4.1 Temperature Dependence of the Electrical Resistance of the Mixture Sample before the Sudden Resistance Change
During heating, tellurium–copper screws pressurized the carbon fibers in the mixture; therefore, in all experiments, the resistance of the mixture was measured without any loss of contact between the carbon fibers and tellurium–copper screw electrodes until the end of the measurements.
The electrical conduction in random mixtures of conductors and insulators is well explained by percolation theory [13, 16, 17]. In a mixture of a conductive filler and insulator, as the filler concentration increases, the filler particles begin to form clusters and contact each other. Above a certain filler concentration (percolation threshold), the growing clusters reach a size that allows contact between them, and a continuous network structure of the conductive filler is formed. The formation of this network was detected as a rapid increase in the electrical conductivity. At volume fractions above this percolation threshold, the electrical conductivity of the mixture increased dramatically and then plateaued. This plateau can be explained by the saturation of the conductive network within the mixture [18].
The electrical conductivity of the n-alkanes was approximately 1 × 10− 14 S cm− 1 and that of the carbon fiber used in this study was 6.667 × 103 S cm− 1. The electrical resistance of the carbon fiber/n-alkane mixture sample before heating was 0.805–21.365 Ω, including the contact resistance, and the diameter and length of the mixture sample were 0.09 cm and 2 cm, respectively. Thus, the electrical conductivity of the mixture sample before heating was 14.71 to 390.5 S cm− 1. However, the actual electrical conductivities were greater than these values when contact resistance was considered. The volume fraction of the carbon fiber filler (0.460–0.637, see Sect. 4.3.2) in these samples was well above the percolation threshold, and the electrical conductivity of the mixture was presumed to reach the plateau state.
As the coefficient of thermal expansion of PTFE is ~ 1.51×10− 4 K− 1 [19], for example, when the temperature of the sample increases by 205 K (see Fig. 3h), its volume expansion is ~ 9%. The change in the volume fraction of the carbon fiber filler owing to the thermal expansion of PTFE was 0.041–0.057; therefore, there was no significant change in the volume fraction of the filler during heating. Hence, during heating, the electrical conductivity of the mixture was maintained at a plateau state, and the resistance of the mixture changed little with increasing temperature.
Therefore, the temperature coefficient of the electrical resistance of the mixture can be considered to depend on the filler (carbon fiber) and should have a negative value [20]. However, Figs. 2 and 3 show that the resistance of both samples monotonically increases from the time heating begins. The increase in the resistance of the mixture with increasing temperature can be explained as follows: the coefficient of thermal expansion of PTFE is on the order of 10− 4 K− 1 [Ref. 19] and that of graphite is on the order of 10− 6 K− 1 [21]; thus, the PTFE tube expands more than the carbon fiber with increasing temperature. Therefore, it can be assumed that as the temperature increases, the contact pressure between the mixture and the telluric–copper electrode decreases, thereby increasing the contact resistance of the sample and the measured resistance value.
Although the resistance of the sample mixture itself decreased with increasing temperature, the measured values increased with increasing temperature because the contact resistance increased owing to the thermal expansion of PTFE. Therefore, the increase in the measured resistance was mostly due to the contact resistance.
4.3 Temperature at which the Resistance of the Mixtures Increases Suddenly
It cannot be concluded that the sudden increase in the electrical resistance of the pitch-based carbon fiber/n-alkane mixture observed in this study was due to the transition from a superconducting state to a normal conducting state. Instead, it should be considered that the temperature of the alkane in the mixture reached its boiling point; that is, the sudden increase in resistance was caused by the sudden increase in the contact resistance between the carbon fiber/n-alkane mixture and the tellurium–copper electrode, which, in turn, was due to the rapid expansion of the alkane, which occurred when the temperature of the alkane in the mixture reached its boiling point. It is possible that a sharp increase in resistance may have occurred because the carbon fiber network broke as the alkane boiled. Thus, whether the alkanes in the mixture reached their boiling point when the resistance rose sharply should be investigated for all samples based on the results from temperature measurements using the thermocouple and by measuring the amount of Joule heat generated in the mixture by the current used to measure the resistance.
For the three carbon fiber/n-octane, two carbon fiber/n-nonane, two carbon fiber/n-decane, two carbon fiber/n-dodecane, carbon fiber/n-tridecane, and three carbon fiber/n-hexadecane samples, the values of Tincrease can be obtained from Figs. 2 and 3. Table 3 lists these values, together with the boiling point (Tboil) of the n-alkanes injected into each sample and the differences between these two values, i.e., Tboil − Tincrease.
The most reasonable explanation for the abrupt increase in resistance observed for all the carbon fiber/n-alkane mixtures is the sudden expansion of the alkane in the mixture when the temperature of the mixture reached the boiling point of the alkane. In other words, the increase in resistance can be explained by the rapid deterioration of the contact between the mixture and tellurium–copper electrode owing to the vaporization of the alkane. However, in all the experiments performed in this study, the temperature at which the sudden increase in resistance occurred was lower than the boiling point of the alkane injected into the sample (Table 3).
Additionally, there was a 20–40 K difference in Tincrease for samples containing the same alkane, even though the pressures applied by the screws were the same. If Tincrease is equal to the boiling point of the alkane, it should be the same for all the samples injected with the same alkane. Hence, the observed differences in Tincrease for the samples injected with the same n-alkanes suggest that the sudden increase in resistance was not caused by the boiling of n-alkanes in the mixtures.
Table 3
Temperature at which the electrical resistance of the carbon fiber/n-alkane mixture increases suddenly (Tincrease), boiling point of the n-alkane (Tboil), and Tboil − Tincrease for each sample.
Sample name | Tincrease (K) | Tboil (K) | Tboil –Tincrease (K) |
n-octane | 2a | 367.77 | 398.76 | 30.99 |
2b | 379.41 | 398.76 | 19.35 |
2c | 386.38 | 398.76 | 12.38 |
n-nonane | 2d | 379.34 | 423.93 | 44.59 |
2e | 406.73 | 423.93 | 17.20 |
n-decane | 3a | 387.22 | 447.26 | 60.04 |
3b | 424.95 | 447.26 | 22.31 |
n-dodecane | 3c | 402.90 | 489.43 | 86.53 |
3d | 442.94 | 489.43 | 46.49 |
n-tridecane | 3e | 453.97 | 508.54 | 54.57 |
n-hexadecane | 3f | 470.81 | 560.05 | 89.24 |
3g | 498.98 | 560.05 | 61.07 |
3h | 504.24 | 560.05 | 55.81 |
Table 3 shows that according to the temperature measured by the thermocouple in contact with the PTFE tube, Tincrease did not reach Tboil for any of the samples. As noted above, the samples were heated by immersing them in n-heptadecane (boiling point: 574 K) in a glass container heated using a hot plate. Because n-heptadecane in the container was heated while stirring with a brass blade, it can be considered that the temperature of n-heptadecane was uniform. The mixture in the PTFE tube was heated by n-heptadecane outside the tube, indicating that the actual temperature of the mixture inside the tube was lower than the temperature indicated by the thermocouple in contact with the tube. Thus, if it is assumed that the temperature of the alkane in the mixture has reached its boiling point, the possibility that the temperature of the mixture had increased owing to Joule heating caused by the current used to make the resistance measurements (AC, 10 mA) should be considered. Because the measured resistance data were stored in the computer memory every second, it was possible to determine the total amount of Joule heating generated in the mixture from the time when the heating of the sample began until a sudden increase in the electrical resistance occurred.
Thus, to examine the effect of heating produced by the measured current, we calculated the upper limit of the temperature increase in the carbon fiber/n-alkane samples using the flowchart shown in Fig. 5a. First, as shown on the right-hand side of the chart, the total Joule heat (Qtotal) generated in the sample was obtained by determining the integrated value of the resistance from the start of the sample heating until a sudden increase in resistance (using the values stored in the computer), which was then multiplied by the square of the measured current. Additionally, as shown on the left-hand side of Fig. 5a, the temperature distributions of the carbon fiber/alkane mixture and tellurium–copper screws were calculated (see the supplementary information for details). The sample mixture and tellurium–copper screws were contained in the PTFE tube, which was immersed in n-heptadecane, and the thermal conductivities of the mixture and tellurium–copper screws were approximately three orders of magnitude greater than those of the PTFE tube and n-heptadecane. Thus, it was expected that the temperature distributions of the mixture and tellurium–copper screws would be uniform. This was, in fact, generally true for the derived temperature distributions. The heat capacity of the mixture (Cfib + alk) was calculated, as was the total heat capacity of the mixture and the tellurium–copper screws (Cfib + alk + tel, see the supplementary information for details). Based on the assumption that the mixture and tellurium–copper screws were completely thermally insulated, the temperature increase (ΔT = Qtotal/Cfib + alk + tel) produced by the measured current from the start of heating until a sharp increase in resistance occurred was determined. Tboil – Tincrease was then compared with ΔT. If Tboil – Tincrease > ΔT, the temperature of the alkane in the mixture does not reach the boiling point of the alkane. The results of these calculations are listed in Table 4.
4.3.1 Temperature Distribution in the Pitch-Based Carbon Fiber/N-Alkane Samples
Because the thermal conductivities of the pitch-based carbon fibers and tellurium–copper screws were several orders of magnitude higher than those of the PTFE tubes and n-heptadecane, it can be considered that the mixtures consisting of pitch-based carbon fibers and alkanes, as well as the two tellurium–copper screws, were almost completely thermally insulated during heating (see the supplementary information for details). Thus, to examine whether the temperatures throughout the carbon fiber/alkane mixtures and two tellurium–copper screws were uniform during the heating of the sample and whether it was reasonable to treat them as a single body, their temperature distributions during the heating of the samples were calculated. Details of the calculation process and results are provided in the supplementary information. The thermal conductivity of the pitch-based carbon fiber/n-alkane (see Sect. 1.3.1 in the supplementary information) and tellurium–copper screws (table S1 in the supplementary information) was approximately three orders of magnitude higher than that of PTFE (Sect. 1.1 in the supplementary information). Thus, within the pitch-based carbon fiber/alkane mixtures and tellurium–copper screws, the heat flow in the radial direction of the PTFE tube was negligible compared with that in the axial direction. Hence, the analysis of heat conduction in the carbon fiber/alkane mixtures and tellurium–copper screws was treated as a one-dimensional problem.
As mentioned earlier, the electrical resistance of the sample was mainly due to the contact resistance between the carbon fiber and the tellurium–copper screws. Therefore, most of the heat generation should occur at the boundaries between the carbon fiber and tellurium–copper screws. The model based on the calculation of the temperature distributions is illustrated in Fig. 5b (see the supplementary information for further details). Figure 5b shows the temperature distribution immediately after the heat is first generated at the boundaries and how the heat generated at the boundary propagates into the carbon fiber/alkane mixture and tellurium–copper screws. In the situation illustrated in Fig. 5b, it can be assumed that the generated temperature distribution is symmetrical around point c (the center of the carbon fiber/alkane mixture). The one-dimensional non-steady-state heat conduction equation (Equation (S5) in the supplementary information) was solved for the mixture (the region between points b and c in Fig. 5b) under boundary conditions S3 and S4 and initial condition S6 given in the supplementary information. An analytical solution (S8) was obtained for the temperature distribution T(x, t) (0 ≤ x ≤ L1, t > 0) covering the region between points b and c (see supplementary information). We also derived the temperature difference, T(x, t) − T(L1, t), between any location in this region and the point with the lowest temperature (point c, x = L1; see equation (S10) in the supplementary information).
The temperature distribution in the tellurium–copper screws was obtained by solving the one-dimensional non-steady-state heat conduction equation for the region between points d and e in Fig. 5b under boundary conditions S11 and S12 and initial condition S14 given in the supplementary information. For the temperature distribution T(x, t) (0 ≤ x ≤ L2, t > 0) covering the region between points d and e, an analytical solution was obtained (see equation (S16) in the supplementary information). The change in the temperature difference T(x, t) − T(L2, t) between any location in this region and the point with the lowest temperature (point e, x = L2) over time was obtained from an analytical solution of T(x, t) (see equation (S17) in the supplementary information).
According to the graphs showing the relationship between the temperature and AC resistance shown in Figs. 2 and 3, the maximum resistance measured for any of the alkanes from the start of heating to the point at which the resistance suddenly increased, Rmax, was 81.62 Ω for hexadecane (Fig. 3f). Because the current used to make the resistance measurements (i) was 10 mA, the largest calorific value generated in any of the samples per unit time (\({Q}_{max}={R}_{max}{i}^{2}\)) was 8.162 × 10−3 W. Accordingly, the temperature distributions of the carbon fiber/alkane mixture and tellurium–copper screws were calculated for this case.
Because the amount of heat generated at the two surfaces b and d per unit time is equal owing to symmetry and as the inner diameter (d1) of the PTFE tube is 0.9 mm, the amount of heat flowing from the heated surface into the mixture per unit time is given as follows:
$$\frac{q}{2}=\frac{{Q}_{max}}{4}÷\left(\frac{\pi {{d}_{1}}^{2}}{4}\right)=3.207 \times {10}^{-3} \left[W{mm}^{-2}\right]$$
1
.
In the case of n-hexadecane, for which the maximum amount of heat was generated (Fig. 3f), the temperature distribution of the carbon fiber/n-hexadecane mixture was calculated based on its physical properties at room temperature.
As described in Sect. 1.3.1 of the supplementary information, using the relationship between the volume fraction of the fibers and thermal conductivity of the mixture proposed by Demain and Issi [25] for pitch-based carbon fiber/polycarbonate mixtures, the thermal conductivity of the pitch-based carbon fiber/n-hexadecane mixture k1 was estimated as 86.5 W m− 1 K− 1 = 0.0865 W mm− 1 K− 1. As explained in Sect. 1.3.2 of the supplementary information, the product of the density and specific heat of the carbon fiber/n-hexadecane mixture, ρ1cp1, was set to 1.615 × 10− 3 J mm− 3 K− 1 for calculation. By substituting q/2 = 3.207 × 10− 3 W mm− 2, k1 = 0.0865 W mm− 1 K− 1, α1 = k1/ρ1cp1 = 53.6 mm2 s− 1, and L1 = 10 mm into the analytical solution T(x, t) − T(L1, t) (Eq (S10) in the supplementary information), the change in the temperature difference between any location in this region and the point with the lowest temperature (point c, x = L1) with time (i.e., T(x, t) − T(L1, t)) was calculated (see the supplementary information for the details of the calculation). The results are shown in Fig. 5c. The steady state reached 0.8 s after the start of heating. The points in the carbon fiber/n-hexadecane mixture with the highest temperatures corresponded to heated surfaces b and d, and the lowest temperature occurred at the center of the mixture at point c. Because the difference between the maximum and minimum temperatures during the steady state was 0.185 K, the temperature difference between any two points in the carbon fiber/n-hexadecane mixture after the start of heating did not exceed 0.185 K.
Similarly, the temperature distributions were determined for the other mixtures, and the difference between the maximum and minimum temperatures was less than 0.185 K in each case. Thus, it can be assumed that the temperature distribution of the carbon fiber/n-alkane mixture in the PTFE tube was almost uniform during the heating of the samples.
The temperature distribution in the tellurium–copper screws was then calculated for n-hexadecane, i.e., the alkane for which the maximum calorific value was generated (Fig. 3f). The change in the temperature difference T(x, t) − T(L2, t) between any location in this region and the point with the lowest temperature (point e, x = L2) with time was obtained using the analytical solution (S17) provided in the supplementary information. For the carbon fiber/n-hexadecane mixture, q/2 was set to 3.207 × 10− 3 W mm− 2. The density, specific heat, thermal conductivity, and thermal diffusivity of tellurium–copper used in the calculations were room temperature values, as listed in Table S1 of the supplementary information. The change in the temperature difference T(x, t) − T(L2, t) between any location in this region and the point with the lowest temperature (point e, x = L2) with time was derived by substituting q/2 = 3.207 × 10− 3 W mm− 2, k2 = 0.355 W mm− 1 K− 1, L2 = 6 mm, and α2 = 103.14 mm2 s− 1 into the analytical solution (S17). The change in temperature distribution in the tellurium–copper screws over time was also calculated. The results are shown in Fig. 5d. As shown in Fig. 5d, the steady state is reached 0.15 s after the heating begins; the highest temperature in the tellurium–copper screws is found at the heat-generating surface at d, and the lowest temperature occurs at the interface between the n-heptadecane liquid and tellurium–copper at e. The difference between the maximum and minimum temperatures for the tellurium–copper screws is 0.027 K. Thus, the temperature difference between any two positions in the tellurium–copper screws does not exceed 0.027 K. Accordingly, the temperature of the tellurium–copper screws can be regarded as approximately equal to the temperature of the interface between the carbon fiber/n-hexadecane mixture and the tellurium–copper screws. Based on the above results, it is reasonable to consider the carbon fiber/n-alkane mixture and two tellurium–copper screws as one body, and it can be assumed that the temperature of the mixture and the tellurium–copper screws increases uniformly.
4.3.2 Temperature Increase of the Carbon Fiber/N-Alkane Mixtures
The results of the calculations presented in the previous section show that the temperature throughout the carbon fiber/alkane mixture and the two tellurium–copper screws increased uniformly as the temperature increased, which implies that, with respect to the heating process, they can be treated as one body. In addition, the thermal conductivities of PTFE and heptadecane are approximately three orders of magnitude lower than those of carbon fiber/n-alkane mixtures and tellurium–copper. Thus, in each case, the temperature increase in the carbon fiber/n-alkane mixture was obtained by assuming that the mixture and tellurium–copper screws were completely thermally isolated during the experiment. Therefore, it was assumed that all of the Joule heat generated in the sample from the start of heating to the time when the sudden resistance increase occurred was used to increase the temperature of the carbon fiber/n-alkane mixture and tellurium–copper screws. Based on this assumption, the temperature increase in the carbon-fibre/n-alkane mixture was calculated.
First, to determine the heat capacity of each carbon fiber/n-alkane sample, the temperature at which the measured resistance was equal to the average resistance, Tm, was calculated from the relationship between the temperature and resistance obtained for each experiment (Figs. 2 and 3).
The density of each alkane and carbon fiber at Tm (ρalk and ρfib, respectively), as well as the specific heat of constant pressure of the alkane and carbon fiber at Tm (cp,alk and cp,fib, respectively), were determined from the known relationships between the density and temperature and between the specific heat of constant pressure and temperature for each alkane and carbon fiber [26–29] (see supplementary information for details on how to obtain these values). Table S2 lists the values of ρalk, cp,alk, ρfib, and cp,fib at Tm. Thus, for the volume fractions of the carbon fiber in the mixture (Vfib) of 0.460 (meaning that the weight of the fiber packed in the PTFE tube = 0.013 g) and 0.637 (weight of the fiber = 0.018 g), the density (\({\rho }_{fib + alk}^{\left(k\right)}\)) and specific heat (\({c}_{fib + alk}^{\left(k\right)}\)) at Tm of the carbon fiber/n-alkane mixture were determined as
$${\rho }_{fib + alk}^{\left(k\right)}={V}_{fib}{\rho }_{fib}+(1-{V}_{fib}){\rho }_{alk}$$
2
,
and
$${c}_{p,fib + alk}^{\left(k\right)}=\left[{V}_{fib}{\rho }_{fib}{c}_{p,fib}+\left(1-{V}_{fib}\right){\rho }_{alk}{c}_{p, alk}\right]/{\rho }_{fib + alk}^{\left(k\right)}$$
3
.
respectively. Here, k = 1 and 2 correspond to cases where Vfib = 0.460 and 0.637, respectively. The values of \({\rho }_{fib+alk}^{\left(k\right)}\) (k = 1, 2) and \({c}_{fib+alk}^{\left(k\right)}\) (k = 1, 2) for each carbon fiber/n-alkane mixture are listed in Table S3 of the supplementary information. In each case, the table also gives the value of the product of the density and specific heat of the mixture (\({\rho }_{fib+alk}^{\left(k\right)} {c}_{fib+alk}^{\left(k\right)}\)) as well as the product of the density and specific heat of the tellurium–copper (ρtel ctel), based on the room-temperature values of ρtel and ctel (Table S1 in the supplementary information).
The heat capacity when the carbon fiber/n-alkane mixture and two tellurium–copper screws were considered as one body, \({C}_{fib + alk +tel}^{\left(k\right)}\) (k = 1, 2), was calculated as follows (also provided in the supplementary information):
\({C}_{fib + alk + tel}^{\left(k\right)}={\rho }_{fib + alk}^{ \left(k\right)} {c}_{fib + alk}^{\left(k\right)}{V}_{fib + alk}+{\rho }_{tel}{c}_{tel}{V}_{tel}\) (k = 1, 2). (4)
The values of \({C}_{fib + alk + tel}^{\left(k\right)}\) (k = 1, 2) for each experiment (see Figs. 2 and 3) are listed in Table S4 in the supplementary information.
Q total denotes the total heat generated inside the PTFE tube from the start of heating to the point where a sudden increase in resistance occurs and is obtained by multiplying the time integral of the resistance over that period by the square of the current (10 mA). Table 4 presents the calculated Qtotal values for each experiment. Dividing Qtotal by the heat capacity of the mixture and two tellurium–copper screws, \({C}_{fib + alk + tel}^{\left(k\right)}\) (k = 1, 2), gives the overall temperature increase that occurred, i.e., \({\varDelta T}_{fib + alk + tel}^{\left(k\right)}\) (k = 1, 2).
The values of \({C}_{fib + alk + tel}^{\left(k\right)}\) (k = 1, 2), Qtotal, \({\varDelta T}_{fib + alk + tel}^{\left(k\right)}\) (k = 1, 2), Tboil, Tincrease, and Tboil – Tincrease for each experiment are shown in Table 4.
Table 4
Overall heat capacity of the carbon fiber/alkane mixture and tellurium–copper screws, \({{C}}_{{f}{i}{b} + {a}{l}{k} + {t}{e}{l}}^{\left({k}\right)}\) (k = 1, 2), total Joule heat generated in the sample (Qtotal), temperature increase of the mixture and the tellurium–copper screws from the start of the heating to the sudden increase in resistance (\({\varDelta {T}}_{{f}{i}{b} + {a}{l}{k} + {t}{e}{l}}^{\left({k}\right)}\) (k = 1, 2)), boiling point of the alkane (Tboil), temperature when the resistance suddenly increased (Tincrease), and Tboil – Tincrease for each experiment.
Sample name | \({T}_{m}\)(K) | \({C}_{fib + alk + tel}^{\left(1\right)}\) (J K− 1) | \({C}_{fib + alk + tel}^{\left(2\right)}\) (J K− 1) | \({Q}_{total}\) (J) | \({\varDelta T}_{fib + alk + tel}^{\left(1\right)}\) (K) | \({\varDelta T}_{fib + alk + tel}^{\left(2\right)}\) (K) | \({T}_{boil}\) (K) | \({T}_{increase}\) (K) | \({T}_{boil}-{T}_{increase}\) (K) |
n-octane |
2a | 323.09 | 4.712 × 10− 2 | 4.751 × 10− 2 | 0.3552 | 7.54 | 7.48 | 398.76 | 367.77 | 30.99 |
2b | 342.20 | 4.794 × 10− 2 | 4.857 × 10− 2 | 0.2750 | 5.74 | 5.66 | 398.76 | 379.41 | 19.35 |
2c | 335.29 | 4.765 × 10− 2 | 4.819 × 10− 2 | 0.0655 | 1.38 | 1.36 | 398.76 | 386.38 | 12.38 |
n-nonane |
2d | 340.78 | 4.807 × 10− 2 | 4.862 × 10− 2 | 0.3651 | 7.59 | 7.51 | 423.93 | 379.34 | 44.59 |
2e | 357.71 | 4.878 × 10− 2 | 4.954 × 10− 2 | 0.4067 | 8.34 | 8.21 | 423.93 | 406.73 | 17.20 |
n-decane |
3a | 347.30 | 4.853 × 10− 2 | 4.910 × 10− 2 | 0.3062 | 6.31 | 6.24 | 447.26 | 387.22 | 60.04 |
3b | 369.85 | 4.946 × 10− 2 | 5.031 × 10− 2 | 0.1063 | 2.15 | 2.11 | 447.26 | 424.95 | 22.31 |
n-dodecane |
3c | 356.60 | 4.922 × 10− 2 | 4.981 × 10− 2 | 0.4076 | 8.28 | 8.18 | 489.43 | 402.90 | 86.53 |
3d | 371.07 | 4.983 × 10− 2 | 5.059 × 10− 2 | 0.2235 | 4.49 | 4.42 | 489.43 | 442.94 | 46.49 |
n-tridecane |
3e | 372.84 | 5.003 × 10− 2 | 5.077 × 10− 2 | 0.4770 | 9.53 | 9.40 | 508.54 | 453.97 | 54.57 |
n-hexadecane |
3f | 394.47 | 5.124 × 10− 2 | 5.212 × 10− 2 | 2.3310 | 45.49 | 44.72 | 560.05 | 470.81 | 89.24 |
3g | 423.57 | 5.237 × 10− 2 | 5.357 × 10− 2 | 1.8582 | 35.48 | 34.69 | 560.05 | 498.98 | 61.07 |
3h | 440.54 | 5.297 × 10− 2 | 5.435 × 10− 2 | 0.6328 | 11.95 | 11.64 | 560.05 | 504.24 | 55.81 |
Table 4 shows that when the volume fractions of the carbon fibers are 0.460 and 0.637, there is almost no difference (within 1–3%) in the magnitudes of \({\varDelta T}_{fib + alk + tel}^{\left(1\right)}\) and \({\varDelta T}_{fib + alk + tel}^{\left(2\right)}\). The Tboil, Tincrease, and Tboil – Tincrease values are also shown for each alkane. Comparing Tboil – Tincrease and \({\varDelta T}_{fib+alk+tel}^{\left(k\right)}\) (k = 1, 2), it can be observed that the former is considerably larger than the latter for all samples. This indicates that Tincrease was below Tboil in all the experiments, although there was some heating owing to the current used to measure the resistance. These results confirm that the observed rapid increase in resistance was not caused by the rapid increase in the contact resistance between the carbon fiber/n-alkane mixture and tellurium–copper screw occurring as the alkanes in the mixtures boiled. Moreover, the alkanes remained in the liquid state from the start of heating until a sudden increase in resistance occurred. Thus, these results also confirm that the sudden increase in resistance observed in the carbon fiber/alkane mixtures cannot be explained by the breaking of the carbon fiber network at the glass transition temperature or the melting point of the polymer, as observed for conductive carbon fiber-reinforced polymers.
4.4 Causes of Sudden Increase in Resistance of Carbon Fiber/N-Alkane Mixtures
In Sect. 4.3.2, it was shown that the sudden increase in the resistance of the carbon fiber/n-alkane mixtures was not due to the discontinuous increase in the contact resistance arising from the breaking of the contact between the mixtures and tellurium–copper screw electrodes, as with the n-alkanes in the boiled samples. Thus, in each case, it can be considered that the sudden increase in resistance was due to the rapid increase in the resistance of the carbon fiber/alkane mixture. Therefore, it can be said that Tincrease is the critical temperature at which the resistance of the mixture suddenly increases. Such a sudden increase in resistance occurs during the transformation from the superconducting state to the normal conducting state. However, because our measurements were made using the two-terminal method, it was not possible to determine whether the resistance of the mixture at temperatures below Tincrease was zero, owing to the problem of contact resistance. Thus, in this section, it is shown that n-alkanes have the effect of dramatically reducing the resistance of the mixture by comparing the amount of scatter in the change in resistance for a sudden increase in resistance with that in the measured resistance at temperatures below Tincrease.
Figure 4 shows that the magnitude of the change in the resistance at Tincrease (Rc2 – Rc1) varies by more than two orders of magnitude, even for samples with the same composition, but the variation in Rs and Rc1 measured at temperatures lower than Tincrease is small. The carbon fibers used in the samples were pitch-based carbon fibers and were extremely brittle owing to their high graphitizability [15]. Therefore, when the carbon fibers were packed into the PTFE tube using a metal wire, the fibers were broken into irregularly shaped pieces. As a result, the average aspect ratio of the fibers varied significantly between the samples.
For carbon fiber/polymer mixtures, it has been shown that the percolation threshold decreases and saturation conductivity increases as the fiber aspect ratio increases and that the saturation conductivity increases by approximately two orders of magnitude as the average aspect ratio increases by a factor of 4–5 [12, 13]. This suggests that relatively small variations in the fiber aspect ratio result in variations of approximately two orders of magnitude in the resistance of the mixture. In this study, the size of these variations was almost equal to the observed scatter in the size of the change in resistance at the time of the sudden increase in resistance. Therefore, it can be considered that even in the case of pitch-based carbon fiber/alkane mixtures, a large scatter in Rc2 − Rc1 occurs owing to the variations in the aspect ratios of the fibers. This extremely large scatter in the amount of change in resistance can be regarded as a characteristic of the brittle pitch-based carbon fiber/alkane mixtures being used. As the aspect ratios of the fibers that are the main influence on the size of the resistance of the mixtures should not change after the carbon fibers are packed into the PTFE tube, there must be a surprising reason why the resistance immediately before heating the sample (Rs) and the resistance immediately before the sudden increase in resistance (Rc1) do not exhibit such a large scatter.
As discussed in Sect. 4.1, the monotonic increase in the resistance of the sample before the sudden increase in resistance was due to the increase in contact resistance caused by the thermal expansion of the Teflon tube. As the measured resistance of the pitch-based carbon fiber/alkane mixture sample with sample name n is a function of the temperature T, it can be expressed by Rmeasure(n, T), i.e., the sum of the actual resistance of the mixture Ractual(n, T) and contact resistance Rcontact(n, T), as follows:
$${R}_{measure}\left(n,T\right)={R}_{actual}\left(n,T\right)+{R}_{contact}\left(n,T\right).$$
5
The measured resistance of the mixture with sample name n just before the rapid increase in resistance (Rc1(n)) and the measured resistance of the mixture with sample name n at the end of the sudden increase in resistance (Rc2(n)) are expressed by Equations (6) and (7), respectively, as follows:
$${R}_{c1}\left(n\right)={R}_{measure}\left(n,{T}_{increase}\right)={R}_{actual}\left(n,{T}_{increase}\right)+{R}_{contact}\left(n,{T}_{increase}\right),$$
6
$${R}_{c2}\left(n\right)={R}_{measure}\left(n,{T}_{increase}+\varDelta {T}_{increase}\right)$$
$$={R}_{actual}\left(n,{T}_{increase}+\varDelta {T}_{increase}\right)+{R}_{contact}\left(n,{T}_{increase}+\varDelta {T}_{increase}\right).$$
7
According to the experimental data (see Table 1), 0.15 K ≤ ∆Tincrease ≤ 3.01 K. Because Rc1 and Rc2 are resistance values measured at almost the same temperature, we can regard Rcontact(n, Tincrease) = Rcontact(n, Tincrease+∆Tincrease). The magnitude of the change in resistance at Tincrease (Rc2 – Rc1) is the actual change in the resistance of the mixture at Tincrease without the effect of the contact resistance. This is because the contact resistance components in Rc1 and Rc2 are identical. Thus, the contact resistance values cancel each other out, and the ultimate value is given by Eq. (8).
\({R}_{c2}\left(n\right)-{R}_{c1}\left(n\right)={R}_{actual}\left(n,{T}_{increase}+\varDelta {T}_{increase}\right)-{R}_{actual}\left(n,{T}_{increase}\right)\) (8)
Because Ractual(n, Tincrease) is the actual resistance of the mixture just before the sudden increase in resistance, the actual resistance of the mixture when the sudden increase in resistance ends (Ractual(n, Tincrease+∆Tincrease)) can be expressed by Eq. (9).
$${R}_{actual}\left(n,{T}_{increase}+\varDelta {T}_{increase}\right)={R}_{c2}\left(n\right)-{R}_{c1}\left(n\right)+{R}_{actual}\left(n,{T}_{increase}\right)$$
9
As discussed in Sect. 4.1, when the volume fraction of carbon fiber is 0.460–0.637, the resistances of all the mixture samples reach a plateau from the heating start temperature to the temperature Tincrease at which a sudden increase in resistance occurs and can be regarded as almost constant. Furthermore, the electrical resistance of the carbon fiber has a negative temperature coefficient (–0.528 × 10− 3 K− 1) [30]. For example, in Fig. 3h, where the temperature increase is the largest in the experiment, the temperature rise of the sample from the start of heating until the rapid increase in resistance occurs is 205 K. In this case, from the temperature coefficient, the resistance of the mixture can be estimated to decrease by 10.8% from the initial resistance value until the temperature reaches Tincrease. In this experiment, the maximum estimated decrease in resistance from the temperature Ts just before heating to the temperature Tincrease, where the resistance rises sharply, is approximately 10%. Thus, it was assumed that Ractual(n, Tincrease) ≈ Ractual(n, Ts).
We then obtained the variances of the actual resistance values of the mixture samples for sample names 2a–2e and 3a–3e at temperatures Ts, Tincrease, and Tincrease+∆Tincrease, denoted as var[Ractual(n, Ts)], var[Ractual(n, Tincrease)], and var[Ractual(n, Tincrease+∆Tincrease)], respectively. Furthermore, we derived the relationships between the standard deviations. First, because Ractual(n,Tincrease) ≈ Ractual(n, Ts ), var[Ractual(n, Tincrease)] ≈ var[Ractual(n, Ts)]. At Ts, the following relationship holds between the measured resistance of the mixture, actual resistance of the mixture, and contact resistance.
$${R}_{measure}\left(n,{T}_{s}\right)={R}_{actual}\left(n,{T}_{s}\right)+{R}_{contact}(n,{T}_{s})$$
10
Because Rcontact(n,Ts) > 0, the following inequality can be derived from Eq. (10):
$$var\left[{R}_{measure}\left(n,{T}_{s}\right)\right]>var\left[{R}_{actual}\left(n,{T}_{s}\right)\right]\approx var\left[{R}_{actual}\left(n,{T}_{increase}\right)\right].$$
11
The relationship between the inverse of each term in Eq. (11) is as follows:
$$1/var\left[{R}_{actual}\left(n,{T}_{s}\right)\right]\approx 1/var\left[{R}_{actual}\left(n,{T}_{increase}\right)\right]>1/var\left[{R}_{measure}\left(n,{T}_{s}\right)\right].$$
12
Because Ractual(n, Tincrease) > 0, the following inequality can be obtained from Eq. (9):
$${R}_{actual}\left(n,{T}_{increase}+\varDelta {T}_{increase}\right)>{R}_{c2}\left(n\right)-{R}_{c1}\left(n\right).$$
13
Therefore, the following inequality holds for the variances of Ractual(n, Tincrease+∆Tincrease) and Rc2(n)–Rc1(n).
$$var\left[{R}_{actual}\left(n,{T}_{increase}+\varDelta {T}_{increase}\right)\right]>var\left[{R}_{c2}\left(n\right)-{R}_{c1}\left(n\right)\right]$$
14
From inequalities (12) and (14), we obtain inequality (15):
$$\frac{var\left[{R}_{actual}\left(n,{T}_{increase}+\varDelta {T}_{increase}\right)\right]}{var\left[{R}_{actual}\left(n,{T}_{s}\right)\right]}\approx \frac{var\left[{R}_{actual}\left(n,{T}_{increase}+\varDelta {T}_{increase}\right)\right]}{var\left[{R}_{actual}\left(n,{T}_{increase}\right)\right]}>\frac{var\left[{R}_{c2}\left(n\right)-{R}_{c1}\left(n\right)\right].}{var\left[{R}_{measure}\left(n,{T}_{s}\right)\right]}$$
15
From Table 2, for samples 2a–2e and 3a–3e, the standard deviation of the size of the change in resistance when a sudden increase in resistance occurs \(\sqrt{var\left[{R}_{c2}\left(n\right)-{R}_{c1}\left(n\right)\right]}\) and the standard deviation of the measured resistance immediately before heating \(\sqrt{var\left[{R}_{measure}\left(n,{T}_{s}\right)\right]}\) are 17.73 and 0.3935, respectively. The following relationship is obtained for the ratio of the standard deviation of the actual resistance of the mixture immediately after the sudden increase in resistance ends to those of the actual resistances of the mixture immediately before heating and immediately before the sudden increase in resistance.
$$\frac{\sqrt{var\left[{R}_{actual}\left(n,{T}_{increase}+\varDelta {T}_{increase}\right)\right]}}{\sqrt{var\left[{R}_{actual}\left(n,{T}_{s}\right)\right]}}\approx \frac{\sqrt{var\left[{R}_{actual}\left(n,{T}_{increase}+\varDelta {T}_{increase}\right)\right]}}{\sqrt{var\left[{R}_{actual}\left(n,{T}_{increase}\right)\right]}}>45.06$$
16
From Eq. (16), it can be seen that the scatter in the actual resistance of the mixture sample increases sharply, bordering Tincrease. This indicates that adding n-alkanes drastically reduced the resistance of the carbon fibers in the PTFE tubes at temperatures below Tincrease and that the contact between the carbon fibers and alkanes may have reduced the resistance to zero. The resistance range of the samples, covering two orders of magnitude, is a normal characteristic of pitch-based carbon fiber/alkane mixtures packed into PTFE tubes. However, in these experiments, this characteristic was not observed at temperatures below Tincrease.
Superconductors rapidly change into normal conductors when the critical temperature is exceeded. The rapid increase in the resistance observed when the PTFE tubes were filled with a mixture of pitch-based carbon fibers and alkanes was very similar to the change in the resistance of superconductors at their critical temperatures. Given the effect of alkanes in drastically reducing the resistance of the mixtures, the sudden increase in resistance and simultaneous sudden appearance of large fluctuations in the resistance of the pitch-based carbon fiber/alkane mixtures suggest that a transition from the superconducting phase to the normal conducting phase occurred. The resistances measured at the start of heating and immediately before the sudden increase in resistance can be assumed to be the values of the contact resistance only, and the relatively small fluctuations in these values can be interpreted as fluctuations in the contact resistance.
A previous study [10] suggested that a superconducting state can be obtained by bringing graphite into contact with an alkane, based on the demonstration of a ring current flowing continuously without attenuation at room temperature in a circular PTFE tube filled with graphite flakes and alkanes. The drastic reduction in the resistance of pitch-based carbon fibers resulting from the use of alkanes, as confirmed in the present study, supports the results of this earlier study. In addition, it may be argued that the temperature at which the resistance increased rapidly in the experiments conducted in the present study is the critical temperature for superconductor-like materials obtained by placing graphite in contact with n-alkanes.
In the remainder of this paper, the observed Tincrease is referred to as the critical temperature Tc.
4.5 Relationship between Critical Temperature, Tc, and the Magnitude of the Increase in Resistance at Tc
Figures 2, 3a–d, and 3f–h show that the critical temperature increases as the size of the change in the resistance decreases at the point where the sudden increase in resistance occurs. Figure 6a illustrates the relationship between the critical temperature Tc and the magnitude of the sudden increase in the resistance at Tc for n-octane and n-hexadecane. A nearly linear relationship was observed between the critical temperature and the magnitude of the increase in resistance.
Based on previous research [13, 14], it can be assumed that the larger the average aspect ratio of the carbon fibers, that is, the longer the average length of the fibers, the smaller the sudden increase in the resistance of the mixture. Thus, the longer the fiber is, the higher the critical temperature. The critical temperature is affected not only by the physical properties of the alkane but also by the characteristics of the carbon fiber, such as the fiber aspect ratio. The sides of the fiber consist of the basal surface of graphite, and the fracture surface of the fiber consists of the edge surface. This indicates that the larger the ratio of the area of the basal surface to that of the edge surface is, the higher the critical temperature, suggesting that the graphite basal surface plays an important role in superconductivity.
Figure 6b shows the relationship between the carbon number and critical temperature Tc for n-alkanes. In this case, the critical temperature for each alkane corresponds to Tc when the magnitude of the sudden increase in resistance is 2.358 Ω, i.e., the magnitude of the change in resistance at Tincrease (453.97 K) measured for n-tridecane (see the case of n-tridecane in Fig. 3e and Table 1). Figure 6a shows that there is a nearly linear relationship between Tc and the magnitude of the change in resistance at Tc. Thus, for n-alkanes with 8–10, 12, and 16 carbon atoms, the Tc value for an increase in resistance of 2.358 Ω can be obtained by linear interpolation.
Figure 6b shows that the higher the carbon number of the n-alkanes is, the higher the critical temperature. The boiling point of n-alkanes increases at a constant rate with the number of carbon atoms. In addition, the thermal motion of n-alkane molecules becomes more intense as the temperature approaches the boiling point. It is presumed that the conductive state of the mixture of graphite and n-alkanes is destroyed by the thermal motion of the n-alkane molecules. The information shown in Fig. 6b forms the basis of the hypothesis that the use of n-alkanes with 16 or more carbon atoms may result in superconductors with a critical temperature of over 500 K.
4.6 Evidence of the Superconductivity of Carbon/N-Alkane Mixtures and Future Experiments to Verify This
To confirm the existence of materials that exhibit superconductivity at room temperature, graphite flakes with an average thickness of 0.1 mm and average diameter of 1 mm were obtained from highly oriented pyrolytic graphite, and n-octane was packed into a Teflon tube to form a ring consisting of a graphite flake/n-octane mixture [10]. In our experiment [10], the Teflon tube was initially formed into a ring shape by heating it with a hairdryer. The tubes were then filled with graphite flakes and n-octane. Next, the two ends of the Teflon tube were connected, and the resultant Teflon ring was fixed using a glass tube packed with graphite flakes and n-octane (see Fig. 1(a) in Ref. [10]). An electric current is applied to the ring via electromagnetic induction. The magnetic field produced by the ring current was maintained for 50 days without any decay; thus, the current flowed through the graphite flake/n-octane mixture without any resistance. The radius of the coil and inside diameter (diameter of the mixture wire) of the PTFE tube were 0.8 cm and 0.96 mm, respectively. It was assumed that the size of the ring current decreased by 99% over the 50-day period [10]; therefore, the calculated resistance of the wire was less than 7.36 × 10− 17 Ω. Based on the length of the coil (1.6 cm × π) and cross-sectional area of the wire ((0.048 cm)2 × π), the estimated electrical resistivity of the mixture was less than 1.060 × 10− 19 Ω cm. This value was only approximately 1/1013 of the resistivity of silver (1.62 × 10− 6 Ω cm), suggesting that the resistance of the graphite flake/n-octane mixture was essentially zero.
Esquinazi et al. noted that the concept of thermomagnetic hysteresis, that is, the difference between the zero-field cooling and field cooling curves for different constant applied magnetic fields, is very useful in determining the presence of superconductivity in materials with only a few superconducting regions, such as granular superconductors [31–33]. Esquinazi et al. [31, 32] found that high-quality graphite powders treated with water and n-heptane exhibited clear and reproducible granular superconducting behavior at critical temperatures above 300 K. Furthermore, they found that over a wide range of magnetic field strengths and temperatures, the thermomagnetic hysteresis of a Y123 (YBa2Cu3O7 − x) powder with O vacancies is similar to that of graphite powder treated with n-heptane [33]. Thus, the existence of a superconducting region for a graphite powder/n-alkane mixture was confirmed. The persistent current flow in the ring-shaped Teflon tube filled with graphite flakes and n-alkanes, as described above, suggests that the mixture in this study is not composed of isolated granular superconductors and thus constitutes a continuous superconductor with interconnected superconducting regions.
At room temperature and atmospheric pressure, we observed that a very small magnetic field (with a strength of a few millimeters) was completely shielded by the graphene on which the n-alkane was dropped [34, 35]. This observation suggests that the Meissner effect, which is definitive evidence of superconductivity, occurred in the graphite/alkane mixture. We are currently working on improving our experiments to improve reproducibility.
It is clear that this study obtained experimental results confirming the superconductivity of carbon/n-alkane mixtures, and we will continue to conduct research on the occurrence of zero resistance and sudden changes in the electrical resistance in these materials. We will also study the superconducting transition using magnetization measurements, identify superconductor structures, and aim for consistent reproducibility of the results.