In the Laboratory
Refractive Index Determination of Transparent Polymers: Experimental Setup for Multi-Wavelength Determination and Calculation at Specific Frequencies Using Group Contribution Theory
W
Jay Dlutowski, Andres M. Cardenas-Valencia,* David Fries, and Larry Langebrake Center for Ocean Technology, Marine Science College, University of South Florida, St. Petersburg, FL 33626; *
[email protected] Owing largely to their processing capabilities, the use of plastics as optical materials in micro-total-analysis systems has spread enormously (1–6). Polymers have been used to either facilitate the incorporation or fabricate functional optical devices (from optical wave guides to variable focal-length lenses). The knowledge of a material’s optical properties enables the modeling and optimization of optical wave guides. Refractive indices, n, for the most common plastics are known, or can be calculated using the fundamental chemical composition only at certain wavelengths (7–9). However, some applications may require n-values as function of frequency (each electromagnetic frequency has an unequivocal wavelength correspondent). Also the method used for processing and depositing the materials dictates its optical properties (9, 10), making necessary the measurement of processed materials’ properties.
Standard techniques for the determination of refractive index in solid materials include the prism-coupling method, ellipsometry, and photometry (11–13). These techniques can be elaborate (coupling method and photometry) and sometimes expensive (especially ellipsometry). Of these methods, photometry is capable of measuring n as a function of wavelength. Bozlee et al. (14) described a laboratory using a spectroscopic method, but they required, as in other references (15), the use of solvents of known refractive indices. Angular dependence of reflection coefficients to determine n-values was reported by Jung and Rhee (15) (a laser light source was used) and is the basis of our approach as well. This method is illustrated in Figure 1. Transmission measurements can be related to the material’s refractive index through Fresnel reflection theory. The parallel and perpendicular Fresnel components (R1p and Rls, respectively) of reflected light that leaves a medium with a refractive index of no and enters another with a refractive index equal to n are given as a function of the incidence angle, θ (defined in Figure 1), by
− R lp =
n1 no n1 no
2
n1 no
cos θ + 2
n1 no
cos θ +
cos θ −
n1 no
cos θ +
n1 no
R ls =
2
2
2
− sin θ 2
2
− sin θ
2
− sin θ 2
2
− sin θ
For unpolarized light the total reflection loss coefficient, Rl, is given by
Rl = Figure 1. Schematic of experimental setup. Bottom figures illustrate need for correction factor owing to refraction of the beam.
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1 R lp2 + R ls2 2
(
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In the Laboratory
Group Contribution Theory Various attempts have been proposed in the literature to calculate various properties, including the refractive index, of polymers based on their chemical structure (16). For nonabsorbing materials, this calculation relies on an additive property, called the molar refraction, R. Three of the most popular definitions of the molar refraction are the Lorentz– Lorenz equation RLL =
Figure 2. Photo of experimental setup.
The transmitted power, Tp, can be related to the reflection coefficient by Tp = C (1 − R l )
(3)
r 2 a cos C =
d cc 2
− πr
r2 −
d cc 2
2
where dcc is defined in Figure 1 and r is the radius of the detector (same as the beam radius for this experiment). The method described here differs from those used by other works, in that a filament white-light source is used. Improved precision and thickness measurement of the substrate may be obtained by using a highly coherent source, (15); however, laser sources are generally expensive and require more laborious alignment and the results are limited to the radiation wavelength provided by the laser. The experiment presented here enables students to determine the index of refraction at various wavelengths. Two optical polymers, poly(dimethyl siloxane) (PDMS) and poly(methyl methacrylate) (PMMA), are proposed as examples. Molar refractivities are provided for PMMA only (as PDMS values could not be found) to compare to the index of refraction calculation using group contribution theories. From this calculation, students discover the simplicity of its use. This experiment would be suitable for a course in organic chemistry or any course discussing the optical properties of polymeric materials as it allows the students to discover that the index of refraction varies with wavelength. The experimental protocol is versatile and can help explain optical concepts (refraction and reflection) and the relationship with refractive index, a very important physicochemical constant. In addition, procedure modifications to the present lab are possible. For instance the curing method and polymer兾curing agent ratio can be varied to study effects on n. Also, replicate measurements could be performed to study measurement statistics to predict accuracies and precisions.
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M = (n − 1)Vm ρ
(6)
and the Vogel equation R V = nM
(7)
where ρ is the density of the solid in question, Vm is the molar volume, and M is the molar mass. The molar refraction of a polymer, Rtotal, can be calculated as the group contribution of all the chemical groups that comprise the polymer’s repetitive unit and the molar refraction functional group values, ri (16)
(4)
2
(5)
the Gladstone and Dale equation RGD = (n − 1)
2
where C is a constant that accounts for the refraction of the beam as it passes through the air–slide interface, which effectively deviates part of the beam outside the detector. C can be expressed as
d cc 2 r
n2 − 1 n2 − 1 M = Vm n2 + 1 n2 + 1 ρ
Rtotal =
∑ ηi ri
(8)
where ηi is the number of times the chemical group appears in the chemical formula. The group contributions of chemical functional groups have been determined by regression by Goedhart (17), who analyzed more than a thousand substances. A polymer refractive index can then be calculated by using the above formulas and Rtotal. Summary of Procedure A variable path-length cuvette holder (CUV-Var), a fiber-optics spectrophotometer, (USB-2000), a visible light source (DH2000), and optical fibers (P400-2SR) were acquired from Ocean Optics, Inc. (Dunedin, Florida). Optical mounting components were acquired from Thor Labs (Newton, NJ). Any similar setup that enables the angle of incidence to be accurately adjusted is acceptable. It is imperative that the optical bench is level, that the beam passes through the center of the sample directly over the rotation axis, and that the turntable is properly attached so that the angle indicator is accurate. The actual experimental setup used is shown in Figure 2.
Sample Slide Preparation Two different materials are characterized in this experiment, PMMA and PDMS. PMMA is readily available as slides or plates (other transparent plastics such as polycarbonate or polystyrene could also be used). PMMA (acquired in 12 × 12-in. plates from United States Plastic Corp.) was cut into small pieces (approximately 2.0 × 5.0-cm) for the measurements. PDMS material is prepared from a commercial resin by mixing 1 part silicone-elastomer curing agent
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In the Laboratory Table 1. Properties of PMMA and PDMS and Characteristics of the Slides Polymer
Structure
Molar Mass/ (g/mol)
Density/ (g/cm3)
Refractive Index
Slide Thickness/mm
100.1
1.17
1.49
1.6
068.0
1.00
1.43
1.0
CH3
PMMA
CH2
C COOCH3
n
CH3
PDMS
Si O CH3
n
with 10 parts silicone-elastomer base (Sylgard 184, Dow Corning). The uncured resin is then poured into plastic molds, which could be weighing plates, covered to prevent contamination, and allowed to cure overnight. The use of PDMS material is optional, but provides the instructor the opportunity to vary the material composition and study the effects on the optical properties. Characteristics of the slides used for experimentation and some of the polymer’s properties are shown in Table 1.
Experimental Procedure The tungsten lamp is allowed to reach the proper temperature (waiting time of ∼45 min) then the reference (no sample) and dark measurements are made. Both slides were cleaned with deionized water and left to dry before proceeding with the measurements. Next, the slide is placed in the sample holder and transmission readings are recorded at the following angles of incidence: 0⬚, 40⬚, 55⬚, and 70⬚. Five replicates were measured on each material, with dark and refer-
Figure 3. Solid lines are simulated reflection curves vs incidence angle for various indexes of refraction passing through a 1.0-mm thick slide. Discrete data points constitute example data points taken for a PDMS slide (1.0-mm thick) at various wavelengths.
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ence readings taken at the beginning of each data set. Students are asked to find the transmission from 400 nm to 700 nm at intervals of 50 nm. Upon taking the average value of the readings, they are asked to plot their data on the curves provided and then determine the index of refraction based on which curve most closely matches their data. Experimental data for PDMS and PMMA at various wavelengths are shown in Figures 3 and 4. The solid-line curves shown in Figures 3 and 4 are generated using eqs 2 and 3. In a more advanced course students can be asked to perform a nonlinear regression of the generated data to estimate a refractive index value by numerical optimization. Hazards No significant hazards are involved with the polymeric plates. Materials safety data-sheets (MSDS) references are included in the Supplemental Material.W Care must be taken while using incandescent filament lamps as they heat up, representing a burn risk.
Figure 4. Solid lines are simulated reflection curves vs incidence angle for various indexes of refraction passing through a 1.6-mm thick slide. Discrete data points constitute example data points taken for a PMMA slide (1.6-mm thick) at various wavelengths.
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In the Laboratory
Table 2. Comparison of Experimental and Theoretical Results Polymer
Experimental Value
PMMA PDMS
Theoretical Values Lorentz & Lorenz
Gladstone & Dale
Vogel
1.50
1.484
1.485
1.475
1.43
----
----
---
Note: Data obtained at 589 nm.
Figure 5. Experimental results, transmission vs wavelength at various incidence angles for PDMS (left) and PMMA (right): 夽 0°, 䊊 40°, 䊐 55°, 䉭 70°.
Results and Discussion The experimental results of transmission versus wavelength and angle for the PMMA and PDMS slides are shown in Figure 5. Transmission values then can relate the refractive index at a given angle via the provided calibration curves shown in Figures 3 and 4. The student is asked to use these figures to obtain refractive index results and to plot the values of refractive index obtained as a function of wavelength, as it has been done in Figure 6. Calculated values of refractive index using group contribution theories with molar refractivity values, given in the Supplemental Material,W are presented in Table 2. Conclusions A simple pedagogical laboratory that can determine n(λ) values of transparent polymers is presented. The method’s usefulness is shown, as molar refractivities of some materials can not be readily found in the literature. W
Supplemental Material
Instructions for the students and notes for the instructor are available in this issue of JCE Online. Literature Cited 1. Huikko, K.; Kostiainen, R.; Kotiaho, T. Eur. J. Pharm. Sci. 2003, 20, 149–171. 2. Metz, S.; Holzer, R.; Renaud, P. Lab Chip 2001, 1, 29–34. 3. Janowiak, M.; Cardenas-Valencia, A. M.; Hall, M; Fries, D. P. Meas. Sci. Technol. 2005, 16, 729–737.
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Figure 6. Estimated refractive indices at various wavelengths: PDMS (䉬) and PMMA (䊊).
4. Duffy, D.; MacDonald, J. C.; Schueller, J.; Whitesides, G. Anal. Chem. 1998, 70, 4974–4984. 5. Dasgupta, P. K.; Eom, I. Y.; Morris, K. J.; Li, J. Anal. Chim. Acta 2003, 500, 337–364. 6. Cardenas-Valencia, A. M.; Fries, D.; Ding, X.; Broadbent, H.; Langebrake, L. In Micro Total Analysis Systems 2004, Vol.1. Proceedings of µTAS 2004, 8th International Conference on Miniaturized Systems for Chemistry and Life Sciences, Malmö, Sweden, Sept 26–30, 2004; Laurell, T., Nilsson, J., Jensen, K., Harrison, D. J., Kutter, J. P., Eds.; Royal Society of Chemistry: Cambridge, United Kingdom, 2004; pp 114–116; Special Publication No. 296. 7. Sultanova, N. G.; Nikolov, I. D.; Ivanov, C. D. Opt. Quantum Electron. 2003, 35, 21–34. 8. Schnyder, B.; Lippert, T.; Kötz, R.; Wokaun, A.; Graubner, V.; Nuyken, O. Surf. Sci. 2003, 532, 1067–1071. 9. Polymer Handbook, 3rd ed.; Brandrup, J., Immergut, E. H., Eds.; Wiley Interscience: New York, 1989. 10. Hoshino, K.; Shimoyama, I. J. Micromech. Microeng. 2003, 13, 149–154. 11. Morris, I. L.; Jenkins, T. E. J. Phys. E: Sci. Instrum. 1989, 22, 27–30. 12. Bass, J. D.; Weidner, D. J. Rev. Sci. Instrum. 1984, 55, 1569–1573. 13. Reisinger, A. R.; Morris, H. B.; Lawley, K. L. Opt. Eng. 1981, 20, 111–114. 14. Bozlee, B. J.; Exharos, G. J.; Jimenez, A. E.; van Swam, S. L. J. Chem. Educ. 2002, 79, 619–622. 15. Jung, C.; Rhee, B. K. Appl. Opt. 2002, 41, 3861–3865. 16. Van Krevelen, D. W.; Hoftyzer, P. J. Properties of Polymers— Correlations with Chemical Structure; Elsevier: Amsterdam, 1972. 17. Goedhart, D. J. Proceedings of the Gas-Phase Chromatography International Seminar, Monaco, Oct 12–15, 1969; p 12.
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