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A new optical method for studying the viscoelastic behaviour of nematic liquid crystals
Volume 5, number 5
OPTICS COMMUNICATIONS
A NEW OPTICAL METHOD
FOR STUDYING
OF NEMATIC
THE VISCOELASTIC
August 1972
BEHAVIOUR
LIQUID CRYSTALS
J. WAHL and F. FISCHER
Physikalisches lnstitut der Universitiit, 44 Milnster, Germany Received 3 May 1972 Simple shear flow of the nematic liquid crystal MBBA is observed between crossed polarizers. The optical anisotropy is found to be a pure function of the product of shear velocity and film thickness. This agrees with the continuum theory of Leslie. The constants kl 1/k33, hi/h2, and ( h 2 - h I )/k33 are determined.
The viscoelastic behaviour o f nematic liquid crystals is examined by studying the shear flow o f an oriented liquid crystal. As a model let us consider a homeotropic layer o f thickness d between two plane glass plates (optic axis and director n perpendicular to the boundary). One gets a simple shear flow by moving one o f the plates with the velocity o parallel to the other. Then the director n forms an angle O(x) with the film normal, x is the position between the plates, going from 0 to d. The boundary conditions are 0(0) = O(d) = 0. Leslie [1] has derived a differential equation and applied it to a homogeneous layer [0(0) = O(d) = ½70. The general solutions o f 0(x, d, v) contain 6 viscosity constants btl . . . . ,/a6, which are reduced to 5 independent parameters by Parodi's relation [2]. 0 also contains Franck's elastic constants k 11 and k33. The results can easily be adapted to our homeotropic layer. Putting the layer between crossed polarizers and using monochromatic light (~0 = 546 nm) normal to the layer one can measure the average optical anisotropy (ne(x)-n o) of the layer as a function o f the shear velocity v. In order to keep stationary boundary conditions at the edge o f the layer also we use two circular glass plates, one rotating against the other with the constant angular frequency co (ranging from 4 × 10 - 5 to 4 X 1 0 - 2 s e c - l ) . In this way we cover a whole range o f shear velocities 0 ~< o ~< wR, where the plate radius R = 3 cm. The thickness d ranges from 50 to 500 tim and can be determined to within + 2/lm.
Fig. 1. Interference pattern arising from optical anisotropy caused by simple shear flow of an initially homeotropic layer of MBBA, observed between crossed polarizers. The shear velocity increases with the radius r. The experimental parameters for this exposure are: monochromatic light of wavelength ho = 546 nm, temperature T ---22 ° C, film thickness d = 150 ~um,relative angular velocity of the boundaries to = 8.36 X 10-4sec -1. Because of the small shear rates (7 < 25 sec- 1) the heat o f friction within the layer can be neglected. Also centrifugal forces can be ignored. So we are allowed to maintain the formulas of simple shear flow given by Leslie [ 1]. One observes a black cross and concentric dark rings. Fig. 1 shows one of the contact exposures which we have got from MBBA (clearing point 40°C) at 22 + 0.5°C. The number of rings increases with co to
341
Volume 5, number 5
OPTICS COMMUNICATIONS
a limiting value which is proportional to the film thickness d. The ring order m determines the average optical anisotropy (ne (x)-n o) =mXo/d as a function of the local shear velocity v = cor. By over-exposing the film these ring radii r can be measured very accurately. By a dimensional analysis applied to the average optical anisotropy, similar to that done by Ericksen [3] we find that mko/d is a pure function of od. This relation is confirmed by the experimental data. Plotting mid against vd, we find that the experimental points from exposures with different co and d all fall on the same curve shown in fig. 2. The mutual deviations of sets of data obtained from different exposures are given by the vertical bar. Within one set the fit to the curve is much better.
kll/k33
August 1972
= 0.86 + 0.04.
For this purpose the main refractive indices nil = 1.555 and n I = 1.787 have been interpolated (for 22°C and 546 nm) from the data on MBBA given by Brunet-Germain [4]. h 1 and 3`2 are usual combinations of viscosity constants: 3`1 =/22 - / 2 3 and )~2 =/25 - / 1 6 . For vd higher than about 7 × 10-8m2sec -1 the average optical anisotropy approaches a saturation value, as can be seen from fig. 2. The whole layer except for two thin boundary films is oriented by the shear flow. With the aid of Leslie's differential equation [ 1] we derive for this limiting case
mid = (m/d)o [ 1 - c o n s t " (vd)- 1/2]
.
.10~m-~
This is confirmed very well by the experiment. (m/d)o is found to be 4.14 X 105m -1 . The saturation value of the anisotropy corresponds to a limiting angle 00. From the index ellipsoid one can obtain the relation
MBBA 22°C i
I
J~
i
i
10 5 velocity x thickness
0 (11
I
~
i
i
t
Fig. 2. m/d, proportional to the average optical anisotropy, as a function of the product of velocity and thickness. We have expanded
mid = a2(od)2
mid in a Taylor series of od:
+ aa(od) 4 + . . . .
a2
n,,
(1 _ n2 ](3`2 - ~.1 2
~60X 0 5
n-~l]\k 3 - - - - - ~ )
a~ 3`0
a4:7nH(l~n2/n2)~7kll-k33 k33
'
4)
By plotting (m/d)/(od)2 as a function of(od) 2 one gets a straight line. By fitting this straight line to our experimental points (for od < 3 X 10-10m2sec - 1 ) we determine a 2 and a 4. From this we calculate the following constants of the nematic liquid crystal MBBA at 22°C: (3,2 - 3`1)/k33 = (2.98 + 0.06) X 1010m-2sec, 342
n2/n2).
where w = (m/d)oCao/n0 + 1. The continuum theory of Leslie [1] yields cos 200 = Xl/;k2. Using these equations we get for MBBA at 22°C: X1/X 2 = - 0 . 9 5 9 -+ 0.005, 00
Applying the formulas of Leslie (eqs. 6.11 to 6.15 in [1]) to our homeotropic layer we get =
cos 200 = 1 - 2(1 - w-2)/(1 -
15 x10-8m2/s
= 81.8 + 0.5 ° .
As far as we know this is the first experiment which proves the validity of Leslie's equations applied to simple shear flow. We believe that our optical method is quite accurate for determining the viscoelastic constants of nematic liquid crystals. A detailed version of this work will be published elsewhere.
References [1] [21 [3] [4]
F.M. Leslie, Arch. Rat. Mech. Anal. 28 (1968) 265. O. Parodi, J. Phys. (Paris) 31 (1970) 581. J.L. Ericksen, Trans. Soc. Rheol. 13 (1969) 9. M. Brunet-Germain, Compt. Rend. Acad. Sci. (Paris) 271B (1970) 1075.
OPTICS COMMUNICATIONS
A NEW OPTICAL METHOD
FOR STUDYING
OF NEMATIC
THE VISCOELASTIC
August 1972
BEHAVIOUR
LIQUID CRYSTALS
J. WAHL and F. FISCHER
Physikalisches lnstitut der Universitiit, 44 Milnster, Germany Received 3 May 1972 Simple shear flow of the nematic liquid crystal MBBA is observed between crossed polarizers. The optical anisotropy is found to be a pure function of the product of shear velocity and film thickness. This agrees with the continuum theory of Leslie. The constants kl 1/k33, hi/h2, and ( h 2 - h I )/k33 are determined.
The viscoelastic behaviour o f nematic liquid crystals is examined by studying the shear flow o f an oriented liquid crystal. As a model let us consider a homeotropic layer o f thickness d between two plane glass plates (optic axis and director n perpendicular to the boundary). One gets a simple shear flow by moving one o f the plates with the velocity o parallel to the other. Then the director n forms an angle O(x) with the film normal, x is the position between the plates, going from 0 to d. The boundary conditions are 0(0) = O(d) = 0. Leslie [1] has derived a differential equation and applied it to a homogeneous layer [0(0) = O(d) = ½70. The general solutions o f 0(x, d, v) contain 6 viscosity constants btl . . . . ,/a6, which are reduced to 5 independent parameters by Parodi's relation [2]. 0 also contains Franck's elastic constants k 11 and k33. The results can easily be adapted to our homeotropic layer. Putting the layer between crossed polarizers and using monochromatic light (~0 = 546 nm) normal to the layer one can measure the average optical anisotropy (ne(x)-n o) of the layer as a function o f the shear velocity v. In order to keep stationary boundary conditions at the edge o f the layer also we use two circular glass plates, one rotating against the other with the constant angular frequency co (ranging from 4 × 10 - 5 to 4 X 1 0 - 2 s e c - l ) . In this way we cover a whole range o f shear velocities 0 ~< o ~< wR, where the plate radius R = 3 cm. The thickness d ranges from 50 to 500 tim and can be determined to within + 2/lm.
Fig. 1. Interference pattern arising from optical anisotropy caused by simple shear flow of an initially homeotropic layer of MBBA, observed between crossed polarizers. The shear velocity increases with the radius r. The experimental parameters for this exposure are: monochromatic light of wavelength ho = 546 nm, temperature T ---22 ° C, film thickness d = 150 ~um,relative angular velocity of the boundaries to = 8.36 X 10-4sec -1. Because of the small shear rates (7 < 25 sec- 1) the heat o f friction within the layer can be neglected. Also centrifugal forces can be ignored. So we are allowed to maintain the formulas of simple shear flow given by Leslie [ 1]. One observes a black cross and concentric dark rings. Fig. 1 shows one of the contact exposures which we have got from MBBA (clearing point 40°C) at 22 + 0.5°C. The number of rings increases with co to
341
Volume 5, number 5
OPTICS COMMUNICATIONS
a limiting value which is proportional to the film thickness d. The ring order m determines the average optical anisotropy (ne (x)-n o) =mXo/d as a function of the local shear velocity v = cor. By over-exposing the film these ring radii r can be measured very accurately. By a dimensional analysis applied to the average optical anisotropy, similar to that done by Ericksen [3] we find that mko/d is a pure function of od. This relation is confirmed by the experimental data. Plotting mid against vd, we find that the experimental points from exposures with different co and d all fall on the same curve shown in fig. 2. The mutual deviations of sets of data obtained from different exposures are given by the vertical bar. Within one set the fit to the curve is much better.
kll/k33
August 1972
= 0.86 + 0.04.
For this purpose the main refractive indices nil = 1.555 and n I = 1.787 have been interpolated (for 22°C and 546 nm) from the data on MBBA given by Brunet-Germain [4]. h 1 and 3`2 are usual combinations of viscosity constants: 3`1 =/22 - / 2 3 and )~2 =/25 - / 1 6 . For vd higher than about 7 × 10-8m2sec -1 the average optical anisotropy approaches a saturation value, as can be seen from fig. 2. The whole layer except for two thin boundary films is oriented by the shear flow. With the aid of Leslie's differential equation [ 1] we derive for this limiting case
mid = (m/d)o [ 1 - c o n s t " (vd)- 1/2]
.
.10~m-~
This is confirmed very well by the experiment. (m/d)o is found to be 4.14 X 105m -1 . The saturation value of the anisotropy corresponds to a limiting angle 00. From the index ellipsoid one can obtain the relation
MBBA 22°C i
I
J~
i
i
10 5 velocity x thickness
0 (11
I
~
i
i
t
Fig. 2. m/d, proportional to the average optical anisotropy, as a function of the product of velocity and thickness. We have expanded
mid = a2(od)2
mid in a Taylor series of od:
+ aa(od) 4 + . . . .
a2
n,,
(1 _ n2 ](3`2 - ~.1 2
~60X 0 5
n-~l]\k 3 - - - - - ~ )
a~ 3`0
a4:7nH(l~n2/n2)~7kll-k33 k33
'
4)
By plotting (m/d)/(od)2 as a function of(od) 2 one gets a straight line. By fitting this straight line to our experimental points (for od < 3 X 10-10m2sec - 1 ) we determine a 2 and a 4. From this we calculate the following constants of the nematic liquid crystal MBBA at 22°C: (3,2 - 3`1)/k33 = (2.98 + 0.06) X 1010m-2sec, 342
n2/n2).
where w = (m/d)oCao/n0 + 1. The continuum theory of Leslie [1] yields cos 200 = Xl/;k2. Using these equations we get for MBBA at 22°C: X1/X 2 = - 0 . 9 5 9 -+ 0.005, 00
Applying the formulas of Leslie (eqs. 6.11 to 6.15 in [1]) to our homeotropic layer we get =
cos 200 = 1 - 2(1 - w-2)/(1 -
15 x10-8m2/s
= 81.8 + 0.5 ° .
As far as we know this is the first experiment which proves the validity of Leslie's equations applied to simple shear flow. We believe that our optical method is quite accurate for determining the viscoelastic constants of nematic liquid crystals. A detailed version of this work will be published elsewhere.
References [1] [21 [3] [4]
F.M. Leslie, Arch. Rat. Mech. Anal. 28 (1968) 265. O. Parodi, J. Phys. (Paris) 31 (1970) 581. J.L. Ericksen, Trans. Soc. Rheol. 13 (1969) 9. M. Brunet-Germain, Compt. Rend. Acad. Sci. (Paris) 271B (1970) 1075.