I
WLADlMlR PHILIPPOFF The Franklin Institute, Philadelphia, Pa.
Stress-Optical Analysis of Fluids Concept of stress-optical coefficient, well known in the stress-optical analysis of solids, can also be applied to fluid flow
T H E stress analysis of viscoelastic fluids ( 3 ) in stationary laminar flow requires the assumption that they show elastic effects. The concept of elasticity without a yield value (which prohibits a static measurement), though necessary, is not obvious. A clear-cut reason for the existence of such an elasticity in fluids consists of the change in configuration or distribution of particles in flow which can be detected by direct mechanical means or by the influence of the particles on the optical properties of the solutions: stress or flow birefringence. Elastic stresses in flowing liquids can also be understood as arising from the balance of continuous stressing and simultaneous stress relaxation as introduced in Maxwell’s original theory. Whereas in liquids in laminar flow, disregarding a hydrostatic pressure, the stress-tensor degenerates into a single shear stress, T , in elastic bodies the normal and shear stresses are coupled by the properties of the material and form a spatial distribution by a stress-tensor.
Relations Used The stress birefringence law (Equation 1) states that for solids the magnitude of the birefringence, An, should be proportional to the applied stress-specifically, to the difference of the principal stresses, AP (the difference between the stresses determined by the stress-tensor in directions where there are no shear stresses), in plane shear or, equivalently, to the ~ which ~ is~a t maximum shear stress, T an angle of 45’ to the principal stresses. With r the shear stress and xn the angle between the largest tensile principal stress and the direction of shear, the relation is:
An
=
CAP
=
2C7,,,.
=
2Cr sin 2xm
(1 1
This relationship has been used extensively for the stress analysis of solid transparent models. Constant C is the stressoptical coefficient measured in units of sq. cm. per dyne or brewsters (Br). Measurement of An and x,
,
determines the magnitude and direction of AP (or T , , , ~ ~ . ) . C, calculated for rubbers consisting of coiled molecules by Kuhn and Grun ( I ) , is independent of their molecular weight. Raman and Krishnan (7) calculated C for pure liquids. Recently Lodge ( 2 )suggested applying the same reasoning to fluids. The further claim is made that flow birefringence makes possible a stress optical analysis of fluids in stationary laminar flow in other geometries than a concentric cylinder viscometer. The “isochromatics” give, as in solids, the locus of points of equal stress. The extinction angle, xo,in flow birefringence is then identified with x,. The use of basic principles of deformation mechanics leads further to the formulation of an angle xo (between principal strain and direction of shear) in terms of an elastic or recoverable shear strain, s, in plane simple shear for any amount of s (4) s = 2 cot 2x,
(2)
IN TOLUENE ( 2 5 O C . I 150
m I
P I
100
a
I
1
6000
/
5000
< I P
E
4000
CI
c 3000 ,j (0
BO
W
2000
1000
t; % w
I
0 0
200
400
RATE OF
600
800
o 1000
w
SHEAR,D ( S E C T 1 1
Figure 1. Birefringence An, extinction angle, xo,and shear stress, T , as functions of rate of shear, D, for a silicone solution
Figure 2. Birefringence was measured as a function of ~ for~ seven ~ solutions .
T
Numbers in parentheses are stress-optical coefficients, C, in Br
VOL. 51, NO. 7
JULY 1959
883
IO
‘RECOIL BIREFRINGENCE RHEOGONIOMETEF
A
x 0
I,000,000 1200
--
A
R=O.O0953cm. X R=0.00147cm.
80
I .
fn
500,000
u“
u)
a w r W
n 350,000 d a
I
W
w -J m
250,000
dW
6 LL
>
150,000
E LL
100,000
0
0 w
:
50,000
0.I
I
I
80
I
S H E1A N/CM?) 0 R STRESS, r ( D YIO0
1000
Figure 3. Recoverable shear vs. shearing stress Was measured with three methods for two solutions
40
0
20
40
t
c- RECOVERABLE SHEAR, S
60
80
100
120
RATIO ( L / R )
Figure 4. Recoverable shear was determined from capillary measurements Parameter is rate of shear
The theory of large elastic deformations of isotropic bodies in plane simple shear gives (for any stress-strain relation) : P11
- PZZ = 7 s
=
27 Cot 2Xm
(3)
where P11 - P ~ isP the difference of the normal stresses parallel to the direction of shear, P11, and perpendicular to the plane of shear, P ~ z . Weissenberg ( 8 ) introduced the equality for any material in laminar flow from symmetry considerations : Pzz
=
Pss
(4)
where P33 is the normal stress in the direction perpendicular to both PII and PZZ. Equations 1 through 4 are the foundation for the experimental determination of the stresses or the stress analysis of fluids in stationary laminar flow, as they show that the conditions are described at a specific rate of shear D by only two parameters, r and s. However, they become important only if
xo = X m
= XD.
Experimental Results
The stress-optical law can be investigated, combining measurements of the with those of the flow curve [T = f(D)J flow birefringence: An = f ( D ) and x, = f ( D ) . Using D as the independent variable, three nonlinear curves for the three properties are obtained as shown in Figure 1. Figure 2 shows a number of measurements of the combined nonlinear curves, plotted according to Equation l as An = f ( T ~ ~ using ~ . ) the assumption x1 = xm. The proportionality required by Equation 1 is fulfilled in a very broad range of =100:1 in 7max.and An. This confirms the identification of xo = xm. Equation 2 can be tested by combining mechanical measurements of s (or XO), xo, and xn from rheogoniometer
884
measurements ( 6 ) (Figure 3). s determined according to Equations 2 and 3, or directly by the “recoil” of the rotational viscometer, is identical even in the nonlinear ‘T - s range. s is the total amount of rotation (in shear units) in the reverse direction upon sudden unloading of the inner cylinder of a rotational viscometer. This confirms the assumption that flowing viscoelastic solutions determine a condition of elastic stress and proves xn = x0 = xm. The coincidence of the measurements means that the rheogoniometer, which measures x , ~ ,does not stand alone as an experimental method but can be successfully correlated with, or supplemented by. more conventional instruments such as the rotational viscometer. Further proof of the existence of elasticity in stationary flow is given in capillary experiments (5). If an elastic energy is present in a fluid with a finite relaxation time flowing through a capillary. it is transported out and influences the pressure distribution along the capillary length as does kinetic energy. This reasoning leads to the use of a plot of pressure us. length or length-radius of the capillary at constant rate of shear to determine the recoverable shear, s, at any rate of shear available in the capillary viscometer (both at the wall of the capillary). This has been extended to 1,000,000 sec.+, compared to about 1 to 100 set.-' available in the Weissenberg rheogoniometer. Figure 4 plots such a relationship for a 3 7 , solution of polyisobutylene in white oil. Conclusions
The mathematical formulation of the equations cannot give the range of these relationships; this is left for the experiments. In flowing polymer solutions, values of s in the hundreds can occur.
INDUSTRIAL AND ENGINEERING CHEMISTRY
This extends the idea of the recoverdble shear to hitherto unmeasurable values. The occurrence of recoverable shears of this magnitudr means according to Equation 3 that the elastically determined normal stresses will completely change the stress pattern peculiar to Newtonian liquids. The measurements shown demonstrate the presence in a flowing liquid of three tensors. all oriented at the same angle, xm = xo = XD. to the direction of shear: (1) the stress tensor determined by PI^ P Z Zand xm; (2) the strain tensor determined by s and xD, and (3) the optical tensor detrrmined by An and x,. They are not of the same shape, but Equation 1 expresses a proportionality between the difference of the principal axes of the stress and optical tensors, whereas Hooke’s law, if valid, requires a proportionality between s and r. The experimental evidence is representative of much more complete results obtained in recent years, and constitutes a new approach to the rheology of viscoelastic bodies, esprcially polymer solutions.
-
literature Cited (1) Kuhn, W., Griin, F., Kolloid-Z. 101,
248 11942). (2) Lodge, k. S., Trans. Faraday Soc. 52, 120 (1956). (3) Philippoff, W., Acta Rheol, in press. (4) Philippoff, W., J . Appl. Phys. 27, 984 (1956). (5) Philippoff, W., Gaskins, F. H., Trans. SOC. Rheol., in press. ( 6 ) Philippoff, W., Gaskins, F. H., Brodnyan, G. G., J . Appl. Phys. 28, 1118 (1957). (7) Raman, C . V., Krishnan, K. S., Phil.M a g . (7) 5,769 (1928). (8) Weissenberg, K., Proc. Ist Intern. Congr. Rheology, Amsterdam, 1948, 3, 36 (19493. RECEIVED for review January 15, 1959 ACCEPTED April 20, 1959
