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A dielectrophoretic study of the carbon nanotube chaining process and its dependence on the local electric fields

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Abstract

The chaining process of a system of interacting carbon nanotubes (CNTs) under an alternating current electric field is investigated at two regions of different electric field characteristics. For the region of uniform electric field (far from the electrodes), a two-dimensional multiparticle approach based on the dielectrophoretic (DEP) theory and classical mechanics is proposed to investigate the CNT rotational and translation motion. For this scenario, CNT rotation and alignment along the electric field direction occurs first, followed by the translation and chaining processes which were found to be highly dependent on the CNT-to-CNT initial configuration. On the other hand, the presence of high electric field gradients governs the CNT chaining at regions near the electrodes. DEP forces caused by such gradients were computed by finite element analysis and compared to the magnitude of the CNT-to-CNT interacting forces at zones of uniform electric fields. A critical distance of CNT-to-CNT separation was estimated, which determines if a CNT is attracted towards the electrode or if it is attracted by other CNTs away from the electrodes. Experimental evidence of CNTs dynamic motion under electric fields is presented to support the predicted trends.

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Acknowledgements

Technical support of Carlos Falla and José Bante (CINVESTAV) in the experimental section is strongly appreciated.

Funding

This work was supported by the “Fondo Sectorial de Investigación para la Educación” through the SEP-CONACYT grant No. 235905 (A.I. Oliva-Avilés). V.V. Zozulya acknowledges additional support from CONACYT project No. 256458.

Author information

Correspondence to A. I. Oliva-Avilés.

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The authors declare that they have no conflict of interest.

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Supplementary material 2 (MP4 1577 kb)

Supplementary material 3 (MP4 1714 kb)

Supplementary material 4 (MP4 2284 kb)

Supplementary material 5 (MP4 29525 kb)

Supplementary material 1 (MP4 372kb)

Supplementary material 2 (MP4 1577 kb)

Supplementary material 3 (MP4 1714 kb)

Supplementary material 4 (MP4 2284 kb)

Supplementary material 5 (MP4 29525 kb)

Appendix A. Expressions for the frictional and dielectrophoretic terms of the governing motion equations

Appendix A. Expressions for the frictional and dielectrophoretic terms of the governing motion equations

According to the proposed model for the dynamics of CNT rotation, a frictional (Ti,fr) and DEP terms (Ti,DEP) are required, see Eq. (2). The frictional torque for CNT-i is defined as,

$$T_{i,fr} = \, 8\pi\eta\left( {r_{e} } \right)^{3} K_{r} \frac{{d\theta_{i} }}{dt}$$
(10a)
$$r_{e} = \, \left( a \right)^{1/3} \left( b \right)^{2/3}$$
(10b)
$$K_{r} = \frac{{4p_{{}}^{2} \left( {1 - p_{{}}^{4} } \right)}}{{3\left[ {\frac{{2p_{{}}^{2/3} \left( {2 - p_{{}}^{ - 2} } \right)}}{{K_{t} }} - 2} \right]}}$$
(10c)
$$K_{t} = \frac{{\sqrt {\left( {1 - p_{{}}^{ - 2} } \right)} }}{{p_{{}}^{ - 2/3} ln\left[ {p\left( {1 + \sqrt {\left( {1 - p_{{}}^{ - 2} } \right)} } \right)} \right]}}$$
(10d)
$$p \, = \, a/b$$
(10e)

In Eq. (10a), η is the viscosity of the suspending fluid, re is the equivalent radius of a sphere of volume equal to that of the CNTs, see Eq. (10b), and Kr and Kt are, respectively, the rotational and translational friction coefficients for prolate ellipsoids [52, 53], with p as the CNT aspect ratio (length/diameter), see Eq. (10e). As can be noticed, the frictional torque depends on the instantaneous angular velocity of the CNT (i/dt). On the other hand, according to the DEP theory, the induced dipole on the CNT-i interacts with the external electric field and tends to align them along the field direction [33, 34] following the expression,

$$T_{i,DEP} = \frac{1}{4}\,V_{CNT}\epsilon_{m} E^{2} Re[\alpha^{*}]sin(2\theta_{i} )$$
(11a)
$$\alpha^{*} = \frac{{\left( {\varepsilon_{eq}^{*} - \varepsilon_{m}^{*} } \right)^{2} }}{{\left[ {\varepsilon_{m}^{*} + \left( {\varepsilon_{eq}^{*} - \varepsilon_{m}^{*} } \right)L} \right]\left( {\varepsilon_{eq}^{*} + \varepsilon_{m}^{*} } \right)}}$$
(11b)
$$\varepsilon_{eq}^{*} = \varepsilon_{lay}^{*} \left[ {\frac{{\varepsilon_{CNT}^{*} + \frac{\delta }{2a}\left( {\varepsilon_{CNT}^{*} - \varepsilon_{lay}^{*} } \right)}}{{\varepsilon_{lay}^{*} + \frac{\delta }{2a}\left( {\varepsilon_{CNT}^{*} - \varepsilon_{lay}^{*} } \right)}}} \right]$$
(11c)
$$\varepsilon^{*} = \varepsilon - j\frac{\sigma }{2\pi f}$$
(11d)
$$L = \frac{{ln\left( {2p} \right) - 1}}{{p^{2} }}$$
(11e)

In Eq. (11a), VCNT is the volume of the CNTs (VCNT= 4πab2/3), εm is the real permittivity of the suspending fluid, E is the electric field magnitude, and α* is the complex DEP depolarization factor for ellipsoidal particles under rotation, see Eq. (11b), with L as the longitudinal axis depolarization factor, see Eq. (11e) [23, 31]. As can be noticed from the expressions, the complex polarization factor depends on the interphase CNT/fluid layer properties [23, 31] through the complex equivalent permittivity (εeq*, see Eq. (11c)), which depends on the complex permittivity of the CNTs (εCNT*), of the interphase layer (εlay*) and of the thickness of such a layer, δ. From Eq. (11d), the complex permittivity depends on the real permittivity (ε) and conductivity (σ); thus, the expressions of εCNT*, εlay* and εm* can be obtained from Eq. (11d). For the translational motion, the contribution of the drag forces of the surrounding medium needs to be considered, see Eq. (5a) and (5b). When the CNTs are considered as prolate ellipsoids, the frictional terms for the x- and y-motion are expressed as [31, 52, 53],

$$\left( {F_{i,fr} } \right)_{x} = \, 6\pi \eta r_{e} K_{t} \frac{{dx_{i} }}{dt}$$
(12a)
$$\left( {F_{i,fr} } \right)_{y} = \, 6\pi \eta r_{e} K_{t} \frac{{dy_{i} }}{dt}$$
(12b)

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Oliva-Avilés, A.I., Alonzo-García, A., Zozulya, V.V. et al. A dielectrophoretic study of the carbon nanotube chaining process and its dependence on the local electric fields. Meccanica 53, 2773–2791 (2018). https://doi.org/10.1007/s11012-018-0869-4

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Keywords

  • Carbon nanotubes
  • Dielectrophoresis
  • Classical electrodynamics
  • AC electric fields
  • Dynamic motion