Dielectrophoretic modeling of the dynamic carbon nanotube network formation in viscous media under alternating current electric fields
Introduction
Since early reports on carbon nanotube (CNT) manipulation in liquids using electric fields [1], [2], the electric field-guided assembly of CNTs has attracted much attention due to its non-invasive and potentially scalable characteristics, as well as its relatively facile implementation. The CNT’s anisotropic structure and its electrical properties render a rapid response to applied electric fields, resulting in CNT orientation along the field direction and the formation of an aligned network with stable electrical connectivity between electrodes [1], [2], [3]. Alignment and precise control of CNT networks are highly desirable characteristics for the fabrication and optimization of high performance devices such as bio-, chemical and piezoresistive sensors, field-effect transistors, AFM tips, force transducers and nano-electromechanical switches, among many others [3], [4], [5]. Although both direct current (DC) [6], [7] and alternating current (AC) [2], [3], [6], [8], [9] electric fields are able to induce CNT alignment, AC fields have proven to be more efficient to that aim. Theoretical [7], [10], [11], [12] and experimental [7], [8], [10], [13], [14], [15], [16], [17], [18] studies on CNT alignment and manipulation by electric fields have been reported by several authors. When an electric field is applied to a viscous medium containing dispersed CNTs, an effective dipole is induced on the CNTs as a result of the different electrical properties between the CNTs and their surrounding medium. The interaction between the dipoles induced on the CNTs and the applied electric field results in rotational and translational CNT motion through a mechanism called dielectrophoresis (DEP) [19], [20], [21]. Although a complete understanding of the dynamics of CNT networks formation under electric fields is still unclear, it is generally accepted that the main governing mechanisms are: (i) CNT rotation and alignment along the electric field direction, (ii) CNT-to-CNT Coulombic interactions, and (iii) CNT migration towards an electrode. When a dipole is induced on a CNT, a torque will tend to orient the CNT along the field direction [6], [7], [12], [13], [14], [23]. Due to the induced dipoles, contiguous CNTs will be mutually attracted promoting a head-to-head contact and the formation of aligned structures [7], [9], [11], [21], [24]. Furthermore, when a CNT reaches an electrode, a remarkable change of the electric field distribution occurs at the vicinity of the CNT apex, inducing a dielectrophoretic force which tends to attract neighboring CNTs [1], [9], [11], [14], [23]. However, the sequence of these events and the time scale at which they occur are yet unclear. Chen et al. [14] reported on a controllable interconnection of singlewall carbon nanotubes (SWCNTs) under AC electric fields between microelectrodes, concluding that the DEP force and the electric field redistribution at the CNT apex are responsible for the interconnection phenomenon. Farajian et al. [22] theoretically investigated the polarization and aggregation of CNTs in suspensions under electric fields based on two levels of organization, viz. CNT alignment and CNT-to-CNT contact (chaining). They modeled the CNTs as solid rods with hemispherical ends and used an exact numerical approach based on self-consistent Coulombic interactions with classical electrostatics. Recently, Monti et al. [7] investigated the alignment of SWCNTs by a DC electric field by considering three involved phenomena, SWCNT rotation, translation and migration towards the positive electrode. They proposed simplified differential equations for each mechanism and provided rough estimates of the time required by each studied mechanism. Other experimental reports have concluded that better CNT alignment, dispersion and control can be achieved by increasing the electric field frequency [2], [8], [16], [23], [25], [26], [27], [28]. In a recent experimental investigation, Oliva-Avilés et al. [13] reported on the dynamic formation of a network of electric field-aligned multiwall carbon nanotubes (MWCNTs) in liquids of different viscosity, showing that AC frequencies at and above the kHz range yield faster formation of aligned networks than lower frequencies. These authors also observed that the morphology of the final MWCNT network presents an important dependence on the electric field frequency. Thus, among other factors involved, the electric field frequency is expected to play a key role on the different mechanisms involved on the CNT behavior under the application of an AC electric field. However, the dependence of the individual mechanisms governing the CNT network formation on the frequency remains unclear, and the correlation with experiments is often contradictory. Understanding the sequence of events and the role of the electric field frequency, CNT’s concentration and aspect ratio in the different phenomena occurring in a viscous medium containing CNTs would represent an important step towards tailoring such a CNT network.
Given this motivation, a modeling scheme consisting in a set of three DEP-based nonlinear differential equations corresponding to the major accepted mechanisms taking place in the dynamics of the CNT network formation by the application of an AC electric field is proposed herein. A layer at the CNT/liquid interface is included in the DEP models and investigated in terms of its electrical conductivity, permittivity and thickness. Only by considering such an interface layer, the experimentally observed effect of the AC frequency on the CNT network formation is adequately captured. Finally, a description of the sequence of dynamic events occurring during the formation of the CNT aligned network is proposed.
Section snippets
Modeling
A dielectrophoretic approach using electrodynamics is developed here to describe the dynamic motion of CNTs dispersed in a liquid under the application of an AC electric field. The set of governing equations corresponding to CNT rotation, CNT-to-CNT Coulombic interactions and CNT migration towards an electrode are presented in the following sections.
Determination of the interface layer parameters and effect of frequency
The effect of the electric field frequency on the dynamics of the three investigated mechanisms is captured by the complex permittivity term, Eq. (1). The DEP terms describing the mechanisms of CNT rotation, CNT-to-CNT Coulombic interaction and CNT migration towards an electrode are proportional to Re[α∗], |β∗|2 and Re[β∗], respectively, see Eqs. (2), (6), (9). It was noted that the trends associated to variations in the interface layer properties observed for Re[α∗], |β∗|2 and Re[β∗] as
Discussion
According to the models proposed here, the consideration of an interface layer in a CNT/liquid system allows to capture the trends observed in several experiments regarding the role of the electric field frequency on the formation of CNT networks. The presence of an interface layer in a particle/liquid system under the application of electric field has been proposed by several authors [17], [18], [21], [36], [37], [38], [39], [40], [41]. Although the nature and composition of such a layer are
Conclusions
The formation of an aligned CNT network in a viscous medium under the application of an AC electric field was modeled by a classic dielectrophoretic approach considering three mechanisms: CNT rotation towards the electric field direction, CNT-to-CNT Coulombic interaction and CNT migration towards an electrode. A CNT/liquid system including an interfacial layer of electrical conductivity and permittivity intermediate between those of the CNT and liquid and thickness equal to the CNT diameter (10
Acknowledgments
This work was supported by CONACYT-CIAM project No. 188089 of Dr. Avilés. A.I.O. acknowledges CONACYT for his PhD scholarship and the additional financial support for his visiting position at Virginia Tech through the “Becas Mixtas” program. A.I.O. would also like to thank Adarsh Chaurasia at Virginia Tech for many fruitful discussions.
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