BRIEF COMMUNICATIONS

R E L A X A T I O N S P E C T R A AS A M E A N S

R U B B E R I N H O M O G E N E I T Y

V . G . E p s h t e i n , L . E . V i n i t s k i i , a n d L . G . E v s e e v a

O F E S T I M A T I N G

UDC 678"539.381

The r e su l t s of an invest igat ion of the ef fec t of rubbe r inhomogeneity on the nature of the re laxa t ion spec t r a a r e p resen ted . The re is a cor re la t ion between the spec t rum p a r a m e - t e r s and the inhomogeneity cha r ac t e r i s t i c s of the rubbe r s .

All po lymer m a t e r i a l s a r e cha r ac t e r i z ed by a broad range of re laxa t ion t imes . The re laxa t ion log H (¢)and r e t a rda t ion log L(~) spec t r a of ce r t a in l inear and lightly c ross l inked po lymer s have been examined in a s e r i e s of publicat ions, notably [1, 2]. Relaxat ion spec t ra of unfilled v u l c a n i z a t e s - n a t u r a l (NK), poly- butadiene (SKB), bu tad iene - s ty rene , and butadiene-ni t ro le rubber s - w e r e analyzed in [3].

Obviously, in view of the s t rongly e x p r e s s e d thixotropy of b lack-f i l led rubbe r s and the difficulty of in te rpre t ing the r e su l t s , l i t t le of a fundamental na ture has been published concerning the re laxat ion spec t r a of technical vu lcaniza tes . At the s a m e t ime, as compared with unfilled po lymer s (linear, branched, and cross l inked) f i l led rubber s have a g r e a t e r va r i e t y of s t ruc tura l e l ement s . Consequently, in mechanica l , t he rma l , e lec t r i ca l and other s t r e s s f ields rea l iza t ion of the equi l ibr ium conformat ions co r responds to a b roade r r a t e in terva l .

Thus, d i f fe rences in the dis locat ion and distr ibution of i n t e rmolecu la r and interbundle bonds, i n t r a - molecu la r and i n t r a s t ruc tu r a l in terac t ions , and r u b b e r - f i l l e r and f i l l e r - f i l l e r bonds, together with the p r e s e n c e of di f ferent s t ruc tu ra l fo rmat ions , coils , entanglements , e tc . , account for the b road range of re laxa t ion t imes and hence the s t ruc tu ra l inhornogeneity of technical r u b b e r s .

We have a t t empted to es tab l i sh a co r re la t ion between the p a r a m e t e r s of the re laxat ion spec t rum and the cha r ac t e r i s t i c s of r u b b e r inhomogeneity.

If the molecu la r and s t ruc tu ra l r e a r r a n g e m e n t s that take place in the p r e sence of mechanical or o ther fo rces a re coopera t ive , we may a s sume the exis tence of a ~ continuous re laxat ion spec t rum in b lack- f i l l ed rubbe r s with a developed th ree -d imens iona l s t ruc tu re . In this case the dis t r ibut ion function log H(¢) may be r e g a r d e d as continuous and consis t ing of three regions: the region of the max imum, a t rans i t ion region

~0

8

6 - - -

4 -q

Fig. 1.

\ \ \

\

-2 0

IgT 2

Relaxat ion spec t rum (NK rubber ) .

and a pla teau region (Fig. 1). I t should be kept in mind that for b l ack - fi l led rubbe r s the concept of "equi l ibr ium" is a conditional one. The region of the max imum was cons t ruc ted on the bas i s of E' (w) data,* obtained f rom the vibroinsula t ion curves [4]. The end of the t rans i t ion region and the beginning of the pla teau were obtained on the bas i s of c o m p r e s s i v e s t r e s s re laxat ion m e a s u r e m e n t s made with a specia l device that excludes c reep [5]. F r o m these data we calculated values of ET(~), E' (w), and E (T), the components of the continuous spec t rum for NK black-f i l led rubber (see Fig . 1) calcula ted in the f i r s t approx i -

*E' and E" a re the r ea l and imag ina ry components of the complex dynamic modulus E* = ~F(E~) 2 + ( E " ) 2, r e spec t ive ly .

Yaros lav l ' Technological Inst i tute . T rans l a t ed f rom Mekhanika Po l imerov , No. 5, pp. 916-919, Sep tember -Oc tober , 1969. Original a r t i c le submit ted July 24, 1968.

© 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

808

8,~0 6

6./0~

2-10 ~

t06

8"t0: 5.(0~

4.i0

2 8¢0

6.t0

q,fO l l 0 t 2 5 4

Idyn/c m 2 ~ ~

-; i 2 3

Fig. 2. Relaxat ion modulus as a function of log T for homogeneous {a) and inhomogeneous (b) rubbe r s : 1) SKS-30 ARM, x = 0.13; 2) NK, x = 0.10; 3) NK, a f t e r aging, x = 0.30; 4) the s a m e , x = 0.50.

3,0

2,0 - -

"~ ~ ~ - ~ i l ....... 6=0,3~,,

z,o i ~ i

t O t l / ~ " t ~ !

02 04 o6 0 02 03 o6

Fig . 4. Height of re laxa t ion spec t rum (Alog I-l) as a function of the inhomogeneity (x) for va r ious deformat ion r e g i m e s : a) c o m p r e s s i o n [1) e = 0 . 2 ; 2) 0 .3 ; 3) 0.5]; b) tension [1) ~= 2 .0 ; 2) 3.0].

8 algN a

7 -~----o 3 $ -_ Z

6 ~

5 . . . . . ( g r -t 0 ! 2 3 V

,o

8 - - - - I

-33 -27 -2,i -t,8

Fig. 3. Height of re laxa t ion spec t rum as a function of: a) the volume of the rubbe r e l emen t (@ = const; 1) V = 1 cm3; 2) 6 cm3; 3) 31 cm3); b) the shape of the bea r ing s u r - face (1) e l l ipse , 40x20 ram; 2) rec tangle , 30 x 20 ram.

mat ion . The val idi ty of using E' (w) instead of E*(w) is based on the assumpt ion that under our loading conditions E' >> E" [1]. The points for calculat ing E'(w) at w =10-30 Hz w e r e obtained on a h y s t e r e s i s machine at the Central Scientific R e s e a r c h Insti tute of the Minis t ry of Means of Communicat ion [6]. As usual, the inhomogeneity was de te rmined f r o m the d i spers ion of the s t rength c h a r a c - t e r i s t i c s on the bas i s of s ta t i s t ica l m e a s u r e m e n t s . F o r pu rposes of e s t ima t ing the inhomogeneity we used the coeff icient of va r i a t ion (x), that is , the ra t io of the s tandard deviat ion to the mean breaking s t r e s s r e f e r r e d to the c ross sec t ion a t b r e a k

ff x = = . (1)

x

The value of x va r i ed f rom 0.08 to 0.5 depending on the type of r ubbe r o

Specimens of unfilled NK rubbe r subjected to t h e r m o - oxidative aging [7] were l e a s t homogeneous .

As the inves t iga ted p a r a m e t e r of the re laxa t ion s p e c t r u m we took the height Alog H, i .e . , the d i f fe r - ence {along the ordinate axis) between the m a x i m u m and min imum values obtained f r o m the end of the t rans i t ion reg ion and the pla teau r eg ions . This p a r t of the tog H curve was se lec ted because the region of the m a x i m u m was thoroughly inves t iga ted in [3].

In o rde r to obtain the re laxa t ion s p e c t r a we used the approx imate graphic different ia t ion method [2] based on the de te rmina t ion of the f i r s t de r iva t ive of the s t a r t ing function. (In our case it was undes i rable to employ higher approx imat ions s ince the de te rmina t ion of the second and higher der iva t ives would have introduced a cons iderable e r r o r . ) In accordance with the r ecommenda t ions of [8], we s t a r t ed out f rom the s t r e s s re laxa t ion curve . F r o m this curve we cons t ruc ted the dependence of the re laxa t ion modulus E(T), defined as the ra t io of the s t r e s s to the initial s t ra in , on log t ime . In the f i r s t approximat ion the function log H(T) is equal to the slope of the curve obtained a t each point taken with the opposi te sign.

The cour se of the c o m p r e s s i v e and tens i le re laxa t ion curves is e ssen t i a l ly the s a m e . Accordingly, in our general ana lys i s we used r e su l t s obtained in both r e g i m e s .

If i t is a s s u m e d that inhomogeneous rubbe r s a r e c h a r a c t e r i z e d by a b r o a d e r range of s t ruc tu ra l e l ements , then obviously for such rubbe r s the individual va lues of the re laxa t ion t imes (points on the log H curve) should differ more sha rp ly . Hence the values of Alog H and the nature of the re laxa t ion modulus cu rves should be different .

I t is c l ea r f r o m Fig. 2 that in E (T)-log 7 coordinates the nature of the re laxat ion cu rves depends impor tan t ly on the homogenei ty of the rubbe r . Homogeneous rubbe r s a r e cha r ac t e r i z ed by a concave curve (Fig. 2a), whe reas for inhomogeneous rubbe r s x -< 0.13 the cu rva tu re changes sign (see Fig. 2b).

809

TABLE 1 Thus, in the course of the exper iments it was noted that for homogeneous rubbe r s (x - 0.08) log H is a s t ra igh t l ine para l l e l to the axis of a b s c i s s a e or

Shape factor I~//4 dyne/era ~ inclined at a smal l angle to that axis . The nature of the spec t rum does not depend e i ther on the volume (Fig. 3a) of the spec imen or on the shape of the bear ing

0,1 10 su r face (Fig. 3b). The difference Alog H i n c r e a s e s percept ib ly with x, r each ing 0,4 8 0,7 7 3.5-4 decimal o rde r s , whereas for homogeneous rubbe r s it does not exceed 1-1.5

o r d e r s .

The re la t ions p resen ted in Fig. 4, obtained f rom expe r imen t s on the en t i re range of rubbe r s in compres s ion and tension, a t tes t to the val idi ty of the above

r e m a r k s : there is a l i nea r co r re l a t ion between the inhomogeneity, e s t imated f r o m the coefficient of va r ia t ion of the e l a s t i c - s t r e n g t h c h a r a c t e r i s t i c s , and Alog H.

The s p e c t r u m p a r a m e t e r -- the di f ference of the heights of the re laxa t ion t ime dis t r ibut ion - is thus also capable of se rv ing as a m e a s u r e of inhomogeneity. It is c l ea r f r o m Fig. 4 that, as ~ i n c r e a s e s , Alog H fa l l s . Consequently, it may be a s s u m e d that the level of s t r a in (and hence s t r e s s ) affects the inhomoge- nei ty. In tension, as a r e su l t of orientat ion, o rde r ed regions a r e fo rmed , while in compres s ion m i c r o - c r a c k s and m i c r o b r e a k a g e s a r e evidently "healed." Under the influence of mechanica l s t r e s s e s the s t r u c - tura l e lements a r e m o r e uni formly dis t r ibuted.

The re la t ion between the spec t rum p a r a m e t e r s , ~, and the inhomogeneity of the rubbe r spec imen is approx imate ly desc r ibed by an empi r i ca l exp res s ion of the type*

Fig . 5. Relaxat ion spec t r a at t = 70 ° C; 1) SKS-30 ARM, x = 0.13; 2) butyl rubber , x = 0.20; 3) lXTK, a f t e r aging, x = 0.30; 4) x = 0.50.

0,S

20 60 ~0o T°C

AL9H

*

"t

!gv 2

Fig. 6 Fig . 7

F ig . 6. Inhomogenei ty (x) of unfil led vulcanizates based on natura l (1,1a) and ch loroprene (2,2a) rubbe r s as a function of t empe ra tu r e : 1,2) without soaking; la ,2a) a f te r soaking for 20 rain.

Fig. 7. Height of re laxa t ion spec t r a as a function of the volume of the rubbe r e l ements (¢ = const): 1) V = I cm 3, Alog H = 1.0; 2) 6 cm ~, 2.5; 3) 31 cm 3, 4.5.

*In der iv ing Eq. (2) we s t a r t ed out f rom the assumpt ion that the coordinate or igin is the pole of the bundle of s t ra igh t l ines . In fact , owing to measu r ing e r r o r s (10-12%), the ordinate of the pole is s l ightly higher .

810

A lg H= (a-b~)x, (2)

where a and b are constants that characterize the properties of the rubber and the deformation regime.

As the test temperature r ises to 70 ° C, the inhomogeneity, estimated from the value of Alog H, falls (Fig. 5). The observed changes are consistent with the results of previous measurements of the tempera- ture dependence of x estimated from the dispersion of the elastic-strength characterist ics. Using non- equilibrium conditions (curves I, 2 in Fig. 6), owing to the competing influences of orientation (crystall- ization) and rising temperature, beyond 70 ° C the coefficient of variation increases sharply. Conversely, under near-eqmlibrium deformation conditions, owing to the molecular rearrangements that reduce the rubber to the equilibrium state, x falls (curves la, 2a in Fig. 6).

As distinct from homogeneous rubbers, for which log H(T) depends only slightly on geometry, the log H(7) spectra for inhomogeneous rubbers depend strongly on the volume of the relaxing specimen (Fig. 7). The shape of the bearing surface (see Fig. 3b) and the shape factor - ratio of bearing to lateral surface (see table) - do not have much effect on the spectrum parameters .

lo 2. 3. 4. 5.

6o 7.

8.

L I T E R A T U R E C I T E D

J. D. Ferry , Viscoelastic Properties of Polymers, Wiley, New York (1961). A. V. Tobolsky, Properties and Structure of Polymers, Wiley, New York (1960). Yu. V. Zelenev and A. P. Molotkov, VMS, 6~ 1426 (1964). L. E. Vinitskii, G. V. Butakov, L. G. Evseeva, and N. V. Ilyukhin, Mekhan. Polim., No. 6, 935 (1966). V. N. Shlyapina and G. G. Popov, Tr. Vsesoyuzn. Nauchn.-issled. Inst. Zh.-d Transporta, No. 322, 102 (1966). L. B. Belikov and L. E. Vinitskii, Zavod. Lab., 33. 769 (1967). M. A. Tsydzik, L. E. Vinitskii, and S. A. Ivanova, in: Aging and Stabilizing Rubbers [in Russian], Goskhimizdat (1958), p. 87. T. Alfrey, Mechanical Behavior of High Polymers, Interscience, New York-London (1948).

811