ふく射伝熱モデルに基づくCNT導線の破断条件推定指標の提案
Proposal of Estimation Index for CNT Conductor Fracture Condition Based on Reflective Heat Transfer Model

上柳太一, 遠藤央, 中村裕司, 佐藤廉, 源元颯人, Qu Yijun, 森伸介, 田中真平 / Taichi Ueyanagi, Mitsuru Endo, Hiroshi Nakamura, Ren Sato, Hayato Minamoto, Qu Yijun, Shinsuke Mori, Shimpei Tanaka

令和5年電気学会全国大会 IEEJ2023 Program

2-605

Nagoya University, March 15 - 17, 2023.

• IEEJ Electronic Library

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背景 - Background

カーボンナノチューブ(CNT: Carbon Nanotube) は優れた特性から，銅線に代わる導線の材料として注目されている．単体のCNTはナノマテリアルであるため，複数のCNTを束ねて撚ったCNT糸として導線に用いる．関口らはCNT 糸が負の抵抗温度係数を持つことを示した[1]. これらの温度特性は温度が明らかでないと利用できない．CNT糸は直径が数十μm であり，温度の計測は困難である．そこで，本稿では温度を直接計測することなく，温度特性について議論できる指標を提案する．また，その指標を用いてCNT糸が破断する条件を明らかにする．

Carbon Nanotubes (CNT: Carbon Nanotube) are gaining attention as potential replacements for copper wires due to their superior properties. Individual CNTs are nano-sized materials, so multiple CNTs are bundled and twisted to form CNT threads for use as conductors. Sekiguchi and others have shown that CNT threads have a negative temperature coefficient of resistance[1]. These temperature characteristics cannot be utilized if the temperature isn’t clear. CNT threads have a diameter of several tens of micrometers, making temperature measurement challenging. Therefore, this paper proposes an indicator that can discuss temperature characteristics without directly measuring the temperature. Furthermore, we clarify the conditions under which CNT threads break using this indicator.

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2　手法 / Methods

準備実験から電流密度によって時間経過に伴う抵抗値の変化は大きく異なることを確認した．導線として利用する場合，CNT糸の破断は避ける必要がある．通電時のCNT糸の破断に至る過程において温度を用いずに議論するために，通電時CNT糸の発熱とふく射伝熱をモデル化する．通電時のふく射伝熱の式，CNT糸の発熱の式より，単位時間単位表面積あたりのふく射熱を評価値$K$として用いることで，破断時の電流密度($J_{break}$)は以下のように表される．

From preliminary experiments, we confirmed that the change in resistance value over time greatly varies depending on the current density. When used as a conductor, it’s essential to avoid breaking the CNT thread. To discuss the process leading to breaking the CNT thread during electrification without using temperature, we model the heat generation and radiation heat transfer of the CNT thread when electrified. Using the formula for radiation heat transfer during electrification and CNT thread heat generation, we use the radiation heat per unit time per unit surface area as the evaluation value $K$. The current density at the time of breaking ($J_{break}$) is expressed as follows.

\[J_{break} = 2\sqrt{\frac{\left( \sigma_o + \alpha^4 \sqrt{\frac{1}{K_{break}}} \right) K_{break}}{d}}\]

通電実験からモデルを検証する．通電実験では4つのサンプルを用いる．表(1)に各サンプルの仕様を示す．抵抗値は配線抵抗，接触抵抗の影響を受けない4 端子法を用いて計測した．サンプル1において破断するまで電流密度を段階的に増加させ，電圧と電流を測定した．図1(a)から評価値$K$が3×105 W/m2 程度でCNT糸が破断したことが確認できる．CNT 糸では耐熱温度が同程度であると仮定すると，他のサンプルも同じ$K$の値で破断すると考えられる．

We validate the model through electrification experiments. In the electrification experiment, we use four samples. Table (1) shows the specifications of each sample. The resistance value was measured using the 4-terminal method, which is not affected by wiring resistance and contact resistance. For Sample 1, the current density was gradually increased until it broke, and voltage and current were measured. From Figure 1(a), it can be confirmed that the CNT thread broke at an evaluation value $K$ of about 3×10^5 W/m^2. Assuming that the heat resistance temperature is similar in CNT threads, it is believed that the other samples would also break at the same value of $K$.

Table 1 Specification of samples

No.	Diameter of yarn [μm]	Length of yarn [cm]
1	69.7	2.7
2	56.1	2.7
3	40.0	2.7
4	102.4	2.5

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結果 - Result

計測した破断する評価値$K$の値($K_{break}$)，式(1)から他のサンプルの破断する電流密度を推定する．サンプル2～4に対して破断するまで電流密度を1 A/mm2から段階的に上げて通電した．図1(b)～(d)に通電実験の結果を示す．横軸に電流密度$J$，縦軸に$K$を示している．破断しない測定点は黒色，破断した測定点は赤色で示している．式(1)における$\sigma_0$，$\alpha$は各サンプルに異なる電流密度で通電し，最小二乗法で求めた．図1(b)～(d)より多少の誤差はあるが概ね推定できていることが確認できる．図1から定数$\sigma_0$, $\alpha$を求める際の測定点数の増加による，破断する電流密度の推定値の変化が確認できる．サンプル2, 3において，初めの4点から推定した破断する電流密度$J_{break}$（青色の点線）を超えていた点が，全ての測定点から推定した$J_{break}$（緑色の一点鎖線）以下となっている．そして$J_{break}$以下の値においては破断せずに計測を終了した．したがって，測定点数の増加によって推定精度が向上したと考えられる．これらのことより，本稿で提案する評価指標Kは温度の値を用いずに，破断する電流密度をよく推定できていると言える．

The value of $K$ at which the rupture occurs, referred to as $K_{break}$, was measured, and the breaking current density for the other samples was estimated from Equation (1). For Samples 2 to 4, the current density was increased stepwise from 1 A/mm^2 and electrified until they broke. Figure 1(b) to (d) show the results of the electrification experiments. The horizontal axis represents the current density J, and the vertical axis represents $K$. Measurement points that did not break are shown in black, and those that did break are shown in red. In Equation (1), $\sigma_0$ and $\alpha$ were determined by electrifying each sample at different current densities and using the least squares method.

From Figures 1(b) to (d), although there are some deviations, it can be generally observed that our estimations align with the actual measurements. From Figure 1, we can also observe changes in the estimated breaking current density values with increasing measurement points when determining the constants $\sigma_0$ and $\alpha$. For Samples 2 and 3, points that initially exceeded the estimated $J_{break}$ current density $J_{break}$ (shown by the blue dotted line) based on the first four points, now lie below $J_{break}$ (shown by the green dashed line) estimated from all measurement points. And they ended their measurements without breaking at values below $J_{break}$. This suggests that the accuracy of estimation improves with an increase in the number of measurement points.

From these findings, it can be concluded that the evaluation indicator K proposed in this paper can accurately estimate the breaking current density without the need for temperature values.

Fig.1 通電実験の結果と推定した電流密度 / Results of the conduction experiment and estimated current density

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結論 - Conclusion

本研究ではCNT糸の導線としての実用化を目指し，本稿では温度の値を用いない温度特性についての議論に取り組んだ．温度の評価指標$𝐾$を提案し，その指標を用いてCNT糸が破断する電流密度を推定した．推定値を実験で確認し，指標の有効性を示した．この指標を用いることでモータ設計における電流密度の上限を定めることができ，現実的なモータ設計が可能になる．また，この指標は被膜の耐熱温度などの温度に上限がある状況にも応用可能である．

In this research, we aimed for the practical application of CNT threads as conductors. In this paper, we tackled the discussion on temperature characteristics without relying on actual temperature values. We introduced an evaluation index $K$ for temperature and used this index to estimate the current density at which CNT threads would break. The validity of this estimate was confirmed experimentally, demonstrating the effectiveness of our index. Using this index enables us to set an upper limit on the current density in motor design, paving the way for realistic motor designs. Moreover, this index can be applied in situations with temperature constraints, such as when considering the heat resistance temperature of coatings.

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参考文献 - Reference

[1] 関口貴子・スンダラムラジャシュリ：「軽量配線材としてのカーボンナノチューブ-銅複合材料の開発」, 応用物理, vol.90, no. 1, pp. 40-44(2021), in IEEE Trans Ind Electron, vol. 64, no. 10, pp. 7972- 7983 (2017)

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