Frequency and temperature dispersions of high polymers

JOURNAL OF COLLOID SCIENCE 11, 214-225 (1956)

FREQUENCY AND TEMPERATURE DISPERSIONS OF HIGH POLYMERS Shigeharu Onogi and Kyuzaburo Ui Department of Textile Chemistry, Faculty of Engineering, Kyoto University, Kyoto, Japan Received March 20, 1956

INTRODUCTION The frequency and temperature dependences of viscoelasticity of high polymers are of practical as well as theoretical interest. The relaxation spectra of amorphous and plasticized high polymers differ more or less from those of highly oriented crystalline polymers, such as cellulosic materials, particularly on the long-time side (1). The spectra of oriented crystalline polymers are very great, and little subject to the time scale, whereas those of typical amorphous polymers are small at long times and become greater with the shortening of the relaxation time from about 1 second, until they approach almost the same order of magnitude as those of oriented crystalline polymers at around 10-7 to 10-8 second. In this region, therefore, the elastic dispersion with respect to frequency can be observed on amorphous polymers (2). Low temperatures produce the same state of affairs as high frequencies, changing amorphous polymers from the rubbery to the glassy state. Thus the dispersion with respect to temperature occurs in the vicinity of the glass temperature (3, 4). Other high polymers show intermediate behaviors between these two extremes, and the relaxation time at which the maximum of the spectrum appears shifts to the long-time side to some extent in accordance with the nature of the polymer. These two dispersions, "frequency dispersion" and "temperature dispersion," are quite similar to each other, suggesting that there exists some intfinate relationship between them. The quantitative discussion of such relationships must be made on the basis of the experimental data obtained for a given material by varying extensively both the frequency and temperature, but the available data on this are very scanty at present. This paper is concerned with the application of the recently proposed general law relating to the time scale and temperature for the viscoelastic behavior in amorphous polymeric systems. From the experimental curves of the dynamic response as a function of temperature at a given frequency, 214

FREQUENCY AND TEMPERATURE DISPERSIONS OF HIGH POLYMERS

215

the response as a function of frequency at a specified temperature can be computed. This is done by making an assumption concerning the relaxation spectrum and thereby obtaining the "characteristic temperature" for the material. This procedure has been tested by means of the data of the dynamic experiments o n two types of polyvinylidene-vinyl chloride copolymer; it has also been tested with the published data on polyvinyl chloride-dimethyl thianthrene, Between the computed and the measured value good agreement is obtained in the former case, and excellent agreement in the second. REDUCTION OF VARIABLE-TEMPERATURE DATA TO VARIABLE-FREQUENCY DATA

As mentioned by Ferry and Fitzgerald (5), the assumption that all relaxation times depend identically on temperature is supported by numerous data on various linear polymers. According to these authors, the customary exponential expression between the relaxation time and the absolute temperature with a constant activation energy is not valid, but the following algebraic one is fit to describe quite well the temperature dependence in various substances, such as a wide variety of polymers, polymer solutions, organic glass-forming liquids, and inorganic glasses

(6, 7): log aT

=

--Cl(T

--

[1]

T~)/(c2 + T -- T.O,

where a~, represents the ratio of any relaxation time at temperature T to its value at a reference temperature T~, which is suitably chosen for each system and lies about 50°C. above the glass transition temperature; cl and c2 are the universal constants and found to be 8.86 and 102. Employing this, the relaxation time r at temperature T can be given by log r = log r~ - c~(T -

T~)/(c~ + T -

[2]

T~),

where r~ is the value of r at T~. When one uses any other reference temperature To than the characteristic temperature T,, r must be given by the following equation in terms of To and the corresponding relaxation time TO:

l o g r = logr0-- clc~(T -

To)/(c2 + T - -

T~)(c2 + To -- T~).

[3]

Accordingly, the real and imaginary parts of the complex dynamic modulus and some other dynamic properties measured in variable-temperature experiments at a given frequency v0 call be reduced to the variablefrequency data at a specified temperature To by plotting them against the following "reduced frequency" v: logv = l o g v 0 - c l c ~ ( T -

To)/(c2 +

T--

T~)(c.2 + T o -

T~).

[4]

216

SHIGEHARU ONOGI AND KYUZABURO UI

In reverse, of course, the variable-frequency data can also be reduced to the variable-temperature data. Moreover, in performing this reduction, it is possible to determine the distribution of relaxation times from the variable-temperature dynamic data as well as from the variable-frequency data. : Thus, for instance, the frequency dispersion can be predicted from the temperature dispersion, and so far as the above algebraic expression for the temperature dependence of r is valid, the predicted dispersion should be the same as the actual one obtainable from variable-frequency experiments. For practical performance of these reduction processes it is required that the characteristic temperature T8 be known. It can suitably be chosen from the shift factor ar as a function of temperature, which can be determined from th e experimental results of the frequency dependences of the complex dynamic modulus and some other viscoelastic properties at various tempera?~ures. As will be shown later , T~ can also be determined from the experimental curves of the real and imaginary parts of the complex dynamic modulus as a function of temperature at a given frequency, though only approximate. EXPERII~IENTAL METHODS

The real and imaginary parts of the complex dynamic modulus, E ~ and E ~r, of two kinds of commercial polyvinylldene-vinyl chloride fiber of 250 deniers, one of which (1) contains only plasticizer while the other (2) contains plasticizer and pigment, were measured by means of the vibrating reed: method at constant frequency (100 ± 2 cycles per second)and over a temperature range from - 2 0 ° to 80°C. In addition to this, the frequency dependence of E ~ of the fiber 1 was measured with the same method at various temperatures from - 1 2 to 42°C. and over a frequency range from about 20 to 120 cycles per second. The details of the apparatus and measuring techniques have been described elsewhere (8-10), and the vibrator, in th e present case, was set in the thermostat to perform the experiments at constant temperature. The variable-frequency experiments at constant temperature were carried out quite independent of the experiments on variable-temperature described above, by employing four experimental methods according to the frequency range. 1 In the frequency range from about 0.2 to l0 cycles per second, the free damping oscillation method was employed; from 50 to 220 cycles per second, the stretch-vibrometer similar to that of Lyons (11) ; from 1 to 5 kc., the longitudinal-wave resonance, and in higher range, the longitudinal-wave transmission method originating from Nolle (12, 13) were 1 These experiments were performed b y courtesy of Assistant Professor H. Kawai of the Department.

FREQUENCY AND TEMPERATURE

DISPERSIONS OF HIGH POLYMERS

2~17

used, respectively. The details of the apparatus and measuring techniques have also been described elsewhere:(1). RESULTS The real and imaginary parts of the complex dynamic modulus, E' and E", of the polyvinylidene-vinyl chloride fiber 1 and 2 obtained l~y the variable-temperature experiments are plotted against the reciprocal absolute temperature x ( = l / T ) in Figs. 1 and 2, respectively. These results show typical temperature dispersions over the temperature range about - 2 0 to 60°C., and the maximum values of E" for these samples are found at 14.4 and 7.9°C., respectively. Figure 3 shows the frequency dependence of E' of the fiber 1 at various temperatures. Shifting each curve in the figure along the abscissa by a suitable distance, which corresponds to log aT, a composite curve referred

VINYLIDENE CHLORIDE/VINYLCHLORIDE COPOLYMER, t. I00 cps xlO 'o

x I 0 I~

15

E °

l

-3

0

I0 tO

U •

I

¢

II

5

t

3.0 FIG.

1. E '

and

E"

vs.

3.5 reciprocal

I

I

I

,

I

&O

,

0

x 1 0 "z

"

absolute temperature for polyvinylidene-vinyl c h l o r i d e f i b e r 1. ' . ., ........

218

SttlGEI-IARU ONOGI AND KYUZABURO UI

VINYLIDENE CHLORIDE/VINYLCHLORIDECOPOLYHER,2

.:

I00 cps

A3

x/Om

15 :

'

.:.:

2

G) "0

i 0

0

3.0

3,5

4.0

x 1~3

.~, I/OK FxG. 2. E' and E" vs. reciprocal absolute temperature for polyvinylidene-vinyl chloride fiber 2.

to a specified temperature can be obtained. Figure 4 shows such a composite curve referred to 23°C., and in Fig. 5 the evaluated values 2 of log a r are plotted (with open circles) against temperature in contrast with those calculated from Eq. [1] with assuming T, = 312 (solid circles and curve). Good agreement can be seen between them. The results of the variable-frequency experiments for the fiber 1 at 23.5°C. are dotted in Fig. 6. The values of E ' and E " disperse over the whole frequency range covered, and the m a x i m u m value of the latter appears at around 3 kc. The value for 3°C. is not certain but rather arbitrary.

FREQUENCY .A~NDTEMPERATURE DISPERSIONS OF HIGH POLYMERS 219

i

I

Il

IIit

~

I0"

"

i0

°

~

e

i

m

21

- 1 2 °C 3

8

'2

I

~23 ~ 30

!01°! I I0

I

2

I IIIII z~

6

I 8 /00

2

FREQUENCY, cps FIG. 3. Frequency dependence of E ' for polyvinylidene-vinyl chloride fiber 1 at nine temperatures as indicated.

I012

~

10~

j

j

, 4206 * 35

® o G e

bj l Ot'

e3

30 23 20 15

tO

IOq

O. ~

I

10

102

I0~

10'

10 5

106

FREQUENCY. cps. FIG. 4. Real part of the complex dynamic modulus reduced to 23°C., plotted against reduced frequency. DISCUSSION

It is found at once from Figs. 1 and 6 that the maximum values of E" of the polyvinylidene-vinyl chloride fiber 1 measured by the variable-frequency and variable-temperature experiments are quite equal. Moreover, it seems that the limiting values of E' at both sufficiently high frequency and low temperature are also equal. Such a special feature can also be found

220

SHIGERARU l

ONOGI AND KYUZABURO UI '

l

l

l

l

l

6

l

l

l

l

• Co.l, cLtLcded(Ts=3/2

5 4 3 ~2 I

0 -/ -2 -3 I I -20 -I0

I

I

I

0

I

I

I

I0

I

.

20

I

I

30

I

I

40

50

TENPERATURE, °C FIG. 5. Logarithm of reduction factor ar plotted against temperature (open circles, observed; solid circles, calculated with T, = 312). x 10~°15

3 x I010

Ts = 305 ~. 6o

o

\

/ "~" 5

~ o

.

0

,

f

O.f

.

[

]0

I0 2

:

I

lO3

I

/0 4

I

JO5

I

106

. . . . F R E Q U E N C Y , cps :., FzG. 6. Frequency dependences of E ' and E" for the polyvinylidene-vinylchloride fiber 1 (dots : obtained from variable-frequency experiments; lines : reduced from the variable-temperature data in Fig. 1).

FREQUENCY , ~ D TEMPERATURE DISPERSIONS: OF HIGH POLYMERS

-

I0 %

=

PVC-DMT

221

IX'°'

2 0 0 cps

8-

n4

7,

,

I/

.3

5

h\ I-

I II

2,. \~.

i

0 3.s

3.6

3.7

0

3.8

3.q X,

1/°K

t~.O

4.1

/+.2 X

163

FIo. 7. G' a n d G" vs. reciprocal absolute t e m p e r a t u r e for 10% P V C - D M T gel (After E. R. F i t z g e r a l d a n d J. D. Ferry),

in the dynamic data on the polyvinyl chiofide-dimethyl thianthrene gel (10 % PVC by volume) obtained by Fitzgerald and Ferry (14), who carried out dynamic measurements with the electromagnetic transducer method in the frequency range of about 30 to 5000 cycles per second and at -23.55 to 25.00°C. In Fig. 7, the real and imaginary parts of the complex dynamic shear modulus, G' and G", at 200 cycles per second are plotted against x, and in Fig. 8, those measured by varying the frequency at - 0.2°C. are dotted. The maximum values of G't in these data are quite equal to each other. So far aS these substances are concerned, therefore, it seems prope r to consider that:the height of the relaxation spectrum in the frequency-dispersion region is independent of temperature. The only remarkable difference between both the dispersions is that the temperature dispersion takes place rather sharply over a comparatively narrow temperature range, whereas the frequency dispersion takes place gradually over an extensively wide range of frequency. And it is very significant to ascertain whether the two dispersions are reducible to each other according to Eq. [4] or not. The two pairs of solid and dashed curves in Fig. 6 show the results of

222

S H I G E H A R U O N O G I AND K Y U Z A B U R O U I

/0,o

I0 % PVC-DNT - 0.2 °C

~

6' _.

... I 0 q

6"

(11 ¢..

-o l 0 s

g L"t

=
I0

10 2

10 9

10 4.

IO s

,, I0 e

I I 0 '/

I0 a

FREQUENCY, cps FIG. 8. Frequency dependences of G' and G" for the 10% PVC-DMT gel (dots: obtained from variable-frequency experiments; lines: reduced from the variabletemperature data in Fig. 7).

reduction of the temperature dependences of E ' and E " for the polyvinylidene-vinyl chloride fiber 1 shown in Fig. i according to Eq. [4]. The reference frequency ~0in this case is 100 cycles per second, and the reference temperature To is 296.5°K. The solid curves in the figure show the results of reduction with T, = 305, which is calculated from Eq. [5] mentioned below with the assumption of a w i d e box distribution, while the dashed curves represent the results reduced with T8 = 312, which has been determined from av shown in Fig. 5. As mentioned above, the dots in the figure show the results of the variable-frequency experiments at 23.5°C. (= To). The above value of T, of 305 was calculated from the variable-temperature data according to the following equation:

T~ = c2 --

[5]

Xm

This can be derived by assuming for the complex dynamic modulus the box distribution of relaxation times and the above-mentioned temperature dependence of r, namely, Eq [1]. Then, E' and Ett can be given as a function of temperature as follows: Eo

2 2

2f(x)

1 + ~o r.,~ e c0~7"21seef(x) '

E p = E, + -~- In 1 -~E"

=

E0(tan -1

w r m : :(=) -

tan-l~rl.,e/(=)),

[6a] [6/)]

FREQUENCY AND TEMPERATURE

DISPERSIONS OF HIGH POLYMERS

223

where E~ is the static modulus, E0 is the height of the box distributiou assumed, r~ and r,~, are, respectively, the lower and the upper limits of the distribution at T~, and f ( x ) is given from Eq. [1] as follows: f(x) = -2.303c1(T-

T~)/(c2 A- T -

= --2.3030(1 -- T~x)/[1 ~- (c~ -

T~) T~)x].

fT]

E 't has a maximum value,

Em~x

7t" =

~ Eo

[8]

(r,~ >> rts),

at reciprocal temperature x,~ which satisfies e+(='~) = ~-l(v~,~vz~) -'/2.

[9]

On the other hand, the slope of the tangent of the E' - x curve at x,~, m, is given as (r,~ >> r~:)

m = Eof'(x)

[101

= EoAHa(x,,)/R

= 2.303 oc~.Eo/[1 + (c2 -

T,)xm] 2,

where R is the gas constant and AHa(x) is the apparent activation energy for the relaxation processes. This results immediately into Eq. [5]. The above value of T~ of 305 is obtained by using E 0 ( = ~r-2E : ~ )

= 1.43 X 101°, m = 3A7 X 1014,

and

x~ = 3.48 X 10-3.

Leaving the subject of the derivation of Eq. [5] and turning back to Fig. 6, it is to be noticed that both the solid and dashed curves agree fairly well with the variable-frequency data in the whole frequency range covered, with the exception of small difference in E' and E" at the low and high frequency ends, respectively. The lower T , in general, the sharper the reduced curve becomes, and therefore the solid curve is sharper than the dashed one. But the difference is rather small and the former agrees with the measured values as well as the latter. With the assumption of a single relaxation time instead of the above box distribution, one obtains Eq. [5] with m replaced by 2m. In this ease E0 is given as 2E". . . . and T~ is found to be 284°K. This value is too small to yield good agreement. It appears that the sharper the relaxation spectrum, the smaller the calculated value of T~ is, and that the value of T~ calculated from Eq. [5] is always somewhat smaller than the true one, since the actual spectrum is always broader than the box distribution assumed. But it is

22z~

SHIGEHARU ONOGI AND KYUZABURO UI

much bette r to assume a box distribution than to assume a single relaxation time. Thus,!we cannot expect the correct value of T~ from Eq. [5], but when we have the variable-temperature data alone and no available data concerning the shift factor at, Eq. [5] is very useful to calculate T~ and hence to reduce the data to the variable-frequency data. The same state of affairs as mentioned above call be found in the case of PVC-DMT gel mentioned above. The value of T~ calculated from Eq. [5] is 288, whereas that reported in Ref. 6 is 293. Here again the calculated value is smaller by 5 °, but the results of reduction do not differ so much. The two curves in Fig. 8 show the results of reduction of the temperature dependences of G' and G r' shown in Fig. 7. The values To, T., and ~0in this case are, respectively, 272.8°K. (-0.2°C.), 293°K, and 200 cycles per second. Here again the dots in the figure show the values of G' and G" measured actually at -0.2°C. by varying the frequency. They lie almost on the corresponding curves, showing that the frequency dispersion predicted from the temperature dispersion coincides quite well with the actual one, and that the above algebraic expressions for the temperature dependence of r and for the reduced frequency are valid. SUM2dARY

The real and imaginary parts of the complex dynamic modulus of two kinds of commercial polyvinylidene-vinyl chloride fiber have been meas± ured by means of several measuring methods at - 2 0 ° to 80°C~ over the frequency range 0.2 to 105 cycles per second. Both the variable-temperaturd and variable-frequency data show typical dispersions in the experimental ranges covered, being similar to each other. By employing the algebraic expression for the temperature dependence of the relaxation time given by Ferry and his schools, the variable-temperature data can be reduced to the variable-frequency data, and vice versa. For polyvinylidene-vinyl chloride fiber and PVC-DMT gel, the frequency dispersion predicted from the temperature dispersion exhibits its agreement with the actual case. ACI~NOWLEDGMENTS The authors wish to express their sincere gratitude to Professor M. ttorio for his kind guidance and encouragement for their studies. They also wish t o acknowledge their indebtedness to Assistant Professor H. Kawai of the Department for his experimental support, and to Professor J. D. Ferry of the Department of Chemistry, University of Wisconsin, for his helpful discussion. They are also indebted to the Research Institute for Synthetic Fibers, Japan, for financial support. REFERENCES 1. FTJJINO, K., KAWAi, I~I., AND HORINO, T., Textile Research J. 25, 722 (1955). 2. See, for example, IV~Y, D. G., MROWC.a, B. A., _aND GUTH, ]~., J. Appl. Phys. 20, 486 (1949).

FREQUENCY AND TEMPERATURE DISPERSIONS OF HIGH POLYMERS 225 3. NOLLE, A. W., J. Polymer Sci. 6, 1 (1950). 4. SCtIMIF,DER, K., AND WOLF, K., Kolloid-Z. 127, 65 (1952); ibid. 134, 149 (1953). 5. FERRY, 5. D., AND FITZGERALD,]~. R., Proc. Pnd Intern. Congr. Rheology, p. 144 (1954). 6. WILLIAMS,M, L., J, Phys. Chem. 59, 95 (1955). 7. WILLIAMS,M. L., LANDEL, 1~. F., AND FERRY, J. D., J. Am. Chem. Soc. 77, 3701 (1955). 8. Honlo, M., O.nOGI, S., NAKAYAMA,C., AND YAMAMOTO,K., J. Appl. Phys. 22, 966 (1951). 9. HORIO, M., AND ONOGI, S., J. Appl. Phys. 22, 971 (1951). 10. Homo, M., AND OnOGI, S., J, Appl. Phys. 22,977 (1951). 1t. LYons, W. J., Textile Research J. 19, 123 (1949). 12. NOLLE, A. W., J. Acoust. Soc. Amer. 19, 194 (1947). 13. NOLLE, A. W., J. Appl. Phys. 19, 753 (1948). 14. FITZGERALD,]~. R., AND FERRY, J. ~)., J. Colloid Sci. 8, 1 (1953).