Model for the temperature dependent transmission of optically transparent poly(methyl methacrylate) composites
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James O. Staffer Department of Chemistry, Graduate Center for Materials Research, University of Missouri-Rolla, Rolla, Missouri 65401-0249 (Received 18 February 1992; accepted 21 October 1992)
An infinite cylinder model was used to predict the optical transmission of transparent composites containing unidirectionally aligned glass fibers. The parameters used in the model are the volume content and diameter of the glass fibers, the refractive index of the fiber and matrix, the nonwet fiber content, the thickness of the composite, and the temperature coefficient of the refractive index of the matrix. The transmission calculated from the model agreed well with the measured temperature-dependent transmission of a composite from 20 to 70 °C for thin specimen ( 0 and e 0 = 1, x = 2irRg/ \ and A is the wavelength in the PMMA matrix, m = ng/nP is the relative refractive index of the glass fiber (ng) and PMMA (nP), Jn is a Bessel function of the first kind, Hn is a Hankel function of the second kind, and the primes denote differentiation of the functions with respect to their arguments. The refractive index of PMMA at any temperature above 25 °C and below Tg (100 °C) is nP = 1.4913 1.1 X 10" 4 (r - 25), provided that a refractive index, nD, of 1.4913 at 25 °C and a dn/dT = - 1 . 1 X 10" 4 /°C is assumed. In comparison, the change in refractive index for the glass fiber is negligible (dn/dT =
+3.5 X 10~ 6 /°C) over this small temperature range. The relative refractive index of the glass fiber (nD = 1.4907) and PMMA at any temperature below Tg of the PMMA is m = m(T) = 1.4907/[1.4913 - 1.1 X 10~ 4 (r - 25)]. (9) The scattering efficiency of a composite, Qg = Qg(T), and the transmission is a function of temperature because of the temperature-dependent relative refractive index. Equation (1) can be rewritten as: It{T)/h =
(10)
exp(-
According to Eq. (1), the optical transmission for a PMMA composite containing glass fibers will vary exponentially with the volume fraction of fibers (V/,g), the radius of the glass fibers (Rg), the optical path length (h, thickness of the composite), and the temperature of the composite. III. EXPERIMENTAL The glass fiber for reinforcing the PMMA matrix was produced from an optical glass (BK10, Schott Glass Technologies Inc., PA) whose refractive index, nD, in bulk form is 1.49776 ± 0.00001. The glass was remelted and fired in a resistance-heated Pt/Rh ten-hole bushing at 1250 °C for 45 min to remove bubbles. When fibers were drawn at 1204 °C and at a speed of 4.0 m/s, the average diameter was 17 ^im with a standard deviation of 2.3 (iim. Because of rapid cooling, these fibers had a lower refractive index (nD) of 1.4907, with a standard deviation of 0.0002. Strips of fibers were prepared by moving a collection drum at a controlled speed in a direction normal to the pulling direction for the fibers. No coating was applied to the fibers. Composite specimens with unidirectionally aligned glass fibers were prepared by a sheet lamination method. A strip of fibers was rinsed with MMA (methyl methacr
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