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Mechanical properties of vapor-grown carbon fiber
Carbon Vol. 33, No. 9, pp. 1217-1221,1995 Copyright 0 1995 Else&r Science Ltd
Pergamon
Printed in Great Britain. All rights reserved 00X-6223/95 $9.50 + 0.00
000th6223(95)00057-7
MECHANICAL
PROPERTIES OF VAPOR-GROWN FIBER
CARBON
T. M. TRITT,’ J. R. GUTH,’ A. C. EHRLICH~ and D. J. GILLESPIE’ ‘Applied Sciences, Inc., P.O. Box 579, Cedarville, OH 45314, U.S.A. ZMaterials Physics Branch, Naval Research Laboratory, Washington, DC 20375, U.S.A.
R. L. JACOBSEN,’
(Received 3 February 1995; accepted in revised form 24 February 1995) Abstract-The elastic properties of individual vapor-grown carbon fibers have been studied via the lowstrain, vibrating-reed technique. The average Young’s modulus is found to be 680 GPa, with some indication that the intrinsic modulus of CVD carbon may be much higher. This value exceeds those found by earlier high-strain, static-pulling methods. Although the amplitude of vibrational response for these fibers is linear in the driving force, the shape of the frequency response deviates from Lorentzian as the amplitude increases. This behavior has not been previously reported in other materials. Possible
explanations based on the fibers’ microstructure are discussed. Key
Words-Vapor-grown
carbon
fiber,
vibrating-reed,
modulus,
internal
friction,
non-Lorentzian
resonance.
1. INTRODUCTION
Vapor-grown carbon fiber (VGCF) possesses many physical properties which are outstanding, even by the high standards of the graphitic fiber industry[ 11. Heat treated VGCF can have electrical resistivity as low as 50 @-cm[2]. Its room temperature thermal conductivity of nearly 2000 W/m-K is second only to diamond[3]. The Young’s modulus of VGCF has also been reported to be high, in the range of 360-600 GPa [ 41. However, mechanical measurements on individual fibers are somewhat difficult because the material is quite brittle and susceptible to damage from handling. Previous measurements have been made by static pulling techniques [ 4-61, which introduce high strains into the samples and may result in cracking, thereby compromising the measurement. This has led to a wide range of reported values that are perhaps lower than the true VGCF modulus. In this work we investigate the elastic properties of individual vaporgrown carbon fibers using an extremely low-strain vibrating-reed technique. We find a higher average value of modulus, with a more modest spread in the data. In addition, an unanticipated dynamic phenomenon, apparently unique to VGCF, is revealed. 2. EXPERIMENTAL
TECHNIQUE
2.1 Fiber microstructure VGCF is produced by chemical vapor deposition (CVD) of carbon from pyrolysis of methane, to build up the diameter of a core carbon filament. This creates a fiber with a distinctive “tree-ring” microstructure of nested, concentric cylindrical shells which are highly graphitized by heat treatment to about 2900°C[7]. Unlike many other carbon fibers, little or no cross-linking exists between the concentric graphene shells.
The core filament of VGCF is formed by catalytic reaction between methane and iron particles. Such filaments can be made arbitrarily long while being of negligible thickness, on the order of 10 nm. Fibers with this catalytic core consist almost exclusively of CVD carbon, and are simply referred to as VGCF. Some fibers were grown by deposition onto commercial PAN-type carbon fiber of about 5.1 Lcrnthickness (Amoco T-650 PAN fiber). Fibers with a PAN-type core are identified as “hybrid” fibers[ S].
2.2 Vibrating-reed method The fibers were studied using the vibrating-reed technique[9]. In this method an individual fiber is mounted as a cantilever, with one end solidly fixed and the other end free. The mechanical resonant frequencies of the fiber, co,, is then determined by its longitudinal Young’s modulus, Y, density, p, and dimensions [ lo].
where L is the fiber’s length, K the radius of gyration of its cross section, and CI, is a constant that depends on the mode of vibration, n, and the boundary conditions. Here, t10 (fundamental) is equal to 3.516, and CI~(first overtone) is 22.03. Dimensions of the fibers can be measured by microscopy or diffraction techniques, and thus the modulus can be determined directly from the resonant frequency. In this sort of measurement, fiber integrity is vital. If a particular fiber is flawed (e.g. cracked, has non-uniform crosssection, or bad boundary condition) this can be detected by comparing the frequency of the fundamental to the first, and if necessary higher, overtones. The resonant frequency is measured with a helical-resonator vibrating-reed apparatus, described in detail elsewhere[ 111. The fiber is driven into 1217
R. L.
1218
JACOBSEN
vibration by a piezoelectric crystal. The voltage supplied to this crystal is the drive amplitude. The response amplitude of the vibrating reed is detected through the variation in capacitance between the reed and a nearby electrode. This signal is augmented by a lock-in amplifier, referenced to the drive frequency. When the driving frequency is swept through a resonance of a vibrating reed, the response amplitude, R, normally follows a characteristic shape known as a Lorentzian [ 121,
“-J&
12)
This is the response curve for any oscillatory system with a linear restoring force, and a frictional dissipation that is linear in velocity. Here, A is the peak response amplitude and fi is a damping factor that describes the reeds internal resistance to motion. For each resonance observed, a non-linear least-squares fit of this form is made to the data, with A, /3 and w, as adjustable parameters. The resonant frequency, and therefore the modulus, are thereby determined. From the fitted value for fl one can also calculate the reed’s internal friction, sometimes denoted as the inverse of the resonance quality factor, Q[ 121. The internal friction is a measure of how quickly energy is dissipated by the vibrational motion. Such losses are generally indicative of internal degrees of freedom that can respond to the stresses inside the fiber, such as mobile dislocations or defects, or thermal flows. The vibrating-reed measurements are always performed under vacuum to assure that any dissipation is due to processes internal to the fiber, and not friction with the surrounding atmosphere. The vibrating-reed technique offers certain advantages over the usual static stress-strain devices commonly used for measuring the elastic properties of carbon fibers. First, since it is a dynamic measurement, it is capable of obtaining internal friction information, as described above, which is not accessiTable 1. Geometry
and modulus
ble to static techniques. Second, while stress-strain devices must typically distort the fiber by at least a few tenths of a percent, the vibrating-reed method is inherently low-distortion, with local strains rarely exceeding 10~‘. Thus, there are fewer concerns about stress-induced damage to the fiber, and non-linear behavior.
3. RESULTS 3.1 Modulus measurements Several fibers were tested and the results are summarized in Table 1. Fibers of pure VGCF have a “V” in their designations; hybrid fibers begin with an “H”. Fiber lengths were measured with a traveling optical microscope, and diameters were determined by either electron microscopy or diffraction techniques. The fiber density of heat-treated VGCF is known from other measurements to be about 2.15 g/cm3 [ 131. The mean densities of the hybrid fibers are figured based on VGCF vs PAN fiber content. Values of modulus are calculated from these parameters and the measured fundamental resonance frequency. With such tiny specimens, it is difficult to perfectly control small perturbations that might effect the resonances. Such perturbations might include slight variations in reed cross-section, less than ideal boundary conditions, or a bit of dust on the reed. The effect of these perturbations, more often than not, is to reduce the resonant frequency. Usually, any effect will be greater on the fundamental than on the overtones. As a result, one can detect and largely correct for these perturbations by monitoring the frequency ratio between the fundamental and first overtone. This ratio has an ideal value of 6.267[ 111. If the ratio for a particular reed is high, this generally indicates that the measured fundamental frequency, and therefore the calculated modulus is low. Figure 1 shows calculated modulus plotted against frequency ratio, for the fibers of Table 1. It is indeed evident that a downward trend exists in calculated modulus with increasing frequency ratio. The slope of the least-squares fit line can be used to correct the data for VCCF
and hybrid
fibers
Density tg/emY
Modulus
7.6 11.4
2.15 2.15
550 450
6.46 6.48
655 566
5.18 4.45
19.9 19.1
2.15 2.15
515 685
6.66 6.24
129 670
v7 V8 v9 Vll v12 v13
4.76 5.03 4.39 4.73 4.00 5.25
12.7 28.1 13.1 12.7 19.1 8.2
2.15 2.15 2.15 2.15 2.15 2.15
750 720 950 525 445 670
6.07 6.21 6.39 6.52 6.29
583 683
Hl H3 H9
5.36 5.54 3.04
9.0 5.3 22.4
2.03 1.80 2.13
550 190 630
6.41 6.14 t
628
Reed designation
Length (mm)
Vl v2
4.01 2.62
v3 V6
1-First overtone
Diameter (@ml
eta/.
not measured.
(GPa)
w,, G
Corrected modulus
643 717 1017
t
Mechanical
properties
of vapor-grown
carbon
fiber
1219
3.2 Non-Lorentzian behavior
01 6.0
I
I
I
I
6.2
6.4
6.6
6.8
Frequency
Ratio
Fig. 1. Modulus, as calculated from the fundamental resonance frequency, plotted vs first overtone to fundamental frequency ratio. The solid line is a least squares fit. The data includes most of the fibers from Table 1.
of each fiber to a value more closely representative of an unperturbed reed. These values are listed in the last column of Table 1. The measurements indicate that the CVD carbon in VGCF has an extremely high intrinsic modulus. All pure VGCF fibers for which a corrected modulus could be calculated had a modulus of at least 550 GPa. The average value was 685 GPa. Our range of modulus overlaps part of the range previously reported by Koyama et al.[4], but extends much higher. One fiber (V9) exhibited a corrected modulus in excess of 1000 GPa. This specimen was wellbehaved, and we have no reason to disbelieve the measurement. The wide range of current, and previous, modulus measurements may be due to the random introduction of weaknesses by the handling and high-strain measurement of this brittle material. The 1000 GPa measurement may be an observation of the intrinsic modulus of CVD graphite, thanks to a combination low-strain technique and fortuitously gentle handling of the specimen. One of the hybrid fibers (H3) responded with a modulus of only 190 GPa, near the value for the core PAN-type fiber [S]. However this fiber had only a very thin coating of CVD carbon, accounting for only 7% of the value of K’, which is the appropriate measure of contribution to modulus. Of the other hybrids, Hl had a VGCF contribution to ICYof 70% and a corrected modulus of 628 GPa, while in H9 VGCF accounted for 95% of K~, leading to an uncorrected modulus of 630 GPa. It should be noted that, while we have corrected for the portion of small perturbations that alter the frequency ratio between the fundamental and overtone, we cannot correct for a proportional shift in both modes. Any errors of this nature will again tend cause us to calculate low values of modulus. Thus our modulus results, while fairly accurate, should be taken to indicate a lower bound on the true modulus. The above measurements were made at the lowest possible drive amplitudes. This was necessitated by a remarkable finding that appears to be unique to VGCF fibers, as discussed below. modulus
These experiments have shown that VGCF, unlike other tested carbon fibers[14], to have a nonLorentzian frequency response with a deviation that increases with the drive amplitude. Figure 2 shows the response of the fundamental for a VGCF fiber at various drive levels. The solid curves are best-fit Lorentzians for each case. Normally, a deviation of this form, the peak leaning to lower frequency, is indicative of a so-called soft non-linearity. (See for example Fig. 14.10 in Morse and Ingard[15].) To describe this theoretically, the usual linear form of Hooke’s law is modified by adding a term to the restoring force that is cubic in the displacement, i.e. F = Yx + Y’x3. For a soft non-linearity, Y’ is negative. However, attempts to fit the data with a Lorentzian form that contains correction terms to account for a soft non-linearity yielded little improvement in the quality of the fit. Therefore a new form was sought. We discovered that the data is very well fit by using the normal Lorentzian form, but plotting the data on a Cartesian coordinate system for which the y-axis is tilted at an angle from the vertical. Or, equivalently, one can add a term in the denominator of the Lorentzian formula which makes the center frequency proportional to the response amplitude. The modified Lorentzian is as follows:
“=J&
(3)
where E is also now an adjustable fitting parameter. This form was used to generate the excellent dashedline fits in Fig. 2. This modified Lorentzian form was then used to characterize the dependence of VGCF vibrational response upon drive amplitude. Interestingly, the
1440
1450
Drive
1460
Frequency
1470
1480
(Hz.)
Fig. 2. Vibrational response amplitude vs frequency for a VGCF fiber at various drive amplitudes. The drive amphtude ratio for the curves is 1: 5 : 20: 50 from bottom to top. Open circles represent the data. Solid lines show the bestfit Lorentzian. Dashed lines are a fit with the modified Lorentzian form of eqn 3. The data is from fiber V7.
R. L. JACOBSENet
1220 a m
4000
.?Y
a
z
3000
1
I
I
I .
0.007
I
.* k 1 .
. .
.
.
1
0.006
3
i 0.005
0.004
\ 0.003 D
2
0.002
0
0.001
Y 0
2
0
I
0
I
I
I
AApli
0.000
I
i
6
Fig. Peak response amplitude (open circles) and internal friction (filled circles) vs drive amplitude. The data is from fiber V7.
peak response amplitude, A, shown in Fig. 3, appears to be linear in the drive amplitude. This is perhaps surprising. As the frequency response curve is patently non-Lorentzian, we cannot be dealing with a system described by a single restoring force that is linear in displacement, and therefore there is no reason to expect a linear relationship between drive and response. Figure 3 also shows that the internal friction increases with drive amplitude in manner that tends to level out at high amplitude. By way of comparison, internal friction is independent of amplitude in a system with regular Lorentzian response. The behavior of Fig. 3 has been verified on several VGCF reeds, and occurs on overtones as well as the fundamental.
4. DISCUSSION An important question to answer is what kind of physical model corresponds to the behavior described by eqn 3. The answer may lead to a better understanding of VGCF microstructure and dynamic behavior. We have seen that, despite a resemblance in the response, a standard Lorentzian form. modified to account for a nonlinear restoring force, does not explain the data. How then, is the form which does fit the data different? In a system with a slightly nonlinear restoring force, the modulus depends on the instantaneous displacement. This makes the frequency response dependent on amplitude. However, since the restoring force varies during the course of the oscillation, the system does not have a well-defined center frequency of the resonance, except in the zero amplitude limit. By contrast, the modified Lorentzian does have a well-defined center frequency at all times. This center frequency, w,+sR, varies with the rms response amplitude, but not with the instantaneous displacement of the reed. This leads to the question of what mechanism could cause the value of the resonant frequency to vary with rms response amplitude. Referring to eqn 1, this behavior might occur if any of the parameters on the right hand side of this equation were to vary
al
with rms amplitude. At the small amplitudes of a vibrating reed, it seems reasonable to immediately eliminate G and p from suspicion. But we may question whether the unique morphology VGCF and properties of graphite could result in unanticipated behavior in the effective value of Y, K, or r*. For example, in the derivation of eqn 1, it is assumed that all deflection in the vibrating reed is due to bending of the reed[ 111. However, if a material’s shear modulus, G, is small enough compared to Y, some of the reed’s deflection will be due to shear distortion[ 161. It is known that dislocation motion on the basal planes of graphite can reduce G between planes[ 171. If an increasing amplitude of vibration were to activate similar motion of dislocations, or perhaps relaxations associated with other defects such as impurities between the cylindrical shells of VGCF[lS], this could manifest itself in an effective reduction of Y. Note that the concept of activation with increasing rms amplitude is important here, as defects that are equally mobile at all rms amplitudes would simply contribute to a constant reduction of Y. A pedagogical analogy here might be made to a number of large ball bearings resting in the dimples of a waffle iron. The bearings only become free to move when the waffle iron is agitated above some given rms level. The bearings remain free until agitation drops below this level, at which point they settle back into the dimples. A distribution of dimple depths would lead to activation which is gradual with increasing agitation amplitude. In this scenario the activated, mobile entities could also act as a source of dissipation, accounting for the increase in internal friction with vibration amplitude. The fact that the internal friction appears to level off at high amplitude is consistent with the intuitive picture that the overall mobility of any population of defects must eventually saturate. Another possibility might lie in the existence of an instability in the concentric-cylinder microstructure that leads to a change in the cross-sectional shape, and thus K, during vibration. If one tries to bend a tin can or a rolled-up piece of paper, one finds that the structure rapidly fails in a manner that collapses the diameter along the direction of bending. Such failure does not occur in a solid rod. Tendency of the VGCF cylinders toward this type of distortion with increasing vibration amplitude could reduce the effective value of K and thus, the resonant frequency. Here, the growing internal friction would result from energy being transferred, via this distortion, out of the normal vibrational mode and into other lattice degrees of freedom (phonons). One fault with this view is that the distortion of the cross-sectional shape should depend on the instantaneous, and not rms, displacement of the reed. Therefore the effects of the distortion should probably behave more like a standard nonlinear restoring force. This flaw could perhaps be repaired by imagining that some visco-elastic phenomenon slows the fiber’s shape change to a time-
Mechanical
properties
of vapor-grown
scale longer than a cycle of vibration (typically on the order of a millisecond). Then the average value of IC would tend toward the value it has at the rms level of displacement, rather than its equilibrium value at zero displacement. Either of the above explanations could probably be equally well cast, in mathematical terms, as an amplitude dependence in the value of a, because in either case the nature of the vibrational mode would be slightly altered. A nice feature of these explanations is that neither is inconsistent with a peak response amplitude which depends approximately linearly on the drive amplitude. Most explanations based on restoring forces which are non-linear in instantaneous displacement would fail here.
5. CONCLUSION We have investigated the elastic properties of VGCF, using the low-strain vibrating-reed technique as an alternative to static pulling methods. The average fiber modulus is 680GPa, and there are indications that the intrinsic modulus of CVD carbon may approach 1000 GPa. Resonances of VGCF reeds display a unique distortion of their frequency response curves with increasing amplitude. This distortion is accompanied by an increase in the fibers internal friction. This behavior may be caused by activation of defect motion between the concentric graphite cylinders that make up the fiber, or by some other feature of this morphology.
carbon
fiber
1221
REFERENCES 1. For a general review, see M. S. Dresselhaus, G. Dresselhaus, K. Surihara, I. L. Spain and J. A. Goldberg, Graphite Fibers and Fihwnents. Springer-Verlag, Berlin (1988). 2. J. Heremans, Carbon 23,477 (198.5). J. Heremans and C. P. Bee@ Jr, Phys. Rev. B 32, 1981 (1985). T. Koyama, M. Endo and Y. Hishiyama, Jpn. J. Appl. Phys. 13, 1933 (1974). N. Kandani, M. Coulon and L. Bonnetain, Extended Abstracts Carbone ‘84, p. 142. Bordeaux (1984). W. E. Yetter, C. P. Beetz, Jr and G. W. Budd, Extended Abstracts 17th Biennial Conference on Carbon, p. 291. Lexington, KY (1985). G. G. Tibbetts, M. Endo and C. P. Bee@ Jr, SAMPE J. 22-5 (1986). 8. J. R. Gaier, M. L. Lake, A. Moinuddin and M. Marabito, Carbon 30, 345 (1992). 9. T. Tiedie. R. R. Haerina and W. N. Hardv. J. Acoust. Sot. Am. 65, 1171 (19793. 10. P. M. Morse, Vibration and Sound, p. 151. AIP, New York (1976). 11. X.-D. Xiang, J. W. Brill and W. L. Fuqua, Rev. Sci. Instrum. 60, 3035 (1989). Classical Dynamics of Particles and 12. J. B. Marion, Systems, 2nd edn, p. 117. Academic, New York (1970). and M. M. Slabe, Synch. 13. J. R. Gaier, P. D. Hambourger Met. 31, 229 (1989). A. C. Ehrlich and D. J. 14. T. M. Tritt, R. L. Jacobsen, Gillespie, Carbon (submitted). 15. P. M. Morse and K. U. Ingard, Theoretical Acoustics, p. 851. McGraw-Hill, New York (1968). Phil. Mag. 41, 744 (1921). 16. S. Timoshenko, Industrial Carbon and Graphite. 17. J. H. W. Simmons, p. 511. Sot. of Chem. Industry, London (1957). 18. M. Dresselhaus (private communication).
Pergamon
Printed in Great Britain. All rights reserved 00X-6223/95 $9.50 + 0.00
000th6223(95)00057-7
MECHANICAL
PROPERTIES OF VAPOR-GROWN FIBER
CARBON
T. M. TRITT,’ J. R. GUTH,’ A. C. EHRLICH~ and D. J. GILLESPIE’ ‘Applied Sciences, Inc., P.O. Box 579, Cedarville, OH 45314, U.S.A. ZMaterials Physics Branch, Naval Research Laboratory, Washington, DC 20375, U.S.A.
R. L. JACOBSEN,’
(Received 3 February 1995; accepted in revised form 24 February 1995) Abstract-The elastic properties of individual vapor-grown carbon fibers have been studied via the lowstrain, vibrating-reed technique. The average Young’s modulus is found to be 680 GPa, with some indication that the intrinsic modulus of CVD carbon may be much higher. This value exceeds those found by earlier high-strain, static-pulling methods. Although the amplitude of vibrational response for these fibers is linear in the driving force, the shape of the frequency response deviates from Lorentzian as the amplitude increases. This behavior has not been previously reported in other materials. Possible
explanations based on the fibers’ microstructure are discussed. Key
Words-Vapor-grown
carbon
fiber,
vibrating-reed,
modulus,
internal
friction,
non-Lorentzian
resonance.
1. INTRODUCTION
Vapor-grown carbon fiber (VGCF) possesses many physical properties which are outstanding, even by the high standards of the graphitic fiber industry[ 11. Heat treated VGCF can have electrical resistivity as low as 50 @-cm[2]. Its room temperature thermal conductivity of nearly 2000 W/m-K is second only to diamond[3]. The Young’s modulus of VGCF has also been reported to be high, in the range of 360-600 GPa [ 41. However, mechanical measurements on individual fibers are somewhat difficult because the material is quite brittle and susceptible to damage from handling. Previous measurements have been made by static pulling techniques [ 4-61, which introduce high strains into the samples and may result in cracking, thereby compromising the measurement. This has led to a wide range of reported values that are perhaps lower than the true VGCF modulus. In this work we investigate the elastic properties of individual vaporgrown carbon fibers using an extremely low-strain vibrating-reed technique. We find a higher average value of modulus, with a more modest spread in the data. In addition, an unanticipated dynamic phenomenon, apparently unique to VGCF, is revealed. 2. EXPERIMENTAL
TECHNIQUE
2.1 Fiber microstructure VGCF is produced by chemical vapor deposition (CVD) of carbon from pyrolysis of methane, to build up the diameter of a core carbon filament. This creates a fiber with a distinctive “tree-ring” microstructure of nested, concentric cylindrical shells which are highly graphitized by heat treatment to about 2900°C[7]. Unlike many other carbon fibers, little or no cross-linking exists between the concentric graphene shells.
The core filament of VGCF is formed by catalytic reaction between methane and iron particles. Such filaments can be made arbitrarily long while being of negligible thickness, on the order of 10 nm. Fibers with this catalytic core consist almost exclusively of CVD carbon, and are simply referred to as VGCF. Some fibers were grown by deposition onto commercial PAN-type carbon fiber of about 5.1 Lcrnthickness (Amoco T-650 PAN fiber). Fibers with a PAN-type core are identified as “hybrid” fibers[ S].
2.2 Vibrating-reed method The fibers were studied using the vibrating-reed technique[9]. In this method an individual fiber is mounted as a cantilever, with one end solidly fixed and the other end free. The mechanical resonant frequencies of the fiber, co,, is then determined by its longitudinal Young’s modulus, Y, density, p, and dimensions [ lo].
where L is the fiber’s length, K the radius of gyration of its cross section, and CI, is a constant that depends on the mode of vibration, n, and the boundary conditions. Here, t10 (fundamental) is equal to 3.516, and CI~(first overtone) is 22.03. Dimensions of the fibers can be measured by microscopy or diffraction techniques, and thus the modulus can be determined directly from the resonant frequency. In this sort of measurement, fiber integrity is vital. If a particular fiber is flawed (e.g. cracked, has non-uniform crosssection, or bad boundary condition) this can be detected by comparing the frequency of the fundamental to the first, and if necessary higher, overtones. The resonant frequency is measured with a helical-resonator vibrating-reed apparatus, described in detail elsewhere[ 111. The fiber is driven into 1217
R. L.
1218
JACOBSEN
vibration by a piezoelectric crystal. The voltage supplied to this crystal is the drive amplitude. The response amplitude of the vibrating reed is detected through the variation in capacitance between the reed and a nearby electrode. This signal is augmented by a lock-in amplifier, referenced to the drive frequency. When the driving frequency is swept through a resonance of a vibrating reed, the response amplitude, R, normally follows a characteristic shape known as a Lorentzian [ 121,
“-J&
12)
This is the response curve for any oscillatory system with a linear restoring force, and a frictional dissipation that is linear in velocity. Here, A is the peak response amplitude and fi is a damping factor that describes the reeds internal resistance to motion. For each resonance observed, a non-linear least-squares fit of this form is made to the data, with A, /3 and w, as adjustable parameters. The resonant frequency, and therefore the modulus, are thereby determined. From the fitted value for fl one can also calculate the reed’s internal friction, sometimes denoted as the inverse of the resonance quality factor, Q[ 121. The internal friction is a measure of how quickly energy is dissipated by the vibrational motion. Such losses are generally indicative of internal degrees of freedom that can respond to the stresses inside the fiber, such as mobile dislocations or defects, or thermal flows. The vibrating-reed measurements are always performed under vacuum to assure that any dissipation is due to processes internal to the fiber, and not friction with the surrounding atmosphere. The vibrating-reed technique offers certain advantages over the usual static stress-strain devices commonly used for measuring the elastic properties of carbon fibers. First, since it is a dynamic measurement, it is capable of obtaining internal friction information, as described above, which is not accessiTable 1. Geometry
and modulus
ble to static techniques. Second, while stress-strain devices must typically distort the fiber by at least a few tenths of a percent, the vibrating-reed method is inherently low-distortion, with local strains rarely exceeding 10~‘. Thus, there are fewer concerns about stress-induced damage to the fiber, and non-linear behavior.
3. RESULTS 3.1 Modulus measurements Several fibers were tested and the results are summarized in Table 1. Fibers of pure VGCF have a “V” in their designations; hybrid fibers begin with an “H”. Fiber lengths were measured with a traveling optical microscope, and diameters were determined by either electron microscopy or diffraction techniques. The fiber density of heat-treated VGCF is known from other measurements to be about 2.15 g/cm3 [ 131. The mean densities of the hybrid fibers are figured based on VGCF vs PAN fiber content. Values of modulus are calculated from these parameters and the measured fundamental resonance frequency. With such tiny specimens, it is difficult to perfectly control small perturbations that might effect the resonances. Such perturbations might include slight variations in reed cross-section, less than ideal boundary conditions, or a bit of dust on the reed. The effect of these perturbations, more often than not, is to reduce the resonant frequency. Usually, any effect will be greater on the fundamental than on the overtones. As a result, one can detect and largely correct for these perturbations by monitoring the frequency ratio between the fundamental and first overtone. This ratio has an ideal value of 6.267[ 111. If the ratio for a particular reed is high, this generally indicates that the measured fundamental frequency, and therefore the calculated modulus is low. Figure 1 shows calculated modulus plotted against frequency ratio, for the fibers of Table 1. It is indeed evident that a downward trend exists in calculated modulus with increasing frequency ratio. The slope of the least-squares fit line can be used to correct the data for VCCF
and hybrid
fibers
Density tg/emY
Modulus
7.6 11.4
2.15 2.15
550 450
6.46 6.48
655 566
5.18 4.45
19.9 19.1
2.15 2.15
515 685
6.66 6.24
129 670
v7 V8 v9 Vll v12 v13
4.76 5.03 4.39 4.73 4.00 5.25
12.7 28.1 13.1 12.7 19.1 8.2
2.15 2.15 2.15 2.15 2.15 2.15
750 720 950 525 445 670
6.07 6.21 6.39 6.52 6.29
583 683
Hl H3 H9
5.36 5.54 3.04
9.0 5.3 22.4
2.03 1.80 2.13
550 190 630
6.41 6.14 t
628
Reed designation
Length (mm)
Vl v2
4.01 2.62
v3 V6
1-First overtone
Diameter (@ml
eta/.
not measured.
(GPa)
w,, G
Corrected modulus
643 717 1017
t
Mechanical
properties
of vapor-grown
carbon
fiber
1219
3.2 Non-Lorentzian behavior
01 6.0
I
I
I
I
6.2
6.4
6.6
6.8
Frequency
Ratio
Fig. 1. Modulus, as calculated from the fundamental resonance frequency, plotted vs first overtone to fundamental frequency ratio. The solid line is a least squares fit. The data includes most of the fibers from Table 1.
of each fiber to a value more closely representative of an unperturbed reed. These values are listed in the last column of Table 1. The measurements indicate that the CVD carbon in VGCF has an extremely high intrinsic modulus. All pure VGCF fibers for which a corrected modulus could be calculated had a modulus of at least 550 GPa. The average value was 685 GPa. Our range of modulus overlaps part of the range previously reported by Koyama et al.[4], but extends much higher. One fiber (V9) exhibited a corrected modulus in excess of 1000 GPa. This specimen was wellbehaved, and we have no reason to disbelieve the measurement. The wide range of current, and previous, modulus measurements may be due to the random introduction of weaknesses by the handling and high-strain measurement of this brittle material. The 1000 GPa measurement may be an observation of the intrinsic modulus of CVD graphite, thanks to a combination low-strain technique and fortuitously gentle handling of the specimen. One of the hybrid fibers (H3) responded with a modulus of only 190 GPa, near the value for the core PAN-type fiber [S]. However this fiber had only a very thin coating of CVD carbon, accounting for only 7% of the value of K’, which is the appropriate measure of contribution to modulus. Of the other hybrids, Hl had a VGCF contribution to ICYof 70% and a corrected modulus of 628 GPa, while in H9 VGCF accounted for 95% of K~, leading to an uncorrected modulus of 630 GPa. It should be noted that, while we have corrected for the portion of small perturbations that alter the frequency ratio between the fundamental and overtone, we cannot correct for a proportional shift in both modes. Any errors of this nature will again tend cause us to calculate low values of modulus. Thus our modulus results, while fairly accurate, should be taken to indicate a lower bound on the true modulus. The above measurements were made at the lowest possible drive amplitudes. This was necessitated by a remarkable finding that appears to be unique to VGCF fibers, as discussed below. modulus
These experiments have shown that VGCF, unlike other tested carbon fibers[14], to have a nonLorentzian frequency response with a deviation that increases with the drive amplitude. Figure 2 shows the response of the fundamental for a VGCF fiber at various drive levels. The solid curves are best-fit Lorentzians for each case. Normally, a deviation of this form, the peak leaning to lower frequency, is indicative of a so-called soft non-linearity. (See for example Fig. 14.10 in Morse and Ingard[15].) To describe this theoretically, the usual linear form of Hooke’s law is modified by adding a term to the restoring force that is cubic in the displacement, i.e. F = Yx + Y’x3. For a soft non-linearity, Y’ is negative. However, attempts to fit the data with a Lorentzian form that contains correction terms to account for a soft non-linearity yielded little improvement in the quality of the fit. Therefore a new form was sought. We discovered that the data is very well fit by using the normal Lorentzian form, but plotting the data on a Cartesian coordinate system for which the y-axis is tilted at an angle from the vertical. Or, equivalently, one can add a term in the denominator of the Lorentzian formula which makes the center frequency proportional to the response amplitude. The modified Lorentzian is as follows:
“=J&
(3)
where E is also now an adjustable fitting parameter. This form was used to generate the excellent dashedline fits in Fig. 2. This modified Lorentzian form was then used to characterize the dependence of VGCF vibrational response upon drive amplitude. Interestingly, the
1440
1450
Drive
1460
Frequency
1470
1480
(Hz.)
Fig. 2. Vibrational response amplitude vs frequency for a VGCF fiber at various drive amplitudes. The drive amphtude ratio for the curves is 1: 5 : 20: 50 from bottom to top. Open circles represent the data. Solid lines show the bestfit Lorentzian. Dashed lines are a fit with the modified Lorentzian form of eqn 3. The data is from fiber V7.
R. L. JACOBSENet
1220 a m
4000
.?Y
a
z
3000
1
I
I
I .
0.007
I
.* k 1 .
. .
.
.
1
0.006
3
i 0.005
0.004
\ 0.003 D
2
0.002
0
0.001
Y 0
2
0
I
0
I
I
I
AApli
0.000
I
i
6
Fig. Peak response amplitude (open circles) and internal friction (filled circles) vs drive amplitude. The data is from fiber V7.
peak response amplitude, A, shown in Fig. 3, appears to be linear in the drive amplitude. This is perhaps surprising. As the frequency response curve is patently non-Lorentzian, we cannot be dealing with a system described by a single restoring force that is linear in displacement, and therefore there is no reason to expect a linear relationship between drive and response. Figure 3 also shows that the internal friction increases with drive amplitude in manner that tends to level out at high amplitude. By way of comparison, internal friction is independent of amplitude in a system with regular Lorentzian response. The behavior of Fig. 3 has been verified on several VGCF reeds, and occurs on overtones as well as the fundamental.
4. DISCUSSION An important question to answer is what kind of physical model corresponds to the behavior described by eqn 3. The answer may lead to a better understanding of VGCF microstructure and dynamic behavior. We have seen that, despite a resemblance in the response, a standard Lorentzian form. modified to account for a nonlinear restoring force, does not explain the data. How then, is the form which does fit the data different? In a system with a slightly nonlinear restoring force, the modulus depends on the instantaneous displacement. This makes the frequency response dependent on amplitude. However, since the restoring force varies during the course of the oscillation, the system does not have a well-defined center frequency of the resonance, except in the zero amplitude limit. By contrast, the modified Lorentzian does have a well-defined center frequency at all times. This center frequency, w,+sR, varies with the rms response amplitude, but not with the instantaneous displacement of the reed. This leads to the question of what mechanism could cause the value of the resonant frequency to vary with rms response amplitude. Referring to eqn 1, this behavior might occur if any of the parameters on the right hand side of this equation were to vary
al
with rms amplitude. At the small amplitudes of a vibrating reed, it seems reasonable to immediately eliminate G and p from suspicion. But we may question whether the unique morphology VGCF and properties of graphite could result in unanticipated behavior in the effective value of Y, K, or r*. For example, in the derivation of eqn 1, it is assumed that all deflection in the vibrating reed is due to bending of the reed[ 111. However, if a material’s shear modulus, G, is small enough compared to Y, some of the reed’s deflection will be due to shear distortion[ 161. It is known that dislocation motion on the basal planes of graphite can reduce G between planes[ 171. If an increasing amplitude of vibration were to activate similar motion of dislocations, or perhaps relaxations associated with other defects such as impurities between the cylindrical shells of VGCF[lS], this could manifest itself in an effective reduction of Y. Note that the concept of activation with increasing rms amplitude is important here, as defects that are equally mobile at all rms amplitudes would simply contribute to a constant reduction of Y. A pedagogical analogy here might be made to a number of large ball bearings resting in the dimples of a waffle iron. The bearings only become free to move when the waffle iron is agitated above some given rms level. The bearings remain free until agitation drops below this level, at which point they settle back into the dimples. A distribution of dimple depths would lead to activation which is gradual with increasing agitation amplitude. In this scenario the activated, mobile entities could also act as a source of dissipation, accounting for the increase in internal friction with vibration amplitude. The fact that the internal friction appears to level off at high amplitude is consistent with the intuitive picture that the overall mobility of any population of defects must eventually saturate. Another possibility might lie in the existence of an instability in the concentric-cylinder microstructure that leads to a change in the cross-sectional shape, and thus K, during vibration. If one tries to bend a tin can or a rolled-up piece of paper, one finds that the structure rapidly fails in a manner that collapses the diameter along the direction of bending. Such failure does not occur in a solid rod. Tendency of the VGCF cylinders toward this type of distortion with increasing vibration amplitude could reduce the effective value of K and thus, the resonant frequency. Here, the growing internal friction would result from energy being transferred, via this distortion, out of the normal vibrational mode and into other lattice degrees of freedom (phonons). One fault with this view is that the distortion of the cross-sectional shape should depend on the instantaneous, and not rms, displacement of the reed. Therefore the effects of the distortion should probably behave more like a standard nonlinear restoring force. This flaw could perhaps be repaired by imagining that some visco-elastic phenomenon slows the fiber’s shape change to a time-
Mechanical
properties
of vapor-grown
scale longer than a cycle of vibration (typically on the order of a millisecond). Then the average value of IC would tend toward the value it has at the rms level of displacement, rather than its equilibrium value at zero displacement. Either of the above explanations could probably be equally well cast, in mathematical terms, as an amplitude dependence in the value of a, because in either case the nature of the vibrational mode would be slightly altered. A nice feature of these explanations is that neither is inconsistent with a peak response amplitude which depends approximately linearly on the drive amplitude. Most explanations based on restoring forces which are non-linear in instantaneous displacement would fail here.
5. CONCLUSION We have investigated the elastic properties of VGCF, using the low-strain vibrating-reed technique as an alternative to static pulling methods. The average fiber modulus is 680GPa, and there are indications that the intrinsic modulus of CVD carbon may approach 1000 GPa. Resonances of VGCF reeds display a unique distortion of their frequency response curves with increasing amplitude. This distortion is accompanied by an increase in the fibers internal friction. This behavior may be caused by activation of defect motion between the concentric graphite cylinders that make up the fiber, or by some other feature of this morphology.
carbon
fiber
1221
REFERENCES 1. For a general review, see M. S. Dresselhaus, G. Dresselhaus, K. Surihara, I. L. Spain and J. A. Goldberg, Graphite Fibers and Fihwnents. Springer-Verlag, Berlin (1988). 2. J. Heremans, Carbon 23,477 (198.5). J. Heremans and C. P. Bee@ Jr, Phys. Rev. B 32, 1981 (1985). T. Koyama, M. Endo and Y. Hishiyama, Jpn. J. Appl. Phys. 13, 1933 (1974). N. Kandani, M. Coulon and L. Bonnetain, Extended Abstracts Carbone ‘84, p. 142. Bordeaux (1984). W. E. Yetter, C. P. Beetz, Jr and G. W. Budd, Extended Abstracts 17th Biennial Conference on Carbon, p. 291. Lexington, KY (1985). G. G. Tibbetts, M. Endo and C. P. Bee@ Jr, SAMPE J. 22-5 (1986). 8. J. R. Gaier, M. L. Lake, A. Moinuddin and M. Marabito, Carbon 30, 345 (1992). 9. T. Tiedie. R. R. Haerina and W. N. Hardv. J. Acoust. Sot. Am. 65, 1171 (19793. 10. P. M. Morse, Vibration and Sound, p. 151. AIP, New York (1976). 11. X.-D. Xiang, J. W. Brill and W. L. Fuqua, Rev. Sci. Instrum. 60, 3035 (1989). Classical Dynamics of Particles and 12. J. B. Marion, Systems, 2nd edn, p. 117. Academic, New York (1970). and M. M. Slabe, Synch. 13. J. R. Gaier, P. D. Hambourger Met. 31, 229 (1989). A. C. Ehrlich and D. J. 14. T. M. Tritt, R. L. Jacobsen, Gillespie, Carbon (submitted). 15. P. M. Morse and K. U. Ingard, Theoretical Acoustics, p. 851. McGraw-Hill, New York (1968). Phil. Mag. 41, 744 (1921). 16. S. Timoshenko, Industrial Carbon and Graphite. 17. J. H. W. Simmons, p. 511. Sot. of Chem. Industry, London (1957). 18. M. Dresselhaus (private communication).