Preparation of Graded-Index Plastic Optical Fibers by the Diffusion

A poly(methyl methacrylate) (PMMA) base gradient-index plastic optical fiber (GI-POF) was fabricated by the diffusion-assisted coextrusion process usi...
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Ind. Eng. Chem. Res. 2002, 41, 2418-2424

Preparation of Graded-Index Plastic Optical Fibers by the Diffusion-Assisted Coextrusion Process In-Sung Sohn and Chang-Won Park* Department of Chemical Engineering, University of Florida, Gainesville, Florida 32611

A poly(methyl methacrylate) (PMMA) base gradient-index plastic optical fiber (GI-POF) was fabricated by the diffusion-assisted coextrusion process using diphenyl sulfide (DPS) as a refractive index modifying dopant. In the diffusion-assisted coextrusion process, PMMA containing DPS and pure PMMA were coextruded concentrically as the core and cladding, respectively. Subsequently, DPS contained in the core was allowed to diffuse in the diffusion zone of a coextrusion die forming a gradually varying concentration gradient in the radial direction. Thereby, a refractive index profile was created. The refractive index profile was then characterized by measuring the concentration profile of DPS using FT-IR spectroscopy. The measured profiles showed a good agreement with the predictions of the theoretical model reported previously (Ind. Eng. Chem. Res. 2001, 40, 3740). Because the concentration profile of the dopant is strongly affected by its diffusivity, the refractive index profile could be controlled by adjusting the processing temperature and/or the residence time of the materials in the coextrusion die. The results of this study suggest that the diffusion-assisted coextrusion process is a viable method to fabricate a GI-POF with a bandwidth higher than 600 Mbits/s at a distance of 100 m. 1. Introduction Gradient-index plastic optical fibers (GI-POFs) have drawn significant interest as high-bandwidth datacommunication media for local area networks or home networks. Despite their higher attenuation than glass optical fibers (GOFs), POFs are perceived to be more suitable for the short-haul applications because of their flexibility and durability. These properties of polymers make it possible for POFs to have a much larger core diameter in the order of 1 mm compared to 5-10 µm of single-mode GOFs.1 Because a large diameter makes splicing of optical fibers much easier and allows the use of lower cost light sources, the economic advantage is very significant. A GI optical fiber, which has a continuously varying refractive index profile across the fiber radius, provides a much higher bandwidth than step-index (SI) optical fibers. Especially when the refractive index profile is nearly parabolic, the modal dispersion of the input light signal is minimized, making it possible to transmit data at a very high rate.2 For these reasons, there have been considerable research efforts to develop fabrication methods for GI-POFs with the optimum refractive index profile.3-12 In our previous paper,13 a method called the diffusion-assisted coextrusion process was proposed for the fabrication of GI-POFs with a bandwidth higher than 400 Mbits/s at 100 m (or 400 Mbps/100 m). In this process, two polymers containing refractive index modifying additives are fed separately into a coextrusion die (Figure 1). In the coextrusion die, the polymers are formed into a concentric core-cladding structure, and subsequently the additives contained in the polymers diffuse in the radial direction as the materials proceed downstream in the diffusion zone. As the material exits the diffusion zone, it is drawn into a fiber. Because the * To whom all correspondence should be addressed. Tel.: (352) 392-6205. Fax: (352) 392-9513. E-mail: [email protected].

Figure 1. (a) Schematic of the diffusion-assisted coextrusion process. (b) Schematic of the initial and downstream additive concentration profiles in the diffusion zone.

material is solidified in a nonequilibrium state, a continuously varying additive concentration profile (or the refractive index profile) is created in the radial direction. We have indicated previously that the refractive index profile created by the diffusion-assisted coextrusion process can provide a bandwidth substantially higher than that obtainable with SI-POFs although the refractive index profile may not be of an optimal shape. In the present study, an experiment has been conducted to confirm the theoretical predictions, thereby demon-

10.1021/ie010902o CCC: $22.00 © 2002 American Chemical Society Published on Web 04/17/2002

Ind. Eng. Chem. Res., Vol. 41, No. 10, 2002 2419

Figure 2. Chemical structures of additives: (a) DPS (MW ) 186, refractive index n ) 1.633); (b) Oil Red EGN (MW ) 418).

strating that the diffusion-assisted coextrusion process is a viable method to fabricate GI-POFs. 2. Experiment Two different additives have been used for the present study: (1) diphenyl sulfide (DPS) and (2) Oil Red EGN. DPS, whose molecular structure is shown in Figure 2, is a refractive index modifying dopant that has been used in other studies.14 It has a much higher refractive index (1.633) than poly(methyl methacrylate) (PMMA; 1.491), thus having good latitude to modify the refractive index of PMMA with only a small amount. It is desirable to keep the dopant concentration as low as possible because it acts not only as a plasticizer, decreasing the maximum service temperature of the GIPOF, but also as a source of light scattering, increasing the attenuation of the fiber. DPS is known to have good compatibility with PMMA and has a relatively high diffusivity. It is also known to have good thermal stability at an elevated temperature. The thermal stability of a GI-POF is important because its refractive index profile may degrade upon long-term usage at an elevated temperature. Sato et al.14 has suggested that the thermal stability of the refractive index is affected by the diffusivity and the plasticization effect of a dopant, and the materials with polar groups attached to aromatic rings are most appropriate. In their study, a PMMA base GI-POF prepared by the interfacial gel polymerization method using DPS as a dopant showed little change in the refractive index profile or in the bandwidth characteristics after 12 000 h of aging at 85 °C. Oil Red EGN, whose molecular structure is also shown in Figure 2, is not a refractive index modifying dopant but a red colorant. It was adopted to study the diffusional characteristics visually and by using a UVvisible spectrophotometer. In the following, experimental procedures to make a fiber as well as the method to measure the additive concentration profile are described separately for each additive (i.e., DPS and Oil Red EGN). DPS-Doped GI Fiber. To ensure good dispersion of the additive in PMMA (Cyro Ind. Acrylite H-15) for the core material, a master batch (i.e., PMMA chips or pellets containing the additive at a high concentration) was first prepared by dissolving both DPS (Fluka) and PMMA in toluene followed by 3 h of mixing. The mixture was then cast into a sheet and dried in a vacuum oven for 7 days. The dried plaque or sheet was broken into small chips of about 3 mm in its characteristic size. The DPS concentration in the master batch was 20 wt %. For the core material, the master batch was dry-blended with PMMA pellets at 32.5 wt %, making the final

concentration of 6.5 wt %. Pure PMMA was used as the cladding material. The core and the cladding materials were extruded using two 3/4 in. extruders. Because PMMA is hygroscopic, absorbing moisture up to 0.3 wt % at room temperature,15 the materials were dried under vacuum at 70 °C for at least 24 h prior to extrusion. Otherwise, the absorbed moisture could generate numerous bubbles during extrusion. The total extrusion output was varied between 75 and 87 g/h to vary the residence time of the material in the diffusion zone. The melt temperature was also varied between 188 and 216 °C. As Figure 1a indicates, the diffusion zone is a heated tube attached at the end of a typical coextrusion die. Its dimension was 50 cm in length and 7.67 mm in diameter. The core and the cladding materials are formed into a concentric structure in the coextrusion die and proceed into the diffusion zone where the additive diffusion takes place at the core-cladding interface, forming a concentration gradient in the radial direction (Figure 1b). As the material leaves the diffusion zone, it may be drawn into a fiber of a desired diameter. While the typical diameter of a POF is between 0.75 and 1 mm, a 3 mm fiber was made in the present study to make the measurement of the concentration profile easier. The concentration profile of DPS in the fiber (and refractive index profiles) was measured using an FTIR spectrometer mounted with an optical microscope (Nicolet Magna FT-IR 760). Thin cross-sectional fiber samples were prepared and scanned across the radius by the following procedures. Disks of 1 mm thickness were first cut from the 3 mm fiber using a wire saw. The disks were then ground into 100 µm thick circular films using abrasive papers (400 and 2000 grit). The fine scratches on the film surface made by the abrasive papers would cause scattering of light, causing serious measurement error, and they had to be removed by polishing with an aluminum oxide slurry (20 wt % aqueous solution of 50 nm Al2O3 at pH 4). The films were then washed with deionized water and dried in a vacuum. The 100 µm thick sample was then placed on a gold mirror under a microscope attached to the spectrometer, and focused light from a Nernst glower through a 10× object lens was lit on the sample. The light coming out of the surface includes both light directly reflected from the sample and light that passes first through the sample and is then reflected back from the interface between the mirror and sample. It was detected with an MCT detector. A total of 128 spectra within the spectral range of 4000-650 cm-1 were collected and averaged for each point on the sample. This procedure was repeated while scanning along the radial direction of the 3 mm wide sample at a 50 µm interval. Thus, the spatial resolution of the DPS concentration profile in the radial direction was 50 µm with 30 data points across the radius. The relationship between the additive concentration and transmitted light can be obtained from Beer’s law:16

A ≡ log(I0/I) ) abc

(1)

Here A is the absorbance defined as the logarithm of the ratio of the incident light to the transmitted light intensities. a, b, and c are the absorptivity or extinction coefficient, the path length of the sample, and the concentration of the additive in grams per liter, respec-

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Figure 3. FT-IR spectra of DPS-doped GRIN fiber at four different radial positions (Q ) 87 g/h, z ) 50 cm, melt temperature ) 188 °C).

constant, it was difficult to achieve and the radial position of the core-cladding interface (Rf) varied between 0.43 and 0.6. Nevertheless, the influence of the interface position Rf on the diffusional characteristics of the additive is negligible, and the data in Figure 4 show an agreement with the expected trend. Figure 4 indicates that the magnitude of the slope of the concentration profile at the inflection point, which represents the degree of radial diffusion of the additive, decreases with increasing melt temperature. Because the diffusivity increases with temperature, this trend is expected. The concentration profile for Figure 4a is the steepest because of the lower diffusivity at a lower temperature despite the larger residence time resulting from a lower flow rate than other data sets. As will be described later in detail, the solid lines in Figure 4 represent the theoretical predictions described in our previous paper13 and the goodness of fits for all curves were greater than 0.99. The additive concentration profile is directly related to the refractive index profile through the LorentzLorenz equation:17

n)

x11+-2φφ

(2)

where

∑i φ)

(ni2 - 1)xi

(ni2 + 2)Fi xi

(3)

∑i F

i

Figure 4. Concentration profiles of DPS in the DPS-doped GRIN fiber (symbols, measured data; solid lines, theoretical predictions): (a) Q ) 74.5 g/h, melt temperature ) 188 °C, DA ) 1.01 × 10-6 cm2/s; (b) Q ) 80.9 g/h, melt temperature ) 204 °C, DA ) 1.95 × 10-6 cm2/s; (c) Q ) 80.7 g/h, melt temperature ) 216 °C, DA ) 2.85 × 10-6 cm2/s.

tively. The concentration of the additive at each radial position could be calculated from the above equation because a linear relation exists between the absorbance and the concentration. In Figure 3, FT-IR spectra of the DPS-doped fiber at four different radial positions are plotted for comparison. The peak appearing at 1581 cm-1 is due to the phenyl group of DPS molecules, and that at 1749 cm-1 is for the ester group of PMMA molecules. The peak height at 1581 cm-1 gets smaller with increasing radial position from the center of the fiber sample, while the ester peak shows little variation, indicating a decrease in the DPS concentration with the radial position. The height of the phenyl peak was measured and normalized using the corresponding ester peak as a reference, and the DPS concentration at each radial position was calculated based on Beer’s law. The normalized DPS concentration profiles were plotted in Figure 4 for three different flow rates of 74.5, 80.9, and 80.7 g/h. These flow rates are equivalent to the average residence time in the diffusion zone of 21.5, 19.7, and 19.8 min, respectively. For each flow rate, the melt temperature was also varied at 188, 204, and 216 °C, respectively. While it was attempted to maintain the flow rate ratio between the core and the cladding

Here ni, xi, and Fi are the refractive index, mass fraction, and density of each component in the mixture, respectively. Unless the density difference between the mixture components is very large, the Lorentz-Lorenz equation shows a near-linear relationship between the refractive index and the concentration. Thus, the concentration profiles given in Figure 4 match closely to the refractive index profiles for most cases of our interest. Oil Red EGN-Doped GI Fiber. A master batch containing 0.465 wt % Oil Red EGN in PMMA was also prepared following the same procedure as that of the DSP master batch. For the core material, the master batch was dry-blended with pure PMMA at 5 wt %, making the final concentration 230 ppm. For the cladding, the same pure PMMA was used. Although the final concentration of the Oil Red EGN is much lower than that of DPS, it was high enough to produce a color gradient detectable by a UV-visible spectrophotometer. The color gradient was even recognizable to the naked eyes. The core and the cladding materials were extruded using the same 3/4 in. extruders. The materials were also dried under vacuum prior to the extrusion to eliminate any moisture in them. The extrusion rate (or the flow rate) was fixed at 87 g/h, which resulted in a residence time of 18.6 min in the diffusion zone. The melt temperature was varied within the same range as the DPS experiment, which was between 188 and 216 °C. The radial concentration profile of the Oil Red EGN in the fiber was measured with a Zeiss Axiotron microscope spectrophotometer. Following the same

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Figure 5. UV absorption spectra of (a) PMMA and (b) 0.465 wt % Oil Red EGN containing PMMA.

sample preparation procedure, 100 µm thick samples of 3 mm in diameter were prepared. These samples were then placed on a motorized stage, and UV light from a mercury lamp was focused on the sample using a 20× object lens. The intensity of the transmitted light was detected by a photomultiplier tube. Because Oil Red EGN has a maximum absorption at 520 nm, as shown in Figure 5, incident light at the 520 nm wavelength with a 20 nm spectral bandwidth was selected by passing the light source through a grating monochromator. The samples were scanned radially with a spatial resolution of 10 µm on the motorized stage, and five scans were taken and averaged for each sample to reduce the error caused by light source fluctuation and detector noise. In Figure 6, the normalized absorbance of the Oil Red EGN-doped PMMA fiber was plotted as a function of the normalized radial position of the fiber. Assuming that the absorbance at 520 nm is proportional to the concentration of the additive, the absorbance curve is equivalent to the radial concentration profile. Parts a-c of Figure 6 are the profiles obtained at melt temperatures of 188, 204, and 216 °C, respectively. Despite attempts to keep the relative flow rate of the core to the cladding material constant, it varied slightly depending on the melt temperature. The position of the core-cladding interface, Rf, was 0.526, 0.587, and 0.474 for parts a-c of Figure 6, respectively. As observed in Figure 4, diffusion of Oil Red EGN took place to a greater extent when the melt temperature was higher. Also overlaid in Figure 6 as a solid line is the theoretical prediction described in our previous paper,13 and the goodness of fit was greater than 0.99 for all curves. The concentration profiles of Oil Red EGN (Figure 6) are steeper than those of DPS (Figure 4) because of the the lower diffusivity of Oil Red EGN. Thus, the refractive index profile of a GI-POF and its bandwidth characteristics can be controlled by the choice of dopant materials as well as by varying the melt temperature and the residence time. Detailed quantitative analyses of the diffusivities and their temperature dependences are described in the following sections. 3. Comparison between Measured and Predicted Concentration Profiles Theoretical prediction of the additive concentration profile given in our previous paper requires the diffusivity of the additive to be known. Following the free

Figure 6. Concentration profiles of Oil Red EGN in the Oil Red EGN-doped GRIN fiber (symbols, measured data; solid lines, theoretical predictions; Q ) 87 g/h): (a) melt temperature ) 188 °C, DA ) 4.90 × 10-7 cm2/s; (b) melt temperature ) 204 °C, DA ) 1.17 × 10-6 cm2/s; (c) melt temperature ) 216 °C, DA ) 2.30 × 10-6 cm2/s.

volume theory,18,19 the diffusivity of a small molecular species in a polymer matrix can be estimated once its diffusivity at one temperature is known. Thus, in comparison of the measured concentration profiles with the predicted ones, the diffusivity of the additive at the lowest experimental temperature (i.e., 188 °C) was determined by fitting the data to the theoretical prediction using the diffusivity as an adjustable parameter. Once the diffusivity at 188 °C was determined, the diffusivities at other temperatures (i.e., 204 and 216 °C) were estimated by the free volume theory.

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Figure 7. Diffusivities of DPS and Oil Red EGN in a PMMA matrix (a, DPS; b, Oil Red EGN). Solid line: estimation by the free volume theory using the diffusivities at 188 °C (closed symbols) determined by curve fitting of the data in Figures 4a and 6a. Open symbols: diffusivities predicted by the free volume theory and used for the solid lines in Figures 4b,c, and 6b,c.

The diffusivities of DPS and Oil Red EGN determined by nonlinear regression using the data at 188 °C were 1.01 × 10-6 and 4.9 × 10-7 cm2/s, respectively (i.e., Figures 4a and 6a). Figure 7 shows the temperature dependence of the diffusivities of DPS and Oil Red EGN determined by the free volume theory using the diffusivities at 188 °C (closed symbols in Figure 7). The open symbols represent the diffusivity values that have been used for the solid lines in Figure 4 for data sets b and c and Figure 6b,c. The agreement between the measured and predicted concentration profiles appears to be excellent, with the goodness of fit greater than 0.99. Determination of the Arrhenius plot (i.e., Figure 7) is described in the appendix section for interested readers. 4. Estimation of the Bandwidth for a DPS-Doped GRIN Fiber The bandwidths of the GI-POFs with the refractive index profiles given in Figure 4 were estimated by the ray analysis described by Sohn and Park.13 The bandwidth of a POF is affected by the modal and material dispersions as well as mode coupling and differential mode attenuation (DMA).20-22 The pulse energy injected into an optical fiber is distributed among various allowed modes. Because different modes have different group velocities, the light input spreads as it propagates along the optical fiber. This pulse spreading is called the modal dispersion. The material dispersion represents pulse spreading caused by a finite distribution of the wavelength of the incident light and the wavelength dependence of the refractive index of a material. Mode coupling occurs because of the energy exchange between higher and lower modes, whereas DMA occurs because the input signal may not have uniformly distributed modes affecting the bandwidth characteristics of an optical fiber. The material dispersion is, however, smaller than the modal dispersion by orders of magnitude unless the modal dispersion is extremely small as in the singlemode optical fibers. Mode coupling is also known to be negligible unless the fiber length is longer than 100 m. While DMA appears to have a significant influence, it acts in a way to increase the bandwidth of POF.21 Thus, the current ray analysis, in which only the modal

Figure 8. Impulse response curves for the 100 m long DPS-doped fibers with the refractive index profiles given in Figure 4 (the refractive indices at the center and at the edge of the fibers are 1.500 and 1.492, respectively).

dispersion has been considered, may provide the lower limit of the bandwidth. Thus, it is expected that the actual bandwidth of the POF fabricated by the current method will be greater than the predicted value in this section. Assuming that the input light has a uniform Lambertian distribution,13 the impulse response curves for 100 m long fibers whose refractive index profiles change from 1.500 at the center to 1.492 following the profiles shown in Figure 4 were calculated, and the results are given in Figure 8. As the melt temperature increases, the refractive index profile gets broader, approaching closer to the ideal parabolic profile (Figure 4). This change in the refractive index profile seems to result in the sharpening of the response curve and the leftward shift toward smaller average travel time. The bandwidths, estimated from the response curves as the inverse of the 4 times the root-mean-square widths of the curves, were 587, 616, and 670 Mbps/100 m for the fibers made at the melt temperatures of 188, 204, and 216 °C, respectively. It may be noted that the bandwidth of 670 Mbps/100 m can be achieved by the diffusionassisted coextrusion process. This bandwidth is larger than that obtainable with an SI-POF by a factor of 2 or more, and it exceeds the IEEE 1394-s400 standard. 5. Summary and Conclusion PMMA base GRIN-POFs were made by the diffusionassisted coextrusion process using DPS and Oil Red EGN as the dopants. While the diffusivity of the additive is shown to have the strongest influence in determining the radial concentration profile (or the refractive index profile), it can be adjusted to some extent by manipulating the melt temperature and the flow rate of the material during extrusion. The concentration profiles, which are linearly related with the refractive index profiles, were characterized by FT-IR and UV-vis spectroscopy, and the results were compared with the prediction of the theoretical model by Sohn and Park.13 The diffusivities of the dopants were found to be in the range of 10-7-10-6 cm2/s, which is consistent with the predictions of the free volume theory. The concentration profiles, which are strongly affected by the diffusivities of the dopants, can be controlled to some extent by adjusting the processing temperature and the material residence time. The

Ind. Eng. Chem. Res., Vol. 41, No. 10, 2002 2423 Table 1. Free Volume Parameters Used for the Estimation of Temperature Dependences of Diffusivities Rf (K-1) 3.2 ×

B0 (K-1)

10-4

6.3 ×

additive

v˜ w

Oil Red EGN DPS

10-4

(cm3/mol)

fg

Tg (K)

v˜ 2/ (cm3/mol)

C′2g

0.0168

378

135

52.5

0(0)

v˜ 1

216.21 102.48

(cm3/mol)

281 133.2

bandwidth estimates by the ray analysis indicate that GI-POFs with bandwidths higher than 600 Mbps/100 m can be fabricated by the diffusion-assisted coextrusion process. Considering that the material dispersion and the mode coupling are negligible for POFs shorter than 100 m and that the DMA tends to increase the bandwidth of POF, the bandwidth of the fabricated fibers is expected to be even greater than the predicted value. Acknowledgment The authors thank Mr. Juan C. Cutie of Cyro Industries for providing PMMA samples. They are also grateful to Mr. Gary Scheiffele of the Engineering Research Center at the University of Florida and Dr. J. Hwang at McMaster University in Canada for their assistance in FT-IR and UV-vis measurements. Appendix (Estimation of Diffusivity by the Free-Volume Theory) Diffusion coefficients of small molecules in the polymer matrix are typically in the range of 10-5-10-10 cm2/s at 200-230 °C and are known to show Arrheniustype temperature dependences.18 One of the most successful theories on the diffusion of small molecules may be Cohen and Turnbull’s free volume theory, which correlates the diffusivity with the Maxwell distribution of the free hole volume and molecular volume of solutes. Vrentas and Vrentas23-25 expanded their theory to molecular diffusion in polymeric matrixes, and in the limit of low solute concentration above the glass transition temperature (Tg), their results can be expressed in terms of the Williams-Landel-Ferry (WLF) equation:26

log aT ) log

C′1g(T - Tg) D(T) ) D(Tg) C′2g + T - Tg

(A1)

where aT is the shift factor in the time-temperature superposition principle and the coefficients C′1g and C′2g are related with the properties of polymer and solute by the following equations:

RfB0 C′1g ) ξ fg ln 10

(A2)

C′2g ) fg/Rf

(A3)

Here ξ is the ratio of the critical molar volume of the solute jumping unit to the critical molar volume of the polymer matrix, and it is related with the ratio of the tracer and the matrix molecular jump sizes and shapes. Rf is the difference in the expansion coefficient above and below Tg, and fg is the fractional free volume at Tg defined as the ratio of the average free volume to the molecular volume. According to Vrentas et al.,24 the parameter ξ can be calculated from the following relation if the solute jumps

A/B

ξL

ξ

C′1g

0.342 0.567

2.08 0.987

0.878 0.691

11.528 9.073

as a single unit:

ξ)

ξL 1 + ξL(1 - A/B)

where ξL )

v˜ 10(0) v˜ 2/

(A4)

Here v˜ 10(0) is the molar volume of the equilibrium liquid solute at 0 K, v˜ 2/ is the critical hole free volume per mole of polymer jumping unit required for the displacement of a penetrant jumping unit, and A/B is the aspect ratio of the penetrant molecule. The zero-point molar volume can be estimated using the approximation by Bondi:27

v˜ 10(0) ≈ 1.3 v˜ w

(A5)

where v˜ w is the van der Waals volume. v˜ 2/ is given as 135 cm3/mol for PMMA.24 By differentiation of eq A1, the temperature dependence of the diffusivities can be given as

C′1gC′2gT 2 d log D )d(1/T) (C′2g + T - Tg)2

(A6)

The aspect ratio of the molecule in eq A4 was calculated as follows for the present study. The most stable conformation of the diffusive molecule was obtained first by minimizing the energy of the molecule with GAMESS (ab initio computation software) along with Chem3D (CambridgeSoft, Inc.), and then two principal axes of that molecule were aligned with X and Y axes in Chem3D space. Length-to-breadth ratios were calculated by drawing rectangles that completely enclose the molecule with minimum areas in X-Y and X-Z planes following Rohrbaugh and Jurs’ approach.28 According to Berens and Hopfenberg,29 the motion of small anisometric molecules through a polymer matrix occurs preferentially in a direction that minimizes the displacement required of the polymer chains. In other words, anisometric molecules are oriented and move primarily along their long axes during diffusion through a polymer. They showed experimentally that the diffusion of anisometric penetrants is governed by the dimension that is smaller than its equivalent spherical diameter. In this perspective, the aspect ratio of a molecule was calculated as the geometric mean of the two length-to-breadth ratios. The free volume parameters determined by the prescribed procedure for the present study are listed in Table 1, and the Arrhenius plots for the diffusivities of DPS and Oil Red EGN are given in Figure 7. The solid line in the figure represents the diffusivity estimate by the free volume theory using the parameters in Table 1 and the diffusivity at 188 °C determined by curve fitting of the data given in Figure 4 (data set a) and Figure 6a. The open symbols in Figure 7 represent the diffusivity values that have been used in Figure 4 for data sets b and c and Figure 6b,c.

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The diffusivities of Oil Red EGN are apparently about 2 times smaller than those of DPS. Considering that the molecular weight of Oil Red EGN is 2.3 times larger than that of DPS, these differences are expected because the diffusivity is approximately proportional to the inverse of the square root of the molecular weight.30 Because of the smaller diffusivity, the Oil Red EGNdoped GRIN POF shows a steeper concentration profile than DPS at the same temperature. In calculation of the diffusivity of DPS, the glass transition temperature was taken to be different from that of the pure polymer because of the plasticization effect of small molecules. The glass transition temperature of a polymer solution can be estimated by31

Tg )

Tgp + (KTgs - Tgp)φs 1 + (K + 1)φs

(A7)

where the subscripts p and s denote polymer and solvent, respectively, φ is the volume fraction, and K is the ratio of the difference in the thermal expansion coefficients of solvent above and below Tg to that of polymer:

K)

Rls - Rgs Rlp - Rgp

(A8)

Tg of PMMA containing DPS at 6.5 wt % and that containing Oil Red EGN at 230 ppm were estimated to be 80 and 105 °C, respectively. Literature Cited (1) Senior, J. Optical Fiber Communications; Prentice Hall International: London, U.K., 1985. (2) Halley, P. Fiber Optic Systems; John Wiley & Sons: New York, 1987. (3) Koike, Y. High-Bandwidth Graded-Index Polymer Optical Fiber. Polymer 1991, 32, 1737. (4) Koike, Y.; Ishigure, T.; Nihei, E. High-Bandwidth GradedIndex Polymer Optical Fiber. J. Lightwave Technol. 1995, 13, 1475. (5) Ishigure T.; Horibe, A.; Nihei, E.; Koike, Y. High-Bandwidth, High-Numerical Aperture Graded-Index Polymer Optical Fiber. J. Lightwave Technol. 1995, 13, 1686. (6) Ishigure, T.; Nihei, E.; Koike, Y. Optimum Refractive-Index Profile of the Graded-Index Polymer Optical Fiber, Toward Gigabit Data Links. Appl. Opt. 1996, 35, 2048. (7) Ishigure, T.; Sato, M.; Nihei, E.; Koike, Y. Graded-Index Polymer Optical Fiber with High Thermal Stability of Bandwidth. Jpn. J. Appl. Phys. 1998, 37, 3986. (8) Van Duijnhoven, F. G. H.; Bastiaansen, C. W. M. Monomers and Polymers in a Centrifugal Field: A New Method to Produce Refractive-Index Gradients in Polymers. Appl. Opt. 1999, 38, 1008. (9) Park, C. W.; Lee, B. S.; Walker, J. K.; Choi, W. Y. A New Processing Method for the Fabrication of Cylindrical Objects with Radially Varying Properties. Ind. Eng. Chem. Res. 2000, 39, 79.

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Received for review November 6, 2001 Revised manuscript received March 7, 2002 Accepted March 7, 2002 IE010902O