. ) onfrequency ta ld σσ =− εε =+ tota ′′ =× os l () εε = c − s log [ f log [ ′′ 10 () () σσ ω ∞ −− ∞ 81 1 ΩΩ − 1 (Hz)] ε + + ´ + σ ωτ ωε ωτ σ ∞ ωτ m ε′ dc ×(210 dc 2 22 ), loss ( index o 22 − ωτ 15 ,. 22 ε ´´ ∞ σ ´ –7 =× ) 19 ε″ ∞ and σ′ and ), real part of conductivity ( of conductivity part ), real 10 −− 6 should be multiplied by multiplied be should 111 m − , , Oxford , Oxford (3.120) (3.117) (3.119) (3.118) 127 σ′ 3/31/2016 1:20:38 PM ), ), - - Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 128 © 2017 part polarization loss. polarization part one obta one 128 Ethylene glycol (C by Zaengel [46]. for voltage power high domain summarized frequency been equipment (HV) has and time in spectroscopy Dielectric importance. of engineering of materials selected properties chapter, consider dielectric we this the in shall outlined To theory of the several demonstrate aspects 3.19 of ε′ decrease of image ror the [44] suggestedpower a“universal” and to law according components of to conductivity according imaginary and real conductivityevaluated the and the of ice sea using afour-electrode measured arrangement recently 4. [45] Chapter Buchanan in described due charge, space to as has polarization interfacial the to is attributed amuch increase and larger is possibly hopping carriers, charge duetemperatures the to due conductivity.to increases also stant A relatively in increase small con dielectric of the part real The believed traps. between to suggest hopping of carriers charge mentioned range is the and temperature, slightly increasing with or decreases constant remains exwhere the from 92.4 ps at room temperature to 48.3 to at55°C, ps 92.4 temperature atroom ps from afactor-of-two nearly reduction. Increasing decreases relaxation time The temperature. on the depend index shows that atfrequencies apeak frequency, increasing loss with the whereas decreases part real The quency temperatures. atvarious data presented here follow here presented investigationsdata their closely. chapter. this The in delineated theory covers of results dielectric ofpresentation their aspects many Sengwa [47] et al. molecules of asystematic these in way, properties have dielectric the studied and weightslar amonomer as (mol. wt., 62.1) or polyethylene glycol weights. molecular higher with for automobiles, textileindustry, the etc. in Ethyleneantifreeze glycol is available several in molecu- forate polyester aplasticizer, as asolvent, resins, as for amedium low-temperature in as coolers as σ′ the total conductivity is given total the by 3.117. Equation simply presented σ often as Results are ( ω In this alternative, format the real part describes conduction and dielectric loss, the imaginary loss, imaginary conduction dielectric the and describes part real the alternative, format this In The conductivity increases from zero atω zero from conductivity increases The substituting Upon Jonscher has compiled the conductivity as a function of frequency in a large number of materials of number alarge materials of in conductivityJonscher frequency afunction the as compiled has Fi ). by gure 3.32gure Taylor DIINL COMMENTS ADDITIONAL ins & ≤ 1 for most materials. The exponent 0.6 The ≤n1for either within ponent be to most is observed materials. Francis shows the dielectric constant and loss and indexshows of constant ethylene dielectric the glycol of fre function as 2 H 6 Group, O 2 ) is a polar liquid( ) is apolar LLC with increasing ω increasing with σ σ μ =2.36 D) intermedi applications achemical has as that ∞ ( σ ω σ = ) =σ * =j = εε = 0 to σ =0to 1 σω os + () ∞ ωε dc ωτ +a τ ( 0 22 − 22 F τ ε ε * igure 3.31igure ω ∞ ∞ atω n = ∞ in a manner similar to the mir the to similar amanner =∞in ). If DC conductivity DC exists, thenIf ε′ Dielectrics in Electric Fields in Electric Dielectrics at low or high frequencies (3.124 (3.122) (3.123) (3.121) ( ω 3/31/2016 1:20:38 PM ) = ) ) - - - - K22709_C003.indd 129 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 essentially Debye type. The smaller the value the of α smaller Debyeessentially The type. dispersion phenomena relaxation and are the that of range 0–0.1, demonstrating the in constant essentially αremains value 3.45 of Equation αin parameter shown The is also for temperature. each 2.28 D. 2.28 2.91 factor acorrelation ( (Equation 2) Chapter obtain in and more closely.a semicircle 2000.) Int. J. R. shown. Sengwa (Adaptedfrom as Polym. al., et temperature Ambient peratures. ecreases with increasing polymerization (mol. polymerization increasing with wt.)ecreases temperature. and ε′ weight molecular weight.tion molecular of effect higher 3.33 is of shown The increased Figure in of ethylene- polymerization having configura glycol coiled ahelically chain, amolecular in results 3.32 FIGURE Dielectric Loss and Relaxation—I Loss Dielectric © d 2017 For calculation of static dielectric constant, Sengwa [47] et al. constant, For of dielectric calculation equation static apply Kirkwood the Figure 3.34 shown Figure in are for arcs several temperatures Cole–Cole by Taylor & Dielec Francis tric constant and loss index of ethylene glycol as function of frequency at various tem various at loss of of index ethylene frequency and glycol function as constant tric Group, ε´, ε´´ ε´, ε´´ ε´, ε´´ 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 0 0 0 05 05 05 LLC τ =73.8ps τ =92.4ps τ =48.3ps 2.0 GHz 4.0 GHz 1.64 GHz Fr Fr Fr 308 K Et 01 20 15 10 01 20 15 10 01 20 15 10 equency (GHz) equency (GHz) equency equency (GHz) equency Loss inde Diele 328 K hylene glycol , the broader will be the Cole–Cole arc, becoming becoming arc, Cole–Cole the be will broader , the 298 K ctric c Loss inde Diele Loss inde Diele x (ε ctric c onstant (ε ctric c ´´ ) x (ε x (ε onstant (ε ´´ onstant (ε g ) ´´ ) of 2.56 and dipole) of and moment 2.56 ( ) ´ ) ´ ) ´ ) for ethylene glycol, the and 25 25 25 , Vol. 49, 599–608, μ ) of 129 3/31/2016 1:20:38 PM - . Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 130 © 2017 vegetable oils, to determine moisture content, contamination, adulteration, etc. adulteration, [50]. content, contamination, moisture vegetable T determine oils,to water (~80). pure as constant haverelaxation [49]. dielectric same the mixtures the point, At freezing ratio. mixture the by varying 88 may obtained and be 40 between constant dielectric Desired for liquids. expected as polar decreases, of mixtures constant dielectric the increases, temperature ethylene glycol adensity about has 11% is moreand strongly polar of water. that than As greater of percentage ethylene increasing with fact glycol, that the decreases constant not withstanding dielectric ages by the weight and of latter. of water constant At is the obtained, 0%, dielectric the Fi 130 c [49]. reasonable geometry in results and length line short 3.35 Figure requires shows dielectric the it[48]. should liquids because have the constant is that dielectric high requirements of One the x the 3.34 FIGURE 3.33 FIGURE 2000.) Int. J. R. Sengwa (From Polym. al., et constant. lower dielectric the temperature increasing references. onstant of mixtures of ethylene water and for percent glycol various of of mixtures onstant temperature functions as gure 3.36gure Dielectric properties of have liquids of employed properties Dielectric been quality of the a tool for as determination is well by constant Debye represented of range dielectric 0.5–108 frequency Over the the MHz, Mixtures of ethylene water and glycolMixtures have power applicationspulse in potential applications -axis on the right side gives right ε onthe -axis by Taylor show the relaxation time in several in vegetableshow relaxation time the [50] oils [51] liquids common and with & Francis Dielectr Cole–Col Group, ic constant ( ic constant e diagram for ethylene glycol at 25°C and 55°C. The intersection of the curves with with for curves ethylene of glycol the 25°C at 55°C.e diagram and intersection The Loss index (ε´´ ) 12 16 20 LLC 0 4 8 0 ε 10 20 30 40 50 60 s 0 , and the left side left gives ε the , and ε′ in ethylene glycol. Increasing molecular weight and molecular ethylene of 1/glycol.T in function Increasing ) as . . . . . ×10 3.4 3.3 3.2 3.1 3.0 02 040 30 20 10 Diele PEG 600 PEG 400 PEG 300 EG (1/T ctric c ) (K onstant (ε –1 α =0.05 ∞ ) 55°C . Note the decrease of ε decrease the . Note ) Et hylene glycol α =0.09 25°C Dielectrics in Electric Fields in Electric Dielectrics 3 50 s with temperature. , Vol. 49, 599–608, able 3.5 and 3/31/2016 1:20:38 PM - K22709_C003.indd 131 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 Note: Acetone Chloroform Methanol Ethyl acetate 1,1,1-trichloroethane Water Ethylene glycol Ethanol Propanol Isopropanol Soybean a Liquid Liquids of Selected Frequency Relaxation 3.5 TABLE Various seedoil Peanut oil Olive oil(ac.0.3%) Corn oil Sunflower oil Olive oil(ac.1.2%) Olive oil(ac.1.6%) Olive oil(ac.4.0%) Castor oil Dielectric Loss and Relaxation—I Loss Dielectric FIGURE 3.35 FIGURE J. Appl. Phys. J. Appl. eye. D. the to (Adaptedfrom B. aguide Fenneman, are lines Dashed of temperature. ages was afunction © 50% soybean +50%sunflower oil. 2017 See by Taylor Figure 3.36 , Vol., 53, 8961–8969, 1982.) a & The stat The Francis for graphicalpresentation. ic dielectric constant of ethylene glycol mixtures for various glycol for weight various percent of ethylene glycol constant mixtures ic dielectric Group, Figure IDNo. Dielectric constant (εs) 40 50 60 70 80 90 – 0– 40 1 90 3.47 19.09 111 83.65.63 32.66 19 18 17 16 57.65.2 78.36 15 32.74.15 3.74 24.57 20.42 14 13 12 10 – 9 8 7 6 5 4 3 2 1 LLC 0– 30 0– 20 07 2.01 20.78 .22.2 2.71 4.82 2.25 6.00 7.20 124.85 41.2 2.43 2.43 3.09 3.10 .52.40 2.39 2.41 3.05 2.40 3.08 2.36 3.11 2.36 3.12 2.36 3.19 2.56 3.19 3.19 4.69 ε Te s mp 01 030 20 10 10 erature (°C 0 ε ∞ ) rqec Mz Reference Frequency (MHz) 52080 10Gregory andClarke [52] 4810 3140 2960 2670 1924 1720 6 Gregory andClarke [51] Gregory andClarke [51] Gregory andClarke [52] 964 488 452 390 372 330 315 309 292 288 259 250 122 (W%) 80% 60% 0% 100% Sengwa etal.[46] Cataldo etal.[51] Cataldo etal.[50] Cataldo etal.[50] Cataldo etal.[50] Cataldo etal.[50] Cataldo etal.[50] Cataldo etal.[50] Cataldo etal.[50] Cataldo etal.[50] Cataldo etal.[50] Cataldo etal.[50] Cataldo etal.[50] Cataldo etal.[50] Cataldo etal.[50] Ellison 2007[53] 131 3/31/2016 1:20:38 PM - Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 132 © 2017 ylene terephthalate [55]ylene terephthalate of range 10 frequency the in by percentage weight increasing and of ethylene temperature glycol.decreasing 132 FIGURE 3.37 FIGURE conductivity.to lowest, It is the for ethylene glycol, at25°C; it is ~1.2 ×10 dissipated are due before they are sustained fields electric internal time the the of duration nifies for mix the 3.36 FIGURE for data. for centages by weight 100%centages to water. of ethylene glycol. corresponds 0% located between 100°C110°C between and located conductivity showing the relaxation process, in two different showed results temperature was the atransition there motion ofthe that Further, polymer chains. 10,10 is governed outby bytransport [56], Lu et al. have carried been carrier that whocharge find relaxation phenomenon on the nylon studies in range. negligible Similar whole the in temperature of conductivity DC atlow However, frequencies. are polarization charge space effects and electrode values due to high high ~100°C, are than higher loss the factors obtained effects. At temperatures (~190°C).perature Emphasis is put investigation on the conductivity conductivity and of electrical nylon (~40°C) melting- and tem 11, temperature transition glass is between range temperature the Valuable have contributions by made Neagu's been [54] group on nylon studies in 11 polyeth and - Figure 3.37 shows the intrinsic time constant defined according to according defined 3.37 constant Figure time shows intrinsic the by Taylor tures varying from pure water to pure ethylene glycol. The intrinsic time constant sig ethylene water constant pure to pure time glycol. from intrinsic varying The tures & Intrinsic time constant of water–ethylene glycol mixtures versus temperature for various per for various versus temperature glycol of water–ethylene mixtures constant time Intrinsic Francis Relaxation frequency of selected liquids. See See liquids. of selected frequency Relaxation Group, Time constant (τ [s]) 0 10 10 10 10 1 LLC 2 3 4 5 6 7 8 –4 –3 –2 –1 9 10 11 12 1000 13 – 0– 40 0030 005000 4000 3000 2000 14 0– 30 Rela 15 Fr equency (MHz) equency xation fre 0– 20 τ Te 16 = mp 01 03 40 30 20 10 10 εε 17 σ or 18 erature (°C quency −2 0 to 10 to T able 3.5 6 ) W% Hz and a wide temperature range. In range. In awide temperature and Hz 60% 19 for label identification and references references and for identification label 0% Dielectrics in Electric Fields in Electric Dielectrics 6000 96% −4 s and increases with with increases sand (3.125) 3/31/2016 1:20:39 PM - - K22709_C003.indd 133 Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 dealt with in Ref. in with dealt [58]. 10 from range the loss of in factorfrequency afunction The as shown [57]. as et al. als, by Lee movement. materi considerations apply for also These carrier charge ferroelectric to mechanisms chain, a branch, or even a main structural unit. or even structural abranch, chain, amain of aggregate molecules, asinglean to aside from unit may vary polarization to contribute that units the field, and the influence electric of the under may displaced not be chain of entire polymers, an case the In conceptof of relaxation times. the distribution to polymers conform in chains molecular example water moleculeThe for is atypical single However, relaxation time. molecules larger and show results only few avery that experimental molecules the as haverule asingle relaxation time. However, relaxation time. ence the exception a is more shape an of assumption aspherical than the of dipoles of not rotation does influ the axis the that spherical, molecules the are that assumption or few polymers have Debye relaxation times, on The the demonstrate, adistribution. equations simple molecules polar may have While parameter. one central is the relaxation time dielectric the treatments, these by relaxation In formulas. various treated theoretically are properties ence the which influ they in manner the and temperature, of and frequency functions are properties These fields. electric alternating in materials dielectric of polar properties the with chapterdeals This 3.20 4.Chapter 9. Chapter in detail greater [73].hydration considered in spectroscopy by dielectric are broadband aspects These of blood stored [72].quality explored been has representation monitor to Davidson–Cole cement the of tool providing prospect determine to adiagnostic the with representation using Cole–Cole of analyzed acircle.of yields arc blood been Temporalalso has an properties of dielectric variation 3.113 Equation with conductivity the expressed acomplex as accordance that in noted quantity It for is range requirements. complying compatibility electromagnetic with sion gigahertz the in [71]. range gigahertz netic loss the have in applicationin noise films - potential suppres Such thin given Refs.are in [67–70]. frequency and of temperature function as properties of [66]. studies dielectric typical 1 kHz Other range of −50°Cto temperature the in 150°C at studied been has films nanocomposite polyimide BaTiO in permittivity up of 5% to prediction dielectric w/v. theoretical study and Experimental organic liquid [65]. The dispersed particles included alumina (Al included liquid[65]. alumina organic particles dispersed The insulating an oil, in paraffin have oils studied been of nanopowder paraffin properties dispersions in yield to [63,64]. results demonstrated useful systems been has oil–paper transformer Dielectric polymer. of the Application plot of aCole–Cole crystallization commercial to increased to uted which is attrib- of PTFE, permittivity dielectric [62]. the increase exposureto is observed Thermal of temperature afunction as and 1MHz to of range 1Hz frequency the haveties, in studied been proper due superior to insulation wire extensively twoare polymers that (ETFE), aircraft in used properties. motion of the improvement molecules. result is an restrict The dielectric in matrix ticles the in polymer [61] studies nanopar the fact that the includedrevealing nanocomposites, has polyimide troscopy, tomography, have popularity. to methods Application related methods and of gained these - spec [60]. water or metal through fabric. pass such time-domain Several It as cannot techniques polymers, wood, paper, organic and such ceramics, as awide of range materials penetrate to ability the has radiation submillimeter the and is nonionizing, radiation range, the frequency terahertz the low-density in measured polyethylene to 70° been has −60° from range the [59]. in temperature In Dielectric Loss and Relaxation—I Loss Dielectric © 2017 The role of the motion of molecules in polymers in determining the type of relaxation has been of been relaxation has type the roleThe motionof of the molecules determining polymers in in Additional examples of application of the outlined theories are considered at the end of considered atthe are Additional examples theories of application outlined of the mag strong exhibiting analysis films is shown applicable be to thin to impedance Cole–Cole The dielectric properties of polytetrafluoroethylene (PTFE) and ethylene tetrafluoroethylene of(PTFE) polytetrafluoroethylene properties dielectric The by OCUIG REMARKS CONCLUDING Taylor & Francis Group, LLC 2 O3 3 ) and Titanium oxide (TiO Titanium ) and −3 −3 to 10 to 6 Hz and and Hz 133 3/31/2016 1:20:39 PM 2 3 - - - - ) - - / Downloaded By: 10.3.98.104 At: 19:17 02 Oct 2021; For: 9781498765213, chapter3, 10.1201/b20223-4 K22709_C003.indd 134 © 2017 REFERENCES mechanisms. polarization interfacial ence to referfollowing astudy of with 4continues relaxation phenomena chapters. 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