Journal of Research of the National Bureau of Standards • Vol. 50, No.1, January 1953 Research Paper 2382 A Simple Calculation of Dielectric Loss from Dielectric Dispersion for Polar Polymers Paul Ehrlich For polar polymers und ergoing orientation polarization it is possible to calc ula te dielec- t ri c losses from dielect ric const an t da ta by use of a single approximation already familiar from its appli cation to mechanical prop erties, if this a pproximation is applied to the real par t of the diel ect ri c constant on ly and if data over a suffi cient ly wide fr equ ency range are availabl e. Data obtained at fr equ encies fr om 102 to 108 cycles per seco nd for Bu tvar and for a copol ymer of sty r ene and met hyl methacr yl ate are a nal yzed and it is found t hat ob se rved and calculat ed values of the diel ect ric losses agree within 10 per cent or b ette r. Equations relating the real and the imaginary parts of th e dielectric co n stant have been giv en , but are not in general use because the equations involved are cumb ersome [1 , 2, 3).1 ",;Yorker s mt erested in the mechanical behavior of high polymers have shown ho v the real and the imaginary par t of the modulus of rigidity are rel ated t hrough a di strib ution function [4] and how this function can be obtained by anapprox- imation m ethod from the real part of the modulu [5 , 6] and from both real and imaginary parts [7]. I t will be shown how the dielectric loss can be calculated from dielectric con tant data by an analo go us ap proximation m ethod . The Debye equations [8 ], generalized fo r a di stri- bution of capacitan ce elements with r esistance ele- ments in parallel [9], each with its characteri st ic rel axat ion time, ma themati cally equivalent to a m e- chanical retardation tim e, are (1 a) (1 b) where e', e" are the real and the imagi nar y part of the dielectric con stant , withe angular frequ en cy, r is the rel axation time and y(r) i the di stribution of relaxa- tion times. For mathematical convenience, we d e- fin e y (r) dr = Y(ln r )d In r. Th e cut-off approxima- tion [5, 6, 7] assuming the integrand s in (la), (lb) to vanish for (tJr > 1 and to reduce to their numerators for wr 1 giv es 1 In .- E' - e ",= f _: Y(lnr)d ln T (2a) 1 In - e" = J _: wrY(lnr )dlllr. (2b) (2a) gives agreement with (la ) well within the graph- ical error involved in the calculations, wh ereas values of e" calculat ed from (2b) are generally 30 to 40 percent low when compared to those obtained from 1 Figures in br acket s indicate the literature refe ren ces at the end of this paper 19 (lb ). W·e therefore use the approximaLion ( 2a ) onl y. obtai ning w dl nw (3) by difl"crentiaLion of (2a) wi th resp ect Lo the upper limi t and carry out thi s operation gra phically. Th e exact eq (1 b) is then used to obtain e" 2 by a graphi cal integration for each valu e of w. Th e integration i done o -r aphically, because even the most widely ap- plicabl e equ ations that have been suggested for y (r) on semi empirical grounds [2, 10] and that might hav e b ee n ex pected to repr esent the results cited do not fi t the d ata over the entire frequ ency range and b e- cause, if the cur ve for Y(ln T) is approximated in sec- tions by simple anal yt ical ex pr ession s, integration often b ecom es impossibl e. Da ta are pr esent ed over a wide frequ ency rano·e, including either side of th e absorption maximum, for Bu tvar (polyvinylbutyral) [11], a m ater ial which, for an unpl astieiz ed polymer, has a fairly sh arp disper- s ion and for SM-2, a co polymer with a broad dis- persion made from 49-mole-percent styrene and 51- mole-percent m ethyl me thacrylate ; bo th polymers h aving n egligible d-c condu ct ivi ty (tabl es 1 a nd 2). In each case the calcul ated valu es of e", which can be obtained for points at least one decade inte ri or TABLE 1. Electrical properties of Butvar 1 F· '" Y I I ," (cal. ___ ' ___ ' ____ _ cul nted) cIs 10' .... ......... ••• .. 3.415 0. 0238 0.0130 2. 80 4X 1D' .. .. ........... 3.394 . 02il . 0152 3. 40 10' . . .. .. ........ . ... 3.382 .0301 . 0200 3.80 0. 032 3X1 D' ... ....... . .... 3.364 . 0387 . 0243 4. 275 . 041 10' .... .. .. .... .... .. 3. 336 . 0550 . 0317 4.80 . 056 3X 10' . _. __ __________ 3.307 . 0788 . 0433 5.275 .073 10' ____ . ____ . ___ . ____ 3. 233 . 104 . 062 5.80 . 094 3X 10' . .. . ____ . ___ ___ 3. 170 . 110 . 073 6. 275 . 107 10'. __ ._ . _________ ._. 3.075 . 111 . 078 6. 80 . 114 3X IO' . _ _. _________ ._ 2.988 . 101 .073 7. 275 . 111 107 •••• •• •.••• •• •• ••• 2.915 . 092 . 063 7.80 . 099 4X 1Q1 _ .• .. . .. . .... .. 2.83 .079 .059 8. 40 lOs ...•.. _ ... _ .. ..• .. 2.77 .072 . 053 8. &> I D ata we re obtained on a 3·terminal Schering b rid ge at fre quencies of 10' to 10' cIs and by resonance methods from l(P to ] 0' cIs . I J. D. Ferry a nd E. R. Fitzgerald (r eported before the Ameri can Physical SOCiety in March 1952 at Columbu s, Ohio) have also made similar calc ula tions, but use the cut·off approximation for both real and imaginary parts and then a pply a secon d a ppr oxi mation requiring a certain analytical formula t ion of their distribll tion funct ion.