Bryce Maxwell, Polymer Materials Program, Department of Chemical Engineering, Princeton University, Princeton, New Jersey

Dynamic Properties of Polymer Melts in an Orthogonal Rheometer

Introduction he time dependent mechanical properties of poly- T meric materials have been studied using many tech-

niques; creep, stress relaxation and dynamic loading. Dynamic oscillatory methods have been used to in- vestigate materials at use conditions but only recently has this technique been applied extensively to the study of polymers in the melt state. The purpose of this paper is to describe a method of studying polymer materials by a technique that involves oscillatory dynamic load- ing and to analyze the resulting information.

Dynamic loadings may be applied to polymer melts by many different methods. The orthogonal rheometer method represents a relatively simple system for ap- plying a dynamic strain over a wide range of frequency and strain magnitude and the measurement of the re- sulting stress in the polymer melt both in-phase with the applied strain and out-of-phase with the applied strain as well as the normal force resulting from these stresses.

Experimental Method The orthogonal rheometer has been described previ-

ously (1) and analyzed in detail by Blyler (2). The following is a brief description of the apparatus suf- ficient for the purpose of analysis of the experimental results.

This work was supported in part by a grant from the Chevron Research Corporation.

Figure 1 depicts schematically the plan and side views of the shearing device. A preformed disc speci- men, A, is placed between two metal discs, B and E. The lower disc, B, is mounted on a shaft D and sup- ported in bearings, C. The upper disc, E, of radius r is also mounted in bearings, G. The axis of bearings C and G can be displaced with respect to each other by a controlled amount, “a”. The lower shaft, D, is driven at controlled angular velocity O. The upper disc rotates freely in its bearings, G.

If the displacement, “a”, is zero, the polymer sample acts simply as a coupling agent between the lower and upper plates and is not subiected to any shearing stress except that required to turn the upper plate in its bearings. When this coupling stress is large compared to the viscosity of the specimen the upper plate may also be positively driven at the frequency w to eliminate this stress on the specimen. When the axis of the lower plate and the upper plate are dkplaced by some dis- tance, “a”, the entire specimen between the plates is subjected to a shear through a circular path of radius, “a”. If we designate the thickness of the specimen as, t, then during each revolution all elements of the speci- men are subjected to a shear, 7, per revolution of mag- nitude

y = 2 r a h

In order to visualize this shearing action consider the action one uses when one hand polishes a surface with a circular motion. The surface being polished is station- ary and the polishing cloth is moved in a circular path of radius “a”. The specimen may be considered as the

POLYMER ENGINEERING AND SCIENCE, lULY, 1967 145

y = 2 x o a / t

In the geometry shown in Figure 1, the same shear- ing is taking place but because in this case the plates and specimen are rotating the forces required to cause this shear are stationary in space. Fx is the force in the direotion of shear while F y is the force in the plane of shear at 90” to F x , and Fn is the force normal to the plane of shear.

If the specimen were a Newtonian liquid the force in the direction of shear; Fx, would be proportional to viscosity. If the specimen were an ideal elastic solid the force, Fy, would be proportional to the elastic shear modulus. For a viscoelastic material both a viscous force and an elastic force would be found in Fx and F y . If one considers a unit volume e!ement of the specimen as it rotates around with the plates one sees that F x and F y are rotating with respect to the coordinates of the unit volume. Therefore the applied strain may be considered as being oscillatory and the results analyzed as a complex modulus, G’, composed of an in-phase component G’ and an out-of-phase component G“.

G’ = 7./a/t

PLAN

I , L

/ - G

W“ mb-a

SIDE VIEW

Figure 1. Schematic drawing of the orthogonal rheometer.

- t 1

Figure 2. The viscous force 0s. displacement “a” as expressed in micrometer settings for a Newtonian liquid of 96,500 centi- poises.

polishing wax between the surface and the cloth. I t is apparent then that. the specimen or wax can be sub- jected to various rates of shear either by holding the circular velocity or w constant and varying the radius “a” or by holding the radius “a” constant and varying

the angular velocity. The shear rate, y, is therefore

where G” = .,/a/t 7% = Fq/irr” 7* = Fx/ nr’

The experimental procedure found most suitable for both Newtonian liquids and viscoelastic materials is as follows:

The axis of the upper and lower plates are alligned, that is “a” is set at zero. The specimen is introduced between the plates and the apparatus is brought up to the desired test temperature and allowed to reach tem- perature equilibrium. The rotation of the lower plate is then started at the desired frequency, O. A small dis- placement “a” is then applied and the forces F x and F y recorded. Then an additional increment of displace- ment is added and the forces again recorded. This pro- cedure is repeated up to the strain magnitude of in- terest to the experiment. The resulting data gives the forces as a function of strain magnitude at constant fre- quency. The same procedure is then repeated at other frequencies, W, of interest.

The results of such a series of tests on a Newtonian liquid of 96,500 centipoise viscosity is shown in Figure 2. Although the apparatus was designed for polymer melts of much higher viscosity the expected straight line relationship was found for each frequency (a = 0.009 c.P.s., b, = 0.037 c.P.s., c = 0.147 c.p.s., etc. to e = 1.42 c.P.s.). The term y-micrometer refers to the displacement “a” as measured from the point of Fx = 0. It should be noted that all the lines go through a single origin in this case.

Since F x is the force in the direction of shear, the viscosity, 11, of the specimen may be determined.

7x ”= - Y

In this case, for example, a viscosity of 96,200 centi- poises was found from the data of line A. Other con- firmations (1) of the method have been found for poly- mer melts in the Newtonian range of frequencies.

146 POLYMER ENGINEERING AND SCIENCE, JULY, 1967

Results with a Polymer Melt Polymer melts are often described by a shear rate

dependent apparent viscosity (3). At low values of shear rate, a high viscosity Newtonian behavior may be found. As shear rate is increased the apparent viscosity decreases often to very low relative value in the non- Newtonian range of shear rates. Since it is possible to vary the shear rate by two different methods in the orthogonal rheometer, that is by holding “a” constant and varying w or by holding o constant and varying “a”, it is possible to investigate whether or not viscosity is really explicitly related to shear rate.

For this study a branched polyethylene of the follow- ing characteristics was used:

Weight average molecular weight = 550,000 Number average molecular weight = 31,400 Degree of branching = 2.0 CH, per 100 C Optical melting point = 112°C Melt index = 0.16

The force in the direction of shear, Fx, as a function of displacement “a” expressed a3 micrometer settings for this material at 150°C is shown in Figure 3. The range of frequencies is from 0.00237 to 1.42 cycles per sec- ond. The following points should be noted:

1. There is a linear relationship between force and displacement, that is, between stress and strain, at con- stant frequency. In other words the material exhibits linear viscoelastic response.

2. The slopes of the constant frequency lines are pro- portional to the frequency dependent out-of-phase shear modulus, G”(o) .

3. In contrast to the case of a Newtonian liquid of low viscosity (Figure 2), the constant frequency lines do not meet at a single origin but rather the origin shifts with increasing frequency. This indicates that the forces exerted by the sheared viscoelastic material dis- tort the force measuring apparatus and indicates that it is necessary to take measurements of Fx at various values of strain, a/t, at each frequency not only to check the linear viscoelasticity but also to determine the slope since the origin is unknown.

From the data of Figure 3 the out-of-phase modulus G”(o) can be determined. This is shown in Figure 4 as a function of frequency. Over the approximately three decades of frequency studied the out-of-phase modulus increases by a factor of four with increasing frequency.

As discussed previously it is also possible to deter- mine the viscosity, 9, from the data of Figure 3 . Since for each frequency there is a linear relationship between force and deformation, there will also be a linear re- lationship between shear stress and shear rate. In other words the response is “Newtonian” at constant fre-

Some of the viscosities so determined are plotted on log scales in Figure 5 as a function of shear rate. It is apparent that there is no explicit relationship between the viscosity so determined and shear rate. On the other hand, if we plot the viscosity, q ( o ) , vs. frequency we obtain the relationship shown in Figure 6. The follow- ing points should be noted:

POLYMER ENGINEERING AND SCIENCE, JULY, 1967

quency, 7 ( 0 ) .

I

Figure 3 . Fx as a function of displacement in the Y direction for branched polyethylene at 150°C.

c t 810

0.004 0.01 0.03 - 1 3 4 SLC.

Figure 5. Viscosity vs. shear rate.

147

I . 1

0 . s.p..

Figure 6. Viscosity us. log frequency.

-. . . y. rnieromchr

Figure 7. In-phase force us. displacement for various fre- quencies.

1. The viscosity shows a smooth, monotonic decrease with frequency.

2. Over only three decades of frequency the viscosity decreases from two million poises to less than fifteen thousand poises.

3. The frequency dependent viscosity, ~ ( o ) , is not dependent on shear rate. For example, at a frequency of approximately 0.01 C.P.S. a value 1,146,000 poises plus or minus 1% was obtained over a three fold range of shear rates. All other points on the curve of Figure 6 also represent a wide range of shear rates.

In contrast to other more common liquids, polymer melts exhibit pronounced e1as:ic behavior. The in-phase force, Fy, vs. displacement is shown in Figure 7 for various frequencies, As previously shown by Mooney (4) working with rubber, this response is also linear viscoelastic. The slopes of these straight lines determine the frequency dependent in-phase modulus, G‘(w), as shown in Figure 8, We see the expected increase in in-phase modulus with increasing frequency. In other words the polymer melt exhibits an increasing elastic behavior at shorter and shorter time scales of loading. This is in some ways surprising when considered in conjunction with the very pronounced decrease in vis- cosity with decreased time scale of loading shown in Figure 6. One might expect that the more elastic the response of the material the more resistant it would be to flow. A possible explanation may be found by con- sidering the changes in internal structural response mechanisms as a function of frequency and strain mag- nitude.

Stress Relaxation With the orthogonal rheometer it is possible to de-

termine the stress re!axation behavior in the polymer melt after steady state shearing. The test is started in the same manner as previously described to obtain either the in-phase force, F y , or the out-of-phase force, Fx. Once steady state conditions have been reached the ro- tation of the lower plate is stopped and the relaxing forces on the upper plate observed. The reduced in- phase relaxation modulus G’( t) /G’o is shown as a func- tion of log time in Figure 9. Time zero is the time of stopping the rotation of the lower plate. The upper curve, A, shows the reduced relaxation modulus after subjecting the polymer melt to shearing at a rotational velocity of 0.00468 C.P.S. or strain rate of 0.0066 re- ciprocal seconds. The lower curve, G, results after shear-

w. c.ps.

Figure 8. In-phase modulus as a function of frequency.

148

Figure 9. Reduced in-phase relaration modulus as a function of log time.

POLYMER ENGlNEERlNG AND SCIENCE, lULY, 1967

ing at a rotational velocity of 1.42 C.P.S. or a straining rate of 1.43 reciprocal seconds. The sequence of curves B, C, D, E, F represent the relaxation behavior as the frequency and shear rate are increased from that of A to that of G.

It is apparent that the relaxation behavior is depend- ent on the previous shearing history. The relaxation is more rapid after shearing at high rotational velocities or high shear rates. In other words the shear experience prior to the relaxation has changed the structure in such a manner that the stress relaxation can take place more rapidly. The higher the frequency or shear rate, the more rapid the relaxation. It may be concluded that shear changes the relaxation behavior and that a poly- mer melt does not have a shgle explicit relaxation spec- trum except when relaxed from an undisturbed or placid state.

Similar behavior is also observed in the stress relaxa- tion of the out-of-phase or viscous force. Curves A, B and C of Figure 10 show the stress relaxation of the out-of-phase modulus, G” ( t ) /G“,, after steady state straining at a frequency of 1.42 C.P.S. Curve A results after a straining rate of 0.323 sec-I. Curves B and C result after straining rates of 0.95 and 1.69 respectively. Although each of these three relaxation curves was de- termined after shearing at the same frequency they show a marked trend toward more rapid stress relaxa- tion as the straining rate is increased by increasing the displacement, “a”. In other words the higher shearing rate even at constant frequency changes the molecular distortional response mechanisms of the polymer melt in such a manner that the stress in direction of flow can relax more rapidly. This seems consistant with the common observation that polymer melt flow is non- Newtonian and shows a decrease of viscosity with shear rate. Since the shearing process changes structure in such a manner that it can relax more rapidly in the di- rection of flow, it would be expected that the higher

I 1 I 1.0 r r d r

Figure 10. Stress relaxation of out-of-phase modulus at vart- ous strain magnitudes and in-phase modulus at one compar- able strain magnitude.

POLYMER ENGINEERING AND SCIENCE, JULY, 1967

the shear rate the more rapid the relaxation, thereby permitting flow to take place with less applied stress, i.e., lower viscosity.

In view of the ability of this experimental method

to vary shear rate, i, by two different methods: by changing displacement, “a”, at constant frequency, or by varying frequency at constant displacement, “a”, it is possible to separate the variables o and y to study

whether or not the shear rate, y, is the controlling fac- tor in the structural changes which determine the stress relaxation. Several examples of constant shear rate, +, with various values of frequency, W, and displacement, “a”, were analyzed and it was found that the stress re- laxation of G”(t ) /G”, was not explicitly a function of shear rate. It would appear that both frequency and strain, d t , affect the structural response mechanisms.

One point remains to be considered. Does the stress in the direction of shear, 7.. relax with the same func- tionality with time as the in-phase stress, T”? Stress re- laxation curve D of Figure 10 presents the time de- pendent response of G’(t) /G’o after steady state strain- ing at a frequency of 1.42 C.P.S. and a displacement a / t of 0.160. Curve C of the same figure is the relaxa- tion of G” ( t ) /G”, at approximately the same conditions of previous shearing. The stress relaxation curves ap- pear to be quite similar. This seems to indicate that the same relaxation mechanisms are involved in the structural distortional mechanisms which give rise to the viscoelastic behavior of both the in-phase and out- of-phase responses.

Conclusions The results of this study may be summarized as

follows:

1. In the orthogonal rheometer a linear relationship is found between shear stress and shear strain, both in- phase and out-of-phase with the direction of shearing. Thk easily permits the de.ermination of whether or not a material exhibits linear visse!astic response.

2. For the out-of-phase response there is a linear re- lationship between shear stress and shear rate thus per- mitting the determination of a “Newtonian” frequency dependent viscosity, 7 ( W) .

3. The frequency dependent viscosity is not a func- tion of shear rate or shear magnitude but rather is only a function of frequency.

4. Shear deformation or flow changes the molecular response mechanisms in the polymer melt producing a structure that can relax more rapidly. The higher the frequency, the more rapid the relaxation. The higher the strain magnitude the more rapid the relaxation.

5. After high frequency and high magnitude shear the stress relaxation in-phase and out-of-phase appear to be quite similar.

References 1. B. Maxwell, and R. P. Chartoff, Transactions of the Society

of Rheology 9: 1, ( 1965), p. 41. 2. L. L. Blyler, Jr., ans S . J. Kurtz, (in press). 3. See, for example, “Polymer Processing” by J. M. McKelvey,

John Wiley and Sons, Inc., New York, 1962, page 36. 4. M. J. Mooney, J. Appl. Phys. 35, (1964), p. 23.

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