Liquid Crystal Reflectivity

Garrett Haun and Benjamin Markham

University of Minnesota

Methods of Experimental Physics Spring 2013

Introduction

Liquid crystals are matter in a state that has properties between those of conventional liquid and those of solid crystal. Unlike normal liquids, which have no positional or orientatial ordering of the constituent molecules, a liquid crystal may have crystal like order in one or both aspects [1]. In this experiment the liquid crystal 65OBC (n-hexyl-4’-n-pentyloxybiphenyl-4-caboxylate)is used and it’s phase is determined or changed by temperature. In materials that form liquid crystals the intermolecular forces hold the molecules in fixed positions in the crystalline solid state, however the forces are not the same in all directions; in some directions the forces are weaker than in other directions. As the temperature of the material is increased the molecules vibrate more vigorously and eventually overcome the forces that hold the molecules in place. The increased molecular motion overcomes the weaker forces first, but its molecules remain bound by the stronger forces, this accounts for the quantized layers in the Smectic A phase. For 65OBC the smectic A phase occurs at temperatures between 67.7℃ and 83.7℃ [2]. Within the quantized layers of the Smectic A phase the molecules can slide around each other, and the layers can slide over one another. This molecular mobility produces the fluidity characteristics of a liquid and is similar to the manner that soap behaves, which is where the term “smectic” originates from [3]. If the temperature rises above 83.7℃ the thermal motion destroys the ordering of the liquid crystal phase, pushing the material into a conventional isotropic liquid phase [4].The smectic A phase is positionaly ordered in layers and the molecules are oriented normal to the layer and as seen in Figure 1 below.

Figure 1: Diagram of alignment in smectic A phase. The smectic A phase is composed of layers of rod shaped molecules that all point in the same direction normal to the layers [3].

Theory

Due to their anisotropic nature liquid crystals are birefringent, which means the perpendicular component of light and the parallel component of light have different refractive indexes. Consider the case of a light beam incident on a thin uniform slab of bifringent material. Light polarized parallel to the plane of incidence has different index of refraction than light polarized perpendicular to the plane of incidence. The reflectivity coefficient R_(||) (R_⊥ ) for the component of light with an electric field parallel(perpendicular) to the plane of incidence is given by [5]:

Where θis the angle of incidence, r_∥ (r_⊥ )is the angle of refraction for the parallel (perpendicular) mode, n_∥ (n_⊥ )is the index of refraction for the parallel (perpendicular) mode [5].

Where k is a wave vector of light and h is the thickness of the film. n_⊥ is independent of θ and equal to n_0, and assumed to be independent of film thickness and equal to the value for bulk material. The film thickness is related to the constant layer spacing, a_0, and the number of layers, N, by

For thin enough films (2≤N≤10), where N is the number of layers. Using the small angle approximation of the sine function, sin⁡〖a_⊥ 〗≈ a_⊥ and Snell's law, equation 2 reduces to

Where λ is the wavelength of light. Using this quadratic nature for small N one can associate a value of N with each R_⊥ such that R_⊥∝N^2. Once each R_⊥ measurement has been assigned an N value these values can be plugged into equation 7, which is h(N) in terms of known/measured quantities with a_0(the quantity we are trying to measure) as the slope.

Apparatus

Due to the complex nature of measuring the thickness of a liquid crystal film using the reflectivity of the parallel component of light we only be measured the reflectivity of the perpendicular component of light using the optical arrangement shown below in Figure 2.

Figure 2: Optical arrangement for measuring perpendicular component of reflection off of a smectic A liquid crystal film.

A 2 mW He-Ne laser(λ = 632.816 nanometers) was sent through a polarizer oriented parallel to the plane of incidence in order to yield light that is perpendicularly polarized to the plane of incidence. The beam was then sent through an optical chopper. The optical chopper turned the signals into 679.35 Hz square waves; this allowed us to take advantage of lock-in amplifiers to amplify the small signals into a RMS voltages with low noise. The beams were then sent through a beam splitter, with one beam sent directly into a photodiode used for reference. The second beam was then sent through a convex lens(to focus to beam down to a small point) and variable polarizer(used for calibration, explained below) towards the oven, passing through the glass plate cover towards film plate containing the 65OBC sample, the film plate is shown below in Figure 3.

Figure 3: View of the film plate and spreader, with bulk liquid-crystal sample and a film spread nearly across the film opening.

A portion of the signal beam dependent on the film thickness is reflected off of the film and is directed to a second photodiode. The photodiodes were connected to pre-amplifiers each with a 50 Ohm terminator that converted the small currents into small voltages. Each pre-amplifier is set to a gain of 2×〖10〗^3, the amplified signals were then sent to lock-in amplifiers that gave RMS voltage readings that were directly proportional to the intensity of light incident of their respective photodiodes. The reference frequencies for the lock-in amplifiers are supplied by the optical chopper. The variable polarizer was used to calibrate the beams going to the film and going to the reference photodiode to be of equal intensity. This was accomplished by orienting the beam splitter so that the intensity of light going to the variable polarizer was greater than the intensity going to the reference photodiode. Then a mirror was placed where the liquid crystal film would be, finally the angle of the variable polarized(acting as an attenuator for the already polarized light) was adjusted until the voltage readings of both lock-in amplifiers were equal. Conservative amounts of 65OBC sample were placed on the path of the glass spreader on the film plate shown in Figure 3. Using only small amounts of 65OBC greatly increased our chances of spreading thin films. On rare occasions we had to add more sample to the film plate, this was due to the increased difficultly of spreading films after the sample was distributed throughout the film plate by the glass spreader.

Data Collection

Before measurements were able to be taken the He-Ne laser was allowed 20 to 30 minutes to allow for its intensity to stabilize. The oven’s thermocouple monitor and voltage source were turned on and the LabVIEW temperature control program was set to a temperature of 82℃, we felt this was the highest temperature we could choose while ensuring the sample would still be in the smectic A phase. The reason for choosing the highest temperature possible is because it is easier to spread thinner films at high temperatures[1]. Worth noting is that once inside the 67.7℃ and 83.7℃ temperature range, the 65OBC is in the smectic A phase and no fluctuations inside of this temperature range effect the value of layer spacing a_0. The oven is given about one hour to ensure the air temperature and sample temperature have both reached 82°C. After the warming period was completed, films were made using the film spreader(operated with a handle external to the oven) shown above in Figure 3. The spreader would pull bulk material over the opening in the center of film plate creating films of quantized thickness as shown in Figure 4 below.

Figure 4: Schematic cross-section through the film plate and liquid-crystal film.

After creating a film, sufficient time(about 4-5 minutes) was allotted to ensure the film had reached a stable N state. An interesting phenomenon to note is that to reach an equilibrium N value, the film would always relax down to a smaller(never a larger)N than the original N after being spread. The RMS voltages were then read from the lock-in amplifiers and recorded. Once the laser has stabilized it's intensity remained relatively constant and thus the reference detector signal remained relatively constant throughout the experiment(though it was still measured for each film) and was ~450mV. The reflectivity detector ranged from (0.6-35)mV depending on film thickness. We also had to take into account for noise. Common to both detectors was the noise measured when the laser was blocked right after the optical chopper, these numbers(measured between every film reflectivity measurement) were typically ~0.1mV. Also, the reflectivity detector had additional noise due to light reflecting off the glass window that covers the oven. The center of beams reflected off the glass window did not go into the detector, but due to the Gaussian nature of the light intensity at a distance from the beams center a small amount of the light inevitably went into the detector. This reflectivity detector noise was measured(again between every film reflectivity measurement) by the reading given lock-in amplifier value when no film was present. This number was typically ~0.15mV, but it is the sum of reflection off of the glass window and the inherent noise common in both detectors(~0.1mV). So the noise due to reflection off of the glass window was about ~0.05mV. Reflectivity and noise measurements were taken for 49 films. These numbers and their respective uncertainties were taken by watching the lock-in amplifier voltage number for about 60 seconds, recording the highest and lowest numbers observed, taking the average of these numbers as the reading, and taking half the difference of these numbers as the uncertainty in the reading. To spread a film the glass spreader is simply pulled over the hole in the plate. One way to control the thickness of the films is the speed in which the spreader is pulled across the film hole. Pulling the spreader slowly will usually make thicker films and spreading it quickly will usually make thinner films[1]. Also in order make an already existing film thinner the spreader can pushed ~(1/3) the way over the film and pulled back quickly[1]. After taking reflectivity and noise measurements on a film a new film is spread. Occasionally after waiting the 4-5 minutes to allow for equilibrium we noticed that the film is identical N to the previous film, in these cases we spread a new film in order to allow for independent measurements.

Analysis

The following formula shows how the noise measurements were used in order to give us the correct value for the reflectivity of perpendicular component of light.

We measured 49 R_Ë” values, then proceeded to assign N values in the 2≤N≤10 range. We started by assigning N=2 to our lowest R_Ë” value(N=1 cannot be observed). Because R_⊥∝N^2 for small N we then took our lowest R_Ë” value and multiplied it by 9/4 and there was a cluster of values there(as expected), which we assigned N=3 to. We then took our lowest R_Ë” value and multiplied it by 16/4 and there was a cluster of values there(again, as expected), which we assigned N=4 to. We continued this up to N=10 after which the R_⊥∝N^2 approximation starts to break down for our values of θ, n_0, λ, and approximate a_0 we were expecting. In this 2≤N≤10 range we had 40 points, 39 fit into a well defined N with the correct R_⊥∝N^2 behavior, 1 of these points did not fit into a well defined N(it was about halfway between N=4 and N=5) and was discarded from the analysis. These R_⊥ values(with assigned N), θ, n_0, and λ, all with their respective uncertainties[(θ = (0.190 ± 0.022) radians, n_0 = (1.50 ± 0.01), and λ = (632.816 ± ~0)nanometers] were plugged into equation 7 to find h(N) and its uncertainty for or each R_⊥ of the 39 data points in the 2≤N≤10 range. A least-squared-fit was performed to a straight line and yielded the slope of h(N), and thus a_0 with its uncertainty.

Results

As stated before a least-squared-fit was performed to a straight line and yielded the slope of h(N), and thus a_0. We found this value of a_0 to be equal to (24.2±0.2) Angstroms. This line is shown below in Figure 5.

Figure 5: 65OBC film thickness as a function of Number of Smectic A Layers with the constant layer spacing a_0 as the slope.

Notice there is a non zero intercept, this is inconsistent with what is expected(obviously h(0) does equal 0). This is possibly due to n_0 being a function of N rather than constant[5]. Once a_0 was determined it was plugged into equation 2 in order to deduce the N values of our higher values of R_Ë”(though not necessary for the analysis). The R_Ë” data is plotted below in figure 6 along with equations 2 & 6 for our values of θ = (0.190) radians, n_0 = (1.50), and λ = (632.816) nanometers, and a_0 = (24.2) Angstroms. Notice as stated above that equations 2 & 6 start to diverge away from each other for N values above 10 and this is why the analysis was performed in the 2≤N≤10 range.

Figure 6: Reflectivity of 65OBC as a function of number of Smectic A layers, notice the small N approximation starts to break down above N=10, that is why the analysis is done in the 2≤N≤10 range.

Figure 7 below shows the χ values for the h(N) points in ascending order(the first few are N=2, the next few are N=3, etc.). No systematic error was evident. No points had a high enough χ value to justify them being thrown out, one point had a χ value above 2 but this is expected considering we had 39 data points. The value of the reduced χ squared was 0.624, this is not ideal but certainly reasonable, it indicates one or more of our uncertainties may have been overstated. We suspect the overstated uncertainty is the uncertainty in n_0. The values we read for typical liquid crystals ranged from 1.49-1.51[1,5], therefore it seemed reasonable to assign the value of n_0 and its associated uncertainty as n_0 = (1.50 ± 0.01), but this seemingly small error in n_0 propagated very strongly.

Figure 7: χ values for the h(N) points in ascending order(the first few are N=2, the next few are N=3, etc. No systematic error was evident and no points were rejected. Reduced χ squared equals 0.624.

Conclusion

In conclusion we used an optical reflectivity measurements to determine the layer spacing in Smectic A 65OBC. Our measured value of the layer spacing was (24.2±0.2) Angstroms, we have no published value to compare this number to, but an approximation can be given by the length of the molecule. Using the bond lengths and bond angles one can show the length of 65OBC is ~25 Angstroms, our data is in good agreement with this number.

Acknowledgements

We would to thank Professor Pawloski for his helpful feedback and for his ongoing encouragement when we encountering problems. We would also like to thank Kurt Wick for his help troubleshooting circuit problems that plagued us for much of the semester. Most of all we would like to thank Professor C.C. Huang for allowing us to use his lab space and equipment. C.C. is an expert in the field of liquid crystals and we would also like to thank him for being a great source of knowledge throughout this experiment. We have much respect for him as a person and after some of our difficulties with this project and reading over some of his publications we have gained much respect for him as an experimentalist.

References

1: Stephan John Pankratz (2000). Optical Reflectivity Studies on Freely-Suspended Liquid-Crystal Films. Ph.D. Thesis. University of Minnesota: United States

2: Gloriex, Christ. "Adiabatic Calorimetry Principles and Applications." Katholiek University Leuven, 18 May 2005. Web.

3: Chandrasekhar, Sivaramakrishna. Liquid Crystal. 2nd ed. Cambridge: Cambridge UP, 1992. Print.

4: Clark, Noel A. "Structure and Dynamics of Freely Suspended Liquid Crystals." Archive.org. NASA, 12 Apr. 2004. Web.

5: C. Rosenblatt and N.M. Amer, Applied Physics Letters, Volume 36, Issue 6, pages 432-434(1980)