CARBON PERGAMON Carbon4l(2003)563-570 Elastic moduli of nanocrystallites in carbon fibers measured by in-situ X-ray microbeam diffraction Dieter Loidl, Herwig Peterlik,, Martin Muller, Christian Riekel, Oskar Paris Institute of Materials Physics, University of vienna, Boltmanngasse 5, A-1090 vienna, Austria Institute of Experimental and Applied Physics, University of Kiel, LeibniestraBe 19, D-24098 Kiel, Germany ESRF. BP 220. F-38043 Grenoble Cedex. Fro Erich Schmid Institute of Materials Science, Austrian Academy of Sciences and Metal Physics Institute, University of Leoben, Jahnstrasse 12.4-8700 Leoben. Austria Received 13 September 2002: received in revised form 14 October 2002; accepted 15 October 2002 Abstract The in-plane Youngs modulus and the shear modulus of carbon nanocrystallites were investigated during in-situ tension tests of single carbon fibers by X-ray diffraction using the shift of the 10 band in the meridional direction and the change in the azimuthal width of the 002 reflection. The limiting value for the young s modulus was found to be 1 140 GPa, which is higher than the value for graphite obtained from macroscopic specimens, but coincides with recent measurements on nanotubes. Furthermore, the shear modulus was evaluated using a uniform stress approach and was found to increase with increasing misorientation of the crystallites. It turns out that both the in-plane Youngs modulus and the shear modulus are not constant, but dependent on the orientation parameter o 2002 Elsevier Science Ltd. All rights reserved Keywords: A. Carbon fibers; C. X-ray diffraction; D. Elastic properties, Microstructure, Lattice constant 1. Introduction dimensional structures and exhibit only a weak cross- sectional texture with differences in skin and core. A Carbon fibers combine high tensile strength and high number of structural models, such as ribbon-shaped and tensile modulus with low weight. They are an ideal elongated layers [13], a basket-weave structure [8or a reinforcing material for lightweight structures, e.g. in model consisting of crumpled and folded sheets of layer aerospace applications. In these applications, either high planes, have been proposed [11]. The interlinking of the tenacity or high Youngs modulus is required layers may be responsible for the (usually) high strength of The structure and morphology of carbon fibers have PAN-based carbon fibers [ 19). On the other hand, for MPP been investigated in detail by electron microscopic meth- (mesophase-pitch-based fibers a wide variety of different ods, i.e. SEM [1-7 TEM and HRTEM [8-12, and by structures is observed, e.g. a three-dimensional arrange- X-ray scattering [13-18]. Carbon fibers consist of stacked ment of crystallites and cross-sectional textures such exagonal carbon layers forming small coherently scatter- onion-like, or radial and radially folded structures [17, 20 ing units of only a few nanometers in size [12]. The high Whereas for PAN-based fibers the heat treatment tempera degree of preferential orientation along the fiber axis of the ture(HTT) is the main parameter used to control the layer planes is mainly responsible for the extraordinarily degree of orientation and thus the mechanical properties. high Young s modulus of the fibers [17]. On the one hand, for MPP fibers there are additional ways to influence the PAn (polyacrylnitrilej-based fibers are usually almost one structure arising from the pitch precursor and from the process used to convert it into a fiber form. Edie [20]gives an overview of the effect of different processing parame- orresponding author. Tel :+43-1-4277-51350 fax: +43.1- ters on the structure. In most cases the fibers obtained are 4277-9513 turbostratic,, i.e. the fibers exhibit no regular stacking E-mail address: herwig. peterlik( @univie. ac at(H. Peterlik order of the layer planes and the distance of the planes is 0008-6223/02/S-see front matter 2002 Elsevier Science Ltd. All rights reserved PII:S0008-6223(02)00359-7
Carbon 41 (2003) 563–570 E lastic moduli of nanocrystallites in carbon fibers measured by in-situ X-ray microbeam diffraction a a, b c d Dieter Loidl , Herwig Peterlik , Martin Muller , Christian Riekel , Oskar Paris * ¨ a Institute of Materials Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria b Institute of Experimental and Applied Physics, University of Kiel, Leibnizstraße 19, D-24098 Kiel, Germany c ESRF, BP 220, F-38043 Grenoble Cedex, France d Erich Schmid Institute of Materials Science, Austrian Academy of Sciences and Metal Physics Institute, University of Leoben, Jahnstrasse 12, A-8700 Leoben, Austria Received 13 September 2002; received in revised form 14 October 2002; accepted 15 October 2002 Abstract The in-plane Young’s modulus and the shear modulus of carbon nanocrystallites were investigated during in-situ tension tests of single carbon fibers by X-ray diffraction using the shift of the 10 band in the meridional direction and the change in the azimuthal width of the 002 reflection. The limiting value for the Young’s modulus was found to be 1140 GPa, which is higher than the value for graphite obtained from macroscopic specimens, but coincides with recent measurements on nanotubes. Furthermore, the shear modulus was evaluated using a uniform stress approach and was found to increase with increasing misorientation of the crystallites. It turns out that both the in-plane Young’s modulus and the shear modulus are not constant, but dependent on the orientation parameter. 2002 Elsevier Science Ltd. All rights reserved. Keywords: A. Carbon fibers; C. X-ray diffraction; D. Elastic properties, Microstructure, Lattice constant 1. Introduction dimensional structures and exhibit only a weak crosssectional texture with differences in skin and core. A Carbon fibers combine high tensile strength and high number of structural models, such as ribbon-shaped and tensile modulus with low weight. They are an ideal elongated layers [13], a basket-weave structure [8] or a reinforcing material for lightweight structures, e.g. in model consisting of crumpled and folded sheets of layer aerospace applications. In these applications, either high planes, have been proposed [11]. The interlinking of the tenacity or high Young’s modulus is required. layers may be responsible for the (usually) high strength of The structure and morphology of carbon fibers have PAN-based carbon fibers [19]. On the other hand, for MPP been investigated in detail by electron microscopic meth- (mesophase-pitch)-based fibers a wide variety of different ods, i.e. SEM [1–7], TEM and HRTEM [8–12], and by structures is observed, e.g. a three-dimensional arrangeX-ray scattering [13–18]. Carbon fibers consist of stacked ment of crystallites and cross-sectional textures such as hexagonal carbon layers forming small coherently scatter- onion-like, or radial and radially folded structures [17,20]. ing units of only a few nanometers in size [12]. The high Whereas for PAN-based fibers the heat treatment temperadegree of preferential orientation along the fiber axis of the ture (HTT) is the main parameter used to control the layer planes is mainly responsible for the extraordinarily degree of orientation and thus the mechanical properties, high Young’s modulus of the fibers [17]. On the one hand, for MPP fibers there are additional ways to influence the PAN (polyacrylnitrile)-based fibers are usually almost one- structure arising from the pitch precursor and from the process used to convert it into a fiber form. Edie [20] gives an overview of the effect of different processing parame- *Corresponding author. Tel.: 143-1-4277-51350; fax: 143-1- ters on the structure. In most cases, the fibers obtained are 4277-9513. ‘turbostratic’, i.e. the fibers exhibit no regular stacking E-mail address: herwig.peterlik@univie.ac.at (H. Peterlik). order of the layer planes and the distance of the planes is 0008-6223/02/$ – see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S0008-6223(02)00359-7
D. Loid/ et al. /Carbon 4(2003)563-570 significantly larger than that of crystalline graphite, al 2. Experimental though in some very high-modulus fibers, a three-dimen- sional graphite structure could be observed Five types of pitch-based fibers and a single type of Different models have been proposed to describe the PAN-based fiber were investigated. The pitch-based fibers relation between the orientation distribution of the basal were chosen so that they exhibited a considerable differ- planes and the elastic properties of carbon fibers: the ence in Youngs modulus. The PAN-based fibers were uniform strain model [21, 22], the uniform stress model exposed to a heat treatment temperature(hTT)to obtain [21, 22] and the unwrinkling model [13, 14, 22]. Experi- the same effect. Four PAN-based fibers were investigated ments have been presented for carbon fibers which strong. as-received, and with a HTT of 1800, 2100 and 2400C ly favored the uniform stress model [21]. In a recent work Some properties of the fibers are shown in Table I [22], different models were compared and a mosaic model The diameter of single fibers was measured by a laser was developed, which could explain the non-linearity in diffraction technique [ 31]. The fibers were then glued into the stress-strain curve of carbon fibers, i.e. the increase in a stretching cell, which was especially designed for in-situ the young's modulus during loading tensile tests of single fibers [24]. The Youngs modulus In the present work, we used a high brilliance synchrot was obtained from the slope of the stress-strain curve ron radiation X-ray microbeam to determine the structural during in-situ experiments. As a precise determination of changes of single carbon fibers during in-situ loading. a the Youngs modulus of single fibers is generally difficult small beam diameter has already been successfully applied due to uncertainties in the fiber diameter and in the fiber to obtain local structural information for different polymer strain, as well as due to the non-linearity of the stress- [23-25], natural [26, 27] and carbon fibers [28-30]. Where- strain curve, additional ex-situ experiments on single fibers as, in laboratory experiments, inevitable tilts in a fiber from the same bundle were performed using the method of bundle cannot be separated from the tilts of the layers laser speckle correlation for direct strain measurements within the fibers, this method allows the determination of [32, 33]. The results for the Youngs moduli of the in-situ the change of the orientation of the basal planes from the and ex-situ experiments were comparable, but the values intensity distribution of the 002 reflection with utmost from the latter were used for further evaluation due to the precision [30]. The measurement of single fibers instead of greater precision of this method. The Youngs modulus of bundles equally increases the reliability of the results, as the fibers increases significantly with increasing load [22] the same fiber is always located in the beam For some MPP-based fibers. the maximum value can be The shift of the 10 band in the meridional direction greater than the initial value by 30%. Thus, in Table I during in-situ loading was evaluated to directly obtain the only the initial value in the limit to small strains is given, Youngs modulus of the crystallites. Their shear modulus and used in the equations. was determined by an indirect method from the reduction Tension tests with in-situ X-ray diffraction were carried of the azimuthal width of the 002 reflection using theoret- out at microfocus beamline ID13 at the European cal models from the literature [21,22 Synchrotron Radiation Facility (ESRF) in grenoble name and ype E(o=o) MPP 3.4 HTA7-AR(Tenax) 332 by SEM and laser diffraction on the same fibers used fo ex-situ tension tests(see text). Due to the non-linearity can be greater than the initial modulus by more than 30% and is usually the one found in data sheets. The initial orientation parameter Ihwtm was deduced from the X-ray diffraction patterns and the density data were supplied by data sheets from the companies
564 D. Loidl et al. / Carbon 41 (2003) 563–570 significantly larger than that of crystalline graphite, al- 2. Experimental though in some very high-modulus fibers, a three-dimensional graphite structure could be observed. Five types of pitch-based fibers and a single type of Different models have been proposed to describe the PAN-based fiber were investigated. The pitch-based fibers relation between the orientation distribution of the basal were chosen so that they exhibited a considerable differplanes and the elastic properties of carbon fibers: the ence in Young’s modulus. The PAN-based fibers were uniform strain model [21,22], the uniform stress model exposed to a heat treatment temperature (HTT) to obtain [21,22], and the unwrinkling model [13,14,22]. Experi- the same effect. Four PAN-based fibers were investigated: ments have been presented for carbon fibers which strong- as-received, and with a HTT of 1800, 2100 and 2400 8C. ly favored the uniform stress model [21]. In a recent work Some properties of the fibers are shown in Table 1. [22], different models were compared and a mosaic model The diameter of single fibers was measured by a laser was developed, which could explain the non-linearity in diffraction technique [31]. The fibers were then glued into the stress–strain curve of carbon fibers, i.e. the increase in a stretching cell, which was especially designed for in-situ the Young’s modulus during loading. tensile tests of single fibers [24]. The Young’s modulus In the present work, we used a high brilliance synchrot- was obtained from the slope of the stress–strain curves ron radiation X-ray microbeam to determine the structural during in-situ experiments. As a precise determination of changes of single carbon fibers during in-situ loading. A the Young’s modulus of single fibers is generally difficult small beam diameter has already been successfully applied due to uncertainties in the fiber diameter and in the fiber to obtain local structural information for different polymer strain, as well as due to the non-linearity of the stress– [23–25], natural [26,27] and carbon fibers [28–30]. Where- strain curve, additional ex-situ experiments on single fibers as, in laboratory experiments, inevitable tilts in a fiber from the same bundle were performed using the method of bundle cannot be separated from the tilts of the layers laser speckle correlation for direct strain measurements within the fibers, this method allows the determination of [32,33]. The results for the Young’s moduli of the in-situ the change of the orientation of the basal planes from the and ex-situ experiments were comparable, but the values intensity distribution of the 002 reflection with utmost from the latter were used for further evaluation due to the precision [30]. The measurement of single fibers instead of greater precision of this method. The Young’s modulus of bundles equally increases the reliability of the results, as the fibers increases significantly with increasing load [22]. the same fiber is always located in the beam. For some MPP-based fibers, the maximum value can be The shift of the 10 band in the meridional direction greater than the initial value by 30%. Thus, in Table 1, during in-situ loading was evaluated to directly obtain the only the initial value in the limit to small strains is given, Young’s modulus of the crystallites. Their shear modulus and used in the equations. was determined by an indirect method from the reduction Tension tests with in-situ X-ray diffraction were carried of the azimuthal width of the 002 reflection using theoret- out at microfocus beamline ID13 at the European ical models from the literature [21,22]. Synchrotron Radiation Facility (ESRF) in Grenoble Table 1 Some parameters of the fibers investigated in this work Fiber name and Type Diameter Density E (s 5 0) P (s 5 0) eff hwhm 3 manufacturer (mm) (g/cm ) (GPa) (deg) K321 (Mitsubishi) MPP 10.48 1.9 136 17.6 E35 (DuPont) MPP 9.7 2.10 197 12.0 E55 (DuPont) MPP 10.18 2.10 358 7.09 FT500 (Tonen) MPP 10.0 2.11 380 6.69 K137 (Mitsubishi) MPP 9.54 2.12 500 3.42 HTA7-AR (Tenax) PAN 6.84 1.77 198 19.0 HTA7-18 (Tenax) PAN 7.3 1.77 273 16.3 HTA7-21 (Tenax) PAN 6.44 1.78 332 11.8 HTA7-24 (Tenax) PAN 6.2 1.91 349 9.64 The four PAN-based fibers differed in their final HTT (as-received, 1800, 2100 and 2400 8C, as indicated). The diameter was determined by SEM and laser diffraction on the same fibers used for the X-ray diffraction experiments. The initial modulus Eeff was obtained from ex-situ tension tests (see text). Due to the non-linearity, the maximum modulus can be greater than the initial modulus by more than 30% and is usually the one found in data sheets. The initial orientation parameter P was deduced from the X-ray diffraction patterns and the hwhm density data were supplied by data sheets from the companies
D Loidl et al./ Carbon 41(2003)563-570 France). The diameter of the microbeam (at wavelength (see Table 1)and the bulk density (i.e. the density A=0.0975 nm) was defined by a 10 um pinhole, and X-ray corrected for porosity) was calculated using the relation atterns were recorded by an area detector(MAR CCD) dooa), with M the atomic mass of carbon The setup corresponded to the standard setup of the and n, the avogadro number. The layer spacing doo? and beamline for scanning microbeam diffraction [34] the in-plane lattice spacing do were derived from the peak The load was increased stepwise, and for each loading positions of the 002 reflection and the 10 band of the step the fiber was centered in the beam and 2d diffraction unstretched fibers, with d=A/2 sin 0), 20 being the atterns were recorded at four positions along the fiber scattering angle of the maximum of the respective peak axis. Fig. I shows a 2D diffraction pattern of one of the fibers(hta7-21)with the meridional direction(fiber axis) rresponding to the loading direction. The diffraction 3. Theoretical models patterns from all investigated fibers showed the charac- teristic features of a turbostratic carbon with high preferred For the evaluation of the strain of the graphene planes, orientation(see Fig. 1)[28, 29]. The two-dimensional ntegration of the 10 band was performed in the meridional diffraction patterns were further processed using the eSre direction in a narrow sector to obtain the signal from those software FIT2D [35]. After background subtraction, one- planes only, which are almost perfectly aligned along the dimensional azimuthal scans of the 002 reflections were load axis. The strain ecr of the crystallite is obtained from obtained by integration of a small radial range around the the shift of the lattice spacing d,o under the applied load Q-value of the size-broadened 002 reflections. Moreover ne-dimensional radial scans of the 10 band in the (1) meridional fiber direction were obtained by integration of a small(+10)azimuthal range around the fiber meridian Peak position and shape were found to be independent of The microscopic Youngs modulus of the graphene the integration width within this range. The density of the planes, ecr=do/deer, is then calculated from the slope of the stress versus strain curve fibers was taken from the data sheets of the manufacture For the evaluation of the shear modulus of the crys tallies, the azimuthal intensity distribution of the 002 reflection is used. From geometrical considerations, a coordinate transformation relates the applied stress to the rotation of the graphene planes of a cubic-shaped crys- 10 tallite via the shear modulus gen, if the fiber is subjected to a tensile strain [21, 22). Different to Ref. [21] we use here the misalignment angle denoted by lI instead of the angle 002 002 f between the layer normal and the fiber axis their relation dor--2g-sin l cos Integration of Eq.(2)leads an I(o)=tan I1(o) (-2x) The decrease of the misorientation angle Il(o)under load can be taken as a measure of the shear modulus of the microscopic crystallites. For the evaluation of gr from the equatorial direction unwrinkling behavior, the azimuthal intensity distribution a(n) of the 002 reflection was fitted with a logistic function, the maximum amplitude of the data normalized he fiber HTA7-21 to 1. ity. The meridional 4(3+2√2)hwhm direction (vertical in the figure) on of the external (1+(3+2√2)hm) pores), the 002 reflection(from the stacking of carbon layers), and This one-parametric (i.e. the half-width at half-maxi- the 10 band (from the essentially 2D crystal structure of the mum II hwhm )distribution function was found to approx carbon layers) mate the varying 002 distributions of all the fibers sig
D. Loidl et al. / Carbon 41 (2003) 563–570 565 (France). The diameter of the microbeam (at wavelength (see Table 1) and the bulk density (i.e. the density l50.0975 nm) was defined by a 10 mm pinhole, and X-ray corrected for porosity) was calculated using the relation Œ] 2 patterns were recorded by an area detector (MAR CCD). rB A 10 002 5 3M/(Nd d ), with M the atomic mass of carbon The setup corresponded to the standard setup of the and NA 002 the Avogadro number. The layer spacing d and beamline for scanning microbeam diffraction [34]. the in-plane lattice spacing d10 were derived from the peak The load was increased stepwise, and for each loading positions of the 002 reflection and the 10 band of the step the fiber was centered in the beam and 2D diffraction unstretched fibers, with d 5 l/(2 sin u ), 2u being the patterns were recorded at four positions along the fiber scattering angle of the maximum of the respective peak. axis. Fig. 1 shows a 2D diffraction pattern of one of the fibers (HTA7-21) with the meridional direction (fiber axis) corresponding to the loading direction. The diffraction 3. Theoretical models patterns from all investigated fibers showed the characteristic features of a turbostratic carbon with high preferred For the evaluation of the strain of the graphene planes, orientation (see Fig. 1) [28,29]. The two-dimensional integration of the 10 band was performed in the meridional diffraction patterns were further processed using the ESRF direction in a narrow sector to obtain the signal from those software FIT2D [35]. After background subtraction, one- planes only, which are almost perfectly aligned along the dimensional azimuthal scans of the 002 reflections were load axis. The strain ´cr of the crystallite is obtained from obtained by integration of a small radial range around the the shift of the lattice spacing d under the applied load: 10 Q-value of the size-broadened 002 reflections. Moreover, d (s) 2 d (0) one-dimensional radial scans of the 10 band in the ´ (s) 5 ]]] 10 10] (1) cr d (0) meridional fiber direction were obtained by integration of a 10 small (6108) azimuthal range around the fiber meridian. The microscopic Young’s modulus of the graphene Peak position and shape were found to be independent of planes, e 5 ds/d´ , is then calculated from the slope of cr cr the integration width within this range. The density of the the stress versus strain curve. fibers was taken from the data sheets of the manufacturer For the evaluation of the shear modulus of the crystallites, the azimuthal intensity distribution of the 002 reflection is used. From geometrical considerations, a coordinate transformation relates the applied stress to the rotation of the graphene planes of a cubic-shaped crystallite via the shear modulus g , if the fiber is subjected to cr a tensile strain [21,22]. Different to Ref. [21] we use here the misalignment angle denoted by P instead of the angle j between the layer normal and the fiber axis, their relation being P 5 p/2 2 j : dP 1 ]5 2 ]sin P cos P (2) ds 2gcr Integration of Eq. (2) leads to s tan P(s) 5 tan P(0) expS D 2 ] (3) 2gcr The decrease of the misorientation angle P(s) under load can be taken as a measure of the shear modulus of the microscopic crystallites. For the evaluation of g from the cr unwrinkling behavior, the azimuthal intensity distribution I(P ) of the 002 reflection was fitted with a logistic function, the maximum amplitude of the data normalized Fig. 1. Two-dimensional diffraction pattern of the fiber HTA7-21. to 1: The diffracted intensity is shown in a pseudo-grey scale, dark Œ] P /Phwhm 4(3 1 2 2) corresponding to low and bright to high intensity. The meridional I(P ) 5 ]]]]]] ] (4) P /P 2 Œ hwhm direction (vertical in the figure) is the direction of the external (1 1 (3 1 2 2) ) load. The small-angle scattering (SAXS) signal (from oriented pores), the 002 reflection (from the stacking of carbon layers), and This one-parametric (i.e. the half-width at half-maxithe 10 band (from the essentially 2D crystal structure of the mum P ) distribution function was found to approxi- hwhm carbon layers) are indicated. mate the varying 002 distributions of all the fibers sig-
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)HTA7-24
566 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
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, consetion 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 paramethis 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
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 assumption 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 explanation 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 explicitspacing 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 differThis 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 proposed (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 crystallite 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
D. Loidl et al. Carbon 41(2003)563-570 0010020.030.04005006 3,48 56 2[A Fig. 8.(a) The effective shear modulus ger obtained from Eq. (8b) increases with increasing orientation parameter. The symbols are defined in Fig. 2. The lines are the values obtained for the single crystallites ge by Eq.(3)for MPP- and PAN-based fibers.(b)The eff shear modulus plotted versus the layer spacing dooz There is an additional increase in the shear modulus geff, calculated from Eq. (8b), with increasing orientation 1. The Young's modulus er of the graphene planes parameter as well as with increasing lattice distance dooz decreases with increasing orientation parameter, but (Fig. 8). This effect is much greater than the relatively shows a constant value of e=1140+ 10 GPa for fibers small increase of the crystallite shear modulus for MPP with the highest preferred orientation. This value is fibers and is equally observed for PAN fibers, where the larger than that of graphite, but is coincident with actual crystallite shear modulus was nearly constant( Fig. 6). The measurements of the Youngs modulus of carbon results may be described by supplementing the effective nanotubes constant shear modulus ger by an additional part g(zo -Zi) 2. The half-width of the 002 reflection decreases with depending on the orientation parameter, gerr=gr+ ng load nearly linearly for PAN-based fibers g(zo -Zi). We give here two possible interpretations of MPP-based fibers exhibit a much stronger and these results, which do not necessarily exclude each other linear decrease, which is not predicted by the One is that cross-links located at the crystallite boundaries models are responsible, which are favored by greater misorient- 3. The mean value of the shear modulus ger of the tion angles of the crystallites. The other possibility is that crystallites was evaluated to be g=4.5+0.2 GPa the fibers are built up as(nano-)composites, which would MPP-based and ger 13.6*0.9 GPa for PAN-based also lead to an increase in the shear modulus. The latter fibers. Whereas nearly no dependence of crystallite assumption is supported by the possible existence of two shear modulus on stress was observed for pan fibers it phases, in particular in MPP fibers [30], and by experimen- increases to a greater extent the higher the orientation of tal observation of the arrangement of the crystallites in the mPp fibers aligned micro-domains as suggested by HR-SEM [ 40. 4. With the additional knowledge of the effective Youngs However, concise theoretical models should be developed modulus of the fibers, the shear modulus g er can be to clarify the role of the crystallites, their elastic properties obtained within the framework of the uniform stress of the intel model, which increases from the value of g. with creasing orientation parameter Cross-links located at the crystallite boundaries or a composite cha 6. Conclusions he material are possible explanations for this behavior. 5. Existing models based on crystallites with identical and Microbeam X-ray diffraction is able to provide detailed onstant modulus and only different orientation should quantitative information on the development of the inner be improved. The experimental results suggest that structure and its relation to the mechanical properties of neither the Youngs modulus of the crystallites nor the carbon fibers during in-situ loading. The following param- effective shear modulus are constant, but depend on the eters show a dependence on the applied stress initial orientation parameter
D. Loidl et al. / Carbon 41 (2003) 563–570 569 Fig. 8. (a) The effective shear modulus geff obtained from Eq. (8b) increases with increasing orientation parameter. The symbols are defined as in Fig. 2. The lines are the values obtained for the single crystallites g by Eq. (3) for MPP- and PAN-based fibers. (b) The effective cr shear modulus plotted versus the layer spacing d . 002 There is an additional increase in the shear modulus g , eff calculated from Eq. (8b), with increasing orientation 1. The Young’s modulus e of the graphene planes cr parameter as well as with increasing lattice distance d decreases with increasing orientation parameter, but 002 (Fig. 8). This effect is much greater than the relatively shows a constant value of e 5 1140610 GPa for fibers cr small increase of the crystallite shear modulus for MPP with the highest preferred orientation. This value is fibers and is equally observed for PAN fibers, where the larger than that of graphite, but is coincident with actual crystallite shear modulus was nearly constant (Fig. 6). The measurements of the Young’s modulus of carbon results may be described by supplementing the effective nanotubes. constant shear modulus g by an additional part g(Z 2 Z ) 2. The half-width of the 002 reflection decreases with cr 0 1 depending on the orientation parameter, g 5 g 1 increasing load nearly linearly for PAN-based fibers. eff cr g(Z 2 Z ). We give here two possible interpretations of MPP-based fibers exhibit a much stronger and non- 0 1 these results, which do not necessarily exclude each other. linear decrease, which is not predicted by the actual One is that cross-links located at the crystallite boundaries models. are responsible, which are favored by greater misorienta- 3. The mean value of the shear modulus g of the cr tion angles of the crystallites. The other possibility is that crystallites was evaluated to be g 5 4.560.2 GPa for cr the fibers are built up as (nano-)composites, which would MPP-based and gcr 5 13.660.9 GPa for PAN-based also lead to an increase in the shear modulus. The latter fibers. Whereas nearly no dependence of crystallite assumption is supported by the possible existence of two shear modulus on stress was observed for PAN fibers, it phases, in particular in MPP fibers [30], and by experimen- increases to a greater extent the higher the orientation of tal observation of the arrangement of the crystallites in the MPP fibers. aligned micro-domains as suggested by HR-SEM [40]. 4. With the additional knowledge of the effective Young’s However, concise theoretical models should be developed modulus of the fibers, the shear modulus g can be eff to clarify the role of the crystallites, their elastic properties obtained within the framework of the uniform stress and the properties of the interface. model, which increases from the value of g with cr increasing orientation parameter. Cross-links located at the crystallite boundaries or a composite character of 6. Conclusions the material are possible explanations for this behavior. 5. Existing models based on crystallites with identical and Microbeam X-ray diffraction is able to provide detailed constant modulus and only different orientation should quantitative information on the development of the inner be improved. The experimental results suggest that structure and its relation to the mechanical properties of neither the Young’s modulus of the crystallites nor the carbon fibers during in-situ loading. The following param- effective shear modulus are constant, but depend on the eters show a dependence on the applied stress: initial orientation parameter
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