FIG. 25. Elastic recovery in the rubber rheometer vs. recovery time. A t the t w o temperatures, 80° and 1 3 5 ° C , the previous shear stresses and rates of shear were, respectively, 1.05 X 106 and .74 X 106 d y n e s / c m .2, and 1.6 and 1.9 s e c .- 1. T h e an-nealing time, ta , is the time between stopping the rotor and releasing it. [From M . M o o n e y , Physics 7, 413 (1936).]
which the deformation was held constant before release. Results with Hevea are shown in Figs. 24 and 25. T h e first figure shows that approximate com-plete recovery at 100° requires a day or longer. T h e second figure shows that the time required for complete recovery is less at a higher temperature.
If the logarithm of the recovery at 320 sec. in Fig. 25 is plotted against ta , the resulting curve is far from linear. Hence decay of recovery b y stress relaxation during annealing is not exponential. T h e initial rate of decay is high in comparison with the later rate. This is not surprising, since stress relaxation in cured elastomers is far from exponential in form.
Since stress relaxation is taking place continuously during any continuous deformation, it is to be expected that subsequent elastic recovery like the, deforming stress during deformation, will increase with the rate of deforma-tion. This has been shown b y Dillon and Johnston,13 w h o measured the
ce R
elastic recovery of an extruded rod as a function of rate of extrusion. S o m e of their results are shown in Fig. 26.
A number of curves of elastic recovery v s . previous shearing torque in the M o o n e y viscometer have been obtained in the author's laboratory. I n m a n y cases the recovery at 1 min. is initially approximately proportional t o the previous shearing torque; but at high torques the points fall a w a y from the linear l o w toward lower recovery values. H o w e v e r , this degree of sim-plicity is not the universal rule. M a n y curves are nonlinear and s h o w various irregularities. T h i s w o r k has n o t reached a stage suitable for a detailed report.
W h i l e the problem of computing absolute viscosity from disk viscometer data has been solved, as in Section I I I , 1, the corresponding problem for elastic recovery in the disk viscometer has n o t been s o l v e d ; and a satis-factory analysis of the recovery irregularities just mentioned is at present impossible.
I 06d / C M2'
F I G . 2 7 . Slip v e l o c i t y , v, at the wall of a capillary extrusion v i s c o m e t e r vs. shear stress at the wall. 1 0 0 ° C . (Parshall and B l a c k )
M a n y elastic recovery measurements with various instruments under arbitrary standard conditions have been m a d e for processing control pur-poses. T h e y will b e discussed later in Section I X .
6. SU R F A C E SL I P
In the paper on laws of extrusion b y M o o n e y and B l a c k39 it was mentioned that some experimental data had been obtained which g a v e consistent evidence of surface slip of raw H e v e a on s m o o t h steel at 100° C . T h e data referred t o , obtained b y C . M . Parshall and anlysed b y S. A . B l a c k , are shown in Fig. 27. T h e test instrument used was an extrusion viscometer, and the m e t h o d of analysis was that depending on the effect of capillary diameter on efflux as developed b y M o o n e y .2 9 T h e points shown in Fig. 27 represent slip velocities c o m p u g e d from smoothed extrusion curves.
T h e slip data are n o t v e r y precise, b u t t w o facts seem t o b e definitely indicated. One is that there is measurable slip. T h e other is that the velocity increases rapidly with shear stress, b u t n o t at an infinite rate. If slip
veloc-1 0 0
T A B L E I
SLIP VE L O C I T I E S , A T 100° C *
Metal Surface^
,, Steel Chromium Cadmium Elastomer Viscosity, M Slip Velocity, cm./sec.
Y108 G R - I 48 0.006 0.016 0.022
Gr-I-18 83 0.004 0.008 0.007
H e v e a 107
—
0.005 0.26X-672 G R - S 47 0.020 0.082 0.002
X-603 G R - S 53 0.014
—
0.72G R - S 1700 126 0.008 0.014 0.17
G R - S ( A ) 124 0.22 0.33 0.47
G R - S ( B ) 152 0.20 0.52 0.87
G R - S ( C ) 166 0.20 0.22 0.22
* Blank spaces correspond t o indicated negative velocities. A v e l o c i t y less than 0.01 c m . / s e c . is of doubtful significance.
t T h e surface shearing stress is roughly 2.3 X 104 M d y n e s / c m .2, M being the vis-cosity in M o o n e y units.
ity, v, is represented b y the power law of equation ( 1 3 ) , the slope in Fig. 2 7 shows that ρ = 7 .
T h e implication is that slip m a y b e similar in its mechanism t o viscous flow; and, like viscous flow, m a y be a thermally activated process.
S o m e data obtained with a s m o o t h and a standard serrated rotor in the M o o n e y viscometer, published b y D e c k e r and R o t h ,41 show in certain instances a lower M o o n e y torque reading with the s m o o t h rotor. T h e difference can only b e attributed t o slippage on the s m o o t h rotor. T h e c o m p u t a -tion of absolute slip velocities b y equa-tions ( 1 4 ) or ( 1 6 ) requires measure-ments with b o t h rotors over a range of rotor speeds. Since the published data are limited t o a single speed ( 2 r.p.m.) the computations in this case cannot be carried out exactly; b u t it is possible t o c o m p u t e approximately single slip values if w e assume values for η and p. W i t h η = ρ = 4 , equa-tions ( 1 4 ) give the slip velocities and shear stresses shown in T a b l e I .
T h e dependence of slip v e l o c i t y on the chemical nature of the surface indicates that slip is dependent t o some extent on h o w well the rubber wets the surface.
7 . RH E O L O G I C A L FL O W UN I T S
W h e n a material containing a continuous, three-dimensional thixotropic structure is subjected t o continuous deformation, the structure must be broken u p . A t first, the broken units m a y b e v e r y large in comparison with molecular dimensions and contain m a n y molecules still held together b y
forces of appreciable strength. If the shear strength of such units is greater then the shear stresses that d e v e l o p between the units as they roll and slide over each other, the units m a y persist indefinitely while the material is con-tinuously being sheared. There are reasons for thinking that such units exist and persist in raw elastomers subjected t o continuous deformation.
I n order t o test this hypothesis a new test procedure with the M o o n e y viscometer was developed b y M o o n e y and W o l s t e n h o l m e .62 A s m o o t h rotor was coated with a thin, dried film of raw elastomer containing a rubber-soluble d y e in high concentration. A preformed elastomer sample was then placed around the rotor and, after warm-up, was subjected t o shear in the viscometer.
If the theory is correct, the thixotropic units, or ''rheological units," will roll continuously and will transfer the die across the shear planes as inking rollers on a printing-press transfer ink. T h e v e l o c i t y of the color front is proportional t o the radial distance from the rotor axis; and, a, the angle of the resulting conical color front is
where D is the mean diameter of the rheological units, Ν is the number of rotor revolutions, and h is the rotor-stator clearance.
After an appropriate shearing time the sample is r e m o v e d , chilled, and sectioned. T h e r e are disturbing factors in the test and valid results are n o t always obtained; b u t usually the expected conical color front is observed.
S o m e mean diameters of rheological units as found b y this m e t h o d are given in T a b l e I I . These measurements are rough, as is the theory of the measurement; b u t presumably the data are correct in order of magnitude.
I n its present f o r m the m e t h o d is applicable only t o transparent or white samples.
T h e various experimental results reviewed in the preceding section reveal a number of effects which could b e the subject of theoretical analysis: the dependence of viscosity on temperature, shear stress and nature of the test material; the magnitude of the elastic recovery and the details of the re-covery-time curve ; the rate and extent of b r e a k d o w n and build-up of thixo-tropic structure. H o w e v e r , only t w o of these effects h a v e so far been suc-cessfully treated theoretically. These are the effects of temperature and of shear stress on rate of shear or the viscosity.
I t was mentioned a b o v e that M o o n e y ' s2 measurements on Pale Crepe shown in Fig. 12 h a v e been interpreted b y S m a l l w o o d48 in terms of the
DN (20)
V I I . T h e o r y o f the Fluidity o f Elastomers
6 2 M . M o o n e y and W . E . W o l s t e n h o l m e , Appl. Phys. 25, 1098 (1955).
T A B L E I I deformation of the sheared elastomer and i m p r o v e d the fit obtained with the experimental data published b y several different authors.
These theories treat the shearing material as being h o m o g e n e o u s and
In the revised theory the macroscopic viscosity is assumed t o result from the friction of the rheological units as they slide over each other. T h e fric-tional force is treated as a consequence of temporary molecular attachments across the boundaries of the units. F r o m the m o m e n t of attachment until release b y thermal activation, the stress on the attachment builds up at a rate which is proportional t o the relative sliding v e l o c i t y of the t w o rheo-logical units and t o the d y n a m i c elastic modulus of the units and is inversely proportional t o the unit diameter and t o the number of cross-attachments per unit area. T h e explicit introduction of the elasticity is something added
6 3 R . S. Spencer, Polymer Sei. 5, 591 (1950).
6 4 M . M o o n e y , Appl. Phys. 27, 691 (1956).
t o the E y r i n g theory of activated flow; b u t it is in a c c o r d a n c e with the M a x w e l l picture of flow as a continuous process of relaxation of continu-ously renewed elastic stress.
T h e basic quantitative postulate of the theory deals with the rate of breakage of the cross-surface attachments as the stress on the attachments increases from zero stress at the time they are formed. T h e assumed law, suggested b y M o o n e y et αΖ.65 is:
dH - - nn k_ TR e~e c o s n ^ (21) ElkT cosh *L (21) where η is the number of attachments left unbroken out of a group which
formed at time t = 0, k is B o l t z m a n n ' s constant, Τ is temperature, h is P l a n c k ' s constant, Ε is the activation energy, / is the force o n the attach-ment, and λ is the distance of molecular force. R is a rate factor, usually expressed in terms of an e n t r o p y of activation. It is the probability that rupture will actually occur w h e n the necessary thermal energy is supplied at the attachment.
Before we undertake t o integrate the a b o v e equation we must take c o g -nizance of the fact that the force on a given attachment varies with time, being zero at the m o m e n t the attachment is formed and increasing linearly with time thereafter. T h e rate of increase is derived from a bit of theory in which the molecular group i n v o l v e d in a local cross-surface a t t a c h m e n t is treated as t w o hemispheres, each lying within its respective rheological unit. D u e t o the sliding relative m o v e m e n t of the t w o units, each hemi-sphere is undergoing a continuously increasing elastic displacement with respect t o its unit. B y analogy with Stoke's l a w6 6 for the force o n a c o n -tinuously m o v i n g sphere in a viscous m e d i u m , the force, / o n a hemisphere displaced momentarily a distance δ in an elastic m e d i u m is 3 π α β δ , where a is the hemisphere radius, and G is the elastic modulus of the m e d i u m , the rheological unit. B y classical h y d r o d y n a m i c s the rotational v e l o c i t y of a rheological unit is g/2.® F r o m these facts it can be shown that, at an inter-face parallel t o the shear planes, the disruptive force o n an a t t a c h m e n t grows according t o the equation
/ = AWAGDGT/4, (22)
where D is the diameter of the rheological units.
W i t h / substituted into equation ( 2 1 ) , integration is possible; and an
6 5 M . M o o n e y , W . E . W o l s t e n h o l m e , and D . S. Villars, / . Appl. Phys. 15,324 (1944).
6 6 H . L a m b , " H y d r o d y n a m i c s , " 6th e d . , p . 598. D o v e r P u b l i c a t i o n s , N e w Y o r k , 1945.
6 7 H . L a m b , " H y d r o d y n a m i c s , " 6th e d . , p . 31. D o v e r P u b l i c a t i o n s , N e w Y o r k , 1945.
explicit law of rate of attachment rupture is obtained. If it is assumed that N, the number of attachments per unit area of interface, is constant in a state of constant rate of shear, then it is possible t o obtain expressions for the mean life of an attachment and for the mean shear stress o n an inter-face. T h e equations are
τ = NkTp/\
ta = AkTh/SirXagDG
τ Γ ( s i n h î A , W
Ρ = h/h
_ SirahXDG EikT
a " mT*R e g
where r is the shear stress and ta is the mean life of an attachment.
T h e theoretical law of flow is the relationship between g and r expressed implicitly b y these equations through the parameter a. T h e equations s h o w that g is proportional t o a, and r is proportional t o ρ which is a function of a. Hence a logarithmic p l o t of experimental data for g versus τ should have the same form as a plotted against ρ on the same logarithmic coordinates.
If the agreement in form is satisfactory, any chosen pair of values for g and r determine a corresponding pair of values of a and p.
In Fig. 28 the solid lines show the fit obtained in applying this theory t o M o o n e y ' s2 data on Pale Crepe H e v e a rubber. T h e d o t t e d lines give the best fit obtainable with Eyring's non-linear equation as used b y Small-w o o d .4 8
These t w o theories have been tested also with the data on H e v e a o b -tained b y Saunders and T r e l o a r ,44 the results being similar t o those shown in Fig. 28. S p e n c e r63 shows no fitted curves; but he claims i m p r o v e m e n t over the Smallwood equation. Presumably, therefore, his fit with experi-m e n t is as g o o d as is obtained in Fig. 28 b y experi-means of the equation ( 2 3 ) .
I t appears from the results referred to that in the experimental range covered b y the data of Fig. 28 or b y the data of Saunders and Treloar, it is possible t o obtain g o o d agreement with the experimental data if the Eyring non-linear theory is slightly modified in form, one or t w o additional adjustable parameters being introduced into the equation. T h e choice b e -tween Spencer's theory and the theory presented here therefore seems t o depend for the present on the existence or non-existence of densely packed rheological units.
ο
/J
Γ t
ο" 2
Ο «ο ο ο <\ι "fr
.001 .01 .1 I 10 100 f. 3 t C " l
F I G. 28. T h e o r e t i c a l curves for shear stress, r, vs. rate of shear, g, fitted t o ex-perimental d a t a b y M o o n e y on pale crepe milled 5 min. Full curves, M o o n e y - E y r i n g t h e o r y . D a s h e d curves, S m a l l w o o d - E y r i n g t h e o r y .
F o r purposes of interpreting the experimental data in terms of the present theory, the values of α, λ, D and G must be assumed or estimated from measurements of l o w precision or of uncertain applicability. T h e values estimated for a and λ are a = 5 X 1 0 ~18 c m and λ = 1 0 ~18 c m . On the basis of w o r k of M o o n e y , and W o l s t e n h o l m e ,62 values of D were assumed t o be 19 μ for M o o n e y ' s H e v e a sample and 10 μ for Saunders and Treloar's sample. T h e value assumed for (?, the same for b o t h rubbers, is M o o n e y ' s value c o m p u t e d from the frequency of free oscillation in the rubber rheom-eter.2
T h e results of the analysis of the flow curves are given in T a b l e I I I . These data show that a major difference between the t w o samples of rubber is in the values of E, the activation energy, the softer sample of Saunders and Treloar having the smaller value. T h e values of ta , the mean life of an attachment, are also appreciably different; but this difference m i g h t be anticipated from the difference in jEJ-values. T h e values of N, the number of attachments per unit area, s h o w that only a v e r y small fraction of the isoprene chain segments in an interface are i n v o l v e d in a cross-surface at-tachment at any one time.
T h e values of R are v e r y small and indicate a large entropy term in the release process. C o n c e i v a b l y there is a caging effect, such that t w o locked chain segments, after temporary separation b y local v o l u m e expansion, m a y return t o their same locked position if, while they are separated, the local chain molecules are t o o slow in their response t o the displacing
T A B L E I I I Ν the number of attachments per unit interfacial area, ta the mean life of an attach-m e n t , R the rate c o n s t a n t .
mechanical stress. A c c o r d i n g t o this v i e w R will be smaller, the larger the p o l y m e r molecular weight and the m o r e c o m p l e x the p o l y m e r structure.
T h e theory presented a b o v e has been found t o fit quite well a large number of flow curves of elastomers, and also of plastics a b o v e their second order transition temperature; b u t quite often the experimental curves differ in shape and fail t o agree with the theory at either high or l o w rates in-fluence considerably the flow behavior. T h e direction of the effect can b e predicted in m o s t cases with little d o u b t . W h e n the breakup of thixotropic structure begins, the broken pieces, besides being large, are presumably quite irregular in shape. T h e assumption that the number of cross-surface attachments per unit area is constant is an o b v i o u s oversimplification. In a raw carbon black stock the effective d y n a m i c modulus of the rheological units p r o b a b l y decreases temporarily with working, as does the d y n a m i c modulus of a curved carbon black stock. T h e d y n a m i c viscosity of the rheological units varies with the p o l y m e r , mean molecular weight, molecular weight distribution, and with impurities of l o w molecular weight. T h e vis-cosity of the units p r o b a b l y affects the value of the rate constant, R, in the manner which has been suggested in the discussion of the caging effect.
T h e nonlinear E y r i n g theory in all of its published variations, including the present theory, expresses the stress effect as an exponential function of which the argument is linear in the stress. Higher order terms should b e included t o describe the flow response of rubbers t o stresses sufficiently high t o cause rupture or tearing at r o o m temperature. Such terms will surely b e required t o explain, for example, an upper stress limit in continuous shear at 100° C .
If the rate of shear is progressively lowered, cross-surface diffusion and entanglement will b e c o m e m o r e and m o r e i m p o r t a n t ; and in the limit the b o u n d a r y surfaces of the rheological units will disappear and the material will b e c o m e h o m o g e n e o u s , with high thixotropic structure. N e w t o n i a n flow at a high viscosity could c o n c e i v a b l y result, as described in the linear E y r i n g theory and as exemplified in F o x and F l o r y ' s53 and V a n Heide and W i l -lisms54 samples of polisobutylene, and as exemplified approximately in s o m e of the l o w viscosity H e v e a samples tested with the Williams plastometer.52 T h e situation in H e v e a is complicated b y the possibility of chemical cross-linking b y free radicals, as pointed out b y Pike and W a t s o n68 and b y W h o r l o w .45
T h e various factors that h a v e been discussed here are p r o b a b l y adequate t o explain most of the peculiarities in the rheological curves that h a v e been presented a b o v e ; b u t the reduction of these factors t o quantitative ex-pression for comparison with the data remains a formidable task.
V I I I . Processing 1. GE N E R A L RE M A R K S
T h e principle operations required in processing raw rubbers were briefly referred t o in Section I. M o s t of the processing difficulties that are encoun-tered and require control tests result from t o o high a molecular weight of the elastomer. B y b r e a k d o w n methods w h i c h h a v e been alluded t o , the mean molecular weight can b e reduced t o any desired v a l u e ; b u t there are limitations on the acceptable degree of b r e a k d o w n . B r e a k d o w n is a costly operation, and excessive b r e a k d o w n injures the physical properties of the cured stock. I t w o u l d seem logical t o polymerize the elastomer t o a lower molecular weight t o begin with; b u t this m e t h o d also results in an inferior cured stock. Polymerization t o a high molecular weight followed b y m e -chanical plastication gives better quality, p r o b a b l y because it p r o d u c e s a m o r e favorable molecular weight distribution.
A n alternative m e t h o d of softening an elastomer is t o add a suitable oil.
This method has advantages and is used; but here also quality requirements impose limitations on the a m o u n t of oil that can b e e m p l o y e d .
A s a consequence of the conflicting needs for a high and for a l o w m o l e c u
-6 8 M . P i k e and W . F . W a t s o n , J. Polymer Sei. 9, 229 (1952).
lar weight, there must be a compromise, and this means that most processing is carried out under conditions which are on the verge of being impossible;
and this means, in turn, that processing controls are required in a rubber factory and are coming into more and more general use as more and better control tests are developed.
T h e following are the principal difficulties encountered in processing.
T h e first three result from excessively high viscosity of the crude elastomer or mixed stock.
(1) Excessive power consumption, overloading the power supply;
( 2 ) Excessive stresses in the processing machinery, possible breakage;
( 3 ) Excessive temperature rise in the stock because of viscous conversion of mechanical energy into heat, possible scorching of the stock.
(4) L o w extrusion rate at a given screw speed.
(5) Longitudinal shrinkage of calendered or extruded stock.
( 6 ) Roughness and distortion of calendered or extruded stock.
(7) Torn edges of extruded stock of sharp angular section.
(8) L o w tack, adhesion failure in building operations.
Processing behavior in a number of respects can be fairly well predicted b y certain laboratory tests. Viscosity and elastic recovery are the tests most frequently used, for they seem to have the greatest predictive value.
2 . BR E A D K D O W N A N D MI X I N G
Experimental work b y C o t t o n69 and Busse70 on the breakdown of Hevea led to the belief that the reduction in molecular weight and the softening of the rubber was initially and primarily due to oxidation. However, later work, beginning with Staudinger and Heuer71 showed that chain molecules in flowing systems could be ruptured b y mechanical stress, free radicals being produced in the process. Subsequent chemical action might include more or less oxidation, depending on the temperature and the availablity of oxygen; or, failing oxidation, the free radicals could react with other poly-mer chains and thus maintain the mean molecular weight unchanged. T h e illuminating study b y Pike and W a t s o n68 showed that conditions during mastication variously affected the molecular weight distribution, chain branching and cross-linking; and furthermore that these properties of the masticated rubber affected, in turn, the measured relationship between M o o n e y viscosity and the intrinsic viscosity in solution. T h e increase in the viscosity of G R - S b y gel formation can take place under the action of heat alone, according to Hägen;72 and some additional chemical mechanism therefore seems to be involved in this case.
6 9 F. H . Cotton, Trans. Inst. Rubber Ind. 6, 487 (1931).
7 0 W . F. Busse, Ind. Eng. Chem. 20, 140 (1932).
7 1 H . Staudinger and W . Heuer, Ber. 67, 1159 (1934).
7 2 H . Hagen, Kautschuk 14, 203 (1938); India Rubber World 108, 45 (1943).
K n o w l e d g e of the effects of fillers and of plasticizers or softeners on the rheology of elastomers is largely empirical. T h e relative effectiveness of m a n y plasticizers has been examined b y B u r b r i d g e73 and J a c o b s .74 T h e viscosity increase and other rheological effects produced b y various fillers have been studied b y v a n R o s s e m and H o e k s t r a ,75 Mullins and W h o r l o w ,7 6 and D r o g i n et al.77 T h e range in the viscosity effects of different fillers is enormous. M a x i m u m viscosity increase, accompanied always b y high thixotropy, is p r o d u c e d b y s o m e of the carbon blacks of smaller particle size. A c c o r d i n g t o K r u s e ,78 the carbon black is not uniformly distributed but is concentrated in clouds of individually separated particles. Such non-uniform distribution m a y be explained as the result of the exclusion of the
K n o w l e d g e of the effects of fillers and of plasticizers or softeners on the rheology of elastomers is largely empirical. T h e relative effectiveness of m a n y plasticizers has been examined b y B u r b r i d g e73 and J a c o b s .74 T h e viscosity increase and other rheological effects produced b y various fillers have been studied b y v a n R o s s e m and H o e k s t r a ,75 Mullins and W h o r l o w ,7 6 and D r o g i n et al.77 T h e range in the viscosity effects of different fillers is enormous. M a x i m u m viscosity increase, accompanied always b y high thixotropy, is p r o d u c e d b y s o m e of the carbon blacks of smaller particle size. A c c o r d i n g t o K r u s e ,78 the carbon black is not uniformly distributed but is concentrated in clouds of individually separated particles. Such non-uniform distribution m a y be explained as the result of the exclusion of the