- Email: [email protected]
Optical and structural changes in perlon fibers due to different stresses
European Polymer Journal 36 (2000) 823±829
Optical and structural changes in perlon ®bers due to dierent stresses I.M. Fouda*, H.M. Shabana Department of Physics, Faculty of Science, University of Mansoura, Mansoura 35516, Egypt Received 22 December 1997; received in revised form 24 February 1999; accepted 3 March 1999
Abstract A mechanical device has been used to measure stress as well as strain, and connected to a polarizing interference Pluta microscope to investigate the opto-mechanical behavior of trilobal nylon 6 (perlon) ®bers. The experimental data obtained for draw ratio, stress, refractive indices and birefringence were used to calculate structural parameters, such as cohesive energy density, average work per chain W 0 , work per unit volume W, reduction in entropy due to the elongation, work stored in the body as strain energy W 00 and other parameters. Calculation of the segment anisotropy gs , the number of network chain per unit volume N and the number of crystals per unit volume Y were also given. Empirical formula was suggested to correlate the changes in Dn , W 0 , N, Y and gs with the draw ratios. The study shows that new orientations occurred due to cold drawing at dierent draw ratios and stresses. # 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction Most polymeric materials which are normally isotropic, because of random molecular orientation, can be made optically anisotropic by the application of mechanical stress. In going from tension to compression the material will change from positive uniaxial to negative uniaxial or vice versa depending on polarizability considerations. In both cases the eective optical axis is the stress direction. The birefringence at any point is proportional to the principal stress dierence s1 ÿ s2 [1]. Deformation due to drawing processes is the predominant means of producing new physical structure of polymeric ®bers, which are so important commercially and are of much industrial interest in the end use. Developing the properties of polymers and ®bers by
* Corresponding author. Fax: +20-50-346781. E-mail address: [email protected] (I.M. Fouda).
means of drawing process is due to change in the molecular arrangement, and hence, the creation of new orientation structure which tends to change polymeric properties are expected. One of the most powerful experimental methods of short-range order determination in polymers utilizes birefringence. So, if a polymer is subjected to an external force, such as mechanical drawing, the crystallites and molecules become oriented. The polymer is stronger in the draw direction, and it will be weakest at 908 [2,3]. In recent years, interferometric methods have been used in studying the thermal, mechanical and chemical properties of natural and synthetic ®bers [4±7]. One of the most readily available techniques for changing the polymeric structure are the mechanical process [8±13]. The study of mechanical properties of textile ®bers is to establish a connection between the molecular structure and these properties, and suggest bene®cial modi®cations in the preparation or processing of the ®ber.
0014-3057/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 4 - 3 0 5 7 ( 9 9 ) 0 0 1 2 1 - 4
824
I.M. Fouda, H.M. Shabana / European Polymer Journal 36 (2000) 823±829
The present work deals with the in¯uence of experimental conditions of stretching on orientation and mechanical parameters in uniaxially stretching nylon 6 (perlon) ®bers using a two-beam Pluta polarizing interference microscope. In addition, dierent parameters aected by the drawing process were evaluated.
2. Theoretical consideration The totally duplicated image position by the doublebeam interference Pluta microscope was used to measure the mean refractive indices n 00 and n? of the examined ®bers. The values obtained for n 00 and n? were used for calculating many optical and orientation parameters, such as the virtual refractive index nv [14], polarizabilities, optical con®guration parameter, . . . . The optical orientation function fD was determined from the Herman's equations [15], and was used to calculate the number N1 of random links between the network junction points (the entanglements) from the following equation [16] fD
a2 ÿ aÿ1 5N1
1
Roe and Krighaum [17] derived an expression for the distribution function of segments at an angle y with respect to the draw ratios. o cos y
1 1 3cos2 y ÿ 1 a2 ÿ aÿ1 2 4N1
2
Before orientation, the segments will be randomly oriented at an angle y with respect to the draw direction. After drawing, the segments be constrained at an angle b given by tan b aÿ3=2 tan y
3
2.1. Determination the number of molecules per unit volume
2.2. Mechanical properties The strain optical coecient is given from the following equation [12] Ce
d Dn de
5
the change in birefringence d Dn and the corresponding change in strain. The value of the cohesive energy density (CED), which represents the energy theoretically required to move a detached segment into the vapor phase, was calculated in terms of the bulk modules B. This in turn is related to the square of the solubility parameter d by the following equation: B 8:04 CED 8:04d2
6
The factor 8.04 arises from Lennard±Jones considerations [19]. Stress±strain relationship for deformable materials have been carried out without the assumption of any speci®c molecular model, according to the Mooney± Rivlin equation [20] s= a ÿ aÿ2 2c1 2c2 1=a
7
Plots of s= a ÿ aÿ2 and aÿ1 are found to be linear, especially at low elongation and the constants c1 and c2 can be calculated. Since the stress is related to the elongation by ÿ1 s 1 e3 N e 3KT 31e
8
where s is the stress, e is the strain, K is the Boltzmann constant, T is the absolute temperature and N is the number of network chains per unit volume. Also, the elongation leads to a reduction in the entropy DS according to the following equation: 1 2 DS KN 1 e 2 ÿ3 9 1 e 2
The dierence of the two main refractive indices Dn of the sample is linked with the dierent in the mean polarizabilities of the macromolecule for the same directions DP P 00 ÿ P ? by the following relation [18] 2 2pNm n 2 00 Dn 4 P ÿ P? 3 n
The average work W 0 per chain for a collection of chains, the work W per unit volume and the storable elastic energy W 00 of the network are obtained using equations derived from the above equations [3]. For uniaxial tensile stress, the birefringence and the retractive stress are related by the stress optical coecient Cs [21]. The value of Cs is dependent upon the chemical structure of the polymer. Also, it depends solely on the mean refractive index and the optical anisotropy of the random link as seen from the following equation [2]
where Nm is the number of molecules per unit volume, n is the mean refractive index of the sample n 00 2n? =3:
ÿ 2 2p n 2 2 b1 ÿ b2 Cs 45KT n
10
I.M. Fouda, H.M. Shabana / European Polymer Journal 36 (2000) 823±829
825
where n is the average refractive index, and b1 and b2 are the polarizabilities along and across the axes of such units. The number N 0 of chains between cross links per unit volume at absolute temperature is determined as follows [22] 2 ÿ N 0 gs n 2 2 2 n ÿn a ÿ aÿ1 90e0 n 00
?
11
where e0 is the permittivity and equals 8:85 10ÿ12 F mÿ1 (mÿ3 kgÿ1 s4 A2), and the segment anisotropy gs is given from the following equation: 2 ÿ gs n2 2 Cs 90e0 KT n
12
Plate 1. Cross-sectional view of the trilobal nylon 6 (perlon) ®bers.
3.1. Measurement of the transverse sectional area
2.3. The birefringence of partially crystalline polymers [22] The average orientation angle on uniaxial stretching extension ratio a is given by # " 1=2 a3 tanÿ1 a3 ÿ 1 2 hcos yc i 3 13 1ÿ a ÿ1 a3 ÿ 1 1=2
Plate 1 shows the cross-section of the trilobal nylon 6 ®bers, seen by high power optical microscopy. According to this plate, the cross-sectional view shows irregular shapes (trilobal). 3.2. Application of a double-beam interferometry Plate 2 shows some of the obtained microinterfero-
If there are Y crystals per unit volume, and these have polarizabilities bc along c-axis and bb ba perpendicular to it, then the crystal contribution to the birefringence of the medium is given by ÿ 2
ÿ n2 2 Y b1 ÿ b2 P2 cos2 yc n ÿn 18e0 n 00
?
14
2.4. Calculation of the optical con®guration parameter The optical con®guration parameter Da is related to the stress-optical coecient Cs by the following equation [23] Da
45KTCs =2pn ÿ 2 n2 2
15
3. Experimental procedure and results A micro-strain device was described elsewhere [4] and used in conjunction with two-beam polarizing interference Pluta microscope [24], has been modi®ed to measure stress as well as strain. The modi®cation enables measuring of the mechanical properties and its correlation with optical properties.
Plate 2. (a±d) Microinterferograms of two-beam interferometry from the totally duplicated image position.
826
I.M. Fouda, H.M. Shabana / European Polymer Journal 36 (2000) 823±829
Table 1 Values of the refractive indices n 00 and n? , the birefringence Dn, virtual refractive index nv, strain and stress optical coecients Ce and Cs a
n 00
n?
Dn 10ÿ3
nv
Ce
Cs 10ÿ9 (Pa)ÿ1
1.00 1.12 1.22 1.32 1.44 1.50
1.6181 1.6227 1.6339 1.6380 1.6413 1.6427
1.5885 1.5849 1.5818 1.5813 1.5808 1.5806
29.6 42.7 52.1 56.7 60.5 62.1
1.5980 1.5984 1.5981 1.5989 1.5995 1.5998
± 0.10 0.10 0.04 0.03 0.03
± 5.88 5.78 5.29 4.48 4.25
grams of a two-beam interferometry from the totally duplicated image position. Plane polarized light of l 546 nm was used for vibrating along and across the ®ber axis, and a liquid of refractive index nL 1:5948 at 258C. The corresponding draw ratios are 1, 2, 2.75 and 3.5, respectively. At dierent draw ratios, n 00 , n? and Dn were calculated, and the results obtained are given in Table 1. The refractive index n 00 and the birefringence increase with increasing the draw ratios, whereas the refractive index n? decreases with increasing the draw ratios. The virtual refractive index was calculated and its values are given in Table 1. 3.3. Mechanical properties The stress±strain device in connection with the Pluta microscope has been used to obtain the values of dierent draw ratios, Young's modulus, compressibility w, strain optical coecient Ce and stress optical coecient Cs. The results obtained for Ce and Cs are summarized in Table 1. The value of bulk modules B was calculated and used to evaluate the CED, which in turn is related to the solubility parameter d: The value obtained for the CED or d2 was found to be 0.31 at Poisson's ratio m 0:5: The constants C1 and C2 of Eq. (7) were ÿ4.33 and 16.17, as determined from the relationship between s= a ÿ aÿ2 and aÿ1 : The obtained values of the above parameters were used for calculating the number of network chains per unit volume N, the work per unit
volume W, the average work per chain W 0 and the storable elastic energy of the network W 00 : The values obtained for N and W ' are given in Table 2. Fig. 1 shows the relationship between the draw ratios and both the stress optical coecient Cs and the number N of network chain per unit volume. Both Cs and N decrease with increasing the draw ratios. The draw ratio in its relationship with both the work per unit volume W and the storable elastic energy of the network W 00 is given in Fig. 2. In contrast to the increase of W 0 and the work W, the storable elastic energy of the network W 00 was found to be decreased with increasing the draw ratios. The reduction in entropy DS, the optical con®guration parameter Da, the segment anisotropy gs , the number of chains between cross links per unit volume N 0 and the number of crystals per unit volume Y were also calculated and their values are summarized in Table 2. Fig. 3 illustrates the relationship between the draw ratio and both the number N 0 of chains between cross links per unit volume and the number Y of crystals per unit volume. Both N 0 and Y were found to be decreased exponentially with increasing the draw ratios. Fig. 4 shows the relationship between the birefringence Dn and both the entropy DS and the work per unit volume W. The entropy is found to be decreased with increasing the draw ratios in contrast to the work W. In addition to the above optical and mechanical parameters, the optical orientation function fD and angle y
Table 2 Values of no. of network chain per unit volume N, average work per chain W' optical con®guration parameter Da, reduction in entropy DS, segment anisotropy gs and the number Y of crystals per unit volume a
N 1030
W' 10ÿ23(J)
Da 10ÿ25(cm3)
DS 107(JKÿ1)
gs 10ÿ45 (s4 A4 kgÿ1)
Y 106 (F mmÿ4)
1.00 1.12 1.22 1.32 1.44 1.50
± 48.8 32.7 26.4 23.7 22.4
± 08.22 26.33 53.07 95.44 120.13
± 0.133 0.131 0.119 0.101 0.096
± ÿ13.5 ÿ28.8 ÿ46.9 ÿ75.6 ÿ90.2
± 14.83 14.56 13.32 11.27 10.69
± 56.26 39.30 33.70 32.00 31.03
I.M. Fouda, H.M. Shabana / European Polymer Journal 36 (2000) 823±829
827
Table 3 Values of the orientation function and angle fD and y, the no. of random links between entanglements N1, the distribution function of segment o cos y, the angle b and number of molecules per unit volume Nm a
fD
y8
N1
o cos y
b8
Nm
1.00 1.12 1.22 1.32 1.44 1.50
0.41 0.59 0.72 0.78 0.84 0.86
38.3 31.2 25.3 22.0 18.9 17.5
± 0.12 0.18 0.25 0.33 0.36
0.50 1.04 1.81 2.05 2.26 2.36
38.30 27.04 19.30 14.91 11.02 09.07
1.00 0.93 0.95 1.00 0.94 0.96
ln Fig. 1. The relationship between the draw ratios and both the stress optical coecient Cs and the number N of network chain per unit volume.
and the number of random links N1 between entanglements were calculated. In addition, the distribution function of segment o cos y at an angle y, the angle b, where the segments is constrained after drawing and the number of molecules per unit volume Nm were also calculated. The values of these parameters are given in Table 3. The optical orientation function fD and the distribution function of segment o cos y are drawn as a function of the draw ratios as in Fig. 5. Where, both the parameters increase with increasing the draw ratios. Empirical formula was suggested to correlate the change in the parameters obtained, such as Dn, W 0 , N, Y and gs with the draw ratios as follows
Fig. 2. The draw ratio in its relationship with both the work per unit volume W and the storable elastic energy of the network W 00 :
DnW 0 ma z NYgs
16
Eq. (16) is represented by Fig. 6. The constants m and z of the above equation were found to be 13.16 and ÿ92.75, respectively. 4. Discussion Drawing process is used to vary the degree of orientation and other physical properties in polymeric materials. Birefringence is one of the methods which give a measure of orientation, which is an average of the amorphous and crystalline regions. Studying the mechanical properties of perlon ®bers aims to establish a connection between orientation and other structural parameters which can aid in the end use. Drawing signi®cantly increased the birefringence Dn of Perlon ®bers, indicating the signi®cantly improved
Fig. 3. The relationship between the draw ratio and both the number of chains between cross links per unit volume N 0 and the number of crystals per unit volume Y.
828
I.M. Fouda, H.M. Shabana / European Polymer Journal 36 (2000) 823±829
Fig. 4. The relationship between the birefringence Dn and both the entropy DS and the work per unit volume W.
overall molecular orientation. The increase in Dn is a result of a large improvement in axial packing n 00 and slightly reduced molecular packing in the radial direction n? ). The constant C2 of Mooney±Rivlin equation has been attributed to energy dissipation resulting from chain interactions during deformation; and in conformity with this view, C2 becomes zero when the elastomer is swollen by solvents [20]. It is interesting to note that the parameter N 0 changes signi®cantly with increasing draw ratios, perhaps due to network required by a transition from ane to pseudo ane deformation. The mechanical treatments are used to vary the degree of orientation, crystallinity and other physical properties in polymeric materials. Also, to explain the
Fig. 6. The relationship between ln DnW 0 =YNgs and the draw ratios.
dierent variations obtained due to mechanical eects, there are several structural processes interfered and should be taken into considerations. These structural processes are discussed elsewhere extensively [6]. In the mechanical vein, elastomers must stretch rapidly under tension with little loss of energy as heat, which could be felt due to the change in entropy. By the application of stress, the easily deformable phase is gradually converted into the less deformable phase. On the release of stress, they recover their original dimensions with rebound. So this recovery achieved and indicated by shrinkage parameters [25]. It is clear that there are isothermal kinetic changes due to drawing process con®rmed from the changes of DS: A change of DS due to the drawing process can change the optical property Dn (birefringence) of the perlon ®bers. Also, the retractive forces of the network are produced by the decrease in entropy of freely jointed chains when stretched. DS must be considered to throw light on the energies, which play a role to clarify the phase boundary between the amorphous and crystalline regions. The irregularity along the ®bers axis in both the ®ber diameter and fringe shift, as shown in Plates 1 and 2(a±d), is due to the fact that the examined ®bers are necked during stretching. 5. Conclusion
Fig. 5. The optical orientation function fD and the distribution function of segment o cos y as a function of the draw ratios.
It is clear that the drawing process and the application of two beam interference methods are useful and quick technique to throw light on the optical, mechanical and structural properties and their related parameters as given in the following report: 1. Application of Mooney±Rivlin equation shows lin-
I.M. Fouda, H.M. Shabana / European Polymer Journal 36 (2000) 823±829
2.
3. 4.
5.
ear relationship. The constants C1 and C2 were found to be ÿ4.33 and 16.17, respectively. Various mechanical parameters of the drawn samples are determined in a simple way by calculations depend on stress±strain, which throw lights on dierent physical phenomena (swelling, thermal, optical, . . . ). It is clear that there are isothermal kinetic changes due to drawing process con®rmed from the changes of DS (see Table 2). As the draw ratio increased, the birefringence Dn increased. The number of random links between entanglements and distribution function of segments were found to increase by the drawing eect. Empirical formula was suggested to correlate the observed changes in Dn, W 0 , N, Y and gs with the draw ratios and its constants were determined.
References [1] Hemsley DA. In: Applied polymer light microscopy. London: Elsevier Science, 1989. p. 146. [2] Sperling LH. In: Introduction to physical polymer science, 2nd ed. New York: Wiley, 1992. p. 161. [3] Williams DJ. In: Polymer science and engineering. UK: Prentice-Hall, 1971. p. 190. [4] Hamza AA, Fouda IM, El-Farhaty KA, Helaly SA. Polym Testing 1987;7:329. [5] Barakat N, Hamza AA. Interferometry of ®brous materials. Hilger: Bristol, 1990. [6] Hamza AA, Fouda IM, Kabeel MA, Seisa EA, ElSharkawy FM. J Polym Sci, Polym Phys Ed 1998;36:555.
829
[7] Fouda IM, Shabana HM. J Appl Polym Sci, in press. [8] Jass B, Oultache AK, Mounach H, Halory JL, Monnerie LM. J Polym Sci, Part B: Polym Phys 1996;34:2007. [9] Wu G, Dongziang J, Tucker PA, Cuculo JA. J Polym Sci, Part B: Polym Phys 1996;34:2035. [10] Lee KG, Barton R, Schultz JM. J Polym Sci, Part B: Polym Phys 1995;33:1. [11] Termonia Y. J Polym Sci, Part B: Polym Phys 1996;34:147. [12] Hamza AA, Fouda IM, Kabeel MA, Shabana HM. Polym Testing 1996;15:391. [13] El-Tonsy MM. J Mater Sci 1991;26:2857. [14] de Vries H, Kolliod Z. Polym Sci 1979;257:226. [15] Hermans PH. Contributions to the physics of cellulose ®bers. Amsterdam: North Holland, 1949 (Chapter 3). [16] Perena JM, Duckett RA, Ward IM. J Appl Polym Sci 1980;25:1381. [17] Roe RJ, Krigbaum WR. J Appl Phys 1964;35:2215. [18] Sarkisyan VA, Asratyan MG, Mkhitoryan AA, Katrdzhyan KKH, Dadivanyan AK. Vysokomal 1985;A27(6):1331. [19] Vinogradov GV, Ozyara EA, Malkin AY, Grcchanovskii VA. J Polym Sci 1971;A2(9):1153. [20] Mooney M. J Appl Phys 1940;11:582; Mooney M. J Appl Phys 1948;19:434; Rivlin RS. Trans R Soc (London) 1948;A241:379; Sperling LH. In: Introduction to physical polymer science, 2nd ed. New York: Wiley, 1992. [21] Statton WO. J Polym Sci: Part C 1967;20:117. [22] Jenkins AD. In: Polymer science hand book, vol. 1. London: Amesterdam, 1972. p. 505±23 (Chapter 7). [23] Riande E, Guzman J. J Polym Sci, Polym Phys Ed 1984;22:917. [24] Pluta M. Optica Acta 1971;18:661. [25] Ward IM. J Polym Sci, Polym Symp 1977;53:9.
Optical and structural changes in perlon ®bers due to dierent stresses I.M. Fouda*, H.M. Shabana Department of Physics, Faculty of Science, University of Mansoura, Mansoura 35516, Egypt Received 22 December 1997; received in revised form 24 February 1999; accepted 3 March 1999
Abstract A mechanical device has been used to measure stress as well as strain, and connected to a polarizing interference Pluta microscope to investigate the opto-mechanical behavior of trilobal nylon 6 (perlon) ®bers. The experimental data obtained for draw ratio, stress, refractive indices and birefringence were used to calculate structural parameters, such as cohesive energy density, average work per chain W 0 , work per unit volume W, reduction in entropy due to the elongation, work stored in the body as strain energy W 00 and other parameters. Calculation of the segment anisotropy gs , the number of network chain per unit volume N and the number of crystals per unit volume Y were also given. Empirical formula was suggested to correlate the changes in Dn , W 0 , N, Y and gs with the draw ratios. The study shows that new orientations occurred due to cold drawing at dierent draw ratios and stresses. # 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction Most polymeric materials which are normally isotropic, because of random molecular orientation, can be made optically anisotropic by the application of mechanical stress. In going from tension to compression the material will change from positive uniaxial to negative uniaxial or vice versa depending on polarizability considerations. In both cases the eective optical axis is the stress direction. The birefringence at any point is proportional to the principal stress dierence s1 ÿ s2 [1]. Deformation due to drawing processes is the predominant means of producing new physical structure of polymeric ®bers, which are so important commercially and are of much industrial interest in the end use. Developing the properties of polymers and ®bers by
* Corresponding author. Fax: +20-50-346781. E-mail address: [email protected] (I.M. Fouda).
means of drawing process is due to change in the molecular arrangement, and hence, the creation of new orientation structure which tends to change polymeric properties are expected. One of the most powerful experimental methods of short-range order determination in polymers utilizes birefringence. So, if a polymer is subjected to an external force, such as mechanical drawing, the crystallites and molecules become oriented. The polymer is stronger in the draw direction, and it will be weakest at 908 [2,3]. In recent years, interferometric methods have been used in studying the thermal, mechanical and chemical properties of natural and synthetic ®bers [4±7]. One of the most readily available techniques for changing the polymeric structure are the mechanical process [8±13]. The study of mechanical properties of textile ®bers is to establish a connection between the molecular structure and these properties, and suggest bene®cial modi®cations in the preparation or processing of the ®ber.
0014-3057/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 4 - 3 0 5 7 ( 9 9 ) 0 0 1 2 1 - 4
824
I.M. Fouda, H.M. Shabana / European Polymer Journal 36 (2000) 823±829
The present work deals with the in¯uence of experimental conditions of stretching on orientation and mechanical parameters in uniaxially stretching nylon 6 (perlon) ®bers using a two-beam Pluta polarizing interference microscope. In addition, dierent parameters aected by the drawing process were evaluated.
2. Theoretical consideration The totally duplicated image position by the doublebeam interference Pluta microscope was used to measure the mean refractive indices n 00 and n? of the examined ®bers. The values obtained for n 00 and n? were used for calculating many optical and orientation parameters, such as the virtual refractive index nv [14], polarizabilities, optical con®guration parameter, . . . . The optical orientation function fD was determined from the Herman's equations [15], and was used to calculate the number N1 of random links between the network junction points (the entanglements) from the following equation [16] fD
a2 ÿ aÿ1 5N1
1
Roe and Krighaum [17] derived an expression for the distribution function of segments at an angle y with respect to the draw ratios. o cos y
1 1 3cos2 y ÿ 1 a2 ÿ aÿ1 2 4N1
2
Before orientation, the segments will be randomly oriented at an angle y with respect to the draw direction. After drawing, the segments be constrained at an angle b given by tan b aÿ3=2 tan y
3
2.1. Determination the number of molecules per unit volume
2.2. Mechanical properties The strain optical coecient is given from the following equation [12] Ce
d Dn de
5
the change in birefringence d Dn and the corresponding change in strain. The value of the cohesive energy density (CED), which represents the energy theoretically required to move a detached segment into the vapor phase, was calculated in terms of the bulk modules B. This in turn is related to the square of the solubility parameter d by the following equation: B 8:04 CED 8:04d2
6
The factor 8.04 arises from Lennard±Jones considerations [19]. Stress±strain relationship for deformable materials have been carried out without the assumption of any speci®c molecular model, according to the Mooney± Rivlin equation [20] s= a ÿ aÿ2 2c1 2c2 1=a
7
Plots of s= a ÿ aÿ2 and aÿ1 are found to be linear, especially at low elongation and the constants c1 and c2 can be calculated. Since the stress is related to the elongation by ÿ1 s 1 e3 N e 3KT 31e
8
where s is the stress, e is the strain, K is the Boltzmann constant, T is the absolute temperature and N is the number of network chains per unit volume. Also, the elongation leads to a reduction in the entropy DS according to the following equation: 1 2 DS KN 1 e 2 ÿ3 9 1 e 2
The dierence of the two main refractive indices Dn of the sample is linked with the dierent in the mean polarizabilities of the macromolecule for the same directions DP P 00 ÿ P ? by the following relation [18] 2 2pNm n 2 00 Dn 4 P ÿ P? 3 n
The average work W 0 per chain for a collection of chains, the work W per unit volume and the storable elastic energy W 00 of the network are obtained using equations derived from the above equations [3]. For uniaxial tensile stress, the birefringence and the retractive stress are related by the stress optical coecient Cs [21]. The value of Cs is dependent upon the chemical structure of the polymer. Also, it depends solely on the mean refractive index and the optical anisotropy of the random link as seen from the following equation [2]
where Nm is the number of molecules per unit volume, n is the mean refractive index of the sample n 00 2n? =3:
ÿ 2 2p n 2 2 b1 ÿ b2 Cs 45KT n
10
I.M. Fouda, H.M. Shabana / European Polymer Journal 36 (2000) 823±829
825
where n is the average refractive index, and b1 and b2 are the polarizabilities along and across the axes of such units. The number N 0 of chains between cross links per unit volume at absolute temperature is determined as follows [22] 2 ÿ N 0 gs n 2 2 2 n ÿn a ÿ aÿ1 90e0 n 00
?
11
where e0 is the permittivity and equals 8:85 10ÿ12 F mÿ1 (mÿ3 kgÿ1 s4 A2), and the segment anisotropy gs is given from the following equation: 2 ÿ gs n2 2 Cs 90e0 KT n
12
Plate 1. Cross-sectional view of the trilobal nylon 6 (perlon) ®bers.
3.1. Measurement of the transverse sectional area
2.3. The birefringence of partially crystalline polymers [22] The average orientation angle on uniaxial stretching extension ratio a is given by # " 1=2 a3 tanÿ1 a3 ÿ 1 2 hcos yc i 3 13 1ÿ a ÿ1 a3 ÿ 1 1=2
Plate 1 shows the cross-section of the trilobal nylon 6 ®bers, seen by high power optical microscopy. According to this plate, the cross-sectional view shows irregular shapes (trilobal). 3.2. Application of a double-beam interferometry Plate 2 shows some of the obtained microinterfero-
If there are Y crystals per unit volume, and these have polarizabilities bc along c-axis and bb ba perpendicular to it, then the crystal contribution to the birefringence of the medium is given by ÿ 2
ÿ n2 2 Y b1 ÿ b2 P2 cos2 yc n ÿn 18e0 n 00
?
14
2.4. Calculation of the optical con®guration parameter The optical con®guration parameter Da is related to the stress-optical coecient Cs by the following equation [23] Da
45KTCs =2pn ÿ 2 n2 2
15
3. Experimental procedure and results A micro-strain device was described elsewhere [4] and used in conjunction with two-beam polarizing interference Pluta microscope [24], has been modi®ed to measure stress as well as strain. The modi®cation enables measuring of the mechanical properties and its correlation with optical properties.
Plate 2. (a±d) Microinterferograms of two-beam interferometry from the totally duplicated image position.
826
I.M. Fouda, H.M. Shabana / European Polymer Journal 36 (2000) 823±829
Table 1 Values of the refractive indices n 00 and n? , the birefringence Dn, virtual refractive index nv, strain and stress optical coecients Ce and Cs a
n 00
n?
Dn 10ÿ3
nv
Ce
Cs 10ÿ9 (Pa)ÿ1
1.00 1.12 1.22 1.32 1.44 1.50
1.6181 1.6227 1.6339 1.6380 1.6413 1.6427
1.5885 1.5849 1.5818 1.5813 1.5808 1.5806
29.6 42.7 52.1 56.7 60.5 62.1
1.5980 1.5984 1.5981 1.5989 1.5995 1.5998
± 0.10 0.10 0.04 0.03 0.03
± 5.88 5.78 5.29 4.48 4.25
grams of a two-beam interferometry from the totally duplicated image position. Plane polarized light of l 546 nm was used for vibrating along and across the ®ber axis, and a liquid of refractive index nL 1:5948 at 258C. The corresponding draw ratios are 1, 2, 2.75 and 3.5, respectively. At dierent draw ratios, n 00 , n? and Dn were calculated, and the results obtained are given in Table 1. The refractive index n 00 and the birefringence increase with increasing the draw ratios, whereas the refractive index n? decreases with increasing the draw ratios. The virtual refractive index was calculated and its values are given in Table 1. 3.3. Mechanical properties The stress±strain device in connection with the Pluta microscope has been used to obtain the values of dierent draw ratios, Young's modulus, compressibility w, strain optical coecient Ce and stress optical coecient Cs. The results obtained for Ce and Cs are summarized in Table 1. The value of bulk modules B was calculated and used to evaluate the CED, which in turn is related to the solubility parameter d: The value obtained for the CED or d2 was found to be 0.31 at Poisson's ratio m 0:5: The constants C1 and C2 of Eq. (7) were ÿ4.33 and 16.17, as determined from the relationship between s= a ÿ aÿ2 and aÿ1 : The obtained values of the above parameters were used for calculating the number of network chains per unit volume N, the work per unit
volume W, the average work per chain W 0 and the storable elastic energy of the network W 00 : The values obtained for N and W ' are given in Table 2. Fig. 1 shows the relationship between the draw ratios and both the stress optical coecient Cs and the number N of network chain per unit volume. Both Cs and N decrease with increasing the draw ratios. The draw ratio in its relationship with both the work per unit volume W and the storable elastic energy of the network W 00 is given in Fig. 2. In contrast to the increase of W 0 and the work W, the storable elastic energy of the network W 00 was found to be decreased with increasing the draw ratios. The reduction in entropy DS, the optical con®guration parameter Da, the segment anisotropy gs , the number of chains between cross links per unit volume N 0 and the number of crystals per unit volume Y were also calculated and their values are summarized in Table 2. Fig. 3 illustrates the relationship between the draw ratio and both the number N 0 of chains between cross links per unit volume and the number Y of crystals per unit volume. Both N 0 and Y were found to be decreased exponentially with increasing the draw ratios. Fig. 4 shows the relationship between the birefringence Dn and both the entropy DS and the work per unit volume W. The entropy is found to be decreased with increasing the draw ratios in contrast to the work W. In addition to the above optical and mechanical parameters, the optical orientation function fD and angle y
Table 2 Values of no. of network chain per unit volume N, average work per chain W' optical con®guration parameter Da, reduction in entropy DS, segment anisotropy gs and the number Y of crystals per unit volume a
N 1030
W' 10ÿ23(J)
Da 10ÿ25(cm3)
DS 107(JKÿ1)
gs 10ÿ45 (s4 A4 kgÿ1)
Y 106 (F mmÿ4)
1.00 1.12 1.22 1.32 1.44 1.50
± 48.8 32.7 26.4 23.7 22.4
± 08.22 26.33 53.07 95.44 120.13
± 0.133 0.131 0.119 0.101 0.096
± ÿ13.5 ÿ28.8 ÿ46.9 ÿ75.6 ÿ90.2
± 14.83 14.56 13.32 11.27 10.69
± 56.26 39.30 33.70 32.00 31.03
I.M. Fouda, H.M. Shabana / European Polymer Journal 36 (2000) 823±829
827
Table 3 Values of the orientation function and angle fD and y, the no. of random links between entanglements N1, the distribution function of segment o cos y, the angle b and number of molecules per unit volume Nm a
fD
y8
N1
o cos y
b8
Nm
1.00 1.12 1.22 1.32 1.44 1.50
0.41 0.59 0.72 0.78 0.84 0.86
38.3 31.2 25.3 22.0 18.9 17.5
± 0.12 0.18 0.25 0.33 0.36
0.50 1.04 1.81 2.05 2.26 2.36
38.30 27.04 19.30 14.91 11.02 09.07
1.00 0.93 0.95 1.00 0.94 0.96
ln Fig. 1. The relationship between the draw ratios and both the stress optical coecient Cs and the number N of network chain per unit volume.
and the number of random links N1 between entanglements were calculated. In addition, the distribution function of segment o cos y at an angle y, the angle b, where the segments is constrained after drawing and the number of molecules per unit volume Nm were also calculated. The values of these parameters are given in Table 3. The optical orientation function fD and the distribution function of segment o cos y are drawn as a function of the draw ratios as in Fig. 5. Where, both the parameters increase with increasing the draw ratios. Empirical formula was suggested to correlate the change in the parameters obtained, such as Dn, W 0 , N, Y and gs with the draw ratios as follows
Fig. 2. The draw ratio in its relationship with both the work per unit volume W and the storable elastic energy of the network W 00 :
DnW 0 ma z NYgs
16
Eq. (16) is represented by Fig. 6. The constants m and z of the above equation were found to be 13.16 and ÿ92.75, respectively. 4. Discussion Drawing process is used to vary the degree of orientation and other physical properties in polymeric materials. Birefringence is one of the methods which give a measure of orientation, which is an average of the amorphous and crystalline regions. Studying the mechanical properties of perlon ®bers aims to establish a connection between orientation and other structural parameters which can aid in the end use. Drawing signi®cantly increased the birefringence Dn of Perlon ®bers, indicating the signi®cantly improved
Fig. 3. The relationship between the draw ratio and both the number of chains between cross links per unit volume N 0 and the number of crystals per unit volume Y.
828
I.M. Fouda, H.M. Shabana / European Polymer Journal 36 (2000) 823±829
Fig. 4. The relationship between the birefringence Dn and both the entropy DS and the work per unit volume W.
overall molecular orientation. The increase in Dn is a result of a large improvement in axial packing n 00 and slightly reduced molecular packing in the radial direction n? ). The constant C2 of Mooney±Rivlin equation has been attributed to energy dissipation resulting from chain interactions during deformation; and in conformity with this view, C2 becomes zero when the elastomer is swollen by solvents [20]. It is interesting to note that the parameter N 0 changes signi®cantly with increasing draw ratios, perhaps due to network required by a transition from ane to pseudo ane deformation. The mechanical treatments are used to vary the degree of orientation, crystallinity and other physical properties in polymeric materials. Also, to explain the
Fig. 6. The relationship between ln DnW 0 =YNgs and the draw ratios.
dierent variations obtained due to mechanical eects, there are several structural processes interfered and should be taken into considerations. These structural processes are discussed elsewhere extensively [6]. In the mechanical vein, elastomers must stretch rapidly under tension with little loss of energy as heat, which could be felt due to the change in entropy. By the application of stress, the easily deformable phase is gradually converted into the less deformable phase. On the release of stress, they recover their original dimensions with rebound. So this recovery achieved and indicated by shrinkage parameters [25]. It is clear that there are isothermal kinetic changes due to drawing process con®rmed from the changes of DS: A change of DS due to the drawing process can change the optical property Dn (birefringence) of the perlon ®bers. Also, the retractive forces of the network are produced by the decrease in entropy of freely jointed chains when stretched. DS must be considered to throw light on the energies, which play a role to clarify the phase boundary between the amorphous and crystalline regions. The irregularity along the ®bers axis in both the ®ber diameter and fringe shift, as shown in Plates 1 and 2(a±d), is due to the fact that the examined ®bers are necked during stretching. 5. Conclusion
Fig. 5. The optical orientation function fD and the distribution function of segment o cos y as a function of the draw ratios.
It is clear that the drawing process and the application of two beam interference methods are useful and quick technique to throw light on the optical, mechanical and structural properties and their related parameters as given in the following report: 1. Application of Mooney±Rivlin equation shows lin-
I.M. Fouda, H.M. Shabana / European Polymer Journal 36 (2000) 823±829
2.
3. 4.
5.
ear relationship. The constants C1 and C2 were found to be ÿ4.33 and 16.17, respectively. Various mechanical parameters of the drawn samples are determined in a simple way by calculations depend on stress±strain, which throw lights on dierent physical phenomena (swelling, thermal, optical, . . . ). It is clear that there are isothermal kinetic changes due to drawing process con®rmed from the changes of DS (see Table 2). As the draw ratio increased, the birefringence Dn increased. The number of random links between entanglements and distribution function of segments were found to increase by the drawing eect. Empirical formula was suggested to correlate the observed changes in Dn, W 0 , N, Y and gs with the draw ratios and its constants were determined.
References [1] Hemsley DA. In: Applied polymer light microscopy. London: Elsevier Science, 1989. p. 146. [2] Sperling LH. In: Introduction to physical polymer science, 2nd ed. New York: Wiley, 1992. p. 161. [3] Williams DJ. In: Polymer science and engineering. UK: Prentice-Hall, 1971. p. 190. [4] Hamza AA, Fouda IM, El-Farhaty KA, Helaly SA. Polym Testing 1987;7:329. [5] Barakat N, Hamza AA. Interferometry of ®brous materials. Hilger: Bristol, 1990. [6] Hamza AA, Fouda IM, Kabeel MA, Seisa EA, ElSharkawy FM. J Polym Sci, Polym Phys Ed 1998;36:555.
829
[7] Fouda IM, Shabana HM. J Appl Polym Sci, in press. [8] Jass B, Oultache AK, Mounach H, Halory JL, Monnerie LM. J Polym Sci, Part B: Polym Phys 1996;34:2007. [9] Wu G, Dongziang J, Tucker PA, Cuculo JA. J Polym Sci, Part B: Polym Phys 1996;34:2035. [10] Lee KG, Barton R, Schultz JM. J Polym Sci, Part B: Polym Phys 1995;33:1. [11] Termonia Y. J Polym Sci, Part B: Polym Phys 1996;34:147. [12] Hamza AA, Fouda IM, Kabeel MA, Shabana HM. Polym Testing 1996;15:391. [13] El-Tonsy MM. J Mater Sci 1991;26:2857. [14] de Vries H, Kolliod Z. Polym Sci 1979;257:226. [15] Hermans PH. Contributions to the physics of cellulose ®bers. Amsterdam: North Holland, 1949 (Chapter 3). [16] Perena JM, Duckett RA, Ward IM. J Appl Polym Sci 1980;25:1381. [17] Roe RJ, Krigbaum WR. J Appl Phys 1964;35:2215. [18] Sarkisyan VA, Asratyan MG, Mkhitoryan AA, Katrdzhyan KKH, Dadivanyan AK. Vysokomal 1985;A27(6):1331. [19] Vinogradov GV, Ozyara EA, Malkin AY, Grcchanovskii VA. J Polym Sci 1971;A2(9):1153. [20] Mooney M. J Appl Phys 1940;11:582; Mooney M. J Appl Phys 1948;19:434; Rivlin RS. Trans R Soc (London) 1948;A241:379; Sperling LH. In: Introduction to physical polymer science, 2nd ed. New York: Wiley, 1992. [21] Statton WO. J Polym Sci: Part C 1967;20:117. [22] Jenkins AD. In: Polymer science hand book, vol. 1. London: Amesterdam, 1972. p. 505±23 (Chapter 7). [23] Riande E, Guzman J. J Polym Sci, Polym Phys Ed 1984;22:917. [24] Pluta M. Optica Acta 1971;18:661. [25] Ward IM. J Polym Sci, Polym Symp 1977;53:9.