Analytical method for calculating dielectric relaxation in polar polymers

METHODS OF INVESTIGATION ANALYTICAL M_ETHOD FOR CALCULATING DIELECTRIC RELAXATION IN POLAR POLYMERS* V. N. NosonY[:K X. A. Ostrovskii Pedagogical Institute, Vinnitsa

(Received 5 May 1969) EEPERIM:Eh'TAL results concerning dielectric relaxation in polar polymers, characterized by symmetrical dispersion, are normally analysed by the Cole-Cole [1, 2] graphical method. This method enables dielectric constants to be derived which are limiting values for this range of relaxation, the spectrum width of relaxation times and other constants, which are well defined and in some cases fairly accurate [3-12]. However, the graphical method becomes unsuitable for investigating polymer samples with a wide range of relaxation times, or when experimental results are limited to a restricted frequency range. The effective component of the dielectric constant e, of the low-frequency end of a given range of relaxation and the effective component of the dielectric constant e ~ of the high-frequency end of the relaxation range, obtained by graphical extrapolation, are subject to considerable error. Considerable difficulties arise also when determining the width parameter of the relaxation spectrum.

i

7"=comf

L

/ ~ ~

~L

~L

es ,~

/

I

z~.~,o

L

1

5.20

I

6.00

I

I

8.~O x: ~"

C

FIG, 1

FIG. 2

FIe. 1. Curve showing syrametrical dispersion for analytical calculation. FIG. 2. Isothermal curve showing the dependence of the dielectric loss factor e'" on the effective component e° of the complex dielectric constant of plasticized polyvinylchloride at 43.5°C. Analysis of experimental data is somewhat simplified and more reliable results obtained by using the analytical calculation which we propose, which follows from the Cole-Cole phenomenological theory. A graph is plotted from values 2----8' and y-----e" obtained experimentally for certain frequenciesf (Fig. 1). With symmetrical dispersion this is a circular arc. According to the graph three points .,%Il, M2 and M~ are chosen which seem to coincide * Vysokomol. soyed. A12: _N'o. 6, I442-1445, 1970. 1639

1640

V . N . XOSOL'~-L--K

most closely with the periphery and whose ordinates, if possible, show a m a x i m u m difference The equation of the circle which passes through the selected points, will take the form ( x - - a ) " + ( y - - b ) : = r ~"

(1}

where a, b are the coordinates of the centre C of the circle, r is its radius. Equation (1) is satisfied for all points M~, ,~I~ and Ma. Therefore, from their coordinates all three parameters of the circle a, b and r can be determined. Substitution of the coordinates of these points yields formulae for determining the centre of the circle ct=-• gt2--g~3 '

b=hl~--gl.,, a ,

(2}

where Xl--~ ~

Xl--X 3

Yl--Y2

Yl--Y3

and h~=½[g~.(xt-{-x~)+(y~+y2)] ;

h~3=½[g~3(x~-l-xa)+(y~-i-y3)]

(4)

and its radius r = V ' ( x l - - a ) ~ + (yl--b)'-

(5)

Considering that for the limiting points H and L of the relaxation range s:udied (Fig. 1) the abscissae are x ~ = e ~ , xr.=es, respectively, from equation (1) b y successive substitution of the coordinates of these points we obtain ~=a--V'r2--b

2

(6)

(7)

e , = a + Vr~-b'~

Parameter fl of the width of the relaxation time spectrum can be determined from t h e gradient of the tangent to the circle at point H. By differentiating equation (1) with respect to x, for point H we have: yH'----tan B

=--7,

hence 2 are tan s

e~--a b

I f arc t a n s is determined in radians, formula (8) becomes arc t a n s 9O The m a x i m u m loss f a c t o r era'" is also simply determined. Taking into consideration that e m " = y (a) and substituting the value x----a in equation (1), we have: e,~'"----r~b .

(9)

The accuracy of these calculations is verified by the ratio [1]: ~'~"=

2

t a n f l - ~ = ( r " - - b 2) t a n p - ~ ,

(10)

and by the method of identical transformations, by substituting coordinates of several other points of the curve in equation (1).

Anal~°cical method for calculating dielectric relaxation in polar pol~nners

164t

:ks an example illustrating this method of calculation we used experimental resulta which we obtained for the dielectric relaxation of a P U ~ I K h P 2024-49 commercial plasticized poly~-inylchloride specimen. This specimen had a wide relaxation spectrum, as shown by the circular diagram (Fig. 2). As the small curvature of similar cLtrves considerably hinders graphical determination of e~, e~, ~, f,~ and e~", the use of the analytical method of calculation is therefore more advisable. I n this case three points were selected for the calculation: M I , M2 and M3. For M1 x=7.02; y=0-229;f----0.2 kc/s; M2=5-85; 0.31; 5.0 and M~=5.02; 0-282; 30, respectively. The following parameters were obtained using equations (2)-(9): e8=8'25; e ~ = 3 . 2 7 ; fl----0.164; e~"-----0.320. A control was established by determining e,(" from equation (1), as previously described [1]. The value of e,('=0.321 calculated from this equation shows satisfactory agreement with the value era"=0"32, obtained from ratio (9). ~ne

Mz 0"5

0"2

0"1

8

J,

'

I

I

I

"~-=29.6 °

I

1

16

I

20

I

I

2q

I

I

28

I

l

I

32

\~

38 ~'

Fro. 3. Isothermal curve showing the dependence of e'" on e" for crystalline hydrogen bromide at 73-1°K (according to experimental results [2]). To explain the agreement of the analytical calculation proposed with the graphical calculation, we compared the results obtained b y Cole by a graphical method and the results we obtained analytically for a narrow relaxation spectrum, where the graphical method gives entirely satisfactory results. Experimental values of e', d" and f for hydrogen bromide at 73.1°K were obtained from another study [2]. A circular diagram y = ~ (x) (Fig. 3) was plotted, which corresponds to the equation

e = ~ - ~ l_b (ieor0)B Parameter fl of the width of the relaxation time spectrum in this study is related to parameter a, which was introduced in a previous study [2] b y the relation #-~ 1--g. Obviously, Fig. 3 shows a mirror image of the true graph e' (e") in a complex plane in relation to the abscissa axis. Figure 3 indicates that points M1, M2 a n d M3 are the most suitable for analytical calculation. For M l - - x = 3 0 . 5 ; y=6.18;f.=O.O5kc/s;-'~/2--16.4; 8.92; 0.5; M3--5.98; 2.46; 10, respectively. The results of the calculation were:

~s e~ f~, kc/s e~

Graphical method [2] 36.35 4.35 0.29 --

Analytical method 36.24 4-26 -9.30

fl= 1 --g r

Graphical method [2] 0.678 --

a

--

b

--

,Xnalytical method 0-671 18-38 20.25

--9.08

1642

V. •.

NOSOLYI:rK

Our investigations confirmed the conclusion t h a t with wide relaxation spectra the graphical method of determining parameter fl according to [1] becomes very inaccurate. Equation (8) in these cases gives only approximate values. To retain Cole's phenomenological theory when analysing experimental results of dielectric relaxation with a wide dispersion the graphical method of determining parameter fl should not be used. Calculation should be carried out analytically, by solving the complex transcendental equation [2] 1 1 + cos T B~ 8 ,° ~ 8 ~ ' "

1 cosh fix+ cos-~ f~

More satisfactory results, corresponding to the theory, are obtained when determining parameter f from the F u o s s - K i r k w o o d semi-empirical relation [13] H (0) H ( x ) = cfl------x osh ' where x-~lnf,n/f,f,, being the frequency corresponding to the m a x i m u m value of e,," of the loss factor and f being the frequency used for the measurement. Values of H (x) and H (0) are determined according to the polarization theory developed b y Kirkwood [14] for polar liquids, using the ratios g'(2+--

1

1

e,," (2-~

H (x)=

H (o)=

era*2 I---ern ,,2

These formulae indicate t h a t when e" <
eosh fix' from which parameter f can be readily determined. CONCLUSIONS (1) An analytical method is proposed to calculate the parameters of symmetrical dielectric dispersion in polar polymers, developed from Cole's phenomenologieal theory. The analytical method has advantages over the graphical m e t h o d when there is a wide spectrum of relaxation times. (2) I t is noted t h a t with a wide relaxation time spectrum, observed in the polyvinylchloride-plasticizer system, the Cole-Cole method of determining p a r a m e t e r f for the relaxation spectrum width is no longer valid. This p a r a m e t e r should in these cases be determined from the F u o s s - K i r k w o o d equation.

Translated by

E. ~EI~IERE

REFERENCES

1. K. S. COLE and R. H. COLE, J. Chem. Phys. 9: 341, 1941 2. R. H. COLE, J. Chem. Phys. 23: 493, 1955

3. K. Z. FATTAKHOV, Zh. tekhn, fiziki 24: 1401, 1954 4. G. P. MIKHAILOV and T. I. BORISOVA, Zh. tekhn, fiziki 28: 132, 1958

Analfc'tical method for calculating dielectric relaxatioa in polar polymers

1643

5. T. I. BORISOVA and G. P. ,~IIK~XLOV, Vysokomol. soyed. 1" 5;4, 1959 (~'ot translated in Polymer Sci. U.S.S.R.) 6. L. ¥. KRASNER and G. P. ~ I I K ~ I L O V , Vysokomol. soyed. 1: 542, 1959 (,%'ot translated in Polymer Sci. U.S.S.R.) 7. G. P. ~IIKHAILOV and T. I. BORISOVA, Vysokomol. soyed. 2: 619, 1960 (Translated in Poll-mer Sci. U.S.S.R. 2." 4, 387, 1961) 8. G. P. ~ ~ O V , T. I. BORISOVA, N. N. IVANOV, A. S. ,N'IGMANKHODZIL~YEV, Vysokomol. soyed. A9: 778, 1967 (Translated in Polymer Sci. U.S.S.R. 9: 4. 869, 1967) 9. G. P. ~ I I ~ O V and L. V. KRASNER, Vysokomol. soyed. Ag: 213, 1967 (Translated in Polymer Sci. U.S.S.R. 9: 1, 233, 1967) 10. R. H. COLE and D. W. DAVIDSON, J. Chem. Phys. 29" 1389, 1952 11. L. DE BROUCKERE and G. OFFERGELD, J. Polymer Sci. 30: 105, 1958 12. S. KASTNER and M. DITTMER, Kolloid-Z. u n d Z. f/lr Polymere 204: 74, 1965 13. R. M. FUOSS and J. G. KIRKV~'OOD, J. :4_,her. Chem. Soc. 63: 385, 1941 14. J. G. KIRKWOOD, J. Chem. Phys. 7: 911, 1939