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Structural transformations induced in graphite by grinding: Analysis of 002 X-ray diffraction line profiles
oax6223/90 $3.00 + .oo Copyright 0 1990 Pergamon Press plc
Carbon Vol. 2% No. 6. pp. 897406. 1990 Printed m Great Bri~am.
STRUCTURAL TRANSFORMATIONS INDUCED IN GRAPHITE BY GRINDING: ANALYSIS OF 002 X-RAY DIFFRACTION LINE PROFILES J. B. ALADEKoMot AND R. H. BRAGG Department of Materials Science and Mineral Engineering, University of California, and Materials and Chemical Sciences Division, Lawrence Berkeley Laboratory. 1 Cyclotron Road, Berkeley. CA 94720, U.S.A. (Received 2 October 1989; accepted in revised form 1 February
1990)
Abstract-The x-ray diffraction line profiles of the 002 reflections of graphite samples ground in a ball mill for periods up to 90 hours were studied using newly developed peak analysis methods that permitted separation of overlapping peaks having different peak positions, intensities, and line-breadths. For 3.354A I d,, s 3.375Aa continuous range of single phases characterized by symmetrical peaks of increasing breadths was observed, but peaks whose centroids fell between 3.375A and 3.55A were asymmetrical and composed of a superposition of symmetrical peaks of increasing breadths characterized by 3.375A. 3.40& 3.44A. and 3.55A spacings. There is some indication of a phase characterized by a 3.491% interlayer spacing. These results confirm previous reports of discrete 002 spacings in carbons and disordered graphite and suggest that elemental carbon(s) are composed of mixtures of these metastable graphite phases. Key WordsX-ray
diffraction, interlayer spacmg grmdmg, natural graphite.
1. INTRODUCTION It has been pointed out that when soft carbon materials are annealed at high temperature, or when graphite is disordered by fast neutron irradiation or mechanical grinding. a graph of the interlayer spacing, &. versus the processing time contains a set of arrests or plateaus[ 11. The aggregate of the data from a variety of sources consists of characteristic interlayer spacings of about 3.38A, 3.40& 3.425& 3.44A, 3.55,&, and one at about 3.68&2-61. Although the overall agreement of these d-spacings is good, there are significant gaps in the data obtained from different sources. Data obtained from the annealing of pyrolytic carbons have an initial plateau at about 3.425 * 0.003A and an arrest at about 3.38A, but the other levels are missing[2.3). Data obtained from studies of natural graphite disordered by prolonged mechanical grinding indicate plateaus of 3.4OA and 3.44A. and an inflexion at 3.38A, but the 3.425A and 3.55A spacings were not reported[6]. A 3.86A spacing is implied in the results of Henson and Reynolds(5], but was not observed in any of the other studies. However. a 3.55A spacing is commonly observed in low temperature (40%500°C) heat treated pitches. as are levels or arrests at 3.38A. 3.4OA (rarely), and 3.441j when they are annealed at much higher temperatures. Also. the 3.55%1 spacing has recently been observed in an as-received pyrolytic carbon processed below 2100°C as well as a 3.49A spacing in the same material annealed at 26OO”C[7].
ton leave from the University of Ife, He-Ife. Nigeria 897
Thus the 3.425 ? 0.003A spacing seems unique to pyrolytic carbon, and it is disturbing that the 3.55A spacing was not found in the grinding experiments. The exact determination of d-spacings from the diffuse diffraction patterns of unordered carbons using a diffractometer sometimes requires that a weak pattern of coherent diffracted radiation be separated from the total measured intensity, which may contain large contributions from Compton (incoherent) scattering, and small angle x-ray scattering, SAXS. The problem is aggravated by the fact that, because of the deep penetration of the x-ray beam below the sample surface, the separated peak of the coherent radiation will usually still be distorted. Both the location of the maximum and centroid will be displaced toward smaller angles, and the intensities at the smallest angles may require correction if an inappropriate (too large) divergence slit is used in the diffractometer. An improved understanding of the effects of the foregoing, and other factors causing distortion of xray diffraction line profiles has been achieved, appropriate correction methods have been developed[8.9]. and it was decided to reexamine the mechanically ground graphite material previous1 studied by Tidjani et aL[6], to search for the 3.55 x spacing in the sample ground 90 hours. The work was extended to the remeasurement and analysis of all the samples that had been ground in a ceramic ball mill when it was discovered that most of the line profiles were not single mode (i.e., they consisted of a superposition of peaks having slightly different dtnz spacings, and different peak intensities and line breadths).
898
J. B. 2. EXPERIMENTAL
ALADEKOMO and
raphite
(X8500
6.708& a = 2.461 w ) supplied by Superior Graphite Company. It was ground in a Shutz O’Neil Company vibratory ball mill fitted with a cylindrical ceramic jar (9 cm diameter, 10 cm height employing 1.8 cm diameter ceramic balls). For a run, the mill was charged with 14 g of graphite, and grinding was continued for periods up to 90 hours without interruption. The ground samples were stored in a dessicator over Drierite when not in use. X-ray diffraction measurements were made with an upgraded GE XRD-5 diffractometer (solid state electronics, Ni filtered Cu K, radiation, proportional counter, with pulse height discrimination). Specimens were heavily pressed into holders about 0.2 cm thick using sufficient material that the resultant transmission of incident K. x-radiation was about 37% (l/e). The sample transmission, thickness, and separate density measurements were used to calculate CL,the linear absorption coefficient, and (k/p) the mass absorp tion coefficient. Preliminary continuous scans were made to survey the important features of the diffraction patterns of each sample, but for calculations, automatic stepscan recording was used to obtain accurate line profilesdatain the (CU2), (lOO)-(101) (004), (llO), (112), and (006) angular regions of the diffraction pattern. In a few instances weak, narrow (002) peaks superimposed upon strong, very broad profiles were found by manual fixed count measurements. The raw data were first corrected for the loss of intensity that occurs when the fixed sample length fails to fully intercept the incident x-ray beam as the mean angle of incidence 0 is decreased below a critical angle[9]. After this “beam spill” correction, the contribution of Compton (incoherent) scattering to the total intensity was subtracted[8], and the resulting intensity was corrected for the beam penetration distortion (hereafter denoted “correction for absorption”)[9]. The data thus corrected were not corrected for the angular variation of the atomic scattering factor, or for the Lorentz-polarization factor. In addition, for samples ground 30 hours or more, the total intensity at small angles contained a significant contribution from SAXS. This contribution was determined by first scaling the intensity in the Porod region, where it decreases as C(sin6)“. in order to obtain C and n. and then using these to calculate and subtract the SAXS from the total intensity. The x-ray data obtained as described above for the angular region spanning a peak is the pure diffraction line profile P(28). The centroid and variance of the (002) profiles for each sample were calculated using c
=
centroid: (20) = /(20) p(ze)dze/]P(2e)dze
BRAGG
variance:
PROCEDURE
The starting material was a natural graphite,
R. H.
(1)
a$ =
I
((28) - (28))Z P(20)d28/
I
P(28)d28
(2)
The (002) line profiles were also analyzed for single or multi-component peaks by least squares fitting the data to the “modified” Lorentzian form , p(2e) = ;
[I +E:(28:-
(2e),)q2
(3)
where 1, is the maximum intensity for the ith peak, 5;: is the “effective” peak shape parameter, and (20), is the centroid of this peak. From k,? the pure diffraction peak shape parameter k, was calculated using 1
q=c
1 _
2*2
g
-
S2 3
+
D?
(4)
where A = 2(AA/A) tan(@), is the Cu I
(P(26),,, - P(26)c*kubteJ )2/(N -P).
(5)
Here N is the total number of intensity measurements made over a profile (typically 40-SO), and p is the total number of parameters (3 x the number of modes r). As needed, the results can be displayed graphically or printed out in any useful form. The computer program was tested by constructing synthetic profiles containing one, two, and three modes with varying intensities, breadths, and centroids in the range 3.354A to 3.55A. It was tolerant (i.e.. gave exact solutions even for poor guesses when
Structuraltransformationinducedby .grinding fitting a single mode), less tolerant (i.e., required better guessesespecially for the centroid for bimodal profiles). and required very close guesses for the trimodal examples-otherwi~ there would be no solution or a false solution (e.g., negative parameters or lz a very large number).
3. RESULTS As the g~ndin~ time increased, the measured diffraction patterns, staling with very sharp (MY) reflections, gradually became more diffuse and ultimately transformed into (0021) and (hk) regions, until at 90 hours only the (002) and (10) bands remained. The patterns were indistinguishable from the diffraction patterns obtained from pitch cokes during graphit~tion. At about 311hours a weak SAXS became evident at small angles and the proportion of this contribution increased with further grinding. The powders also became progressively more porous (as high as 65%). Essentially the sequence of events was the reverse of that generally observed during graphitization. While it was possible to obtain good intensity data for as-received graphite using thin samples, the diffraction patterns became weaker and very diffuse as the grinding time increased, so thicker samples were required in order to obtain useful intensity data. But carbon has very low absorption for CuK, x-rays, and
899
the beam penetration is accentuated in the high porosity specimens. The qualitative effects of these factors are a displacement and distortion towards smaller 26 as well as a decrease in intensity of the observed P(29). The extreme distortion that can occur when weakly absorbing, thick samples of material characterized by intrinsically symmetrical narrow peaks (p =;:0.1” 28) (when measured in thin samples) are studied is shown in Fig. 1. The thin specimen peak shape was ~te~ined ~x~~mentally from measurements made on a sample of as-received graphite dusted onto a glassslide. It can be seen from the thin sample profile (compared to that for the thick sample). that the narrow peak is broadened asymmetrically. the intensity is decreased, and the peak centroid is displaced towards smaller angles. It can be shown that because of beam ~netrati~n, the centroid of the peak is displaced by sin2012@, and the peak variance is increased by (sin2012pR):. where R is the specimen-detector slit distance[9.10]. It was found that the thin simple line profile is well represented by eqn (3) with a single mode at 3.354& and the line profile of the absorption~corrected profile from a thick sample agreed closely with the measured thin sample profile. Table 1 summarizes a comparison of results obtained with thick and thin samples of graphite, and demonstrates that the centroid displacement is predicted accurately. A comparison of
-
Raw data
0 n 0 0 corrected
for
28
Fig. 1. X-ray diffraction W2 line profiles of as-received #85(X) graphite (0 as-measured; 2 corrected for absarption:-thin sample; 0 17041 f 63.2([email protected])z]‘. The curves for absorption corrected data. thin sample, and the least squares profiles coincide,
CM 26:6-I
J. B. ALADEKOMO and R. H. BRAGG
900
Table 1. Comparison of &
derived from thick and thin samples of natural graphite
dOO2, A (Thick Sample)
dm2. A (Thin Sample)
Reflection @W
Centroid
Corrected for Absorption
002
3.376
3.350
0.026
0.004
3.354
004
3.376
3.354
0.022
0.000
3.354
3.354
006 008
3.367 3.359
3.353 3.354
0.014 0.005
0.001 0.000
3.355
3.354
Average
3.354
Average
Calculated Displacement
Error
Centroid
Corrected
3.354
3.353
variances (not shown) was less convincing, even for the case of a single mode, in that the variances were usually underestimated for both the thin and thick samples. At first the observed line profiles as in Fig. 1 were characterized only by their centroids and variances (as a measure of line breadths). However, it was found that after 50 hours grinding the observed line profile appeared to be bimodal even though it is very broad (Fig. 2). In this case the centroid displacement, 0.27 degrees 28, is small compared to the breadth of the profile, about 3.5 degrees 28 (i.e., the peak is displaced but its shape is essentially un-
WO)
changed). Thus this peak clearly contains more than one mode, as was confirmed by the least squares analysis. This observation led to abandonment of the use of centroids and variances as the sole measures of peak position and breadth, and the application of eqn (3) to the profiles of all the samples. Thus each sample is characterized by its centroid and the parameters I, and x: of the component modes. Figure 3 shows a peak of intermediate breadth (p = 1” 20). and demonstrates that the effect of low specimen absorption was to shift the entire profile but leave the peak shape only mildly distorted. While such slightly asymmetrical peaks would have been
2pR = 93 -
Raw data
D-O-O-O Corrected for
50
0
20
25
-
[1+0.060(26-25.16)~2
-
[I +0.511(26-26.39)*]*
30 WW
Fig. 2. X-ray diffraction 002 line profiles of #8500 graphite ground 50 hours. (O raw data; C corrected for absorption; 0 39.4/[1 + 0.060 (2$_25.16)z]z; 0 47.2/[1 + 0.511(2&26.39)z]~). The sum of the least squares calculated profiles and the data corrected for absorption coincide.
35
901
Structural transformation induced by grinding
1
I
I
W5)
I
!
I
I
I
243 = 165
Raw data Comcted
for absorption
j
Fig. 3. X-ray diffraction 002 line profile of #8500 graphite ground 25 hours (0 raw data, 0 corrected for absorption; 0 30.5111 + 1.40(2&26.43)‘]*; 0 84.1/[1 + 17.2(2&26.57)2]2). The sum of the least squares calculated profiles and the curve of the data corrected for absorption coincide.
as single mode in a typical experiment, the analysis using Eqn. (3) revealed that, in fact, this profile cobsists of a superposition of symmetrical peaks with centroids at 3.373 (O.OOl)A and 3.45 (O.O20)A, respectively. At about 30 hours grinding, a small amount of SAXS became evident in the data at small angles. and the proportion of SAXS increased as the grinding time increased. Figure 4 illustrates the most extreme case encountered in this study, 90 hours grinding. The results of corrections described in section 2 are shown in the figure, and it was found that the SAXS decreased roughly as (sine)--’ (i.e., n = - 1). The profiles in this case are very broad, but the SAXS is so strong, and decreases so slowly with increasing 28. that it still makes a significant contribution to the total intensity at 28 = 35”! As the grinding time increases, wear from the mill and the balls becomes evident in the mass absorption coefficient of the samples. This is seen in Fig. 5, which shows that p/p starts at a value greater than that for pure graphite, 4.22 cm’/g for CuK, radiation. decreases initially for about 15 hours, but thereafter increases roughly linearly with time. The linear portion extrapolates back to the pure graphite value. The manufacturer supplied (spectroscopic) analysis indicates many ppm metallic impurities in the asreceived material, and several-Fe. Si. Ca. Na, Al. and Mg-in the 100 ppm range. Because the major impurities contributed by wear are AI,O, and SiO?. both the initial decrease in (p/p) and the subsequent interpreted
increase with time are in qualitative agreement with expectations. The graph of d,, versus time of grinding, Fig. 6, shows both the profile centroid and the component peak spacings for each sample. It starts with the literature value for graphite, 3.354 + 0.00018, (3.354(0.0001),&), an d increases continuously up to 3.362(0.0002)~ at 16 hours. Over this range the centroid and least square component peak positions coincide within experimental error. However, while at 20 hours the centroid of the peak increased to 3.364& this peak was least-squares decomposed into symmetrical component peaks at 3.352(0.001)A and 3.365(0.005),&. This phenomenon of asymmetrical peaks with increasing centroids that could split into two or more symmetrical component peaks continues up to 55 hours grinding time. It should be noted that the only spacings observed that are greater than 3.375A were 3.40(0.01)& 3.45(0.02)& 3.5O(O.Ol)A. and 3.55(O.Ol)A. The slight increases in the spacings near 3.55A beginning at about 50 hours are attributed to errors in the removal of increasing proportions of SAXS from the raw x-ray data. Line breadths of the component peaks show an upward trend with grinding time. More revealing is a graph of (L,) versus u&, Fig. 7. This curve shows that while there is some scatter in the data, there is a strong correlation between (L,) and d,X,2,and there are two important regions. For 3.354A 5 r& 5 3.3751& (L,) drops sharply to a value of about 7OA. For 3.3751%5 &,> 5 3.55A (L,) decreases slowly and
902
J. B. ALADEKOMO and R.H.
BRAGG
P/A = 196 Raw data Corrected for sample length and absorption Compton scattering 51.9
Fully corrected,
[1+0.035(2&25.02)*~ Small angle scattering
-
Fig. 4. X-ray diffraction 002 line profile #85OO graphite ground 90 hours (0 raw data; 0 corrected for sample length and absorption,-Compton scattering -O-O- fully corrected and 51.9/[1 + 0.35(21325.02)‘]; -O-O-O- small angle scattering). The curves for the fully corrected profile and the least squares calculated profile coincide. reaches a limiting minimum value of about lo.&&. It should be noted that the range of (L,) is nearly 50
fold in this experiment. Table 2 summarizes the observed d-spacings, crystallite sizes, and relative peak areas with associated estimates of precision. When the grinding time exceeded 55 hours, the peak at 3.375,& could only be observed by using extremely time consuming, manual, fixed count searches, so only rough estimates of line breadths were obtained for these three peaks. As an aside, when (L,) is about lO.SA and dm is about 3.55& the implication is that the average crystallite is just three unit cells thick!
would have been interpreted as coming from 8OA crystallites for which dm is 3.36A. The least squares analysis shows that, in fact, there are roughly equal amounts of 244A and 68w crystallites for which dam is about 3.355A and 3.374 respectively. Figure 4 shows why it is necessary to correct for SAXS, Compton scattering, and beam spill when significant amounts are present. The SAXS has the effect of displacing the peak towards smaller angles, and withI
I
I
A 4.DISCUSSlON
The importance of the corrections for absorption for single mode profiles is well demonstrated by Fig. 1. In this case, the peak and centroid positions differ by about 0.21 degrees 28 or the interlayer spacings by 0.026A for the (002) reflection (i.e., without the correction 3.354A would have been interpreted as 3.380A)! Figure 2 demonstrates a more important reason for correcting for absorption; it revealed asymmetry when the main causes of asymmetry as an artefact have been removed. In this case a strong asymmetry suggesting two peaks comprising the profile is clearly discernible to the naked eye in the distortion-corrected profile. Figure 3 shows why all peaks should be analyzed for multiple components. In this case the asymmetry is small and this peak
4.22 cm2/g
1 -1
l 313 0
50
100
Grinding time (hours) Fig. 5. Mass absorption coefficient of ground #8500 graphite versus grinding time. The mass absorption coefficient of carbon for CuK. radiation is 4.22 cm*.
Structural transformation
I
I 3.55
_
I
I
I
___$__&__&_
id
---------
z
903
induced by grinding
I 3.55A
0
8
3 P ‘3
3.50 _ _______--
f a-_-_
-- 3.49A __ 0 Components from least squares 0 Centroid 0 Determined by manual search,fixed count _ 1 Standard deviation
!I $ 5 2 = B E
3.45 _ _----__-
------------3.44A
i 0 3.40--------
0
----_-_---_---33.4()*
___g-__@
.*a_
o__o-_-o--3.375~
I
I
I
50
100
Grinding time (hours)
Fig. 6. Interlayer spacings a& of ground #8500 graphite versus grinding time. (0 centroid; l least squares component; 0 determined by manual search; I standard deviation.)
out this correction the peak in Fig. 4 would have been interpreted as 3.7.&. It is the authors’ belief that the 3.6 and 3.87A spacings that have been reported[5,14] probably are overestimates because they were found in situations where the SAXS, Compton scattering, and absorption displacement contributions would be very large. The most important result in the present work is the finding that even relatively narrow profiles as in Fig. 3 were decomposable into symmetrical peaks withspacingsof3.354~d,~3.375,3.40,3.45(0.02), 3.50(0.1), and 3.55& The limitingspacingat 3.375A is in agreement with the arrest at 3.38A that is the most common feature of all graphitization investi-
gations[2,3], and the other discrete spacings, except the 3.425A ? 0.003A spacing confirm those mentioned in section 1. Apart from a questionable 3.87A spacing in the work of Walker and Seeley\l4], there is no disagreement between the present results and those obtained in other studies of grinding[l2,13]. However, in those investigations the dw2 values never exceeded 3.38A and L, did not decrease below MX&. The component peaks at 3.45(0.02).~% and 3.50(0.01@ require special attention. The large error in the determination of the first probably indicates the inability of the computer program to resolve overlapping peaks at 3.44A and 3.49A that are both very broad, L, = 24& given the quality of the input data provided. The 3.5O(O.Ol)w spacing probably reflects the same problem in separating broad 3.49A and 3.55A profiles. The presence of a 3.49A spacing would probably be clearly revealed in an analysis of samples ground 42-43 hours. Similarly, a clear delineation of the expected 3.44A spacing would probably come from analyses of material ground 37-38 hours. The success of the modified Lorentzian line shape function adopted in eqn (3) in analyzing composite peaks is noteworthy. The least squares analyis starts with reasonable but arbitrary guesses Z,, k,:, and (20),, for each mode based on visual inspection of the recorder charts, but the resulting least squares calculated “best” values of the interlayer spacings are in striking agreement with those obtained in situations where presumably only single peaks are
Crystallite
size (b.
A)
Fig. 7. Correlation of interlayer spacing, dm. with mean crystallite height (L,) of ground graphite.
found[4,5]. However, the same centroid would have been found in a least squares analysis, regardless of the assumed shape of the component profiles as long
904
J.
B. ALADEKOMO and
R. H.
BRAGG
Table 2. Summary of crystallographic data for ground graphite Grinding Tim (hour.)
0
3.364 (0.0001)
401 (11)
0.04
0.0
5.312
4
3.364 (0.0001)
4S9 ( 8)
0.08
0.0
3.356
6
3.362 (0.0002)
SS9 (10)
0.0s
0.0
3.352
0.0
3.SS8
0.0
3.569
0.0
3.362
8
3.362 (0.0001)
396 ( 7)
0.08
12
3.564 (0.0004)
SSI (13)
0.06
16
3.S58 (0.0002)
3= (8)
20
3.363 (O.aoOl)
S81 (12)
S.SOS(0.006)
75.0 (10)
0.06
0.20
0.19
0.u
S..%4
26
3.366 (0.0002)
244 (7)
5.572 (0.002)
68.0 ( 2)
0.W
0.10
0.28
0.68
3.371
30
S.Sbs (0.001)
162 (11)
S.4OO(o.ou)
SC.0 ( 4)
0.08
0.26
0.22
0.18
3.402
3s
3362 (0.001)
@7 (S)
s.408 (0.010)
S8.0 ( 3)
O.OS
0.22
0.37
0.48
S.401
40
8.373 (0.001)
77 (8)
S.4U(O.o21)
14.0 ( 2)
0.08
0.08
O.Sl
0.46
3.423
46
3.S72 (0.002)
6’ (2)
s.sO4 (0.010)
16.7 (0.4)
0.08
0.07
0.28
0.72
3.470 3.490 S.627
50
3.377 (O.OOS)
41 (2)
8.540 (0.012)
14.1 (0.4)
0.11
0.08
0.31
0.71
56
5.378 (O.OOD)
S7 (6)
8.64s (0.018)
12.4 (0.S)
0.33
0.07
0.32
0.91
64
3.376 (0.006)
S&S
(0.007)
10.8 (0.2)
-1.0
0.08
S.SSa
74
3.374 (0.006)
S.649 (0.008)
10.0 (0.2)
0.06
91.0
3.640
90
S.376 (0.006)
doA
s.649 (0.007)
10.9 (0.2)
0.0s
-1.0
3.646
as they were bell shaped and symmetrical. The importance of the modified Lorentzian shape function is that it comes very close to matching the observed peak shapes over the entire angular range, far better than Gaussian or Cauchy, for example. In a typical case for which the peak intensity might be about 100 c/s the values of l2 were less than 2. This means that the rms average difference between the observed and calculated intensities were less than 1 c/s at every one of about 50 28 abscissa points! The fit for each peak might have been improved by using a sum of two Gaussians, for example, but it was felt that this would introduce complexity without a commensurate improvement in clarity. The modified Lorentzian function fortuitously has convenient mathematical properties and introduces rigor into the calculation of (L,). Given that the line profile is
p(2e) = [l
+ k?(2: - (2e))q2
the peak area, centroid,
and variance are found to
be n112k, (20). and l/k*. The first result affords a simple means of calculating the integral breadth f3 of a profile. This is defined as:
P=
Peak Area Peak maximum
7FI 71 = 2kl = 2k.
(5)
It can be shown that when p is defined in this way. then
(A) p = (L,) cos(8) where (A) is the centroid of the K, radiation, (15,) = JL,dV/V(ll] (i.e.. (L,) is the volume average of the crystallite height normal to the diffracting planes). Thus no assumption is made concerning crystallite shape and (L,) has a well defined physical meaning. It should be noted that given the form of P(20) as in eqn (1). if (L,) is estimated using the full width at half maximum in the Scherrer equation, as is usually done by most researchers, the result will be about 22% greater than the true value of (L,). It can also
Structural transformation induced by grinding
be shown that the variances of all the factors that contribute to peak variance Ilk? are additive[lO]. Thus, when corrections are first made for absorption, and the magnitudes of the other instrumental factors are made small (and are known), it is a simple matter. using eqn (4). to obtain the pure diffraction line breadth from the peak shape parameter k2. Based upon the present work and the works cited previously. Table 3 has been constructed. It shows that there is a discrete set of d,,,: spacings that occur in pure carbon materials during annealing or are observed in graphite after fast neutron irradiation or prolonged mechanical grinding. Because the 3.55A spacing occurs in the fast neutron irradiation of single crystals, it must be associated with carbon defects, (i.e., interstitials or interstitial clusters). The model of disordered graphite that seems appropriate is that of a mixture of these metastable graphite interstitial phases, akin to the model of grafted interstitials proposed by Maire and Mering[ 151for carbons. It is notable that. with rare exceptions, rhe interlayer spacings obtained in rhe peak-search least squares calculation are accurate to -CO.OOlA or better. The data of crystallite sizes and peak areas (proportional lo crystallite volumes) in Table 2 show that the size distributions for each grinding time are narrow (53% of the mean). and at first the crystals cleave into halves having the same interlayer spacing, but after I6 hours grinding on the average the crystallites cleave into fragments such that (L,J (thinner. &greater) = l/3@ ,,) (thicker, &? smaller). The data of Table 2 also imply that (LJ = (IL.,,). The authors’ interpretation of the data is that the range 3.354A 5 d,,,, 5 3.375A is associated with a continuous solid solution wherein 3.375,& represents the maximum solubility of single. C,. carbon inter-
905
stitials. This 3.375A characterizes a very stable, limiting structure (small amounts are present even after 90 hours of grinding!). Recall that an arrest at about 3.38A is found in all investigations-this work shows that it should be 3.375A. The abrupt increases in d,,,, in Fig. 6 at 3.375, 3.40, 3.44, and 3.55A suggest a pillaring effect similar to that observed in :;n,,,ed swelling clay minerals such as vermiculite[ 161. The model of carbons as mixtures of discrete metastable interstitial graphite phases has important implications for studies of graphitization. The neutron irradiation results of Henson and Reynolds[S] indicate that as the c-parameter takes on only discrete increased values, the u-parameter also takes on only discrete decreased values. Because in pregraphitic carbons at the onset of annealing the x-ray lines are very broad, and most likely bimodal (or trimodal). the apparent (hk) banding is to be expected (e.g.. (100) would be a multiplet, as would (101)) so the (10) band might contain four to six broad components. It also follows that the appearance of the (112) reflection is not a measure of the onset of graphitization so much as it reflects the changes in a- and c-spacings in both the (112) and (006) reflections, and the multiplet structure of each reflection, combined with peak narrowing owing to increases in crystallite size. In fact, while stacking faults are no doubt present in these carbons, it is questionable whether or not random layer stacking is the essential feature of “turbostratic carbon.” It is the authors’ view that graphitization consists primarily of the annealing of metastable carbon interstitial phases. A further implication of this interpretation is that it is very unlikely that steps in the dlllZversus time curve will be observed during graphitization unless the starting materials are very homogeneous (i.e.. a single d,,,, for all crystallites). This brings up the spe-
Table 3. tnterlayer spacings in carbons and graphite
Carbons
Graphite
Neutron Irradiation
Annealing
Pandica
3.44A
3.40A
Fischbach,b PacaultC
3.44A (pitch coke)
(3.425 f .003)
Kawamura et aLd
Henson and Reynoldse
3.55A
3.55A
3.44A
3.44A
3.40A
Grinding
Tidjani et al.f
3.55A 3.43A
3.44A
3.40A
3.40A
3.40A 3.375A 3.354A
3.38A
3.38A
3.375A
3.375A
3.315A
3.354A
3.354A
3.354A
3.354A
3.354A
aRef. [4] bRef. [2]
CRef. [3] dRef. [7]
eRef. [5] fRef. [6]
BThis paper
Aladekomo and Braggg
J.B. ALADEKOMO and R. H. BRAGG
906
cial case of pyrolytic carbon. Since pyrolytic carbon anneals during deposition (typically above 2OOO”C), a variation in doo2with thickness occurs naturally and is unavoidable in as-received material. As a first approximation the typical 3.425A -’ 0.0038, interlayer spacing probably represents a mixture of 3.4OA and 3.44A phases. No amount of annealing of such a bimodal deposit can homogenize the bulk material[3]. Moreover, since three discrete metastable phases (increasing doo2) are associated with increasing amounts of grinding (more disorder, greater energy input) up to 3.44& the reverse annealing (graphitization) process should be characterized by three activation energies when d,, is initially at 3.44& instead of the one generally assumed in graphitization studies. Some success in confirming the three level model has been reported by Bragg and Lachter[l7]. The authors feel that line profile analysis should become standard practice in diffraction characterization of carbon materials. While it can be done in principle with Debye-Scherrer patterns, the diffractometer is far better suited to this endeavor. There is nothing special about the computer analysis-all this work was done on a clone of an IBM-PC. For those not familiar with least squares curve fitting, very readable how-to references are available[l8]. 5. CONCLUSIONS This work shows that in experiments in which graphite is disordered by mechanical grinding, the interlayer spacing increases continuously in the range 3.354h; I doo2 I 3.375& as in a solid solution. At 3.375& further grinding causes discontinuous transformations of the material to metastable phases characterized by spacings of 3.40& 3.44& 3SSA, and possibly 3.49A. Because the 3.55A and 3.49A spacings have been observed in pyrolytic carbon, and the others are observed during the annealing of pitches, it is hypothesized that this same set of spacings is unique in elemental carbons of whatever origin. Thus in graphitization studies, a sample should be characterized by decomposing the x-ray diffraction line profiles into their characteristic discrete components. These components should be used when structure-property correlations are made.
Acknowledgement-The
authors would like to thank Dr. Kolawole Aiyesimoju for assistance in performing the profile fitting analyses. This work was supported bv the Director, Office df Energy Research, OfBce of Basic Energy
Sciences, Materials Sciences Division, of the U.S. Department of Energy under Contract No. DE-AC0376SPOOO98.
REFERENCES
1. J. Lachter and R. H. Bragg, Phys. Rev. B 33, 8903 (1986). 2. D. B. Fischbach, In Chemistry and Physics of Carbon (Edited by P. L. Walker, Jr.), Vol. 7, pp. l-105. Marcel Dekker, New York (1971). 3. A. Pacault, In Chemistry and Physics of Carbon (Edited by P. L. Walker, Jr.), Vol. 7,-pp. 167-154. Marcel Dekker. New York 11971). 4. B. Pan&, In Procdedingi of the Second Conference on Industrial Carbon and Graphite, Society of Chemical Industry, p. 439. Gordan and Breach, New York (1966). 5. R. W. Henson and W. N. Reynolds, Carbon 3, 277 (1965).
6. M. Tidjani, J. Lachter, T. S. Kabre, and R. H. Bragg, Carbon 24, 447 (1986).
7. K. Kawamura, J.’ Lachter, and R. H. Bragg, Lawrence Berkelev LaboratorvTechnical Reoort No. LBL-17982 (to be submitted for publication).’ 8. J. Lachter, L. G. Henry, and R. H. Bragg, J Appl. Cryst. 21, 92 (1988).
9. R. H. Bragg and J. B. Aladekomo, Lawrence Berkeley Laboratory Technical Report No. LBL-27415 (to be submitted for publication). 10. A. J.-C. Wilson, Mathematical Theory ofX-ray Powder Diffractometry. Gordon and Breach,Ndw York (1963). 11. B. E. Warren, X-ray Diffraction, Chapter 13. AddisonWesley Publishing Company, Reading (1969). 12. G. E. Bacon, Acta Cryst. 5, 392 (1952). 13. M. Nakamizo. H. Honda. and M. Inaeaki. Carbon 16. Y 281 (1978). 14. P. L. Walker, Jr., and S. B. Seeley, Proc. 3rd Biennial Carbon Conference, 1957, p. 481. Pergamon Press, New York (1959). 15. J. Maire and J. Mering, In Chemistry and Physics of Carbon (Edited by P. L. Walker, Jr.), Vol. 6, pp. 125190. Marcel Dekker, New York (1970). 16. H. Kim, W. Jin, S. Lee, P. Zhou, T. J. Pinnevaia, S. D. Mahanti, and S. Solin, Phys. Rev. Left. 60, 2168 (1988). 17. R. H. Bragg and J. Lachter, Lawrence Berkeley Laboratory Technical Report No. LBL-23284 (March 1987). 18. J. L. Kuester and J. H. Mize, Optimization Techniques with Fortran. McGraw-Hill, New York (1973), pp. 240-
250.
Carbon Vol. 2% No. 6. pp. 897406. 1990 Printed m Great Bri~am.
STRUCTURAL TRANSFORMATIONS INDUCED IN GRAPHITE BY GRINDING: ANALYSIS OF 002 X-RAY DIFFRACTION LINE PROFILES J. B. ALADEKoMot AND R. H. BRAGG Department of Materials Science and Mineral Engineering, University of California, and Materials and Chemical Sciences Division, Lawrence Berkeley Laboratory. 1 Cyclotron Road, Berkeley. CA 94720, U.S.A. (Received 2 October 1989; accepted in revised form 1 February
1990)
Abstract-The x-ray diffraction line profiles of the 002 reflections of graphite samples ground in a ball mill for periods up to 90 hours were studied using newly developed peak analysis methods that permitted separation of overlapping peaks having different peak positions, intensities, and line-breadths. For 3.354A I d,, s 3.375Aa continuous range of single phases characterized by symmetrical peaks of increasing breadths was observed, but peaks whose centroids fell between 3.375A and 3.55A were asymmetrical and composed of a superposition of symmetrical peaks of increasing breadths characterized by 3.375A. 3.40& 3.44A. and 3.55A spacings. There is some indication of a phase characterized by a 3.491% interlayer spacing. These results confirm previous reports of discrete 002 spacings in carbons and disordered graphite and suggest that elemental carbon(s) are composed of mixtures of these metastable graphite phases. Key WordsX-ray
diffraction, interlayer spacmg grmdmg, natural graphite.
1. INTRODUCTION It has been pointed out that when soft carbon materials are annealed at high temperature, or when graphite is disordered by fast neutron irradiation or mechanical grinding. a graph of the interlayer spacing, &. versus the processing time contains a set of arrests or plateaus[ 11. The aggregate of the data from a variety of sources consists of characteristic interlayer spacings of about 3.38A, 3.40& 3.425& 3.44A, 3.55,&, and one at about 3.68&2-61. Although the overall agreement of these d-spacings is good, there are significant gaps in the data obtained from different sources. Data obtained from the annealing of pyrolytic carbons have an initial plateau at about 3.425 * 0.003A and an arrest at about 3.38A, but the other levels are missing[2.3). Data obtained from studies of natural graphite disordered by prolonged mechanical grinding indicate plateaus of 3.4OA and 3.44A. and an inflexion at 3.38A, but the 3.425A and 3.55A spacings were not reported[6]. A 3.86A spacing is implied in the results of Henson and Reynolds(5], but was not observed in any of the other studies. However. a 3.55A spacing is commonly observed in low temperature (40%500°C) heat treated pitches. as are levels or arrests at 3.38A. 3.4OA (rarely), and 3.441j when they are annealed at much higher temperatures. Also. the 3.55%1 spacing has recently been observed in an as-received pyrolytic carbon processed below 2100°C as well as a 3.49A spacing in the same material annealed at 26OO”C[7].
ton leave from the University of Ife, He-Ife. Nigeria 897
Thus the 3.425 ? 0.003A spacing seems unique to pyrolytic carbon, and it is disturbing that the 3.55A spacing was not found in the grinding experiments. The exact determination of d-spacings from the diffuse diffraction patterns of unordered carbons using a diffractometer sometimes requires that a weak pattern of coherent diffracted radiation be separated from the total measured intensity, which may contain large contributions from Compton (incoherent) scattering, and small angle x-ray scattering, SAXS. The problem is aggravated by the fact that, because of the deep penetration of the x-ray beam below the sample surface, the separated peak of the coherent radiation will usually still be distorted. Both the location of the maximum and centroid will be displaced toward smaller angles, and the intensities at the smallest angles may require correction if an inappropriate (too large) divergence slit is used in the diffractometer. An improved understanding of the effects of the foregoing, and other factors causing distortion of xray diffraction line profiles has been achieved, appropriate correction methods have been developed[8.9]. and it was decided to reexamine the mechanically ground graphite material previous1 studied by Tidjani et aL[6], to search for the 3.55 x spacing in the sample ground 90 hours. The work was extended to the remeasurement and analysis of all the samples that had been ground in a ceramic ball mill when it was discovered that most of the line profiles were not single mode (i.e., they consisted of a superposition of peaks having slightly different dtnz spacings, and different peak intensities and line breadths).
898
J. B. 2. EXPERIMENTAL
ALADEKOMO and
raphite
(X8500
6.708& a = 2.461 w ) supplied by Superior Graphite Company. It was ground in a Shutz O’Neil Company vibratory ball mill fitted with a cylindrical ceramic jar (9 cm diameter, 10 cm height employing 1.8 cm diameter ceramic balls). For a run, the mill was charged with 14 g of graphite, and grinding was continued for periods up to 90 hours without interruption. The ground samples were stored in a dessicator over Drierite when not in use. X-ray diffraction measurements were made with an upgraded GE XRD-5 diffractometer (solid state electronics, Ni filtered Cu K, radiation, proportional counter, with pulse height discrimination). Specimens were heavily pressed into holders about 0.2 cm thick using sufficient material that the resultant transmission of incident K. x-radiation was about 37% (l/e). The sample transmission, thickness, and separate density measurements were used to calculate CL,the linear absorption coefficient, and (k/p) the mass absorp tion coefficient. Preliminary continuous scans were made to survey the important features of the diffraction patterns of each sample, but for calculations, automatic stepscan recording was used to obtain accurate line profilesdatain the (CU2), (lOO)-(101) (004), (llO), (112), and (006) angular regions of the diffraction pattern. In a few instances weak, narrow (002) peaks superimposed upon strong, very broad profiles were found by manual fixed count measurements. The raw data were first corrected for the loss of intensity that occurs when the fixed sample length fails to fully intercept the incident x-ray beam as the mean angle of incidence 0 is decreased below a critical angle[9]. After this “beam spill” correction, the contribution of Compton (incoherent) scattering to the total intensity was subtracted[8], and the resulting intensity was corrected for the beam penetration distortion (hereafter denoted “correction for absorption”)[9]. The data thus corrected were not corrected for the angular variation of the atomic scattering factor, or for the Lorentz-polarization factor. In addition, for samples ground 30 hours or more, the total intensity at small angles contained a significant contribution from SAXS. This contribution was determined by first scaling the intensity in the Porod region, where it decreases as C(sin6)“. in order to obtain C and n. and then using these to calculate and subtract the SAXS from the total intensity. The x-ray data obtained as described above for the angular region spanning a peak is the pure diffraction line profile P(28). The centroid and variance of the (002) profiles for each sample were calculated using c
=
centroid: (20) = /(20) p(ze)dze/]P(2e)dze
BRAGG
variance:
PROCEDURE
The starting material was a natural graphite,
R. H.
(1)
a$ =
I
((28) - (28))Z P(20)d28/
I
P(28)d28
(2)
The (002) line profiles were also analyzed for single or multi-component peaks by least squares fitting the data to the “modified” Lorentzian form , p(2e) = ;
[I +E:(28:-
(2e),)q2
(3)
where 1, is the maximum intensity for the ith peak, 5;: is the “effective” peak shape parameter, and (20), is the centroid of this peak. From k,? the pure diffraction peak shape parameter k, was calculated using 1
q=c
1 _
2*2
g
-
S2 3
+
D?
(4)
where A = 2(AA/A) tan(@), is the Cu I
(P(26),,, - P(26)c*kubteJ )2/(N -P).
(5)
Here N is the total number of intensity measurements made over a profile (typically 40-SO), and p is the total number of parameters (3 x the number of modes r). As needed, the results can be displayed graphically or printed out in any useful form. The computer program was tested by constructing synthetic profiles containing one, two, and three modes with varying intensities, breadths, and centroids in the range 3.354A to 3.55A. It was tolerant (i.e.. gave exact solutions even for poor guesses when
Structuraltransformationinducedby .grinding fitting a single mode), less tolerant (i.e., required better guessesespecially for the centroid for bimodal profiles). and required very close guesses for the trimodal examples-otherwi~ there would be no solution or a false solution (e.g., negative parameters or lz a very large number).
3. RESULTS As the g~ndin~ time increased, the measured diffraction patterns, staling with very sharp (MY) reflections, gradually became more diffuse and ultimately transformed into (0021) and (hk) regions, until at 90 hours only the (002) and (10) bands remained. The patterns were indistinguishable from the diffraction patterns obtained from pitch cokes during graphit~tion. At about 311hours a weak SAXS became evident at small angles and the proportion of this contribution increased with further grinding. The powders also became progressively more porous (as high as 65%). Essentially the sequence of events was the reverse of that generally observed during graphitization. While it was possible to obtain good intensity data for as-received graphite using thin samples, the diffraction patterns became weaker and very diffuse as the grinding time increased, so thicker samples were required in order to obtain useful intensity data. But carbon has very low absorption for CuK, x-rays, and
899
the beam penetration is accentuated in the high porosity specimens. The qualitative effects of these factors are a displacement and distortion towards smaller 26 as well as a decrease in intensity of the observed P(29). The extreme distortion that can occur when weakly absorbing, thick samples of material characterized by intrinsically symmetrical narrow peaks (p =;:0.1” 28) (when measured in thin samples) are studied is shown in Fig. 1. The thin specimen peak shape was ~te~ined ~x~~mentally from measurements made on a sample of as-received graphite dusted onto a glassslide. It can be seen from the thin sample profile (compared to that for the thick sample). that the narrow peak is broadened asymmetrically. the intensity is decreased, and the peak centroid is displaced towards smaller angles. It can be shown that because of beam ~netrati~n, the centroid of the peak is displaced by sin2012@, and the peak variance is increased by (sin2012pR):. where R is the specimen-detector slit distance[9.10]. It was found that the thin simple line profile is well represented by eqn (3) with a single mode at 3.354& and the line profile of the absorption~corrected profile from a thick sample agreed closely with the measured thin sample profile. Table 1 summarizes a comparison of results obtained with thick and thin samples of graphite, and demonstrates that the centroid displacement is predicted accurately. A comparison of
-
Raw data
0 n 0 0 corrected
for
28
Fig. 1. X-ray diffraction W2 line profiles of as-received #85(X) graphite (0 as-measured; 2 corrected for absarption:-thin sample; 0 17041 f 63.2([email protected])z]‘. The curves for absorption corrected data. thin sample, and the least squares profiles coincide,
CM 26:6-I
J. B. ALADEKOMO and R. H. BRAGG
900
Table 1. Comparison of &
derived from thick and thin samples of natural graphite
dOO2, A (Thick Sample)
dm2. A (Thin Sample)
Reflection @W
Centroid
Corrected for Absorption
002
3.376
3.350
0.026
0.004
3.354
004
3.376
3.354
0.022
0.000
3.354
3.354
006 008
3.367 3.359
3.353 3.354
0.014 0.005
0.001 0.000
3.355
3.354
Average
3.354
Average
Calculated Displacement
Error
Centroid
Corrected
3.354
3.353
variances (not shown) was less convincing, even for the case of a single mode, in that the variances were usually underestimated for both the thin and thick samples. At first the observed line profiles as in Fig. 1 were characterized only by their centroids and variances (as a measure of line breadths). However, it was found that after 50 hours grinding the observed line profile appeared to be bimodal even though it is very broad (Fig. 2). In this case the centroid displacement, 0.27 degrees 28, is small compared to the breadth of the profile, about 3.5 degrees 28 (i.e., the peak is displaced but its shape is essentially un-
WO)
changed). Thus this peak clearly contains more than one mode, as was confirmed by the least squares analysis. This observation led to abandonment of the use of centroids and variances as the sole measures of peak position and breadth, and the application of eqn (3) to the profiles of all the samples. Thus each sample is characterized by its centroid and the parameters I, and x: of the component modes. Figure 3 shows a peak of intermediate breadth (p = 1” 20). and demonstrates that the effect of low specimen absorption was to shift the entire profile but leave the peak shape only mildly distorted. While such slightly asymmetrical peaks would have been
2pR = 93 -
Raw data
D-O-O-O Corrected for
50
0
20
25
-
[1+0.060(26-25.16)~2
-
[I +0.511(26-26.39)*]*
30 WW
Fig. 2. X-ray diffraction 002 line profiles of #8500 graphite ground 50 hours. (O raw data; C corrected for absorption; 0 39.4/[1 + 0.060 (2$_25.16)z]z; 0 47.2/[1 + 0.511(2&26.39)z]~). The sum of the least squares calculated profiles and the data corrected for absorption coincide.
35
901
Structural transformation induced by grinding
1
I
I
W5)
I
!
I
I
I
243 = 165
Raw data Comcted
for absorption
j
Fig. 3. X-ray diffraction 002 line profile of #8500 graphite ground 25 hours (0 raw data, 0 corrected for absorption; 0 30.5111 + 1.40(2&26.43)‘]*; 0 84.1/[1 + 17.2(2&26.57)2]2). The sum of the least squares calculated profiles and the curve of the data corrected for absorption coincide.
as single mode in a typical experiment, the analysis using Eqn. (3) revealed that, in fact, this profile cobsists of a superposition of symmetrical peaks with centroids at 3.373 (O.OOl)A and 3.45 (O.O20)A, respectively. At about 30 hours grinding, a small amount of SAXS became evident in the data at small angles. and the proportion of SAXS increased as the grinding time increased. Figure 4 illustrates the most extreme case encountered in this study, 90 hours grinding. The results of corrections described in section 2 are shown in the figure, and it was found that the SAXS decreased roughly as (sine)--’ (i.e., n = - 1). The profiles in this case are very broad, but the SAXS is so strong, and decreases so slowly with increasing 28. that it still makes a significant contribution to the total intensity at 28 = 35”! As the grinding time increases, wear from the mill and the balls becomes evident in the mass absorption coefficient of the samples. This is seen in Fig. 5, which shows that p/p starts at a value greater than that for pure graphite, 4.22 cm’/g for CuK, radiation. decreases initially for about 15 hours, but thereafter increases roughly linearly with time. The linear portion extrapolates back to the pure graphite value. The manufacturer supplied (spectroscopic) analysis indicates many ppm metallic impurities in the asreceived material, and several-Fe. Si. Ca. Na, Al. and Mg-in the 100 ppm range. Because the major impurities contributed by wear are AI,O, and SiO?. both the initial decrease in (p/p) and the subsequent interpreted
increase with time are in qualitative agreement with expectations. The graph of d,, versus time of grinding, Fig. 6, shows both the profile centroid and the component peak spacings for each sample. It starts with the literature value for graphite, 3.354 + 0.00018, (3.354(0.0001),&), an d increases continuously up to 3.362(0.0002)~ at 16 hours. Over this range the centroid and least square component peak positions coincide within experimental error. However, while at 20 hours the centroid of the peak increased to 3.364& this peak was least-squares decomposed into symmetrical component peaks at 3.352(0.001)A and 3.365(0.005),&. This phenomenon of asymmetrical peaks with increasing centroids that could split into two or more symmetrical component peaks continues up to 55 hours grinding time. It should be noted that the only spacings observed that are greater than 3.375A were 3.40(0.01)& 3.45(0.02)& 3.5O(O.Ol)A. and 3.55(O.Ol)A. The slight increases in the spacings near 3.55A beginning at about 50 hours are attributed to errors in the removal of increasing proportions of SAXS from the raw x-ray data. Line breadths of the component peaks show an upward trend with grinding time. More revealing is a graph of (L,) versus u&, Fig. 7. This curve shows that while there is some scatter in the data, there is a strong correlation between (L,) and d,X,2,and there are two important regions. For 3.354A 5 r& 5 3.3751& (L,) drops sharply to a value of about 7OA. For 3.3751%5 &,> 5 3.55A (L,) decreases slowly and
902
J. B. ALADEKOMO and R.H.
BRAGG
P/A = 196 Raw data Corrected for sample length and absorption Compton scattering 51.9
Fully corrected,
[1+0.035(2&25.02)*~ Small angle scattering
-
Fig. 4. X-ray diffraction 002 line profile #85OO graphite ground 90 hours (0 raw data; 0 corrected for sample length and absorption,-Compton scattering -O-O- fully corrected and 51.9/[1 + 0.35(21325.02)‘]; -O-O-O- small angle scattering). The curves for the fully corrected profile and the least squares calculated profile coincide. reaches a limiting minimum value of about lo.&&. It should be noted that the range of (L,) is nearly 50
fold in this experiment. Table 2 summarizes the observed d-spacings, crystallite sizes, and relative peak areas with associated estimates of precision. When the grinding time exceeded 55 hours, the peak at 3.375,& could only be observed by using extremely time consuming, manual, fixed count searches, so only rough estimates of line breadths were obtained for these three peaks. As an aside, when (L,) is about lO.SA and dm is about 3.55& the implication is that the average crystallite is just three unit cells thick!
would have been interpreted as coming from 8OA crystallites for which dm is 3.36A. The least squares analysis shows that, in fact, there are roughly equal amounts of 244A and 68w crystallites for which dam is about 3.355A and 3.374 respectively. Figure 4 shows why it is necessary to correct for SAXS, Compton scattering, and beam spill when significant amounts are present. The SAXS has the effect of displacing the peak towards smaller angles, and withI
I
I
A 4.DISCUSSlON
The importance of the corrections for absorption for single mode profiles is well demonstrated by Fig. 1. In this case, the peak and centroid positions differ by about 0.21 degrees 28 or the interlayer spacings by 0.026A for the (002) reflection (i.e., without the correction 3.354A would have been interpreted as 3.380A)! Figure 2 demonstrates a more important reason for correcting for absorption; it revealed asymmetry when the main causes of asymmetry as an artefact have been removed. In this case a strong asymmetry suggesting two peaks comprising the profile is clearly discernible to the naked eye in the distortion-corrected profile. Figure 3 shows why all peaks should be analyzed for multiple components. In this case the asymmetry is small and this peak
4.22 cm2/g
1 -1
l 313 0
50
100
Grinding time (hours) Fig. 5. Mass absorption coefficient of ground #8500 graphite versus grinding time. The mass absorption coefficient of carbon for CuK. radiation is 4.22 cm*.
Structural transformation
I
I 3.55
_
I
I
I
___$__&__&_
id
---------
z
903
induced by grinding
I 3.55A
0
8
3 P ‘3
3.50 _ _______--
f a-_-_
-- 3.49A __ 0 Components from least squares 0 Centroid 0 Determined by manual search,fixed count _ 1 Standard deviation
!I $ 5 2 = B E
3.45 _ _----__-
------------3.44A
i 0 3.40--------
0
----_-_---_---33.4()*
___g-__@
.*a_
o__o-_-o--3.375~
I
I
I
50
100
Grinding time (hours)
Fig. 6. Interlayer spacings a& of ground #8500 graphite versus grinding time. (0 centroid; l least squares component; 0 determined by manual search; I standard deviation.)
out this correction the peak in Fig. 4 would have been interpreted as 3.7.&. It is the authors’ belief that the 3.6 and 3.87A spacings that have been reported[5,14] probably are overestimates because they were found in situations where the SAXS, Compton scattering, and absorption displacement contributions would be very large. The most important result in the present work is the finding that even relatively narrow profiles as in Fig. 3 were decomposable into symmetrical peaks withspacingsof3.354~d,~3.375,3.40,3.45(0.02), 3.50(0.1), and 3.55& The limitingspacingat 3.375A is in agreement with the arrest at 3.38A that is the most common feature of all graphitization investi-
gations[2,3], and the other discrete spacings, except the 3.425A ? 0.003A spacing confirm those mentioned in section 1. Apart from a questionable 3.87A spacing in the work of Walker and Seeley\l4], there is no disagreement between the present results and those obtained in other studies of grinding[l2,13]. However, in those investigations the dw2 values never exceeded 3.38A and L, did not decrease below MX&. The component peaks at 3.45(0.02).~% and 3.50(0.01@ require special attention. The large error in the determination of the first probably indicates the inability of the computer program to resolve overlapping peaks at 3.44A and 3.49A that are both very broad, L, = 24& given the quality of the input data provided. The 3.5O(O.Ol)w spacing probably reflects the same problem in separating broad 3.49A and 3.55A profiles. The presence of a 3.49A spacing would probably be clearly revealed in an analysis of samples ground 42-43 hours. Similarly, a clear delineation of the expected 3.44A spacing would probably come from analyses of material ground 37-38 hours. The success of the modified Lorentzian line shape function adopted in eqn (3) in analyzing composite peaks is noteworthy. The least squares analyis starts with reasonable but arbitrary guesses Z,, k,:, and (20),, for each mode based on visual inspection of the recorder charts, but the resulting least squares calculated “best” values of the interlayer spacings are in striking agreement with those obtained in situations where presumably only single peaks are
Crystallite
size (b.
A)
Fig. 7. Correlation of interlayer spacing, dm. with mean crystallite height (L,) of ground graphite.
found[4,5]. However, the same centroid would have been found in a least squares analysis, regardless of the assumed shape of the component profiles as long
904
J.
B. ALADEKOMO and
R. H.
BRAGG
Table 2. Summary of crystallographic data for ground graphite Grinding Tim (hour.)
0
3.364 (0.0001)
401 (11)
0.04
0.0
5.312
4
3.364 (0.0001)
4S9 ( 8)
0.08
0.0
3.356
6
3.362 (0.0002)
SS9 (10)
0.0s
0.0
3.352
0.0
3.SS8
0.0
3.569
0.0
3.362
8
3.362 (0.0001)
396 ( 7)
0.08
12
3.564 (0.0004)
SSI (13)
0.06
16
3.S58 (0.0002)
3= (8)
20
3.363 (O.aoOl)
S81 (12)
S.SOS(0.006)
75.0 (10)
0.06
0.20
0.19
0.u
S..%4
26
3.366 (0.0002)
244 (7)
5.572 (0.002)
68.0 ( 2)
0.W
0.10
0.28
0.68
3.371
30
S.Sbs (0.001)
162 (11)
S.4OO(o.ou)
SC.0 ( 4)
0.08
0.26
0.22
0.18
3.402
3s
3362 (0.001)
@7 (S)
s.408 (0.010)
S8.0 ( 3)
O.OS
0.22
0.37
0.48
S.401
40
8.373 (0.001)
77 (8)
S.4U(O.o21)
14.0 ( 2)
0.08
0.08
O.Sl
0.46
3.423
46
3.S72 (0.002)
6’ (2)
s.sO4 (0.010)
16.7 (0.4)
0.08
0.07
0.28
0.72
3.470 3.490 S.627
50
3.377 (O.OOS)
41 (2)
8.540 (0.012)
14.1 (0.4)
0.11
0.08
0.31
0.71
56
5.378 (O.OOD)
S7 (6)
8.64s (0.018)
12.4 (0.S)
0.33
0.07
0.32
0.91
64
3.376 (0.006)
S&S
(0.007)
10.8 (0.2)
-1.0
0.08
S.SSa
74
3.374 (0.006)
S.649 (0.008)
10.0 (0.2)
0.06
91.0
3.640
90
S.376 (0.006)
doA
s.649 (0.007)
10.9 (0.2)
0.0s
-1.0
3.646
as they were bell shaped and symmetrical. The importance of the modified Lorentzian shape function is that it comes very close to matching the observed peak shapes over the entire angular range, far better than Gaussian or Cauchy, for example. In a typical case for which the peak intensity might be about 100 c/s the values of l2 were less than 2. This means that the rms average difference between the observed and calculated intensities were less than 1 c/s at every one of about 50 28 abscissa points! The fit for each peak might have been improved by using a sum of two Gaussians, for example, but it was felt that this would introduce complexity without a commensurate improvement in clarity. The modified Lorentzian function fortuitously has convenient mathematical properties and introduces rigor into the calculation of (L,). Given that the line profile is
p(2e) = [l
+ k?(2: - (2e))q2
the peak area, centroid,
and variance are found to
be n112k, (20). and l/k*. The first result affords a simple means of calculating the integral breadth f3 of a profile. This is defined as:
P=
Peak Area Peak maximum
7FI 71 = 2kl = 2k.
(5)
It can be shown that when p is defined in this way. then
(A) p = (L,) cos(8) where (A) is the centroid of the K, radiation, (15,) = JL,dV/V(ll] (i.e.. (L,) is the volume average of the crystallite height normal to the diffracting planes). Thus no assumption is made concerning crystallite shape and (L,) has a well defined physical meaning. It should be noted that given the form of P(20) as in eqn (1). if (L,) is estimated using the full width at half maximum in the Scherrer equation, as is usually done by most researchers, the result will be about 22% greater than the true value of (L,). It can also
Structural transformation induced by grinding
be shown that the variances of all the factors that contribute to peak variance Ilk? are additive[lO]. Thus, when corrections are first made for absorption, and the magnitudes of the other instrumental factors are made small (and are known), it is a simple matter. using eqn (4). to obtain the pure diffraction line breadth from the peak shape parameter k2. Based upon the present work and the works cited previously. Table 3 has been constructed. It shows that there is a discrete set of d,,,: spacings that occur in pure carbon materials during annealing or are observed in graphite after fast neutron irradiation or prolonged mechanical grinding. Because the 3.55A spacing occurs in the fast neutron irradiation of single crystals, it must be associated with carbon defects, (i.e., interstitials or interstitial clusters). The model of disordered graphite that seems appropriate is that of a mixture of these metastable graphite interstitial phases, akin to the model of grafted interstitials proposed by Maire and Mering[ 151for carbons. It is notable that. with rare exceptions, rhe interlayer spacings obtained in rhe peak-search least squares calculation are accurate to -CO.OOlA or better. The data of crystallite sizes and peak areas (proportional lo crystallite volumes) in Table 2 show that the size distributions for each grinding time are narrow (53% of the mean). and at first the crystals cleave into halves having the same interlayer spacing, but after I6 hours grinding on the average the crystallites cleave into fragments such that (L,J (thinner. &greater) = l/3@ ,,) (thicker, &? smaller). The data of Table 2 also imply that (LJ = (IL.,,). The authors’ interpretation of the data is that the range 3.354A 5 d,,,, 5 3.375A is associated with a continuous solid solution wherein 3.375,& represents the maximum solubility of single. C,. carbon inter-
905
stitials. This 3.375A characterizes a very stable, limiting structure (small amounts are present even after 90 hours of grinding!). Recall that an arrest at about 3.38A is found in all investigations-this work shows that it should be 3.375A. The abrupt increases in d,,,, in Fig. 6 at 3.375, 3.40, 3.44, and 3.55A suggest a pillaring effect similar to that observed in :;n,,,ed swelling clay minerals such as vermiculite[ 161. The model of carbons as mixtures of discrete metastable interstitial graphite phases has important implications for studies of graphitization. The neutron irradiation results of Henson and Reynolds[S] indicate that as the c-parameter takes on only discrete increased values, the u-parameter also takes on only discrete decreased values. Because in pregraphitic carbons at the onset of annealing the x-ray lines are very broad, and most likely bimodal (or trimodal). the apparent (hk) banding is to be expected (e.g.. (100) would be a multiplet, as would (101)) so the (10) band might contain four to six broad components. It also follows that the appearance of the (112) reflection is not a measure of the onset of graphitization so much as it reflects the changes in a- and c-spacings in both the (112) and (006) reflections, and the multiplet structure of each reflection, combined with peak narrowing owing to increases in crystallite size. In fact, while stacking faults are no doubt present in these carbons, it is questionable whether or not random layer stacking is the essential feature of “turbostratic carbon.” It is the authors’ view that graphitization consists primarily of the annealing of metastable carbon interstitial phases. A further implication of this interpretation is that it is very unlikely that steps in the dlllZversus time curve will be observed during graphitization unless the starting materials are very homogeneous (i.e.. a single d,,,, for all crystallites). This brings up the spe-
Table 3. tnterlayer spacings in carbons and graphite
Carbons
Graphite
Neutron Irradiation
Annealing
Pandica
3.44A
3.40A
Fischbach,b PacaultC
3.44A (pitch coke)
(3.425 f .003)
Kawamura et aLd
Henson and Reynoldse
3.55A
3.55A
3.44A
3.44A
3.40A
Grinding
Tidjani et al.f
3.55A 3.43A
3.44A
3.40A
3.40A
3.40A 3.375A 3.354A
3.38A
3.38A
3.375A
3.375A
3.315A
3.354A
3.354A
3.354A
3.354A
3.354A
aRef. [4] bRef. [2]
CRef. [3] dRef. [7]
eRef. [5] fRef. [6]
BThis paper
Aladekomo and Braggg
J.B. ALADEKOMO and R. H. BRAGG
906
cial case of pyrolytic carbon. Since pyrolytic carbon anneals during deposition (typically above 2OOO”C), a variation in doo2with thickness occurs naturally and is unavoidable in as-received material. As a first approximation the typical 3.425A -’ 0.0038, interlayer spacing probably represents a mixture of 3.4OA and 3.44A phases. No amount of annealing of such a bimodal deposit can homogenize the bulk material[3]. Moreover, since three discrete metastable phases (increasing doo2) are associated with increasing amounts of grinding (more disorder, greater energy input) up to 3.44& the reverse annealing (graphitization) process should be characterized by three activation energies when d,, is initially at 3.44& instead of the one generally assumed in graphitization studies. Some success in confirming the three level model has been reported by Bragg and Lachter[l7]. The authors feel that line profile analysis should become standard practice in diffraction characterization of carbon materials. While it can be done in principle with Debye-Scherrer patterns, the diffractometer is far better suited to this endeavor. There is nothing special about the computer analysis-all this work was done on a clone of an IBM-PC. For those not familiar with least squares curve fitting, very readable how-to references are available[l8]. 5. CONCLUSIONS This work shows that in experiments in which graphite is disordered by mechanical grinding, the interlayer spacing increases continuously in the range 3.354h; I doo2 I 3.375& as in a solid solution. At 3.375& further grinding causes discontinuous transformations of the material to metastable phases characterized by spacings of 3.40& 3.44& 3SSA, and possibly 3.49A. Because the 3.55A and 3.49A spacings have been observed in pyrolytic carbon, and the others are observed during the annealing of pitches, it is hypothesized that this same set of spacings is unique in elemental carbons of whatever origin. Thus in graphitization studies, a sample should be characterized by decomposing the x-ray diffraction line profiles into their characteristic discrete components. These components should be used when structure-property correlations are made.
Acknowledgement-The
authors would like to thank Dr. Kolawole Aiyesimoju for assistance in performing the profile fitting analyses. This work was supported bv the Director, Office df Energy Research, OfBce of Basic Energy
Sciences, Materials Sciences Division, of the U.S. Department of Energy under Contract No. DE-AC0376SPOOO98.
REFERENCES
1. J. Lachter and R. H. Bragg, Phys. Rev. B 33, 8903 (1986). 2. D. B. Fischbach, In Chemistry and Physics of Carbon (Edited by P. L. Walker, Jr.), Vol. 7, pp. l-105. Marcel Dekker, New York (1971). 3. A. Pacault, In Chemistry and Physics of Carbon (Edited by P. L. Walker, Jr.), Vol. 7,-pp. 167-154. Marcel Dekker. New York 11971). 4. B. Pan&, In Procdedingi of the Second Conference on Industrial Carbon and Graphite, Society of Chemical Industry, p. 439. Gordan and Breach, New York (1966). 5. R. W. Henson and W. N. Reynolds, Carbon 3, 277 (1965).
6. M. Tidjani, J. Lachter, T. S. Kabre, and R. H. Bragg, Carbon 24, 447 (1986).
7. K. Kawamura, J.’ Lachter, and R. H. Bragg, Lawrence Berkelev LaboratorvTechnical Reoort No. LBL-17982 (to be submitted for publication).’ 8. J. Lachter, L. G. Henry, and R. H. Bragg, J Appl. Cryst. 21, 92 (1988).
9. R. H. Bragg and J. B. Aladekomo, Lawrence Berkeley Laboratory Technical Report No. LBL-27415 (to be submitted for publication). 10. A. J.-C. Wilson, Mathematical Theory ofX-ray Powder Diffractometry. Gordon and Breach,Ndw York (1963). 11. B. E. Warren, X-ray Diffraction, Chapter 13. AddisonWesley Publishing Company, Reading (1969). 12. G. E. Bacon, Acta Cryst. 5, 392 (1952). 13. M. Nakamizo. H. Honda. and M. Inaeaki. Carbon 16. Y 281 (1978). 14. P. L. Walker, Jr., and S. B. Seeley, Proc. 3rd Biennial Carbon Conference, 1957, p. 481. Pergamon Press, New York (1959). 15. J. Maire and J. Mering, In Chemistry and Physics of Carbon (Edited by P. L. Walker, Jr.), Vol. 6, pp. 125190. Marcel Dekker, New York (1970). 16. H. Kim, W. Jin, S. Lee, P. Zhou, T. J. Pinnevaia, S. D. Mahanti, and S. Solin, Phys. Rev. Left. 60, 2168 (1988). 17. R. H. Bragg and J. Lachter, Lawrence Berkeley Laboratory Technical Report No. LBL-23284 (March 1987). 18. J. L. Kuester and J. H. Mize, Optimization Techniques with Fortran. McGraw-Hill, New York (1973), pp. 240-
250.