- Email: [email protected]
Mecanochromatic effect in nematics
23 October 1972
PHYSICS LETTERS
Volume 4 1A. number 5
MECANOCHROMATIC
EFFECT
IN NEMATICS
P.G. De GENNES Liquid Crystal Institute*, Kent State University, Kent, Ohio 44242, USA Received 11 September 1972 We consider a nematic slab, with homeotropic boundary orientations (easy axis z) subjected to two simultaneous perturbations = a uniform shear flow along x and a magnetic field H oriented at 45’ in the xz-plane. We find that, in a certain range of H-values, the orientation of the molecules in the mid plane of a thick slab does not correspond. to a fixed angle = in this region the director rotates by many turns between the walls and the mid plane. In suitable cases, the periodicity may become comparable to an optical wavelength and the sample may become colored.
Let d be the sample thickness. one plate moves with velocity uu; the other one is fixed (fig. I). The equation ruling 0(z) has the form [l] 2
1s(Yt-yZcos2e)
=K~+:X,H2cos20. dz*
(1)
“4 ~~o_.
“/
-jJ/ 192
az
6
-1
~~
d/zL,.
The left-hand side represents the hydrodynamic torques. The right-hand side contains both the elastic and magnetic components. We have assumed for simplicity that the splay and bend elastic constants are equal K, 1 = K,, = K and the shear rate s = un/d is independent of z. Eq. (1) has the same form as in zero field, but with a corrected coefficient:
Fig. 1. Tilt angle of the director under the combined action of a shear flow and a field H. Case a is similar to the zero field situation. Case b leads to large distortions.
Y2 = y:x,H=
Lm = 4rrKly1ds = 4nK/ylvo
,
(2)
when IT21> yl, we thus expect that, in a thick slab, the molecules will tend to align along a direction g such that ~0~28” = - y1/l?21. When Iy2] < yl, we expect a very different region, the qualitative features of which may be seen from the special case where .Y2= 0, i.e. fxaH2 = -y2s
= Iy2is .
(3)
In this case we have d20/dz2 = y&K, e = (5~/4K)(~~-fd~).
(4)
The constant in eq. (4) ensures that 8 = 0 on the walls. Eq. (4) describes a “cycloidal” arrangement with a * Permanent address: College de France, pl. M. Berthelot, Paris 5e, France.
a
b
tlY21>l,t
local period L = ndz/de.
C1?2kr,l
Near the walls L is minimum,
and equal to (3)
For ut, = 3 cm/set, K = lop6 dyne, y1 = 10-l
poise, this gives L, - 0.4 micron. For such a case, Bragg reflections corresponding to the interval L,, and oblique incidence, would occur in the visible spectrum = we would have a coloration induced by flow. In practice u,, (or s) are bounded by the onset of disclinations. Typrcal values of smax q uoted by Fischer [2] are of order 10 see-l. For d = 3 mm this would indeed correspond to u. = 3 cm/set; thus the experiment appears feasible. The author is indebted to Dr. Fischer for a very helpful discussion of shear flow in nematics. References [l] F.M. Le.&e, Arch. Rat. Mech. Analysis 28 (1968) 265. [ 21 Fischer, Private communication; Also: Proc. 4th Liquid crystal Conf., Kent, Ohio.
479
PHYSICS LETTERS
Volume 4 1A. number 5
MECANOCHROMATIC
EFFECT
IN NEMATICS
P.G. De GENNES Liquid Crystal Institute*, Kent State University, Kent, Ohio 44242, USA Received 11 September 1972 We consider a nematic slab, with homeotropic boundary orientations (easy axis z) subjected to two simultaneous perturbations = a uniform shear flow along x and a magnetic field H oriented at 45’ in the xz-plane. We find that, in a certain range of H-values, the orientation of the molecules in the mid plane of a thick slab does not correspond. to a fixed angle = in this region the director rotates by many turns between the walls and the mid plane. In suitable cases, the periodicity may become comparable to an optical wavelength and the sample may become colored.
Let d be the sample thickness. one plate moves with velocity uu; the other one is fixed (fig. I). The equation ruling 0(z) has the form [l] 2
1s(Yt-yZcos2e)
=K~+:X,H2cos20. dz*
(1)
“4 ~~o_.
“/
-jJ/ 192
az
6
-1
~~
d/zL,.
The left-hand side represents the hydrodynamic torques. The right-hand side contains both the elastic and magnetic components. We have assumed for simplicity that the splay and bend elastic constants are equal K, 1 = K,, = K and the shear rate s = un/d is independent of z. Eq. (1) has the same form as in zero field, but with a corrected coefficient:
Fig. 1. Tilt angle of the director under the combined action of a shear flow and a field H. Case a is similar to the zero field situation. Case b leads to large distortions.
Y2 = y:x,H=
Lm = 4rrKly1ds = 4nK/ylvo
,
(2)
when IT21> yl, we thus expect that, in a thick slab, the molecules will tend to align along a direction g such that ~0~28” = - y1/l?21. When Iy2] < yl, we expect a very different region, the qualitative features of which may be seen from the special case where .Y2= 0, i.e. fxaH2 = -y2s
= Iy2is .
(3)
In this case we have d20/dz2 = y&K, e = (5~/4K)(~~-fd~).
(4)
The constant in eq. (4) ensures that 8 = 0 on the walls. Eq. (4) describes a “cycloidal” arrangement with a * Permanent address: College de France, pl. M. Berthelot, Paris 5e, France.
a
b
tlY21>l,t
local period L = ndz/de.
C1?2kr,l
Near the walls L is minimum,
and equal to (3)
For ut, = 3 cm/set, K = lop6 dyne, y1 = 10-l
poise, this gives L, - 0.4 micron. For such a case, Bragg reflections corresponding to the interval L,, and oblique incidence, would occur in the visible spectrum = we would have a coloration induced by flow. In practice u,, (or s) are bounded by the onset of disclinations. Typrcal values of smax q uoted by Fischer [2] are of order 10 see-l. For d = 3 mm this would indeed correspond to u. = 3 cm/set; thus the experiment appears feasible. The author is indebted to Dr. Fischer for a very helpful discussion of shear flow in nematics. References [l] F.M. Le.&e, Arch. Rat. Mech. Analysis 28 (1968) 265. [ 21 Fischer, Private communication; Also: Proc. 4th Liquid crystal Conf., Kent, Ohio.
479