D Loidl et al./ Carbon 4/(2003)563-570 nificantly better than Gaussian or Lorentzian functions influence of some elastic constants Two approaches were chosen to determine the shear carbon fibers, small in the governing modulus of the crystallites. One was to replace Il by the (cos" s)may not be neglected as its half-width at half maximum Iwm in Eqs. (2)and(3)to than 25% for fibers with the greatest misalignment of the obtain ger from the decrease of the half-width with crystallites tested in this work. Thus, Eq.( &a)has to be increasing stress of the fiber. The other was to combine supplemented by Eqs. (3)and (4)by the appropriate coordinate transforma- tion to obtain ger from the change of the whole intensity I (cos" 2)=- distribution at each stress step To correlate the microscopic to the macroscopic prop- erties, orientation parameters characterizing the orientation distribution of the crystallites were defined [21, 22] An overview of the different models may be found in dE pls)"(S)sin(E) Ref. [22], where, additionally, a mosaic model was de- (cos 4)= veloped, which is based on a numerical finite-element ds pls)sin(s) calculation of 256 crystallites with a specific orientation distribution. This model was able to describe the non- with p(sds being the number of es with tilt linearity in the load-displacement curve of carbon fiber but, unfortunately, it cannot be reduced to analy angle f in the interval & E+dE In abl ed terms, Eq (5a)states for the second and fourth Z=(cos $)and Z,=(cos s (5b) 4. Experimental results For perfectly oriented graphene planes, Zo and Z,are zero, and for a random orientation distribution they are 1/3 From the diffraction patterns, the 002 reflection and the and 175, respectively 10 band (see Fig. 1) were evaluated as a function of the ion for the porosity(ratio of the density of bulk applied stress. The 10 band was integrated in the meridion- carbon in the fibers to the opic fiber density )i al direction, therefore only those graphene planes were required to obtain the ma effective Young's recorded which were almost perfectly oriented along the modulus of the fibers [ 3] direction of the applied stress. The strain of these planes can be obtained from Eq. (1), assuming the validity of the 6) uniform stress model. This strain is shown as a function of the applied stress in Fig. 2. The symbols denote the type of This effective macroscopic Young's modulus of the fibers, fiber (filled symbols, MPP-based; open symbols, PAN denoted by a capital letter, can be related to microscopic based fibers) and the different symbols represent the orientation parameters. In the elastic unwrinkling model [14], these parameters are denoted by 1, and m 08 (7) with /=(sin 5)1 and m =(cos- s/sin $)Zo for well oriented graphene planes in the fibers, and k a specific compliance of the unwrinkl The corresponding equation in the uniform stress model [21, also denoted by the series rotable elements model [22], is similar The indices denote the fact that these effective properties re, in principle, obtained from a macroscopic experiment, the measurement of the overall Youngs modulus of the Fig. 2. Strain of the crystallites evaluated from meridional integration of the 10 band as stress fibers. In the uniform stress model the following relation Filled symbols, MPP-based fiber K321,(V)E35,(■)E55 would hold: eer=ecr and get=gcr Eq(8a) is based (◆)FT500,(▲)K137.Ope PAN-based fibers: (O) neglecting higher-order terms, (cos"s), and equally the HTA7-AR,(V)HTA7-18, (O)HTA7-21,(0)HTA7-24566 D. Loidl et al. / Carbon 41 (2003) 563–570 nificantly better than Gaussian or Lorentzian functions. influence of some elastic constants. The latter are, for Two approaches were chosen to determine the shear carbon fibers, small in the governing equations. However, 4 modulus of the crystallites. One was to replace P by the kcos j l may not be neglected as its contribution is more half-width at half maximum P in Eqs. (2) and (3) to than 25% for fibers with the greatest misalignment of the hwhm obtain g from the decrease of the half-width with crystallites tested in this work. Thus, Eq. (8a) has to be cr increasing stress of the fiber. The other was to combine supplemented by Eqs. (3) and (4) by the appropriate coordinate transforma- 111 1 1 2 4 tion to obtain gcr from the change of the whole intensity ]5 1 ]] ] kcos j l 2 kcos j l 5 ] distribution at each stress step. Eeg g E eff eff eff eff eff To correlate the microscopic to the macroscopic prop- 1 1 5 1 ] ](Z 2 Z ) (8b) erties, orientation parameters characterizing the orientation 0 1 e g eff eff distribution of the crystallites were defined [21,22]: An overview of the different models may be found in p/ 2 n Ref. [22], where, additionally, a mosaic model was de- E dj r(j ) cos (j ) sin(j ) n ]]]]]]] 0 veloped, which is based on a numerical finite-element kcos j l 5 ] (5a) p/ 2 calculation of 256 crystallites with a specific orientation E dj r(j ) sin(j ) distribution. This model was able to describe the non- 0 linearity in the load-displacement curve of carbon fibers with r(j ) dj being the number of crystallites with tilt but, unfortunately, it cannot be reduced to analytical angle j in the interval j, j 1 dj. In abbreviated terms, Eq. expressions. (5a) states for the second and fourth moment: 2 4 Z 5 kcos j l and Z 5 kcos j l (5b) 0 1 4. Experimental results For perfectly oriented graphene planes, Z and Z are 0 1 From the diffraction patterns, the 002 reflection and the zero, and for a random orientation distribution they are 1/3 10 band (see Fig. 1) were evaluated as a function of the and 1/5, respectively. applied stress. The 10 band was integrated in the meridion- A correction for the porosity (ratio of the density of bulk al direction, therefore only those graphene planes were carbon in the fibers to the macroscopic fiber density) is recorded which were almost perfectly oriented along the required to obtain the macroscopic effective Young’s modulus of the fibers [3]: direction of the applied stress. The strain of these planes can be obtained from Eq. (1), assuming the validity of the rB E 5 E] uniform stress model. This strain is shown as a function of (6) eff r the applied stress in Fig. 2. The symbols denote the type of fiber (filled symbols, MPP-based; open symbols, PAN- This effective macroscopic Young’s modulus of the fibers, based fibers) and the different symbols represent the denoted by a capital letter, can be related to microscopic orientation parameters. In the elastic unwrinkling model [14], these parameters are denoted by l and m : z z 1 1 ]5 ]l 1 km (7) z z E e eff eff 2 with l 5 ksin j l ¯ 1 and m 5 kcos j /sin j l ¯ Z for well z z 0 oriented graphene planes in the fibers, and k a specific compliance of the unwrinkling. The corresponding equation in the uniform stress model [21], also denoted by the series rotable elements model [22], is similar: 111 ]5 1 ] ]Z (8a) 0 Eeg eff eff eff The indices denote the fact that these effective properties are, in principle, obtained from a macroscopic experiment, Fig. 2. Strain of the crystallites evaluated from meridional the measurement of the overall Young’s modulus of the integration of the 10 band as a function of the applied stress. fibers. In the uniform stress model the following relation Filled symbols, MPP-based fibers: (d) K321, (.) E35, (j) E55, would hold: eeff cr eff cr 5 e and g 5 g . Eq. (8a) is based on (♦) FT500, (m) K137. Open symbols, PAN-based fibers: (s) 4 neglecting higher-order terms, kcos j l, and equally the HTA7-AR, (,) HTA7-18, (h) HTA7-21, (
) HTA7-24