D. Loid et al. Carbon 4/(2003)563-570 0.000010020.030.040.050.06007 Fig. 3. In-plane Youngs modulus of the graphene layers as a function of the orientation parameter Z, -Z. The limiting value Fig. 4. Decrease of the half-width (whm)of the azimuthal for small orientation parameters is ee,=1 140+10 GPa. Symbols ntensity distribution of the 002 reflection as a function of the are the same as in Fig 2 applied stress. Symbols as for Fig. 2. The lines are fits for II fro Eq. (3)with shear modulus 8er = 4.5 GPa for the MPP-based fibers and ge= 13.6 GPa for the PAN-based fibers different fibers. The Youngs modulus of the graphene planes is obtained directly from the inverse slope of Fig. 2, tallies were obtained from this diagram, i.e. ger=4.9 GPa and is shown in Fig. 3 as a function of the orientation for MPP and ge= 14.2 GPa for PAN fibers. Furthermore, parameter Z, -Zi. As stated above, Z, cannot generally be combining the reduction of the azimuthal angle(Eq (3) neglected as suggested in previous reports [21], because its with the orientation distribution(Eq.(4)) was used to numerical value is more than 25% of the value of Zo for calculate the crystallite shear modulus at each stress step the highly misoriented fibers hTA7-AR and K321. Fre The results are depicted in Fig. 6. Whereas for PAN fibers Fig 3 one can clearly deduce that the Young's modulus of there is almost no dependence on the stress(with the the graphene planes shows no dependence on the orienta- tion parameter for highly oriented fibers and gradually quently, the highest orientation), for MPP fibers a general decreases for fibers with an orientation parameter larger trend was observed. The shear modulus increases with than 002. The mean value and the standard error of the stress to a greater extent for fibers with higher orientation Youngs modulus of the graphene planes of the three most Eq.(&a) predicts a linear dependence of the reciprocal highly oriented fibers is eer =1140+10 GPa. As this is effective Youngs modulus of the macroscopic fiber(the only the statistical error neglecting instrumental precision, modulus corrected by porosity )on the this represents a lower limit for the error ter Zo 21]. It can be concluded from Fi According to Eqs. (2)and(3), the shear modulus can be relation is valid, but the orientation obtained by the decrease of the half-width of the azimuthal distribution of the 002 refiection with increasing load. Fig 4 shows the decrease of the half-width of the distributi normalized to its initial value for the different fibers Whereas within the mPP-based and the pan-based fibers the differences are rather small, there is a large difference o lines in Fig 4 are fits of Eq(3), the fitted parameters for0.5 in the numerical values between both types of fibers. The he crystallite shear modulus being ger =4.5+0.2 GPa for -1.0 the MPP-based fibers and g.=13.6+0.9 GPa for the PAN-based fibers. From the fitted lines it is clear that Eq ()describes the results for the PAN fibers very well, but for MPP fibers only in the limit of small stresses. The derivative of the half-width with respect to the strength for 5 small stresses. i.e. aI/do in the limit g-0 can be used to directly validate Eq. (2). Fig. 5 shows a linear dependence of alhwhm/ao on whm(o=0), because in Fig. 5. The change of the azimuthal width all,m /ao of the 002 the observed range of 11<20%, sin(n)cos(n)is nearly eflection in the limit of small stresses is proportional to its initial linear. Similar values for the shear modulus of the crys- value I(0). Symbols are the same as in Fig. 2.D. Loidl et al. / Carbon 41 (2003) 563–570 567 Fig. 3. In-plane Young’s modulus of the graphene layers as a Fig. 4. Decrease of the half-width (P ) of the azimuthal hwhm function of the orientation parameter Z 2 Z . The limiting value 0 1 intensity distribution of the 002 reflection as a function of the for small orientation parameters is e 5 1140610 GPa. Symbols cr applied stress. Symbols as for Fig. 2. The lines are fits for P from are the same as in Fig. 2. Eq. (3) with shear modulus gcr 5 4.5 GPa for the MPP-based fibers and g 5 13.6 GPa for the PAN-based fibers. cr different fibers. The Young’s modulus of the graphene planes is obtained directly from the inverse slope of Fig. 2, tallites were obtained from this diagram, i.e. g 5 4.9 GPa cr and is shown in Fig. 3 as a function of the orientation for MPP and g 5 14.2 GPa for PAN fibers. Furthermore, cr parameter Z 2 Z . As stated above, Z cannot generally be combining the reduction of the azimuthal angle (Eq. (3)) 01 1 neglected as suggested in previous reports [21], because its with the orientation distribution (Eq. (4)) was used to numerical value is more than 25% of the value of Z for calculate the crystallite shear modulus at each stress step. 0 the highly misoriented fibers HTA7-AR and K321. From The results are depicted in Fig. 6. Whereas for PAN fibers Fig. 3 one can clearly deduce that the Young’s modulus of there is almost no dependence on the stress (with the the graphene planes shows no dependence on the orienta- exception of the fiber with the highest HTT and, conse￾tion parameter for highly oriented fibers and gradually quently, the highest orientation), for MPP fibers a general decreases for fibers with an orientation parameter larger trend was observed. The shear modulus increases with than 0.02. The mean value and the standard error of the stress to a greater extent for fibers with higher orientation. Young’s modulus of the graphene planes of the three most Eq. (8a) predicts a linear dependence of the reciprocal highly oriented fibers is e 5 1140610 GPa. As this is effective Young’s modulus of the macroscopic fiber (the cr only the statistical error neglecting instrumental precision, modulus corrected by porosity) on the orientation parame￾this represents a lower limit for the error. ter Z [21]. It can be concluded from Fig. 7 that this linear 0 According to Eqs. (2) and (3), the shear modulus can be relation is valid, but the orientation parameter Z was 0 obtained by the decrease of the half-width of the azimuthal distribution of the 002 reflection with increasing load. Fig. 4 shows the decrease of the half-width of the distribution normalized to its initial value for the different fibers. Whereas within the MPP-based and the PAN-based fibers the differences are rather small, there is a large difference in the numerical values between both types of fibers. The lines in Fig. 4 are fits of Eq. (3), the fitted parameters for the crystallite shear modulus being g 5 4.560.2 GPa for cr the MPP-based fibers and g 5 13.660.9 GPa for the cr PAN-based fibers. From the fitted lines it is clear that Eq. (3) describes the results for the PAN fibers very well, but for MPP fibers only in the limit of small stresses. The derivative of the half-width with respect to the strength for small stresses, i.e. ≠P /≠s in the limit s → 0, can be hwhm used to directly validate Eq. (2). Fig. 5 shows a linear dependence of ≠Phwhm hwhm /≠s on P (s 5 0), because in Fig. 5. The change of the azimuthal width ≠Phwhm /≠s of the 002 the observed range of P , 208, sin(P ) cos(P ) is nearly reflection in the limit of small stresses is proportional to its initial linear. Similar values for the shear modulus of the crys- value P(0). Symbols are the same as in Fig. 2