D Loidl et al./ Carbon 41(2003)563-570 5. Discussion Previous models were frequently based tion that carbon fibers consist of basic structural units 12 (BSU) exhibiting the (constant) elastic properties of 10 graphite and that the only varying parameter is the 8 orientation parameter [3, 21, 22]. Our results do not confirm these assumptions. From Fig. 3 it is clear that there exists a limiting value for the Youngs modulus of the grapher planes for small orientation parameters, but it decreases with increasing orientation parameter. A possible explana- ion is the increased crumpling of the graphene sheets, 2 which was confirmed experimentally by TEM and o[] HRTEM observations [11]. It seems to be a general trend independent of the type of fiber that, with increasing Fig. 6. The crystallite shear modulus ger as a function of the misorientation, the crumpling increases and the crystallite applied stress. Symbols are the modulus decreases. further confirmation is the observation that the coherence length Lc of the crystallites decreases in supplemented by the higher-order term or reasons this case [16. For the three fibers with the highest ready mentioned above. We now use Eq.(8b), with orientation investigated in this work, a constant value of knowledge of the Youngs modulus of the whole fiber, the in-plane modulus of er= 1140+10 GPa was obtained, 1/Eese the modulus eer=eer of the graphene planes and the error being the standard error of only three values and the orientation parameter Zo -Zi to calculate the shear thus a lower limit. This value is, however, considerably modulus ge Fig. &a shows that the shear modulus g eater than the value published for graphite of 1020+10 increases with increasing orientation parameter, starting GPa [36]. It should be noted that this latter value was from a value close to the lines describing the respective measured for macroscopic samples of pyrolytic graphite from Fig. 4. Evaluation according to Eq. (7)for the size of some tenths of nanometers. Recent papers have unwrinkling parameter k shows the same trend as for the presented results on the Youngs modulus of carbon shear modulus, differing only in the numerical values. Fig. nanotubes by static tests in SEM of 1100 GPa [37] and by 8b shows the equivalent results as a function of the layer dynamic tests of 1200*20 GPa [38]. It was stated explicit the 002 reflection. From this diagram it is clear that we are value for graphite. Our results for nanocrystallitesGy spacing doo? deduced from the position of the maximum of ly that this value is greater than the currently accept far from the layer spacing of 3. 354 A for perfect graphite carbon fibers support these observations and that the structure of all our fibers is thus turbostratic The shear modulus of the crystallites exhibited a differ This is further confirmed by the fact that no general hkl ent value for MPP and PAn fibers, but was nearly constant reflections were observed for any of the fibres within each group(see Figs. 4 and 5). We attribute this to a difference in the structure of the crystallites of MPP- and PAN-based fibers. The lower limiting value for the MPP fibers obtained by the different evaluation methods pre 0008 posed(see Figs 4-6)is close to the value of g 4.0 GPa for graphite, where the dislocations were pinned by 0.006 irradiation [39]. Fig. 4 shows that Eq. (3) yields good results for the pan fibers but for mpp fibers a deviation was observed at higher stresses. This is due to the increase 0.004 of the shear modulus with increasing stress for this type of fiber(see Fig. 6), which is not taken into account by the 0002▲ actual models. We suggest that these simple models describing the orientation change of an anisotropic crys 0000 tallite in a constant stress field should be improved 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 account for the fact that the decrease of the width of the orientation distribution is more effectively hindered for higher stresses. One step in this direction can be found in Fig. 7. Reciprocal of the effective Youngs modulus from macro- Ref. [22], where a finite-element model of 256 elements scopic tests of the fibers increases linearly with the orientation with different orientations was developed to describe the parameter Zo-ZI. Symbols are the same as in Fig. 2. non-linearity of the stress-strain curve of carbon fibers.568 D. Loidl et al. / Carbon 41 (2003) 563–570 5. Discussion Previous models were frequently based on the assump￾tion that carbon fibers consist of basic structural units (BSU) exhibiting the (constant) elastic properties of graphite and that the only varying parameter is the orientation parameter [3,21,22]. Our results do not confirm these assumptions. From Fig. 3 it is clear that there exists a limiting value for the Young’s modulus of the graphene planes for small orientation parameters, but it decreases with increasing orientation parameter. A possible explana￾tion is the increased crumpling of the graphene sheets, which was confirmed experimentally by TEM and HRTEM observations [11]. It seems to be a general trend independent of the type of fiber that, with increasing Fig. 6. The crystallite shear modulus gcr as a function of the misorientation, the crumpling increases and the crystallite applied stress. Symbols are the same as in Fig. 2. modulus decreases. Further confirmation is the observation that the coherence length L of the crystallites decreases in C supplemented by the higher-order term Z for reasons this case [16]. For the three fibers with the highest 1 already mentioned above. We now use Eq. (8b), with orientation investigated in this work, a constant value of knowledge of the Young’s modulus of the whole fiber, the in-plane modulus of e 5 1140610 GPa was obtained, cr 1/E , the modulus e 5 e of the graphene planes and the error being the standard error of only three values and eff eff cr the orientation parameter Z 2 Z to calculate the shear thus a lower limit. This value is, however, considerably 0 1 modulus g . Fig. 8a shows that the shear modulus g greater than the value published for graphite of 1020610 eff eff increases with increasing orientation parameter, starting GPa [36]. It should be noted that this latter value was from a value close to the lines describing the respective measured for macroscopic samples of pyrolytic graphite values for g obtained for the crystallite shear modulus [36], whereas our measurements concern crystallites with a cr from Fig. 4. Evaluation according to Eq. (7) for the size of some tenths of nanometers. Recent papers have unwrinkling parameter k shows the same trend as for the presented results on the Young’s modulus of carbon shear modulus, differing only in the numerical values. Fig. nanotubes by static tests in SEM of 1100 GPa [37] and by 8b shows the equivalent results as a function of the layer dynamic tests of 1200620 GPa [38]. It was stated explicit￾spacing d deduced from the position of the maximum of ly that this value is greater than the currently accepted 002 the 002 reflection. From this diagram it is clear that we are value for graphite. Our results for nanocrystallites in far from the layer spacing of 3.354 A for perfect graphite ˚ carbon fibers support these observations. and that the structure of all our fibers is thus turbostratic. The shear modulus of the crystallites exhibited a differ￾This is further confirmed by the fact that no general hkl ent value for MPP and PAN fibers, but was nearly constant reflections were observed for any of the fibres. within each group (see Figs. 4 and 5). We attribute this to a difference in the structure of the crystallites of MPP- and PAN-based fibers. The lower limiting value for the MPP fibers obtained by the different evaluation methods pro￾posed (see Figs. 4–6) is close to the value of g $ 4.0 GPa cr for graphite, where the dislocations were pinned by irradiation [39]. Fig. 4 shows that Eq. (3) yields good results for the PAN fibers, but for MPP fibers a deviation was observed at higher stresses. This is due to the increase of the shear modulus with increasing stress for this type of fiber (see Fig. 6), which is not taken into account by the actual models. We suggest that these simple models describing the orientation change of an anisotropic crys￾tallite in a constant stress field should be improved to account for the fact that the decrease of the width of the orientation distribution is more effectively hindered for higher stresses. One step in this direction can be found in Fig. 7. Reciprocal of the effective Young’s modulus from macro- Ref. [22], where a finite-element model of 256 elements scopic tests of the fibers increases linearly with the orientation with different orientations was developed to describe the parameter Z 2 Z . Symbols are the same as in Fig. 2. non-linearity of the stress–strain curve of carbon fibers. 0 1