Cholesteric Liquid Crystals:

As I discussed before, the molecules of a nematic liquid crystal tend to line up with a director, n. Cholesteric liquid crystals are similar to nematic liquid crystals in that the molecules line up with the director, but in a cholesteric liquid crystal, the director rotates in space. The molecules are twisted through 360 degrees as you travel through the sample, tracing out a helical pattern. The twisting of the molecules is known as chirality. A molecule is chiral if it is non-superimposable on its mirror image (See figure 1).

In the above figure Molecule A is a chiral molecule containing a chiral carbon atom indicated by the asterisk. The carbon atom is bonded to four different atoms indicated by the different colors. A mirror image of Molecule A is also shown. Finally on the far right, Molecule A and its mirror image are shown one on top of the other. The fact that the four atoms on the carbon do not match up means that the two molecules are not superimposable. For this reason, Molecule A is said to be a chiral molecule. Molecule A and its mirror image are said to be "enantiomers".

The chirality of the molecules in a liquid crystal results in a slight "twist" in each plane of molecules with respect to the surrounding planes of molecules. Consequently, the orientation of the director in the liquid crystal changes throughout the sample. To help illustrate the twist of the director in space, it is helpful to think of a nut and bolt (see figure 2).

As a nut screws on a bolt, the nut progresses down the axis of the bolt while it rotates around the threads of the bolt. In a similar fashion, the director traces out a helix in space as we move through the liquid crystal. The distance required for the director to rotate through 360 degrees is known as the pitch length. Referring to the nut and bolt example, the pitch is the length of the bolt that the nut travels while making one full turn. Since the director is the average direction of a large number of molecules, it can be defined as pointing in either direction along a specified direction. This means that the orientation of the director repeats itself after a distance of half the pitch length.

Circularly Polarized Light:

We have already discussed the concept of linearly polarized light when we discussed the optical properties of twisted nematic liquid crystals. Light can also be circularly polarized. In talking about circularly polarized light, we can imagine the superposition of two plane polarized waves, one vibrating in the x-direction and the other in the y direction. The equations describing this scenario are as follows:

If we square the x and y components of the electric field vector and add them, the result is the equation for a circle in the x,y plane. This shows that as the resultant electric field travels through space, it traces out a helix (See figure 3).

If the field rotates clockwise the light is said to be right circularly polarized. If it rotates counterclockwise, it is said to be left circularly polarized. A quarter wave plate can be used to achieve circularly polarized light. A quarter wave plate is simply a birefringent material that produces two waves with a 90 degree phase difference because it has two indices of refraction. The light emerges from the quarter wave plate circularly polarized.

To demonstrate this effect, I placed crossed polarizers on an overhead projector. I inserted the quarter wave plate between the crossed polarizers. In rotating the quarter wave plate I observed that at a certain angle all of the light from the overhead was blocked. Since all of the light is blocked, the axis of polarization of the quarter wave plate must be parallel to one of the polarizers. Rotating the plate through 45 more degrees allowed most of the light to be transmitted through the polarizers. Now the light is circularly polarized by the quarter wave plate, so some light is allowed to pass through the second polarizer.

Color:

As was discussed previously, cholesteric liquid crystals possess a degree of twist and a pitch length p. In addition to the twist characteristic, cholesteric liquid crystals are also either left or right handed as was discussed with circularly polarized light. It is interesting to note that a helix which rotates to the right will transmit left circularly polarized light. Another interesting property of cholesteric liquid crystals is that they selectively reflect bright colors. The equation that describes the selective reflection is given by:

where n is the mean refractive index, p is the pitch, theta is the viewing angle with respect to the surface normal, and lambda is the reflected wavelength. The reflected wavelength does not constitute all of the selected wavelength. If the cholesteric liquid crystal sample is right-handed, most of the right-handed component of the incoming wave is reflected while the left-handed component is transmitted. The opposite is true for a left-handed cholesteric liquid crystal. Using this theory, I was able to calculate the pitch length for two cholesteric samples.

Looking normal to the surface of one of the samples on black paper it appeared blue, see figure 4, so I recorded a value of 450 nm for the wavelength of the reflected light.

Using the above equation I calculated a value for p of 281 nm. I repeated the calculation for the second cholesteric sample. It appeared yellow normal to the surface (see figure 5), so I recorded a wavelength of 650 nm for the wavelength of the reflected light.

The value for the pitch of the second sample was 406 nm. When I observed the samples not normal to the surface, but at an angle of 25 degrees from the surface normal, their colors changed. The blue sample looked violet and the yellow sample looked orange. I calculated the wavelength of the reflected light for each sample again using the above formula and knowing the pitch length for each sample. The wavelength of the reflected light for the violet sample was 407 nm, which is very close to the reported wavelength of violet light (400 nm). I repeated the calculation for the orange sample and found the wavelength of the reflected light was 589 nm, which is also close to the reported wavelength of orange light (600 nm).

Next I viewed the same cholesteric liquid crystal displays on the overhead projector. When viewing the samples normal to the surface, the sample that was originally blue now appeared yellow and the sample that was originally yellow appeared blue. This occurred because instead of viewing the reflected light, I was viewing the transmitted light. Subtracting out the reflected light from white light gave me the color of the transmitted light. Subtracting blue light from white gives yellow and subtracting yellow from white gives blue.

For the last experiment in this series, I constructed a simple apparatus consisting of the blue reflecting liquid crystal display, a polarizer, a quarter wave plate, and a flashlight. The flashlight, quarter wave plate and polarizer were arranged so as to produce circularly polarized light, where the direction of polarized light was at an angle of 45 degrees with respect to the optical axis of the quarter wave plate (see figure 6).

The sample was placed on a black background and illuminated with the circularly polarized light. Initially I saw no reflection at all, but as I rotated the quarter wave plate 45 degrees, I saw total reflection.

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