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F. H. GARNER, A. H. NISSAN,' and JACK WALKER2 The University, Edgbaston, Birmingham 15, England
Experimental Investigations of
I Stresses in Sheared Viscoelastic After 15 years of research, there is still no complete explanation of the Weissenberg phenomenon. This article explains some of the difficulties
ONE
objective of a wartime study of flamethrowers was to improve the cohesion of ejected fuels. Gasolines thickened by aluminum soaps met this requirement but showed marked elasticity and peculiar floiv properties (2, 4, 8, 9 ) . The flow peculiarities were attributed to normal stresses-i.e., pressure acting across or normal to the shear planes (7, 8 ) . Because many polymer concentrates and melts sholv large normal stress effects, these phenomena are well known in the chemical industry. However, no completely satisfactory theory has been evolved. The present work was undertaken to study the influence of physicochemical variables on normal stresses in sheared polyisobutene solutions.
Apparatus n'ormal stresses in fluids sheared by a rotating rod cause them to climb u p the rod. If the rod is enclosed in a sleeve, the sheared fluid will flow down and outward from the annulus to produce a measurable fluid head in an ad1 Present address, Rensselaer Polytechnic Institute, Troy. N. Y . Present address, Rrsearch Department, Imperial Oil Ltd., Sarnia, Ontario, Canada
joining reservoir. The apparatus shoivn operates on this principle. ,4n accurately turned Duralumin cylinder, 1.400 f 0.001 cm. in diameter, was free to rotate inside and exactly concentric with a precision-bore glass cylinder, 3.00 i 0.01 cm. in diameter. Liquid sheared in the annular space could flow, into or out of a vertical manometer connected radially with the annulus. The fluid reservoir, concentric cylinder tubes, and manometer were immersed in a bath \\-hose temperature could be controlled to d ~ 0 . 0 1C.~ Angular velocity of the rotor \vas determined electrically. The head developed-i.e., the difference in levels of the annular meniscus and the manometer meniscusivas measured by a vertical-traveling telescope accurate to AO.001 cm. Measured heads were usually not greater than 1.0 cm. and were independent of direction of rotation. Viscosity was measured by observing the rate of fall of small steel spheres through the fluids. Rates of shear were very low and. in this range, the fluids were substantiall!- Xewtonian.
Materials Polyisobutene of three hreight levels ivas used. h'oinen-
clatiire
HMW 1 HMW 2 LMW
%ate Rubbery solid Rubbery solid Viscous liquid
molecular Mol. TVt . 11.4 X 101 9.7 X lo4 13.8 X 103
Molecular weights were determined by mcasuring relative viscosities of dilute solutions of the polymers (5). Various solution concentrations of the polymers in petroleum ether (100' to 120" C. boiling range). carbon tetrachloride, and tolueix, \yere used in this study.
the meniscus fell to a level, h cm.: below the level of the meniscus in the manometer side arm-Le., the rotary fluid developed a head of h cm. Tbvo studies xvere made on h,. The first ivas a photographic study to determine the cross-sectional shape of the climbing column (Figure 1). Rejecting points near the boundaries, the shape of the column followed an inverse fourth power relationship of the form: z = A F - ~where , A is a constant. Vertical stresses producing the column of liquid thus vary as the fourth power of their radial distance. The second study to correlate h, with h was made with the three polyisobutenes in various solvents. In Figure 2:values of log h are plotted against corresponding log h, for all the different polymers, solvents, temperatures, and concentrations. The data fall in a narrow linear band having a slope of 0.34. The head of fluid, h, is therefore a measure of the climbing effect shown by thcse systems and hence a measure of the normal stresses developed by the sheared liquid. Normal Pressure D u e to Shear. Correlations were observed between h and the following parameters of the system: angular velocity, W, measured in revolutions per minute; concentration of the polymer, C, grams of polymer per 100 grams of solution ; temperature of the system, t o C.; type of solvent used; molccular wcight of solute; viscosity of the system at low rates of shear, ~ 7 : poise. Results are summarized under each of these headings in turn. To
t Z
I
General Observations
Equipment for measuring reversed climbing effect included a fluid rescrvoir, concentric-cylinder tubes, and ma nome te r
$58
Height of Climb on Rotating Rod. M'hrn the inner cylinder of the apparatus was rotated. Lvith the annulus about half full of a solution of polyisobutene, the liquid climbed on the rotating rod to a height h, cm.. above the mean level of the meniscus: and the mean level of
INDUSTRIAL AND ENGINEERING CHEMISTRY
Figure 1, Photographic technique was used to show that z = A r - 4
convert the head h cm. to a pressure. it was multiplied by the fluid density, p , grams per ml. to give ph, grams per sq. cm., labeled pressure head.
Influence of Variables on Pressure Head
Figure 2. Normal pressure head and height of climb are related
I-
Data for all systems fall inside broken lines
W
hc = 1.26
h0.54
LL
0
r c3 I
Angular Velocity. The relationship between angular velocity, ij, and pressure head, ph, had the form:
NORMAL PRESSURE HEAD h c m .
ph = A 3
A is constant for any given system at a definite temperature. For the high molecular weight polymers, H M W 1 and H M W 2 , n fell between 1.9 and 2.1 and was usually very close to 2.0, independent of temperature, concentration, and solvent. For any given concentration of the low molecular weight polymer (LMW) n was independent of temperature but varied between 2.4 and 3.2 as solution concentration changed from 55 to 65%. T h e power law, ph = A&”’, was not valid above a certain temperature or below a certain concentration. Concentration. Arbitrary values were chosen for the other independent variables, so that pressure head could be studied solely as a function of concentration. Chosen temperatures were 20 O, 30°, 4 0 ° , and 50’ C . ; angular velocity was fixed at 200 r.p.m. The data are summarized as follows :
K, was generally independent of temperature. It varied with the solvent:
HMW HMW HMW HMW
1 in 2 in 2 in 2 in
System petroleum ether petroleurn ether toluene carbon tetrachloride
K C
0.46 0.55 0.12 1.15
Temperature. Values of pressure head at a constant angular velocity (200 r.p.m.) are again used for comparison, this time with concentration constant, A law of the form b(1og p h ) / b T = -K , explained the behavior of the systems. K , was constant and, for petroleum ether and toluene solutions, was independent of concentration: System HMW 1 in petroleum ether HMW 2 in petroleum ether HMW 2 in toluene HMW 2 in carbon tetrachloride LMW in petroleum ether
Kt 0.012
0.012 0.011 0.007-0.014 0.035-0.055
Polymer Molecular Weight, Other things equal, a 12Oj, increase in molecular weight resulted in a 500% increase in pressure head. On the other hand, K , and K , were little influenced by the change from H M W 1 to H M W 2. The limiting concentration for onset of normal stresses is much higher (55%)
in the LMW than in the H M W (8 to 9%) solutions in petroleum ether (data not shown). Solvent. Solvents can be written in decreasing order of the slope, K,. CC14: petroleum ether: toluene = 1.15 : 0.55:0.12. The observations and theory of Mark and others (7) that flexible linear polymers exhibit higher intrinsic viscosities in “good” solvents than in “poor” solvents can be extended to explain the present behavior.
Viscosities of Polymer Solutions T o obtain viscosities, Faxen’s walleffect correction was applied to falling ball measurements. Viscosity and Temperature. The slope b(1og q)/bT = --k, (constant) was roughly the same for all H M W systems, as shown in the table below. For the LMW system, kt was appreciably greater. Viscosity and Concentration. An exponential law, b(1og q)/dC = k,, accounted for the variation of viscosity with concentration at constant temperature. Slope k, was independent of temperature in all systems, but varied with solvent and polymer: System HMW 1 in petroleum ether HMW 2 in petroleum ether HMW 2 in carbon tetrachloride HMW 2 in toluene LM W in petroleum ether
kr kc 0.0077 0.227 0.0075 0.232 0.0082 0.569 0.0070 0.053 0.0164 0.070
Solvent Effect. According to the theory of Mark and others ( 7 ) a good or energetically favorable solvent promotes intermolecular contacts between separate polymer chains. The polymer will consequently have a higher viscosity in a good solvent or, stated another way, the slope of the plot of log q us. C will be greater than in a poor solvent. A similar solute-solvent interaction applies to both pressure head and viscosity data. Viscosity and Molecular Weight. For H M W 1 in petroleum ether, viscosities were approximately 2 . 2 times those of H M W 2 solutions. The factor for pressure head changing with this 12% molecular weight increase was five times.
Pressure Head and Viscosity The table shows values of Kc/ko and K t / k t for the various systems:
System HMW 1 in petroleum ether HMW 2 in petroleum ether HMW 2 in carbon tetrachloride HMW 2 in toluene LM W in petroleum ether
Kc/kc K d k t 2.03 1.56 2.37 1.60 2.02 2.27 ,,,
1.34 1.57 2.74
K,/kc is a measure of [d(lnph)/b(lnv)], and is roughly 2.0 for all the systems studied. I t follows that ph = constant X v2 when polymer concentration varies at constant temperature.
Discussion Predictions of a priori theories of normal stresses conflict with many of the data presented here. T h e very existence and direction of the pressure head observed in these experiments are a challenge to general predictions. Another such point arises in the correlation between ph and W2. While this law appears to confirm several predictions, it breaks down completely for the solute systems of lower molecular weight. The observed relationship between ph and the square of the viscosity confirms a prediction of Rivlin (6). O n the other hand, the measured exponential variation with temperature conflicts with another prediction of the same theory. An interesting incompatibility arose when the results from the photographic analysis of the climbing column were compared with those from the study relating h and h,. This incompatibility, however, was explained ( 3 ) when the data were examined in the light of a relaxation mechanism postulated earlier ( 4 ) . Relaxation phenomena are lacking from all the other theories. Literature Cited (1) Alfrey, T., Bartovics, A , , Mark, H., J . Am. C h m . Sac. 64, 1557 (1942). (2) Garner, F. H . , J . Znst. Petrol. 32, 256 (1946).
(3) Garner, F. H., Nissan, A. H., Walker, J., unpublished data. (4) Garner, F. F., Nissan, A . H., Wood, G. F., Phil. T r a w Roy. SOC. A 243, 37 11950)
(5)’ Kemp, A. R., Peters, H., IND.ENC. CHEM.33,1263 (1941); 34, 1192 (1942). ( 6 ) Rivlin, R. S., Trans. Faraday 5‘00. 45, 739 (1948). (7) Weissenberg, K . , Nature 159, 310 (1947). (8) Weissenberg, K . , Freeman, S. M., Zbid., 161, 324; 162, 320 (1948). (9) Wood, G. F., Nissan, A. H., Garner, F. H., J . Znst. Petrol. 33, 71 (1947). RECEIVED for review January 6, 1959 ACCEPTED March 2, 1959 VOL. 51, NO. 7
JULY 1959
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