Iyengar, B R Yathiraja

Jatkar, S K Kulkarni

English

IACS Institutional Repository

51 ANOMALOUS DISPERSION OF DIELECTRIC CONSTANT lh S, K. KUl,KARNI JATKAR /\ND H. R. YA'fHIRAJA IYENGAR ABSTRACT. \ ill'" "qllati,," h,her! olilhe i(l"a' af hindere,! r"lali,," ano prderr("(j orientation in '1l1,t,i-cry,ta:lillt: liqui(ls and <,,,lids ha' heen (Ic!ived ill (mler to ql1ant!tntivel.1' ;t('c')unl for the ,Iispe! ,io!! a nel a I l" 'r!,l ion IIf pular substances, 'I'll .. exi,,! ing relationships of llehye, C',le and l'"k and "tbers h:\"e he('11 critically rc\"iewecl, Tht· Ill'\\' {'xp.essioll for .' th,' ,Iieleclrie cow,tant an,! .' lh" ,Iieledric loss al"(! Thd ne\\" l''1uatioll,s ("r di'''i'er"i(l11 aIHI ,tlJ',orptioll have bCl'n extended to binary solutions of polar cOll1ponents un(\ a gencrale(juation Ins beell derived for the (lielectric constant and dieleC'lric le,,, ill terl11' "f thl' relaxation times of the two component". INTROIlOCTION Thl' phcnolllcnon hy which thl:re occurs a marke(1 dccrease of the dielectric constant with increasing frequencies of the applied field accompanied by a strong <Ibsorptioll of ell'elric waves is referred to as a1l0l11alol1s disper-sion, This effcct in liql1ids was first observed expLrimentally by Drude (18CJ7). Debye (lCJ29) based his explanation for anomalons dispersioll on the idea of the cxi"tence of polar 1l10kcn1cs. At sttfficiently low freqncncies when the period of the field i" large cOlIJpared to T, the time of relaxation of the polat substance, the molecules can folio\\" the field with case and conse· quently the makrial has h;gh static dielednc constant. When the frequency of the field is of a IlJagllitl1dc c()ll1parahk to relaxation time the dielectric C'onstant falls gr<ldnally \I"ilh i1lcrea"ilJg irl'qttel1l'ies. Finally at very high frequencies, i, (", w!Jell the period of the field is vel y small COlli pared to T tl:e dipoies 1 each such ;l state wherein they do nol "huw any reSI)OnSe to the applied field, Thus the polar material here is characterised by a constant high frequency didectri(, cOllstauL The l:xtent ,l11d tIlt: rauge of dielectric dispersion is a function of the internal forces iii the substance- such as the viscosity and also dCI,e1lds \Ipon thl;; naturtc of the dipoll:s of which the molecules of the suh~tallce arc constituted, Dchye (loco cit.) was thl! first to derive a qnantitative expression to account for the dispersion and absorption of polar substances. He wrote for the :I ... 1674P-IO 438 S. K. K. latkar and B. R. Y. Iyengar molecular polansation in the dispersion region as follows PC,,) = ~~I) _~I = {TfBf 0(.+ L_ ~! __ ) (e+2) d 3 \ 3/d I+lwr (1) The equation of Debye IS hased on the validity of the Clausiw,-Mosotti relationship which has been found to be a failure \"hen applied to concentrated solutions aud pure liquids and has got limited application even 111 the case of dilute solutions. TIll!> fuilUle is i11h\:r\:11t ill the assumption of a spherical cavity in evaluating the internal field and the postulate of a dipole having all poshible orienLLtion in space. It is of interest to note th,,[ Dehye has derived a formula for the a"erage electric llJOlIJent of a lIlolt;cule in !>olidh assuming that the lllolecule Chn point with its lIIoment ollly in two dcfilllte uircctioll", the dirLctlOll of F and the opposite. The mean 1ll01TI<;nt of a molecule in the dircction of F is 111= 1 1 + jOlT Il"F ",~i t = __ I 1\'1' 1+ ihlT The expression on right hand side can be d<;rived in a "ery simple manner as follows. The general distribution function in a variable field has been given by Debye as ( \ I fF j= A 1+ --. -; COS () ) J + 1111. 1<1 where fi is the angle' Il' makes with F. Thl Boltzmann's distribution function gives fO! 111 and lI z the number of lllOkcule:o. following the t\\O possible orientations one along and the other opposite to the field 111 = A (I + fF _~_ \ kT 1 + iWT I 11 > = A (I _ J!F _1 _ ) - , kT 1 + iw. obtained by putting cos (j = ± I in Debye's eX)lre~sioJl (::!) 2f\ This expression for 111 for solid:o. diffcl~ from that derived earlier for the case of gases by a factor 1 I 3 since in the former case. only the directions parallel and anti-parallel to the field are considered. Proceeding as before Debye has obtained the same expressions for e' and 1>" as for the case of gases. This is because, firstly he has used the same Clausius-~osotti A nomalous Dispersion of Dielectric Con$iant 439 . . (6-T)1\1 relatJOllSI11P - to hold good and secondly the modified expression for (,,+ 2)d 111 for solids has been eliminated in the initial stages of the derivation by being ( ... - I)M expressed in terms of .... - and hence b<:comes ineffective to modify the (. + 2)d expreSSIOll~ for ,,' and ~;II. THE NnW EQUATION It has beel1 foulld by Jatkar, Iyengar and Sathe, (H;J46) that the polarisation of liquids and solids in a static field is given by (6- I)M/ d = 4i7N(aL + p." I VI') instead of the c1as5ical expression (':=I) ~=_4rrN (aE +p.2/akT). (6+ 2) d 3 'fhe fOlmer <:quation has been derived using a new internal field based 011 the concept of a thin long cylindrical cavity which is in conformity with the anisotropy of the dipol<:, and .1150011 the consid<!ratioll of preferred orientation along and opposite to the direction of the ap)Jlie(l field. This equation has been fOlllHl appliLablL- to a large number of polar liquids and solids. Extend-ing the new relationship to the caSe of a variable field, the polarisation p\w) at frequency w is given by p(w) = (6_-1)~ = 4rrN ( O(s + Il"- _1_. _) d kT l+tWT Deilning we have whence i.e., Writing and .11 _ ("0-6",)WT € - ----I + W~T2 These expressions are very similar to the classical expressions but for a factor i€o.:-_2l which comeS associated with WT in the latter casco It is note-(600 + ~d 440 S" K. K. fatkar and B. R. Y. Iyengar th I I "d . I I (f'" + :d I wor y t lUt t Ie two expressIOns all 1 cutlca \\ H!n = I, i.e., when tIe \'",+ 2) high frequency diclectric constant does not very much difTer from the static' dielectric constant. Wyman's (Iq36) and ()n~ager 's (1936) expr(:!s~ioll~ for tlte polarization kad to the sa11l1: cquation~ for e' and • .," as given by the applicat ion of the new equation. For a 1110dified form of Onsager's equation (€-1) _ (€oo-=-~ -(3 = q lti (bC/)~+2: -I ( ( " + 2) ( "00 + 2 ) t I (. + :2) \ where • q' is an empirical C011stant; !the casc iJ = I corre~p0l1d111g to On"agcr's eqnatioll). Cole (Il)3S) has :-ho\\'n that " = "00+ where (eo_-""') 1+ Z" and ((iO:-f OO ) I -t Z' The application of experimental data has shown that '()' is indeed small, (i. c., Z = WT) so that the expressioll~ for ,=' and ,," corre~l,ond to the ncw equation. The maximum in f;" as well as the maximulTl slope of tIll' .' - (,) C\1rve occur at a frequency W r the ('ondition for which is given accordil1h to (J) Debye's expression as "', T = if- +2'. . C/) (2) ()n~ager s WrT = I, (3) the modIfied (fO + 2' O ' . (dcT nsager s equatIOn ~ = J and LlJ the ncw equatIOn (!JrT = 1. 1-0 Thus the relaxation times ohtainec1 frol11 the ohserved values of ~)r are al\\ays smaller by a factor f(~C/)+ 2l in Debye's Cdse a<; compared with the correct valne cal-eu + 2) culated from the new equation. s () r, (J T I I) N S (aJ Nonpolar: Since the Clausius-Mosotti expression is considered to he valid for dilute solutions III nonpolar solvents, it is expected that Debye'~ dispersion equation derived on the basis of this hypothesis "hould he applicable to tnea~uretnel1ls in dilute soll1tions of nonpolar solvenb. The extension of Debye'.., dIspersion equation to a binary system bas heen made by Williams (T934) who has con-sidered a solution in Which there arc 112 polar molecules/c.c. each lwving electronic {lolarisability O(L~(()I and lloment It, dis<;olved in 111 11101s. / C.c. of the A nomalous Dispersion 0/ Dielectric Constant 441 nonpolar solvent each havinl.! electronic poJarii-ahility O<:lj(OI and zero electric moment. Thus the polal i..,atlOl1 per l.Co 01 thl' nllxture P Il(,) = (f'c- 1) = 4" \ 11J(2 (I +11WT~I} (F12 +2) - iO<:IIIO)11 1 +CX F2 (fI)11_ + 31<'1' ,) (-'ZI / I-1) {("211l) - 1) (tl~~x. ) - 1) I h~(,)+2) ] -! /wT!. ('-12(0) + 2 'I ("'12! £ \ + 2) \ f12 xl = h;gh fl equvl1cy llidectlll c01J~tal1t alld '-'2,(j' -- ..,tatk tlldectric coni-tant of solution find '2 = l<:IaxatlOl1 titl1e of thl' polar ..,olnk molecules. Wnting -IC = '1/- I,,/'and sq)<llatlllg thlle,ll and lI11aglJHI1Y ]lart~ \\ hellce If'J.2I(I)- "!.!.!,r) '-1' \fl.!.IO) -f.l.!. I L l) It\2 1+\2 (t-I.!(t\\ -. 121..:1 I) 1 I \ 2 and These eJ..J>ll~~;Oll~ \\hich .lIe \'l'l~ ..,1I11iJat to lho..,l given hy I>ehye for ga~e~ \\ el e dell \ cd on the !Jd~is ot Clansltls- Ylosott 1 t:XI n eSSlOll for total polallsatiOll. TIlL t1~age of the CUll ecl eX!>l e"''''I011 fOi polansatlOD (, - I)~l/d giVeS cxpl<.s..,ion.., for " aud ," very ~illlilar to those derived abo\'c (. 121 ) -+- 2) except that 111 the fOl11)(:r <'fI~e the factO! -n docs lIot occur. In fact (f. J!(III i 2 \ 111 tilluk soint ion.., of l1onpoJm ,",olvln!::. '12I1"=<' 121 ~: ~o that thl factor ('...'.21 L I 1 .21 (~12111) + 21 £01 tuito\1~ly be:col1le~ umty .\Ild hence: lkbye's cxpres..,ioll coincide~ with the new L1IUcltiOIl. (I» Polar: ()llc1ey (1l)3S) has dttUllptl'd to illtel['ll't the dl"'l'er~ioll of polar bina! y solutions u~ing \\'Yl1lall'~ (1<)36) l'l1Ipirical \:tjuatJOl1 for polarisation. No rigorous thc;oretical tn ... atmcnt of thc ~l1bJ(.ct is available III literatUi e· Oil the ba~is of th\.! llC'\\ e(Illation for 1II0leClliar l)olari~ati()n (. -1 )~I, d, thc expres-sion for the 1J10iecuhu JlolarbatlOll PI~(,' of the ~()lutiol1 at h' frt:(lUellcy can be writtcII al->, where PI(, I aut! p!: 1 are the l11oiecuia! POid! isation., of the t\HI polar components at ((l frequency. Now =h-1)MJ,+ d1 (. ",- 1)~~J2 dl 442 S. K. K. latkar and B. R. Y. Iyengar Writing for the sake of simplicity ill terms of specific polarisatiolls J, (_ (f'r- I ) \ 1'3- - d. ) (-2 - I) - - w. ' III 1 and W.' d. ~ , being weight fractions. i.e. whele (~J-I) = hla:) -I) + I _~1J!.l)_-'l!.~l) and dId, 1+ IIllT 1 d1 (&2(01 -.2(",) dJ T I and T ~ being the relaxation tl1l1~:' and • I (0) amI ~ Jill) the :,tatlc diclech Ie constants ot the two polar components. Snb~tlhlttng thc<,e value!:> of P, (.,j) and P2(W) we get Since and + (el(~ -'1(",» 1+ iWT, W J + (e2(~-(2(",» d1 l+iwTJ Separating 612 into its real anti imagillary part!> and comparing with we n;et A nomalous Dispersion 0/ Dielectric Constant 443 and en" = .(6"12JO) - .e'1~<R!twr .. 1+w2r~2 -+ ~1.!.'L-:eli"'L 1 WT 1 WT. I d 12W I - - • - \" dl II +W2T1') II +w1r/} For a nonpolar solvent 1'1(0) c::.t 61(..,) so that the above equations reduce to the expression derived earlier {or nonpolar solutions. MOLRCULAR RADIUS Debye consider.ed the rotation of a molecule amIdst its surroundings analogous to a sphere carrying a dipole, rotating" in a viscous medium. According to this analogy it i~ seen that a torquc is necessary to rotate the mol<:cuk against the inner frictIOn of the medium in which it is suspended. Stokes has calculatc.d that this inl1er frictional fOI cc i:-. given by ~ == 13mlll' where 1/ is the inner fl iction constant of the surrounding medium in which the :-.phere of radius 'll' i:. rotating. From his considerations of Brownian 111otion, Debye has deduced that the rdaxation times r == ~/2/~1'. Substituting ~;rlla3 for; we get _ 4 lT11aJ T ---kT TI1Us a knowledge of T from the ohserved critical frequency Will enable the calculation of 'll', the radius of the molecule of an assllmed spherical :-.bape rotating in a medium of knowl1 viscosity 1/. It is evident that a considerable Humber of llJolecult's do l10t fall in conformity with the picture of a rotating !:>phl:!re. The long zig-zag chain type of molecules which arc often met \"ith ~coop ont a dIsc-like rather than a spherical cavity when they rotate. It will be thus interestJl1g to inve~tigate the frictional forc(;s in conformity with this picture. In fact, Stokes has dealt with such a two dimel!sional lOtatioll about an axis and bas calculatl.d the frictIOnal torque as 4;rI}a 2 II where' a ' is the radil1~ of the disc cavity and' Iz • its thickness. Here S is equal to T / kT correspondillg to the case of a rotation in one plane, as lias been (Ien\'cd by Debye (ltI13) ill all earlier paper. Thus a knowledge of the thickness' II . of a rotnt1l1g chain enables 011\;.' to determine' a ' the radius of the disc. • (l • thell :;hot1ld corn:~pol1l1 to half tht: length of the molecule or any unit thereof which is executing rotation. 1> 1ST R I 11 P 'I' JON () T" R E L A X A T 1 () N " J 1\1 R III the case of compound~ invo!\'illg complex lllolccuiar structllle. e.g •• the high polymers it \\ as fOllnd that there was a <.Onsidcrably broader frequency range of disperSIon alld absorption together \\ ith a smaller maximum value of ti" than predicted uy Debyc's equation. This discrepancy was thought to be dne to thl:! fact, that the polarisation was wrongly assumed to have a single unique relaxation time. Wagner (1913) assumed a distribution 444 S. K. K. }atkar and B. R. Y. Iyengar of relaxation tune ahout some probable value and proposed the use of a logarithmic Gaussian distribution Yager (H1361 predicted that e"max is < hl- e",1 by all amollnt dependillg upon the t1cn~ity of dbtnbutioD. Fuoss 2 and Kilk\\(Jod (19411 have aI..,o deduced an empilical iormula to ddermine the di..,trilmtion function of rdaxatiou tll11e~. Cole and Cole r HI4I 1 ga\'(: all \.'xprCSSlOlI to! the cOlllpk" dielectnc constant I· o -··r ) 1 + (lfU'" 1 J - , hy theIr ~tudy uf di"I'l'ISiOIl ill tbe light 01 tIll' l\q,:and diaglalll or tile cOlllpiex plalle locu~. If thi" , 1~ resuIvL'd ;11tO its leal and imaginal)' component .'- ,." thun ," = (·0--·"')('''1')1- . 1 + (W1'J 2(1-", J.;,·en on this ba:"i~ the cOll(htion at critical fleQI\(;llcy We rel1lains thc sam.; as caJculatfd OD the basis of the nl'\\ cqnatioll for a hillgiL 1', viz, w, .• = J as b '1 b . I I f' 1 I (/." I .. 1 f t can I.' eaS1 y 0 talDC( )y ltlC lllL: oul t lC ant l'ljl1atlllt.: It to lero. 11 ac, d~J the aSSll1l1pt ion of allY d1stribution co-efficient or parameter docs not change the position of the CI itical fn:ljUellcy but bas thl' effect of changmg the ~hapc of the .' and ,." CUI vc~. Cok and Cok (Iq..j I \ IJa\'C di"cl1~~ld t!;l rcprcscntation of di:"persiol1 data Oll the hdsis of the .\rgalld dlagraIlJ 01 «llllpkx rlanl' Iocu~ in \\hICh ,/, the i11lagilJalY iJLl1t of till.' COlllpleX dlekell ic COll~tallt i" plotted again~t .' the real part at varioll~ ff(:ljuellclc~. TilL' loctl~ of ~uch a plot, 011 the basis of a single relaxation tillle, hah been ~hO\\ll to be d ~en1!-('irl'k \\itiI the centrc 011 thc .' axis hy a con~iderat;on of rectors in complex 1'la1lc. The loclls could as \\tll he tk,\uu;tll1l a vuy ~illlpk IlHHllIel lroll1 the expressiollS for .' alld ." by a mere c1iminatioll of t1.e flujllency factor a~ follow-; and .1, = _("0 -coo) \ I + \ z ...... (.~) ; \ = (1)7" '1 . d . (. + ,) acconll11g to t 1e 1le\\' elllwtlon "II I:" "J7-~ according to Debye. t. u + 21 f.11'.:. = = i.e., (to- f oc / 2 ! 1 T \ ) ~! • \ 2 (to - tot) (r+ (2), ('o-e",)" (I + x-ij"-A nomalous Dispersion of Dielectric Constant 445 neot\\etrieally this condition mc~n'" that the r~ctangh~ contained hy two segments of .1 diameter e(111a1 to (cl - rc>; I amI (FO - ('I I:qllclb the square of u chord ,.I' cutting the diameter perpc!1(licl1larly, This means that .:'_,-" curVl' is a semi-ci!c1e with centre on the .' axis. Cole and eule, howl'ver, found that t!xperimental data do not justify lhis semi-circular plot in a nUl1Jber of cases bllt could be reprl'sentcd hya circular arc charaete! ised I,yan angk rx .. /2 which is a !lJeasnre of the deviation irol1l the semicirde, being equal to the angk bdween the ,.' axi~ and the radius through e~. This circular are again has been given a theo! ctieal hasis by Cole and Coli: using complex plane loci. The significance of this angle can he obtained in a very simple way as folIo\\':". Considerillg the Lircular arc it i~ seen fr0111 fig ure I " E I \ , \ E' , \ , ' I I , , \ I \ I \ I , / , ' " '" '.... ,.",,' ..... _----_ ~~o_-: ,·~t_ = 0 <1 Discarding the negative solution for til,,, tan rx .. /2 • k' being a constant characteristic ot the sub!>tance. Thus the parameter , rx' introduced serves to lI1C'rease the t "",,,\ \'alue whirl! usually fall" short of 3-1°741' ,- 1<> 446 S. K. 1\.. latkar and B. R. Y. Iyengar the vallle(~'.!L-_£QO) as cvaluated on thc basis of a ~ingk relaxation time. Cole 2 and Cole have given on the hasis of their cOlllplex plane locus the equatioll for Film a~ ,'''m = ~ (.'o-~(/') tan (I -()()tr/4 This is identical \dth the eql1ation derived above since (I _ ()() tr = j cos ()(tr / ~ - si!;! ()(tr ~ 4) = (I - 2 cos ~tr / 4 sin ()(It i 4) tan 4 cos ()(tr / 4 + sin ()(/T / 4 (cos2 ()(;r / 4 - sin2 ()(tr /4) (J - sill ()(tr / .2 ) - co~ ()(:;-; .z - -l\ (' K N (l \V I. F. D C ::-.[ F. N T Thanks of til(' authors arc due to the Council of Scientific and Industrial Research for financial assistance for this research, (;~ ,(l!lHI, CHI<.!lIlSTHY SJo('TJo/\, INDIA" INS1'ITI'TIl OF SerF"('I', BAN(: 'I.()RF. ;\. REFERJ<;NCT\S Cole, R. n., (193~), T Cllcl1I. P"y~., 6, ;\R.'. l'oh', K. ~., and Cult·, R. n., (19P) , r. CIICIII. P/II'I., 9, .\11. Dcbye, 1'. ilQt3), /ler. neul,I'IIYI., r;CI .. 15, 77i. J>c1»)t"I'. (I<no) , " Polnr "o!t-culc'," Pl>. 7j-!"S Druuc,l'. (IR97), 1.. PIJ),;. CI"'III , 23, ~r,i. Fuoss, R. 'I 011(1 Krik\\oo<1, J (;. (rfJlll, r II/Ier CIII'III. Soc., 63, lXS ]atkar, S. K. K ,hclIg-al, B.){ Y AIHl Sathe. N." (1910), /. Illd '"~I. ,'iIi., 28A, Port II, \-1, ()llcle~', J. L., (IY,'lXi, T, .111/('1. Cllelli. SOl, 50, Ill:;. Om.ager,l. (1936), J . .11111'1. CIII'III Sor., 58, 14il6. Wagner K. W. (19131, ;I /In. 1>1'1. 1'''y.I., ~l), 81;-, William!'. J. W., (1931), TltlllS. Ftlrad. So,,, 30, i23. \\'y 111 & 11 , I. i!9,'l(»), T. AlIIl'l ('f'~/Il. So, .68, qx.'. Yager, \\'. A. ilf)3S), }'''\,lj'l, 7, 134, 

Anomalous Dispersion of Dielectric Constant

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Anomalous Dispersion of Dielectric Constant

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