Carbon Alloys part 6 ppsx

Computer Simulations 159 Yagi, T. Kimura, G. Dresselhaus and M.S. Dresselhaus, Chemical reaction of intercalated atoms at the edge of nano-graphene cluster, liquid crystal and molecular crystals. Molec. Cryst. Liquid Cryst., 340: 71-76,2000. 43. R. Saito, G. Dresselhaus and M.S. Dresselhaus, Physical Properties of Carbon Nanotubes. Imperial College Press, London, 1998. 44. R. Saito, T. Takeya, T. Kimura, G. Dresselhaus and M.S. Dresselhaus. Finite size effect on the Raman spectra of carbon nanotubes. Phys. Rev. B, 59 2388-2392,1999; S. Roche and R. Saito, Effects of magnetic field and disorder on electronic properties of carbon nano- tubes. Phys. Rev. B, 59,5242-5246,1999. 161 Chapter 10 X-ray Diffraction Methods to Study Crystallite Size and Lattice Constants of Carbon Materials Minoru Shiraishi" and Michio Inagakib "Tokai Universiq, School of High-technology for Human Weqare, Department of Material Science and Technology, Numaru, Shizuoka, 410-0395 Japan; 'Aichi Institute of Technology, Tqota, 470-0392 Japan Abstract: The measurement of crystallite sizes (La and L,) and lattice constants (a,, and c,) is described using X-ray diffraction profiles for graphitizable carbons during graphitization. The method is called the JSPS method and is based on the use of an internal standard of silicon crystal and thin sample (0.2 mm) with several corrections of observed intensity being applied. Crystallite size distributions are obtained from diffraction profiles for carbons heat-treated to relatively low temperatures. Keywords: X-ray diffraction, Carbon materials, Crystallite size, Lattice constant, Crystallite size distribution, JSPS method. 1. Introduction X-ray diffraction has been used for structural analyses of solids for quite some time. X-ray diffractometers use powder samples as well as single-crystal materials. X-ray diffraction is used to study mechanisms of graphitization and the characterization of non-graphitic carbon. Excellent reviews are available [l-31. Modern computer technology makes numerical analyses much easier. X-ray diffraction profiles of a petroleum coke (PC30) and a phenol-formaldehyde resin char (PF20), HTT 3000 and 2000"C, respectively, are shown in Fig. 1. Diffraction peaks from the petroleum coke correspond closely to diffraction lines from graphite crystal. However, for the phenol-formaldehyde resin, although a few diffraction profiles match closely those of graphite for position, they are much broader and are asymmetrical. Structures of PO0 and PF20 are thus quite different and X-ray diffractions can go some way in identifying these differences. In graphitizable carbons, the 00 I diffractions are sensitive to the heat treatment temperatures (€I") some 004 X-ray diffraction profiles being shown in Fig. 2. The 162 Chapter 10 Fig. 1. Typical X-ray diffraction profiles of petroleum coke (PC30) heat-treated at 3000°C and phenol-formaldehyde resin carbon (PRO) heat-treated at 2000°C. diffraction profiles sharpen and move to the high angle side with increasing HTT. The profile shape, position and width at half-maximum intensity (FWHM) are used in the characterization of carbons. There is a need, however, for all researchers to use identical procedures for (a) the measurement of the profile (peak) position, (b) the FWHM of the carbon, and (c) calculating (assessment) of stacking sizes from profile analysis of low-temperature carbons. 2 Measurement Method (JSPS Method) In Japan, the X-ray diffraction method for carbons is the so-called JSPS method (Japan Society for the Promotion of Science) and has been used since 1963 [4]. Samples for these measurements are powdered carbons at various stages of the graphitization process. This method specifies the measurement condition for X-ray diffraction and analysis procedures. These features are summarized as follows: 1. Crystalline powders of high purity silicon are used as an internal standard. 2. The thickness of sample is limited to 0.2 mm. 3. Measurements of each diffraction line, 002,004,110 and 112 peaks, are based on the use of Cu K, X-ray radiation. 4. Each diffraction profile is corrected for unwanted intensity factors, such as absorption and Lorenz-polarization. 5. A single procedure is adopted to determine the position and FWHM of each peak. 6. Lattice constants and apparent “crystallite sizes” are calculated by the same mathematical procedures. X-ray diffraction profiles contain components arising from such effects as Lorenz-polarization factors, absorption factors, doubleting of K, radiation and X-ray DifSraction Methods 163 1640 1810 2050 2290 2520 2800.C I I I I I I I I I 50 51 52 63 54 65 56 57 58 28 (ma) Fig. 2. Peak shift of 004 diffraction of graphitizable carbon with heat-treatment temperature. instrumental broadening. The dependence of the atomic scattering factor of carbon on sin 8l‘h affects the profile, particularly the 002 band at low diffraction angles, 8. It is necessary to correct for these factors to obtain a more meaningful crystallite size from the profiles. Crystallite sizes thus calculated, however, still contain effects caused by lattice distortions in carbon structures. Therefore, it is recommended to regard the crystallite sizes obtained as “apparent” crystallite sizes, La and L,. In the JSPS method, observed diffraction intensities are corrected for the follow- ing factors as set out below. 164 Chapter 10 2.1 Absorption Factors As carbon is a light element, X-rays penetrate deep inside a thick sample and are diffractedhcattered there. The diffracted intensity is strengthened at the low-angle side, and the profile broadens. Therefore, observed profiles of carbons need to be corrected for absorption more than for other materials such as metals and metal compounds. Because X-ray scattering intensities, as measured by the diffractometer, are based on X-rays scattered from that part glanced from the detection counter within an irradiated sample volume, the change of the sample volume irradiated by X-rays at a diffraction angle has to be as small as possible. Hence, for low absorption coefficients and keeping small change of the irradiated sample volume during measurement of each diffraction line, the sample thickness is specified as being 0.2 mm. To correct for absorption, the absorption factor,A, in the following equation is used in the JSPS method [5]: (1) 2t cos e 1 -exp(2p‘tcos ece)) + ___ exp(-2p’tcosece) W 1 2p‘w cos ec28 where 8 is the diffraction angle, w is the width of X-ray radiation on the sample surface and p’ is the linear absorption coefficient of the carbon material mixed with a silicon standard for Cu K,. When the bulk density of the X-ray sample p’ is 1.0, the value of p’ is calculated to be 10 for the sample mixed with 10 wt% Si and 16 with 20 wt% Si, because mass absorption coefficients p/p of carbon and silicon are (cL/p), = 4.219 and (p/~)~~ = 65.32, respectively. 2.2 Profile broadening due to K, doublet Because K, radiation consists of K,, and Ka2 with small differences in wavelength (0.54056 and 0.154439 nm, respectively), all diffraction lines must be composed of two lineswith small differences in diffraction angles. For the lines observed at high angles, two separated diffraction lines are observed, but only broadened profiles without apparent separation are observed at low angles. Jones [6] illustrated angular separa- tion curves as a function of 20 and proposed a separation procedure by assuming that the two profiles for K,, and Kd radiations have the same shape and breadth, but differ in height by a factor of two and a shift by A in 28. His procedure is employed in the JSPS method, the software for which is available in all commercially available X-ray diffractometers. 2.3 Imtrumntal Broadening Diffraction profiles are broadened depending on the instrumental and optical conditions used, the instrumental broadening being related to the breadth of the X-ray source, flat specimen surface and axial divergence of the X-ray beam [7]. The X-ray Difiaction Methods 165 peak width of the silicon (internal standard) shows the instrumental broadening of the X-ray diffractometer, because of its high crystallinity. Jones [6] corrected observed profiles for instrumental broadening by applying a Fourier analysis to a sample which causes broadening due to a small crystal size as well as a standard material which does not show broadening because of its large crystal size and showing only instrumental broadening. Klug and Alexander [7] provided correction curves to the widths of X-ray diffractions arising from instrumental broadening under different conditions. The JSPS method adopts their curves for high-resolution low-angle reflection conditions. 2.4 Lorentz-Polarization Factor andhmic Scattering Factor of Carbon As the Lorentz factor (L) and polarization factor (P) change with diffraction angle, corrections are necessary to obtain a correct profile, particularly for broad profiles. In the JSPS method, corrections of the L and P factors are made for the profiles with FWHMs larger than 0.5", according to the following equations: 1 L= sin e .cos e 1 2 p=-(i+cos2 28) 1 +COS 20 .COS 28' 1+cos2 2w P= (3) (4) Equations (2) and (3) are used when a nickel filter is used, and Eqs. (2) and (4) for a graphite counter monochromator. In Eq. (928 is the diffraction angle of grphite, that is 26.56". Because atomic scattering factors decrease with increasing diffraction angle, it is necessary to correct the diffraction profiles of carbons using the square of the atomic scattering factor of carbon vc), given by the following equation [8]: f, = 226069exp(-226907xs2)+L56165exp(-0.656665~~~) + LO5075 exp(-9.75618 x s2 ) + 0.839259 exp(-55.5949 x s2 ) +0.286977 (5) s=sine/h 2.5 Prizcticizl Procedures 2.5. I Correction of observed X-ray intensity The observed X-ray intensity (lobs) is given by 166 Chapter 10 I,, = ~PAG I q2 = KLPAG f: I FgI2 in which K is a constant, L = Lorentz factor, P = polarization factor, G = hue function, F = structure factor of graphite$, = atomic scattering factor of carbon, and F, = geometrical structure factor of graphite. The factor that relates to the size of crystallites in the carbon is G2. To determine apparent crystallite sizes and lattice constants of carbon, the profile of G2 has to be obtained first, this being achieved by correction of the observed profile using the factors, L, P, A and f,'. Then the peak positions and FWHM can be determined. In the JSPS method, the corrected profile is obtained by dividing the observed intensity by the factors LPAf; for each diffraction angle. Because values of LPAf; decrease monotonously, relative values of LPAf; which make 28 = 30" to be unity for 002 diffractions and 28 =57" for 004 diffractions are used. These relative values (FCT) were calculated for a sample thickness of 0.2 mm mixed with 10 or 20 wt% silicon and approximated by the following series: FCT = C, + C, . (28) + C, . (28)2 + C4 . + C, .(28)4 + C, . (7) Values of the coefficients C,, C,, C, in Eq. (7) are listed in Table 1. The diffraction profile has to be further corrected by a levelling of the background. Table 1 Coefficient of correction Eq. (7) for 002 and 004 diffraction intensities Ni filter Counter monochromator Si 10% Si 20% Si 10% Si 20% 002 Diffmction Cl c2 c3 c4 c5 C6 0.1062200x103 -0.1493785 X Id 0.8948090X 10' -0.2796578X 10-' 0.4500046~ W3 -0.2956388~ lW5 0.1229242~ 10.3 -0.7458276X 10' 0.1753749X 10' -0.1876302X 1C2 0.7662217~ lW5 0.8902088X Id -0.1223796x1O2 0.7201680X 10' 4.2217635 X 10-' 0.3521847X lW3 -0.2285659X it5 0.9028824X Id -0.5160503X 10' 0.1 14250% 10" 4.1149O98x 0.4401877X 0.9350447X lo2 -0.1316116X Id 0.7921825 X 10' -0.2491818X 10-' 0.4038340X lC3 -0.2672795 x lt5 0.4067178 x lo2 -0.1807844 X 10' 0.2791991 X 10-' -0.1476100X lod3 0.7836853 X lo2 4.1078659X lo2 0.6383320X 100 -0.1980132X 10-' 0.3170086~10-~ -0.2074537~ 10-5 0.3747529~102 4.1656186X 10' 0.2550629X 10-' -0.1346840~10-~ X-ray Diffraction Methods 167 2.5.2 Determination of lattice constants (e, and a3 As shown in Fig. 3, the diffraction angle 28,, is obtained from the position of middle point of the segment divided by the profile on the line parallel to the background at a position of 2/3 height from the background. From the diffraction angle of the standard silicon (2e0& and that of carbon, (26,, )c, the corrected diffraction angle, (2eo& of the carbon is obtained from the following equation: where (2eCoJsi are the diffraction angles calculated from the lattice constant (0.543073 nm) of the silicon crystal. Lattice constants along the c-axis (c,) and a-axis (a,) are calculated from the diffraction angles (28,,), of either 002,004 or 006 lines and from the 110 line, respectively. 2.5.3 Determination of apparent crystallite size (L, and LJ Apparent crystallite sizes are obtained from the Scherrer equation using FWHM of the profile corrected for instrumental broadening and the K, doublet. The width of profile at the 1/2 position (half-peak width) of the maximum intensity from the background, namely FWHM, is measured at the angle unit, as shown in Fig. 3. When the profiles do not show apparent separation of the K, doublet, FWHM makes the correction on K, doublet by using the curves shown in Fig. 4, which are referred to the profiles intermediate in shape between Gaussian and Cauchy [6]. This curve is approximated by the following equation: Si I iwn 31 1 Carbon 54 55 56 57 28 (Wa) Fig. 3. Measurement example of diffraction angle and width. 168 1 0.9 Chapter 10 0.7 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 &bo or d/Bo Fig. 4. Correction curve for angular separation of K, doublet. (9) BIB, or blb, = 0.9991090+0.1180823~u-2.447168~u2 +8.254356.u3 -20.96720. u + 17.34026. u where Bo is the experimental FWHM for the carbon, bo is the observed FWHM for the silicon standard, and A is the angular separation of the K, doublet given in Table 2, and u is either NB, or 4b,. From the curve of Fig. 4 or Eq. (9), are obtained FWHMs corrected for K, doublet for carbon and silicon, that is B and b, respectively. When the profiles show apparent separation due to the K, doublet, the procedure written above is applied and the FWHM values of B and b are measured from the profile of K,, radiation of sample carbon and standard silicon, respectively. Table 2 A value of each diffraction peak Carbon Silicon hkl AC) hkl AC) 002 004 110 112 006 0.067 0.147 0.229 0.255 0.271 111 311 331 422 0.072 0.152 0.224 0.275 [...]...X-ray Difiaction Methods 169 0 0.2 0 .6 0.4 08 1 bA3 Fig 5 Correction cuwe for instrumental broadening In the next step, the following approximation (Eq (10)) is used for the instrumental broadening correction (Fig 5): p / B =0.9981 266 -0. 068 1532.~-2.592 769 .~~ 2 .62 1 163 .~~ + -0.9584715.~~ (10) where v = b/B, and p is the corrected FWHM for the carbon being studied The corrected FWHM p... Crystallite Sizes on Artificial Graphite Tanso, 25-28,1 963 5 M.E Milberg, Transparency factor for weakly absorbing samples in X-ray diffractometry J Appl Phys., 2 9 64 -65 , 1958; M Shiraishi and K Kobayashi, On absorption correction for X-ray intensity Tanso, 49-50,1973 6 F.W Jones, The measurement of particle size by the X-ray method Proc Roy SOC (London), 166 A: 16- 43,1938 7 H.P Hug and L.E Alexander, X-ray diffraction... Chem SOC 2575-2578,1973 14 B.E Warren and P Bodenstein, The diffractionpattern of fine particle carbon blacks Acta Crystallogr., 18: 282-2 86, 1 965 15 H Fujimoto and M Shiraishi, Characterization of unordered carbon using WarrenBodenstein’s equation Carbon, 3 9 1753-1 761 ,2001 175 Chapter 11 Pore Structure Analyses of Carbons by Small-Angle X-ray Scattering Keiko Nishikawa Division of Diversity Science,... resin as carbon precursor and polystyrene (10 wt%, grain size 20 nm) as the poreforming resin This polymer blend was spun, stabilized and carbonized into carbon fibers Two different carbons, HTT 500 and 1000°C, Carbons A and B respectively, Chapter 11 184 0.05 0.1 Fig 5 The scattering patterns of Carbon A (SOOT) and Carbon B (lOOO°C) were prepared and studied This combination of resins produced carbons... as shown in Fig 6 From the gradients of the lines, we obtained values of R, of 71 8, (7.1 nm) for Carbon A and 69 8, (6. 9 nm) for Carbon B, respectively The s2-values corresponding to (l/R,)' are shown by broken lines The linearity of the Guinier plots, in the Guinier region, provides 71 8, and 69 A as acceptable R, values 10[ ' ' ' 1 I ' ' " 01 $1 A-2 Fig 6 Guinier plots; the plot for Carbon B is displaced... diameter (2R) and the height (L) becomes about 1:1 .66 It is known that for a column with a ratio of 1:1 .65 the Guinier plot shows linearity in the wide region [11.This is because the Guinier plot for Carbon B is more linear in the wider region than in the Guinier region (Fig 6) Chapter I I 1 86 Table 3 Structure parameters for Carbon A (500°C) and Carbon B (100WC) Rga is the gyration radius obtained... glass-like carbon with heat treatment, studied by small angle X-ray scattering (I) Glass-like carbon prepared from phenolic resin Jpn J Appl Phys, 37: 64 86- 6491, 1998 5 A Mittelbach, Zur Rontgenkleinwinkelstreuung verdunnter kolloider systeme Acta Physica Austriac, 19: 53-102, 1 964 6 H Matsuoka, Small-angle Scattering in Colloid Science IV Chemical Society of Japan (ed.), Tokyo Kagaku Dozin, Tokyo, 19 96 (in... Physics of Carbon, Vol 3 Marcel Dekker, New York, pp 289-288,1 968 2 W Ruland, X-ray Diffraction Studies on Carbon and Graphite, In: P.L Walker (Ed.), Chemistry and Physics of Carbon, Vol 4 Marcel Dekker, New York, pp 1-84,1 968 3 J Maire and J Mering, Graphitization of Soft Carbons, In: P.L Walker (Ed.), Chemistq and Physics of Carbon, Vol 6 Marcel Dekker, New York, pp 125-190,1970 4 Japan Society for... electrons in the particle and the integral is performed over the volume of the particle Namely, nR; is equal to the second moment of the particle with the weight of electron density p(r) For a homogeneous particle in which p(r) is assumed to be constant p, Eq (6) is rewritten by The relations between shape parameters of the particle and gyration radius are shown in Table 1.For a spherical particle, the... 72 69 k3 4121 5821 192 67 Secondly,R, values were obtained by using the distance distribution function (Eqs ( 16) and (17)), because here the Guinier regions are narrow in the measurable region due to the relatively large values of R, for these carbons The distance distribution functions for Carbons A and B are shown in Fig 8 The calculated R values from the obtained P(r)s and Eq (5) become 72 8, for Carbon . factor of carbon vc), given by the following equation [8]: f, = 2 260 69exp(-2 269 07xs2)+L 561 65exp(-0 .65 666 5~~~) + LO5075 exp(-9.7 561 8 x s2 ) + 0.839259 exp(-55.5949 x s2 ) +0.2 869 77. 0.78 368 53 X lo2 4.107 865 9X lo2 0 .63 83320X 100 -0.1980132X 10-' 0.31700 86~ 10-~ -0.2074537~ 10-5 0.3747529~102 4. 165 6186X 10' 0.255 062 9X 10-' -0.13 468 40~10-~ X-ray Diffraction. -0.18 763 02X 1C2 0. 766 2217~ lW5 0.8902088X Id -0.1223796x1O2 0.720 168 0X 10' 4.221 763 5 X 10-' 0.3521847X lW3 -0.228 565 9X it5 0.9028824X Id -0.5 160 503X 10' 0.1 14250%