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Opto-thermal properties of fibers
Polymer Testing 18 (1999) 235–247
Material Properties
Opto-thermal properties of fibers 6. Evaluation of some optical structural parameters obtained due to annealing nylon 6 fiber I. M. Fouda*, A. H. Oraby Physics Department of Science, Mansoura University, Mansoura, Egypt Received 10 November 1997; accepted 12 February 1998
Abstract In the present work nylon 6 fibers were annealed at a constant temperature 160 ⫾ 1°C and different times. The Pluta polarizing interference microscope and an acoustic method were used previously for measuring some optical parameters, and the densities of these fibers. The results obtained were utilized for calculations of polarizabilities per unit volume, the isotropic refractive index by utilizing the Lorentz– Loranz equation, and the number of molecules per unit volume. The density results were used to calculate the degree of crystallinity and the mean density fluctuation at different annealing conditions. Also, some structural parameters such as the virtual refractive index, the harmonic mean polarizability of the dielectric, the harmonic mean specific refractivity and the number of the monomeric units per unit volume were obtained. Relationships between the structural parameters with different times are given for these fibers. The generalized Lorentz–Loranz equation given by de Vries is used to determine nylon 6 fiber structural parameters. Comparison between Hermans optical orientation function formula and the corrected formula by de Vries are given. Illustration is given using graphs and microinterferograms. 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction The most readily available techniques for changing the physical properties of polymeric structure are annealing and quenching processes.[1–10] Thermal treatments are used to vary the degree * Corresponding author. Fax: ⫹ 20-50-346-781; e-mail: [email protected] 0142-9418/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 9 4 1 8 ( 9 8 ) 0 0 0 1 0 - 5
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of crystalinity and other physical properties in polymeric materials. In order to elucidate the structural variations induced in fibers by any physical or chemical modification, the use of interferometric and acoustic methods are very useful tools. The orientation factors of fibers are well established, and often evaluated from the optical birefringence[11–13] or sonic modulus.[14– 16] The density method[17] is undoubtedly employed most frequently for the determination of fiber crystallinity. Recently, the interferometric methods applied to fibrous materials have been discussed extensively by many authors.[18,19] For optically anisotropic fibers, the refractive index and the double refraction are parameters characterizing the structure of the material. The birefringence of fibers arises from the orientation of the polymer molecules along the fiber axis averaged over the crystalline and non crystalline region of the fiber. This molecular orientation influences not only the mechanical properties but also other physical properties of the yarn, such as density and uptake.[20] In the present work the optical parameters and density results for samples of nylon 6 having different annealing conditions (previously measured using two-beam interferometry and acoustic techniques)[21,22] are utilized to calculate some other structural parameters.
2. Theoretical considerations Using the Pluta interference microscope in the case of totally duplicated image of the fiber to measure the mean refractive indices in parallel and perpendicular directions of nylon 6 was discussed in detail elsewhere.[22] Also, by applying the equations used by Hermans and Wards[23,24] we can obtain the optical orientation parameters as previously discussed in extensive works.
3. Mean polarizability of monomer unit The polarizability of a monomer unit like the polarizability of a simple organic molecule, usually differs in different directions. As the refractive index of a polymer depends on the total polarizability of the molecules, this leads to the Lorentz–Loranz by the following equations:[25] n2储 ⫺ 1 N(1)␣储 ⫽ . n2储 ⫹ 2 3⑀0
(1)
An analogous formula is used for nⲚa , where n储a and nⲚa are the mean refractive indices of the fiber for light vibrating parallel and perpendicular to the fiber axis respectively, and ␣储, ␣Ⲛ are the monomer polarizability of monomer units. n¯2 ⫺ 1 N(1)␣¯ ⫽ n¯2 ⫹ 2 3⑀0
(2)
where n¯ is the average refractive index (isotropic refractive index), ␣¯ the mean polarizability of a monomer unit and ⑀0 is the permitivity of free space ⫽ 8.85 ⫻ 10−12 F m−1. For a bulk polymer
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237
of density and monomer unit molecular weight M, the number of monomer units per unit volume, which also equal the number of carries of the dipole moment N(1) ⫽ (NA)/M where NA is Avogadro’s number 6.02 ⫻ 1023 and M for nylon 6 ⫽ 113.16.[26] Also N is the number of carriers of the dipole moment. De Vries gave a theory on the basis of internal field with the aid of classical electromagnetic theory, in which he generalized the Lorentz–Loranz equation, so for monochromatic light, the well known Lorentz–Loranz becomes[26] n2 ⫺ 1 N(1)␣ ⫽ . n2 ⫹ 2 3⑀0
(3)
The right-hand side of Eq. (3) is proportional to the density , [kg/m3] of the medium and may also be written n2 ⫺ 1 ⫽ ⑀ n2 ⫹ 2
(4)
where ⑀, [m3/kg] is called the specific refractivity of the isotropic dielectric. Writing this equation for fibers in its parallel and transverse components the generalized Lorentz–Loranz equation becomes: n2储 ⫺ 1 N(1)␣储 ⫽ ⫽ ⑀储. n2储 ⫹ 2 3⑀0
(5)
An analogous formula used for nⲚa . Also, de Vries defined the invariant refractive index, which he call the ‘Virtual refractive index’ nv by nv ⫽
冪
1⫹
3[n2储 ⫺ 1][n2Ⲛ ⫺ 1] [n2Ⲛ ⫺ 1] ⫹ 2[n2储 ⫺ 1]
(6)
where the virtual refractive index nv replaces the mean refractive index n¯ ⫽ 13 (n储a ⫹ 2nⲚa ) which is known as isotropic refractive index, and the above equations leads to the harmonic mean polarizability of the dielectric ␣v by the following equation:
␣v ⫽
3⑀0 n2v ⫺ 1 . · N(1) n2v ⫹ 2
(7)
Likewise, for the harmonic mean specific refractivity, we have
⑀v ⫽ −1·
n2v ⫺ 1 . n2v ⫹ 2
(8)
In a recent approach to the continuum theory of birefringence of oriented polymer,[26] it was found that F ⫽
冋 册冋
册
n21n22 n储 ⫹ nⲚ ⌬na · · . n2储 n2Ⲛ n1 ⫹ n2 ⌬nmax
(9)
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This is slightly different from the original simple expression for the degree of orientation in the following equation: (Opt. Ori. Fun.) F⌬ ⫽
⌬n ⌬nmax
(10)
where ⌬nmax is maximum birefringence for fully oriented fiber and ⌬na is the measured mean birefringence of the fiber under investigation, values ranged between ⫹ 1, 0, ⫺ 0.5 according to the state of orientation: perfect, random or perpendicular to the fiber axis, respectively. The value of ⌬nmax has been previously determined to be (0.072).[26] Also, F (cf. Hermans[28] and Platzek and Kratky[29]) can be given by the following equation F ⫽ (1 ⫹ a)F⌬ ⫺ aF2⌬(1 ⫹ a) ⫽
2n21n22 n (n1 ⫹ n2) 3 v
(11)
where n1, n2 are the refractive indices of fully oriented fiber. Using monochromatic light vibrating parallel and perpendicular to the fiber axis it was found that nv ⬵ niso from Eq. (6), the constant (a) was calculated and found to be (0.611).
4. Crystallinity equations[27] The degree of crystallinity, , was calculated from:
⫽ ( ⫺ a)/⌬
(12)
where ⌬ ⫽ (c ⫺ a), c and a are the densities of the noncrystallinity and crystallinity regions, is the experimental measured value of density, c ⫽ 1.23 g/cm3 [30] and a ⫽ 1.11 g/cm2.[30] Volume fraction of amorphous material was determined by the relation 1 ⫺ ⫽ 1 ⫺ ( ⫺ a)/(c ⫺ a)
(13)
where is the volume fraction of crystalline material. For a two-phase structure consisting of amorphous and crystalline regions with densities a and c, respectively, the mean square density fluctuation, 具2典, can be calculated from the following equation.[31] 具2典 ⫽ (a ⫺ c)2 (1 ⫺ ).
(14)
The value of c, , (1 ⫺ ) and 具2典 for nylon 6 fiber are given in Table 2.
5. Calculation of the isotropic refractive index The isotropic refractive index is given by niso(1) ⫽
n储a ⫹ 2nⲚa . 3
(15)
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The specific volume is given by the following equation (niso ⫺ 1)V ⫽ const.
(16)
Also the Lorentz–Loranz equation is used to relate polarizabilities and refractive index. We have:[32] n2iso(2) ⫺ 1 n2iso(2) ⫹ 2
⫽
冋
1 i n2 ⫺ 1 n2Ⲛ ⫺ 1 ⫹ 2 3 n2 ⫹ 2 n2Ⲛ ⫹ 2
册
(17)
where and i are the densities from measurement and of the isotropic polymer respectively, to estimate a relation showing the crystallization inhomogenity for some types of polymers. Obtained values n储a, nⲚa and are used with Eq. (17) to determine the isotropic refractive index values for annealed nylon 6 fibers, where i ⫽ a ⫽ 1.11 ⫻ 103 kg/m3.
6. Generalization of the Lorentz–Loranz equation The generalization of the Lorentz–Loranz equation given by de Vries when he considered the anisotropy of polarization, take in account the anisotropy index S, where S 储 ⫽ S and S Ⲛ ⫽ ( ⫺ 1/2)S, and the bounds of the anisotropy index ⫺ 1 ⱕ S ⱕ ⫹ 2. For fibers, in the parallel direction n2储 ⫺ 1 ⫽ ⑀储 n2储 ⫹ 2 ⫹ S(n2储 ⫺ 1)
(18a)
n2Ⲛ ⫺ 1 ⫽ ⑀Ⲛ n2Ⲛ ⫹ 2 ⫺ S(n2Ⲛ ⫺ 1)
(18b)
and
where ⑀i the specific refractivity in a certain direction i, and is the density.
7. Calculation number of molecules per unit volume The difference of the two main refractive indices of the sample for the beams polarized respectively along the axes is linked with the difference in the mean polarizabilities of the macromolecules for the same direction (P储 ⫺ PⲚ) by the relation[33] ⌬n ⫽
冉 冊
2N2 n¯2 ⫹ 2 (P¯储 ⫺ P¯Ⲛ) n 3
(19)
where N2 is the number of molecules per unit volume, n¯ is the mean refractive index of the sample, and
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P¯储 ⫽
冉 冊
3 n2储 ⫺ 1 4 n2储 ⫹ 2
(20)
where P¯储a is the polarizabilities per unit volume parallel and an analogous equation in the perpendicular direction.
8. Experimental results and discussion 8.1. Annealing of samples For the annealing conditions used we followed the procedures outlined in previous publications.[8,19]
9. Results 9.1. Measurements of transverse sectional area for fibers The cross-section of the nylon 6 fiber was viewed by high power optical microscopy and it was circular. 9.2. Application of two-beam interferometry The totally duplicated image of the fiber obtained with the Pluta polarizing interference microscope was used to calculate the mean refractive indices n储a and nⲚa of nylon 6 fiber.[22,34,35] Fig. 1(a–e) shows a microinterferogram of non-duplicated images of nylon 6 fiber using the Pluta microscope with annealing time at constant temperature 160 ⫾ 1°C. Fig. 2(a–e) shows a microinterferogram of totally duplicated images of nylon 6 fiber using the Pluta microscope with annealing time at constant temperature 160 ⫾ 1°C. Using these, the mean refractive index of the parallel and perpendicular directions at different annealing temperatures and constant annealing time were calculated. The refractive index of the immersion liquid was selected to allow the fringe shift to be small. Figs. 1 and 2 also show that the fringes shifts changed as the time of annealing increases at constant temperature. Fig. 3 shows the relation between the birefringence ⌬n and the refractive indices differences (n储 ⫺ nv, nⲚ ⫺ nv, n储 ⫺ niso, and nⲚ ⫺ niso). From Fig. 3, nv, and ⌬nmax are used to predict the values of refractive indices n1 and n2 for fully oriented fibers. These values are found to be 1.608 and 1.536, respectively, at 28°C, Table 1 give some experimental results for refractive indices and the apparent volume fraction of crystallinity (), which was calculated from Eq. (12) using the calculated density values. The following calculated values at different annealing times are given in Table 2: virtual refractive index (nv), isotropic refractive index (calculated by two methods using Eq. (15) and Eq. (17) respectively), optical orientation function f and the Hermans function f⌬, the number of monomer units per unit volume, which also equal the number of carriers of the dipole moment (N1), and the number of molecules per unit volume (N2). Also in Table 3,
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Fig. 1. (a)–(e) Microinterfrograms of non-duplicated image of nylon 6 fiber which annealed at 160°C ( ⫽ 546 nm).
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Fig. 2. (a)–(e) Microinterfrograms of totally duplicated image of nylon 6 fiber which annealed at 160°C ( ⫽ 546 nm).
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Fig. 3. Relations between the birefringence ⌬n and the refractive indices differences (n储 ⫺ nv, nⲚ ⫺ nv, n储 ⫺ niso and nⲚ ⫺ niso) of nylon 6 fiber at 1 ⫾ 160°C.
Table 1 Annealing time n储a (hr) Unannealed 1 2 3 4 5 6 7 8 9 10
1.576 1.582 1.587 1.589 1.591 1.593 1.595 1.597 1.597 1.597 1.597
nⲚa 1.535 1.531 1.531 1.531 1.531 1.535 1.535 1.537 1.537 1.537 1.537
⌬na 0.041 0.051 0.056 0.058 0.060 0.058 0.060 0.060 0.060 0.060 0.060
1.12 1.14 1.172 1.21 1.19 1.18 1.154 1.235 1.228 1.20 1.17
%
具2典 ⫻ 10−2%
8.85 26.5 54.9 88.5 70.8 61.9 38.9 100 100 79.6 53.1
99.8 99.8 99.7 99.9 99.7 99.7 99.7 100 100 99.8 99.7
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Table 2 Annealing time (hr) Unannealed 1 2 3 4 5 6 7 8 9 10
niso(1) 1.549 1.548 1.550 1.550 1.551 1.554 1.555 1.557 1.557 1.557 1.557
niso(2) 1.543 1.531 1.515 1.496 1.507 1.515 1.529 1.490 1.493 1.507 1.523
nv 1.548 1.547 1.549 1.549 1.550 1.553 1.554 1.554 1.554 1.554 1.554
f⌬
N(1) ⫻ 1021
f
0.5694 0.7083 0.7778 0.8056 0.8333 0.8056 0.8333 0.8333 0.8333 0.8333 0.8333
0.6101 0.7576 0.8280 0.8560 0.8838 0.8494 0.8771 0.8737 0.8737 0.8737 0.8737
5.96 6.06 6.23 6.44 6.33 6.28 6.14 6.57 6.53 6.38 6.22
N(2) 0.972 0.965 0.961 0.960 0.959 0.960 0.959 0.959 0.959 0.959 0.959
Table 3 Annealing time (hr) Unannealed 1 2 3 4 5 6 7 8 9 10
␣储 ⫻ 10−34 ␣Ⲛ ⫻ 10−34 14.75 14.61 14.31 13.90 14.17 14.33 14.69 13.77 13.85 14.17 14.53
13.87 13.54 13.18 12.76 12.98 13.17 13.46 12.62 12.69 12.99 13.32
␣¯ ⫻ 10−34 14.31 14.08 13.74 13.33 13.57 13.75 14.08 13.19 13.27 13.58 13.93
␣v ⫻ 10−34 14.15 13.88 13.53 13.12 13.35 13.53 13.85 12.98 13.06 13.36 13.70
⑀储 ⫻ 10−2 25.49 25.23 24.69 23.97 24.43 24.70 25.32 23.71 23.85 24.40 25.03
⑀Ⲛ ⫻ 10−2 24.18 23.63 22.98 22.26 22.64 22.95 23.47 21.99 22.11 22.63 23.21
⑀¯ ⫻ 10−2 24.84 24.43 23.84 23.12 23.54 23.83 24.39 22.85 22.98 23.52 24.12
⑀V ⫻ 10−2 24.61 24.14 23.53 22.81 23.21 23.51 24.06 22.54 22.66 23.19 23.79
the calculated values of ␣储, ␣Ⲛ, ␣¯ , ␣v and specific refractivity of the isotropic dielectric in parallel and perpendicular directions and its mean value (⑀储, ⑀Ⲛ, ⑀¯ and ⑀v) at different annealing times are given. Fig. 4 shows the relation between corrected values of optical orientation function f and the Hermans function f⌬, from which f/f⌬ is found. Fig. 5 shows the relation between the birefringence ⌬n and the optical orientation function f. 10. Discussion Studies of polymer thermal effects on the optical properties of fibers gives much useful information about molecular structure changes and orientation. Changes in orientation are accompanied by changes in crystallinity due to the annealing process; this indicates mass redistribution within
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Fig. 4. Relation between corrected values of optical orientation function f and the Hermans function f⌬ of nylon 6 fiber at 1 ⫾ 160°C.
the fiber chain. So, information about the effect of annealing on the microstructure and morphology of polymer materials has been a subject of major technological and scientific interest for a number of years. Characterization of the opto-thermal structural parameters of fibers are important for the textile industry and the end use. The present results in Table 2 shows changes in the isotropic refractive indices related to the crystalline and amorphous regions of the fibers and to the density of the sample, which is related to Eq. (17). The results show that the overall changes in density are important influences on the optical parameters, orientation, crystallinity etc. 11. Conclusion From the analysis of the results obtained one can conclude the following. 1. The combination of the principal refractive indices and the density measurements throw light on the variations occurring during the annealing process on the different structural parameters as shown (Tables 1–3). 2. Comparison, between the obtained values of niso1, niso2 and nv show that every equation has its own merit based on considerations of its derivation and applications.
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Fig. 5. Relation between birefringence ⌬na and of optical orientation function f of nylon 6 fiber at 1 ⫾ 160°C.
3. It was found that the value of (a) is constant whatever the annealing time. 4. It is possible to explain with the aid of the obtained structural results the variation in the enduses of textiles which are influenced by optical properties (buster, light reflection, and absorption etc.) and thermal treatments.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
Haward W, Starkweather JR, George EM, Hansan JE, Roder TM, Brooks RE. J Polym Sci 1956;1:201. Wycoff H. J Polym Sci 1962;62:82. William G, Perkins P, Porter SR. J Mater Sci 1977;2355. Decondia F, Vittoria V. J Polym Sci Phys. Ed. 1985;23:1217. Statton WO. J Polym Sci 1972;A–210:1587. Hamza AA, El-Tonsy MM, Fouda IM, El-Said AM. J Appl Polym Sci 1995;57:265. Fouda IM, El-Nicklawy M, Naser EM, El-Agamy RM. J Appl Polym Sci 1996;60:1269. Fouda IM, Seisa EA. J Polym Polym Compos 1996;4(7):489. Hamza AA, Fouda IM, Sokkar TZN, El-Bakary MA. Polymer International 1996;39:129. Fouda IM, Seisa EA. J Polym Polym Compos 1996;4(4):247. Ward IM. Mechanical Properties of Solid Polymers, 2nd edn. New York: Wiley, 1983. Ward IM. Structure and Properties of Oriented Polymers. London: Applied Science, 1975.
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Cunningham A, Ward IM, Willis HA, Zichy V. Polymer 1974;15:749. Morgan HM. Text Res J 1962;32:866. Dumbleton JH. J Polym Sci 1968;A–2:795. Samules JR. Structural Polymer Properties. New York: Wiley, 1974. Wunderlich B. Macromolecular Physics. New York: Academic Press, 1973:1. Barakat N, Hamza AA. Interferometery of Fibrous Materials. Hilger: UK, 1990. Fouda IM, Kabeel MA, El-Sharkawy J. J Polym Polym Compos 1997;5(3):203. Angad Gaur H, De Vries H. J Polym Sci Phys Ed 1975;13:835. Fouda IM, El-Tonsy MM, Shaban AM. J Mater Sci 1990;26:5085. Fouda IM, Seisa EA, El-Farahaty KA. Polymer Testing 1996;15:3. Hermans PH. Contributions to the Physics of Cellulose Fibers. Amsterdam: North Holland, 1946. Ward IM. Proc Phys Soc Lond 1962;80:1176. Jenkins AD. Polymer Science: A Materials Science Handbook 1, Chap 74 and 7. Amsterdam: North Holland, 1972. De Vries H. Z. Colloid, Polym Sci 1979;257:226. Bourvelles GL, Beautemps J. J Appl Polym Sci 1990;39:352. Hermans PH. P Platzek, Kolloid-Z, 1939:88. Kratky O. Kolloid-Z. 1933;64:213. Bodor G. Structural Investigation of Polymers, Chap 7. 1991:326. Fischer EW, Fakirov S. J Mater Sci 1976;11:1041. Zachariodes AZ, Porter S. The Strength and Stiffness of Polymers. Marcel Dekker: New York, 1983:121. Sarkisyan VA, Asratyan MG, Mkhitoryan AA, Katrdzhyan KKH, Dadivanyan AK. Vysokomal. Sayed. 1985;A27(6):1331. [34] Pluta M. J Optica Acta 1971;18:661. [35] Pluta M. J Microsc 1972;96:309.
Material Properties
Opto-thermal properties of fibers 6. Evaluation of some optical structural parameters obtained due to annealing nylon 6 fiber I. M. Fouda*, A. H. Oraby Physics Department of Science, Mansoura University, Mansoura, Egypt Received 10 November 1997; accepted 12 February 1998
Abstract In the present work nylon 6 fibers were annealed at a constant temperature 160 ⫾ 1°C and different times. The Pluta polarizing interference microscope and an acoustic method were used previously for measuring some optical parameters, and the densities of these fibers. The results obtained were utilized for calculations of polarizabilities per unit volume, the isotropic refractive index by utilizing the Lorentz– Loranz equation, and the number of molecules per unit volume. The density results were used to calculate the degree of crystallinity and the mean density fluctuation at different annealing conditions. Also, some structural parameters such as the virtual refractive index, the harmonic mean polarizability of the dielectric, the harmonic mean specific refractivity and the number of the monomeric units per unit volume were obtained. Relationships between the structural parameters with different times are given for these fibers. The generalized Lorentz–Loranz equation given by de Vries is used to determine nylon 6 fiber structural parameters. Comparison between Hermans optical orientation function formula and the corrected formula by de Vries are given. Illustration is given using graphs and microinterferograms. 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction The most readily available techniques for changing the physical properties of polymeric structure are annealing and quenching processes.[1–10] Thermal treatments are used to vary the degree * Corresponding author. Fax: ⫹ 20-50-346-781; e-mail: [email protected] 0142-9418/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 9 4 1 8 ( 9 8 ) 0 0 0 1 0 - 5
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of crystalinity and other physical properties in polymeric materials. In order to elucidate the structural variations induced in fibers by any physical or chemical modification, the use of interferometric and acoustic methods are very useful tools. The orientation factors of fibers are well established, and often evaluated from the optical birefringence[11–13] or sonic modulus.[14– 16] The density method[17] is undoubtedly employed most frequently for the determination of fiber crystallinity. Recently, the interferometric methods applied to fibrous materials have been discussed extensively by many authors.[18,19] For optically anisotropic fibers, the refractive index and the double refraction are parameters characterizing the structure of the material. The birefringence of fibers arises from the orientation of the polymer molecules along the fiber axis averaged over the crystalline and non crystalline region of the fiber. This molecular orientation influences not only the mechanical properties but also other physical properties of the yarn, such as density and uptake.[20] In the present work the optical parameters and density results for samples of nylon 6 having different annealing conditions (previously measured using two-beam interferometry and acoustic techniques)[21,22] are utilized to calculate some other structural parameters.
2. Theoretical considerations Using the Pluta interference microscope in the case of totally duplicated image of the fiber to measure the mean refractive indices in parallel and perpendicular directions of nylon 6 was discussed in detail elsewhere.[22] Also, by applying the equations used by Hermans and Wards[23,24] we can obtain the optical orientation parameters as previously discussed in extensive works.
3. Mean polarizability of monomer unit The polarizability of a monomer unit like the polarizability of a simple organic molecule, usually differs in different directions. As the refractive index of a polymer depends on the total polarizability of the molecules, this leads to the Lorentz–Loranz by the following equations:[25] n2储 ⫺ 1 N(1)␣储 ⫽ . n2储 ⫹ 2 3⑀0
(1)
An analogous formula is used for nⲚa , where n储a and nⲚa are the mean refractive indices of the fiber for light vibrating parallel and perpendicular to the fiber axis respectively, and ␣储, ␣Ⲛ are the monomer polarizability of monomer units. n¯2 ⫺ 1 N(1)␣¯ ⫽ n¯2 ⫹ 2 3⑀0
(2)
where n¯ is the average refractive index (isotropic refractive index), ␣¯ the mean polarizability of a monomer unit and ⑀0 is the permitivity of free space ⫽ 8.85 ⫻ 10−12 F m−1. For a bulk polymer
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237
of density and monomer unit molecular weight M, the number of monomer units per unit volume, which also equal the number of carries of the dipole moment N(1) ⫽ (NA)/M where NA is Avogadro’s number 6.02 ⫻ 1023 and M for nylon 6 ⫽ 113.16.[26] Also N is the number of carriers of the dipole moment. De Vries gave a theory on the basis of internal field with the aid of classical electromagnetic theory, in which he generalized the Lorentz–Loranz equation, so for monochromatic light, the well known Lorentz–Loranz becomes[26] n2 ⫺ 1 N(1)␣ ⫽ . n2 ⫹ 2 3⑀0
(3)
The right-hand side of Eq. (3) is proportional to the density , [kg/m3] of the medium and may also be written n2 ⫺ 1 ⫽ ⑀ n2 ⫹ 2
(4)
where ⑀, [m3/kg] is called the specific refractivity of the isotropic dielectric. Writing this equation for fibers in its parallel and transverse components the generalized Lorentz–Loranz equation becomes: n2储 ⫺ 1 N(1)␣储 ⫽ ⫽ ⑀储. n2储 ⫹ 2 3⑀0
(5)
An analogous formula used for nⲚa . Also, de Vries defined the invariant refractive index, which he call the ‘Virtual refractive index’ nv by nv ⫽
冪
1⫹
3[n2储 ⫺ 1][n2Ⲛ ⫺ 1] [n2Ⲛ ⫺ 1] ⫹ 2[n2储 ⫺ 1]
(6)
where the virtual refractive index nv replaces the mean refractive index n¯ ⫽ 13 (n储a ⫹ 2nⲚa ) which is known as isotropic refractive index, and the above equations leads to the harmonic mean polarizability of the dielectric ␣v by the following equation:
␣v ⫽
3⑀0 n2v ⫺ 1 . · N(1) n2v ⫹ 2
(7)
Likewise, for the harmonic mean specific refractivity, we have
⑀v ⫽ −1·
n2v ⫺ 1 . n2v ⫹ 2
(8)
In a recent approach to the continuum theory of birefringence of oriented polymer,[26] it was found that F ⫽
冋 册冋
册
n21n22 n储 ⫹ nⲚ ⌬na · · . n2储 n2Ⲛ n1 ⫹ n2 ⌬nmax
(9)
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This is slightly different from the original simple expression for the degree of orientation in the following equation: (Opt. Ori. Fun.) F⌬ ⫽
⌬n ⌬nmax
(10)
where ⌬nmax is maximum birefringence for fully oriented fiber and ⌬na is the measured mean birefringence of the fiber under investigation, values ranged between ⫹ 1, 0, ⫺ 0.5 according to the state of orientation: perfect, random or perpendicular to the fiber axis, respectively. The value of ⌬nmax has been previously determined to be (0.072).[26] Also, F (cf. Hermans[28] and Platzek and Kratky[29]) can be given by the following equation F ⫽ (1 ⫹ a)F⌬ ⫺ aF2⌬(1 ⫹ a) ⫽
2n21n22 n (n1 ⫹ n2) 3 v
(11)
where n1, n2 are the refractive indices of fully oriented fiber. Using monochromatic light vibrating parallel and perpendicular to the fiber axis it was found that nv ⬵ niso from Eq. (6), the constant (a) was calculated and found to be (0.611).
4. Crystallinity equations[27] The degree of crystallinity, , was calculated from:
⫽ ( ⫺ a)/⌬
(12)
where ⌬ ⫽ (c ⫺ a), c and a are the densities of the noncrystallinity and crystallinity regions, is the experimental measured value of density, c ⫽ 1.23 g/cm3 [30] and a ⫽ 1.11 g/cm2.[30] Volume fraction of amorphous material was determined by the relation 1 ⫺ ⫽ 1 ⫺ ( ⫺ a)/(c ⫺ a)
(13)
where is the volume fraction of crystalline material. For a two-phase structure consisting of amorphous and crystalline regions with densities a and c, respectively, the mean square density fluctuation, 具2典, can be calculated from the following equation.[31] 具2典 ⫽ (a ⫺ c)2 (1 ⫺ ).
(14)
The value of c, , (1 ⫺ ) and 具2典 for nylon 6 fiber are given in Table 2.
5. Calculation of the isotropic refractive index The isotropic refractive index is given by niso(1) ⫽
n储a ⫹ 2nⲚa . 3
(15)
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The specific volume is given by the following equation (niso ⫺ 1)V ⫽ const.
(16)
Also the Lorentz–Loranz equation is used to relate polarizabilities and refractive index. We have:[32] n2iso(2) ⫺ 1 n2iso(2) ⫹ 2
⫽
冋
1 i n2 ⫺ 1 n2Ⲛ ⫺ 1 ⫹ 2 3 n2 ⫹ 2 n2Ⲛ ⫹ 2
册
(17)
where and i are the densities from measurement and of the isotropic polymer respectively, to estimate a relation showing the crystallization inhomogenity for some types of polymers. Obtained values n储a, nⲚa and are used with Eq. (17) to determine the isotropic refractive index values for annealed nylon 6 fibers, where i ⫽ a ⫽ 1.11 ⫻ 103 kg/m3.
6. Generalization of the Lorentz–Loranz equation The generalization of the Lorentz–Loranz equation given by de Vries when he considered the anisotropy of polarization, take in account the anisotropy index S, where S 储 ⫽ S and S Ⲛ ⫽ ( ⫺ 1/2)S, and the bounds of the anisotropy index ⫺ 1 ⱕ S ⱕ ⫹ 2. For fibers, in the parallel direction n2储 ⫺ 1 ⫽ ⑀储 n2储 ⫹ 2 ⫹ S(n2储 ⫺ 1)
(18a)
n2Ⲛ ⫺ 1 ⫽ ⑀Ⲛ n2Ⲛ ⫹ 2 ⫺ S(n2Ⲛ ⫺ 1)
(18b)
and
where ⑀i the specific refractivity in a certain direction i, and is the density.
7. Calculation number of molecules per unit volume The difference of the two main refractive indices of the sample for the beams polarized respectively along the axes is linked with the difference in the mean polarizabilities of the macromolecules for the same direction (P储 ⫺ PⲚ) by the relation[33] ⌬n ⫽
冉 冊
2N2 n¯2 ⫹ 2 (P¯储 ⫺ P¯Ⲛ) n 3
(19)
where N2 is the number of molecules per unit volume, n¯ is the mean refractive index of the sample, and
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P¯储 ⫽
冉 冊
3 n2储 ⫺ 1 4 n2储 ⫹ 2
(20)
where P¯储a is the polarizabilities per unit volume parallel and an analogous equation in the perpendicular direction.
8. Experimental results and discussion 8.1. Annealing of samples For the annealing conditions used we followed the procedures outlined in previous publications.[8,19]
9. Results 9.1. Measurements of transverse sectional area for fibers The cross-section of the nylon 6 fiber was viewed by high power optical microscopy and it was circular. 9.2. Application of two-beam interferometry The totally duplicated image of the fiber obtained with the Pluta polarizing interference microscope was used to calculate the mean refractive indices n储a and nⲚa of nylon 6 fiber.[22,34,35] Fig. 1(a–e) shows a microinterferogram of non-duplicated images of nylon 6 fiber using the Pluta microscope with annealing time at constant temperature 160 ⫾ 1°C. Fig. 2(a–e) shows a microinterferogram of totally duplicated images of nylon 6 fiber using the Pluta microscope with annealing time at constant temperature 160 ⫾ 1°C. Using these, the mean refractive index of the parallel and perpendicular directions at different annealing temperatures and constant annealing time were calculated. The refractive index of the immersion liquid was selected to allow the fringe shift to be small. Figs. 1 and 2 also show that the fringes shifts changed as the time of annealing increases at constant temperature. Fig. 3 shows the relation between the birefringence ⌬n and the refractive indices differences (n储 ⫺ nv, nⲚ ⫺ nv, n储 ⫺ niso, and nⲚ ⫺ niso). From Fig. 3, nv, and ⌬nmax are used to predict the values of refractive indices n1 and n2 for fully oriented fibers. These values are found to be 1.608 and 1.536, respectively, at 28°C, Table 1 give some experimental results for refractive indices and the apparent volume fraction of crystallinity (), which was calculated from Eq. (12) using the calculated density values. The following calculated values at different annealing times are given in Table 2: virtual refractive index (nv), isotropic refractive index (calculated by two methods using Eq. (15) and Eq. (17) respectively), optical orientation function f and the Hermans function f⌬, the number of monomer units per unit volume, which also equal the number of carriers of the dipole moment (N1), and the number of molecules per unit volume (N2). Also in Table 3,
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241
Fig. 1. (a)–(e) Microinterfrograms of non-duplicated image of nylon 6 fiber which annealed at 160°C ( ⫽ 546 nm).
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Fig. 2. (a)–(e) Microinterfrograms of totally duplicated image of nylon 6 fiber which annealed at 160°C ( ⫽ 546 nm).
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Fig. 3. Relations between the birefringence ⌬n and the refractive indices differences (n储 ⫺ nv, nⲚ ⫺ nv, n储 ⫺ niso and nⲚ ⫺ niso) of nylon 6 fiber at 1 ⫾ 160°C.
Table 1 Annealing time n储a (hr) Unannealed 1 2 3 4 5 6 7 8 9 10
1.576 1.582 1.587 1.589 1.591 1.593 1.595 1.597 1.597 1.597 1.597
nⲚa 1.535 1.531 1.531 1.531 1.531 1.535 1.535 1.537 1.537 1.537 1.537
⌬na 0.041 0.051 0.056 0.058 0.060 0.058 0.060 0.060 0.060 0.060 0.060
1.12 1.14 1.172 1.21 1.19 1.18 1.154 1.235 1.228 1.20 1.17
%
具2典 ⫻ 10−2%
8.85 26.5 54.9 88.5 70.8 61.9 38.9 100 100 79.6 53.1
99.8 99.8 99.7 99.9 99.7 99.7 99.7 100 100 99.8 99.7
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Table 2 Annealing time (hr) Unannealed 1 2 3 4 5 6 7 8 9 10
niso(1) 1.549 1.548 1.550 1.550 1.551 1.554 1.555 1.557 1.557 1.557 1.557
niso(2) 1.543 1.531 1.515 1.496 1.507 1.515 1.529 1.490 1.493 1.507 1.523
nv 1.548 1.547 1.549 1.549 1.550 1.553 1.554 1.554 1.554 1.554 1.554
f⌬
N(1) ⫻ 1021
f
0.5694 0.7083 0.7778 0.8056 0.8333 0.8056 0.8333 0.8333 0.8333 0.8333 0.8333
0.6101 0.7576 0.8280 0.8560 0.8838 0.8494 0.8771 0.8737 0.8737 0.8737 0.8737
5.96 6.06 6.23 6.44 6.33 6.28 6.14 6.57 6.53 6.38 6.22
N(2) 0.972 0.965 0.961 0.960 0.959 0.960 0.959 0.959 0.959 0.959 0.959
Table 3 Annealing time (hr) Unannealed 1 2 3 4 5 6 7 8 9 10
␣储 ⫻ 10−34 ␣Ⲛ ⫻ 10−34 14.75 14.61 14.31 13.90 14.17 14.33 14.69 13.77 13.85 14.17 14.53
13.87 13.54 13.18 12.76 12.98 13.17 13.46 12.62 12.69 12.99 13.32
␣¯ ⫻ 10−34 14.31 14.08 13.74 13.33 13.57 13.75 14.08 13.19 13.27 13.58 13.93
␣v ⫻ 10−34 14.15 13.88 13.53 13.12 13.35 13.53 13.85 12.98 13.06 13.36 13.70
⑀储 ⫻ 10−2 25.49 25.23 24.69 23.97 24.43 24.70 25.32 23.71 23.85 24.40 25.03
⑀Ⲛ ⫻ 10−2 24.18 23.63 22.98 22.26 22.64 22.95 23.47 21.99 22.11 22.63 23.21
⑀¯ ⫻ 10−2 24.84 24.43 23.84 23.12 23.54 23.83 24.39 22.85 22.98 23.52 24.12
⑀V ⫻ 10−2 24.61 24.14 23.53 22.81 23.21 23.51 24.06 22.54 22.66 23.19 23.79
the calculated values of ␣储, ␣Ⲛ, ␣¯ , ␣v and specific refractivity of the isotropic dielectric in parallel and perpendicular directions and its mean value (⑀储, ⑀Ⲛ, ⑀¯ and ⑀v) at different annealing times are given. Fig. 4 shows the relation between corrected values of optical orientation function f and the Hermans function f⌬, from which f/f⌬ is found. Fig. 5 shows the relation between the birefringence ⌬n and the optical orientation function f. 10. Discussion Studies of polymer thermal effects on the optical properties of fibers gives much useful information about molecular structure changes and orientation. Changes in orientation are accompanied by changes in crystallinity due to the annealing process; this indicates mass redistribution within
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Fig. 4. Relation between corrected values of optical orientation function f and the Hermans function f⌬ of nylon 6 fiber at 1 ⫾ 160°C.
the fiber chain. So, information about the effect of annealing on the microstructure and morphology of polymer materials has been a subject of major technological and scientific interest for a number of years. Characterization of the opto-thermal structural parameters of fibers are important for the textile industry and the end use. The present results in Table 2 shows changes in the isotropic refractive indices related to the crystalline and amorphous regions of the fibers and to the density of the sample, which is related to Eq. (17). The results show that the overall changes in density are important influences on the optical parameters, orientation, crystallinity etc. 11. Conclusion From the analysis of the results obtained one can conclude the following. 1. The combination of the principal refractive indices and the density measurements throw light on the variations occurring during the annealing process on the different structural parameters as shown (Tables 1–3). 2. Comparison, between the obtained values of niso1, niso2 and nv show that every equation has its own merit based on considerations of its derivation and applications.
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Fig. 5. Relation between birefringence ⌬na and of optical orientation function f of nylon 6 fiber at 1 ⫾ 160°C.
3. It was found that the value of (a) is constant whatever the annealing time. 4. It is possible to explain with the aid of the obtained structural results the variation in the enduses of textiles which are influenced by optical properties (buster, light reflection, and absorption etc.) and thermal treatments.
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