JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Theory of Measurement of Flow Birefringence by Use of the S6narmont Compensator

C. A. HOLLINGSWORTH AND W. T. GRANQUIST*Department of Chemistry, University of Pittsburgh, Pittsburgh 13, Pennsylvania

(Received July 21, 1961)

The mathematical theory of the operation of the Senarmont compensator for measuring flow birefringenceis developed. Equations are obtained for the motion and intensity of the cross of isocline. It is shown that theequation used by Rich is good for small values of the retardation but does not describe accurately the be-havior of the cross of isocline for large retardations. In particular it is found that the collapse of the crossoccurs when the analyzer is at an angle less than 25. The angle 26 is the position of the analyzer at which thedarkness of the already collapsed cross is a maximum.

THE purpose of this paper is to present a theoreticalT treatment of the scissors-type motion and theintensity of the cross of isocline during the measurementof flow birefringence by Snarmont's method. Thetheory has been treated to some extent by Rich, buthis treatment does not give the equation for the motionof the cross of isocline, and for this reason some aspectsof the behavior of the compensator are not touchedupon. These omissions are especially important whenthe value of the birefringence is high.

When Senarmont's method is used the optical ar-rangement is such that monochromatic light passesvertically upward successively through a polarizer, thesample, a quarter-wave plate with axis at 90°, or 00, tothe axis of the polarizer, and an analyzer. When thesample is a fluid in Couette flow, it exhibits a cross ofisocline. The arms of this cross are at right angles to oneanother when the analyzer is crossed with the polarizer.When the analyzer is rotated, the cross of isoclineundergoes a scissors-type motion and varies in intensityuntil the cross collapses to a single line. When the valueof the retardation is small it can be calculated to a goodapproximation from the angle (of the analyzer) at whichthis collapse takes place. For higher values of bire-fringence the situation is more complicated, as will beshown in the following development.

The angular symbols which are used in the presentderivation are shown in Fig. 1. O-P is the polarizer;O-A is the analyzer in the crossed position; O-A' is theanalyzer at an angle y from the polarizer, or at an angle0(y=0- 9 0) from the crossed position; a is the angle

P

X F FIG. 1. OP represents thepolarizer. OA and OA' repre-sent the analyzer in the crossedposition and an arbitrary posi-tion, respectively. OX repre-

A sents the optical axis of thefluid.

o ~~~A

* National Lead Fellow, Mellon Institute, Pittsburgh, Penn-sylvania.

1 A. Rich, J. Opt. Soc. Am. 45, 393 (1955). Earlier references aregiven in this paper.

between the optical axis O-X (of the fluid) and thepolarizer O-P. The direction of the optical axis variesaround the annulus in such a way that the angle be-tween the stream lines and the optical axis (called theextinction angle) has the same value in all parts of theannulus.2 The phenomenon which we are consideringinvolves the motion of the two arms of the cross ofisocline relative to one another, and does not, therefore,

2.00 66O

'jjEq (10)

1.50

I (%]> / X~I '

1.00-

/ ~~~~~~~~~%Eq. (I) withoc(6)

oso , Eq. (I) with (3)

0.oc,0. 50 l0° 150 20° 25° 30

e

FIG. 2. The light intensity at the arms of the cross of isocline,given by Eq. (1) with condition (3) up to an analyzer angle 0, andby Eq. (10) from 0, to p. Also, the light intensity given by Eq. (1)with condition (6). 5=60'.

depend upon the extinction angle. There is no loss ingenerality if the extinction angle is taken to be zero, inwhich case the value of a at an arm of the cross of iso-cline is equal to the angle between the arm and thepolarizer O-P.

The relative intensity of the light emerging from the

2 For a recent review of the theory of flow birefringence andmany references see H. G. Jerrard, Chem. Revs. 59, 345 (1959).

562

VOLUME 52, NUMBER 5 MAY, 1962

May1962 SENARMONT COMPENSATOR TO MEASURE FLOW BIREFRINGENCE 563

analyzer is the following functions of a, 0, and theretardation 5:

1= sin2 0+sin2 2a cos20 sin 2 6-2 sin2a sin20 sin8. (1)

For given values of 0 and a the isocline is given by thevalue of a, which makes I a minimum and this condi-tion is

which gives(0I/aoaa)O,= 0,

sin2a= tan20/2tan2s,

a=45 °.

(2)

(3)

(4)

Instead of Eqs. (2), (3), and (4) Rich' used a condi-

1(%)

100 200 30 400 50°e

60 70° 80°

FIG. 3. The light intensity at the arms of the cross of isocline,given by Eq. (1) with condition (3) up to an analyzer angle O, andby Eq. (10) from Oe to 25. Also, the light intensity given by Eq. (1)with condition (6). a= 160°.

tion which is equivalent' to

(all/a)0, 5 = 0, (5)which gives

tan20= sin2a sinS/ (cos22a+sin22acos5). (6)

Equation (6) gives that value of 0 which will make theposition corresponding to a given value of a darker thanany other value of 0 can make it. However, this is not a

3 See for example, H. G. Jerrard, J. Opt. Soc. Am. 38, 35 (1948).4 Rich used the angle y in place of 0, and his equations, there-

fore, deal with the occluded light (the light not transmitted) in-stead of the light transmitted. Thus his Eq. (6), which correspondsto Eq. (6) in the present paper, gives the maximum of the occludedlight. The two equations are, therefore, equivalent.

e

I , , I20° 30° 40° 50° 60° 70° 80° 90

53.

FIG. 4. The analyzer angle at which the cross of isocline collapses,0,, and the difference 25-c as a function of -5.

sufficient condition for the given value of a to correspondto the isocline, since that value of a need not correspondto the darkest part of the field for 0 given by Eq. (6).That Eq. (6) does not in general give the isocline isillustrated by the examples shown in Figs. 2 and 3. It isseen that the light intensity along the path given byEqs. (3) and (4) is less than that along the path givenby Eq. (6), except at the two points given by

anda =222, 0= 2tanl(V2tan 2 )

co=45°, 0= 5.

(7)

(8)

Figures 2 and 3 illustrate that the following behaviorof the cross of isocline takes place when the analyzer isrotated: As 0 increases from zero the cross undergoes ascissors-type motion and becomes lighter. The crossreaches its lightest point when a and 0 satisfy conditions(7), at which point the cross is halfway closed. Furtherincrease of 0 causes the cross to become darker and tocontinue closing. Collapse of the cross is complete when0 reaches a value O, given by the conditions

a=45°, 0= tan71(2tan13). (9)

As 0 increases beyond 0d, the cross of isocline remainscollapsed but becomes darker according to the followingequation:

(10)

which is obtained by the use of Eq. (4) in Eq. (1).Maximum darkness is reached when 0= 2Y. The range of0 over which the cross is collapsed but darkening; i. e.,the value of 'S - 0, is shown as a function of 2U in Fig. 4.The difference between O. and 2U is less than 10 for valuesof a less than 250. For such small values of 6 there is verylittle difference between the behavior predicted by theuse of Eq. (3) and that predicted by the use of Eq. (6).However, for large values of 1, Eq. (6) does not lead to aclear picture of the behavior of the cross of isocline.

The intensity of the light as a function of a is shownfor several values of 0 and for two values of 5 in Figs. 5and 6. For large values of , as 0 increases, the cross ofisocline closes slowly but fades and disappears becausethe cross becomes lighter and the background becomes

I= 12EI-cos(20-8)],

C. A. HOLLINGSWORTH AND W.

FIG. 5. The light intensity as a function of position in the an-nulus for several positions of the analyzer when =200. The in-tensity at the cross of isocline (minima in the curves) is nearly zerofor all these curves.

darker. While the cross is invisible it rapidly collapses.It then appears again as a single line and darkens whilethe field becomes brighter as 0 further increases. Forvalues of as low as 200, the background is never verybright. The cross remains dark but the field upon whichthe arms are closing darkens. The rate of closing is morenearly uniform and the process is not complete until 0has almost reached the value 28.

For all cases, as 0 is increased beyond the value 'a thebehavior is the same as that which would occur if theanalyzer were turned back from 'S to zero.

The behavior for negative retardations can be under-stood by noting that according to Eq. (1) the intensityis the same function of (0, -a, -8) or of (-0, a, -) asit is of (0, a, 8). Thus if the retardation is negative, andthe analyzer is turned in the direction of positive 0, thenthe closing motion is in the opposite direction from thatfor positive retardation. Otherwise the behavior is thesame for negative as for positive . Alternatively, if

IOo

0 0 20 30° 40° 50 600 70° 80° 90CC

FIG. 6. The light intensity as a function of position in the an-nulus for several positions of the analyzer when S = 1600. Also, theintensity at the cross of isocline as a function of position in theannulus.

is negative, rotating the analyzer in the direction ofnegative 0 produces exactly the same behavior as posi-tive rotation does for positive 8.

The behavior for values of a greater than 1800 can bepictured by noting that in Eq. (1), =180+A can bereplaced by 6=-(180--A). Also =360+A can bereplaced by = A.

ACKNOWLEDGMENT

The authors thank Dorothy Hollingsworth for mak-ing the numerical calculations.

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564 T. GRANQUIST Vol. 52