Numerical modelling of multiwalled carbon nanotube based supercapacitors

Abstract

Nowadays there are growing need of energy in every aspect of human living. That is the reason of searching for a highly efficient sources of energy. Supercapacitors or ultracapacitors (how supercapacitors are also described) can be used as ones. Main reason is high energy and power density of those devices. The possibility of using supercapacitors grow along with a reduction in their size while maintaining the above mentioned parameters. Accomplishing of that goal imposes the use of modern materials especially nanomaterials during supercapacitors manufacturing. One of materials recently raising scientists’ interest, as material for ultracapacitors electrodes are carbon nanotubes (CNTs). It is due to the great electrical and mechanical properties and high specific surface. This article presents results of parameters calculation based on finite element method, obtained for supercapacitors models. In the described models used for simulation, electrodes are made of carbon nanotubes, obtained by synthesis using LSCVD method.

Keywords

Supercapacitors carbon nanotubes finite element method energy sources

1. Introduction

In recent years, economic and technological development causes an increasing demand for energy. It is used in every area of our lives. In addition, mobility is necessary as an inseparable element of civilization development. This in turn causes a problem with energy storage. Energy storage involves the conversion of energy from difficult to store forms to forms that enable easy and economically justified storage. In order to store electric energy there are used different kinds of batteries, galvanic cell or capacitors [1], which we repeatedly unload and load with electric currents.

Due to chemical reactions occurring in the electrolyte, and at the junction of electrolyte and electrodes, accumulators accumulate energy, which can be released and used in time of need. Capacitors, on the other hand, accumulate charges on the electrodes using electrostatic forces. Depending on the material used during manufacturing process, capacitors show different properties and can be used for different tasks.

A separate group of capacitors are supercapacitors – the electrochemical capacitors with specific construction in comparison with traditional capacitors (three main elements – the electrodes, the electrolyte and the separator). The most noticeable difference is huge surface of electrodes, which because of use of advanced materials (e.g. Carbon nanotubes, graphene sheets or porous carbon) reaches specific surface of 2000 m ${}^{2}$ /m [2, 3]. Using highly specified nanomaterials in electromagnetic devices giving great technological advantage comparing to traditional devices, e.g nanocrystalline foil based sensors [4].

Carbon nanotubes due to great electrical properties (conductivity $\sim$ 5000 S*cm ${}^{-1}$ ) and large surface area (for multiwalled carbon nanotubes – MWCNTs $>$ 430 m ${}^{2}$ g ${}^{-1}$ ) have been used as materials for supercapacitors electrodes [5, 6, 7]. The electrodes are the most important parts of each supercapacitors, which determines its performance parameters – energy and power density. The achievement of such a large surface area of the electrodes results in very large capacities (hundreds of farads). Electric charge is collected within a double layer (electric double layer-ELD), which is formed on the border of the centers of the electrode-electrolyte. That makes supercapacitors devices combining high energy storage capability with the high power delivery capability (typical for conventional capacitors) [1, 8, 9].

Due to these properties, the authors of this publication also used carbon nanotubes as a material for supercapacitor’s electrodes.

2. Results

Multiwalled carbon nanotubes, serving as a prototype for numerical models, for which calculations were made, was synthesized using liquid source chemical vapor deposition (LSCVD) synthesis technique [10, 11, 12]. The choice of technique was dictated by ease of changes to process parameters. To synthesis of the MWCNTs three zoned quartz tube furnace was used. Furnace control system allows temperature control of each of the zones independently. There is also precise gas and liquid dispensing system attached to the furnace, allowing precise catalyst solution and carrying gases dozing. Changing synthesis temperature, carrier gas flow rate and catalyst both concentration and its feed rate leads to different MWCNTs’ parameters [12, 13], such as length, diameter, material purity and growth rate. That allows to control and change crucial for supercapacitor capacitance electrodes area.

Carbon nanotubes obtained from LSCVD technique grows in form of “carpet” – perpendicular to carpet’s base (Fig. 1). As catalyst in experiments authors used ferrocene dissolved in toluene. Carrier gases was mix of argon and hydrogen. Synthesis temperatures range was 1020–1120 K.

Figure 1.

MWCNTs SEM image – nanotubes in form of “carpet”, LSCVD technique, synthesis temperature 1070 K.

2.1 Modelling of nanotubes in electric field

Nanotubes are often modeled for FEM in the electrostatic field. In the paper [14] the charge distribution on a biased finite length one nanotube cylinder, with diameter of 20–60 nm, above an infinite grounded plane is investigated. The relationship between charge distribution and the geometry of nanotubes is studied in detail by classical electrostatics using full 3D numerical simulation.

In the paper [15] a nonlinear analysis for singly and doubly clamped nanotube based nano-electromechanical system NEMS devices has been reported. In this article, analysis of the charge distribution arising from the electrostatic field is presented.

In the work [16], the electrostatic coupling between singled-walled carbon nanotube SWCNT networks/arrays and planar gate electrodes in thin-film transistors is analyzed both in the quantum limit with an analytical model and in the classical limit with finite-element modeling.

Whereas, in the paper [17], a 3-D simulation is performed using the finite-element method (FEM) to model the wraparound gate carbon nanotube field-effect transistor with multi-CNT channels. Capacitances are extracted from the proposed FEM model. The extracted capacitances are in excellent agreement with the capacitances obtained from the analytical model presented in this study.

Authors of the work [18] present the electrochemical behavior of vertically aligned multi-walled carbon nanotube based supercapacitors as a function of nanotube length, nanotube axial compression and electrode/current collector interface design.

In very interesting papers [24, 25, 26, 27, 28], was investigated the effect of the Electrical Double Layer (EDL) on the voltammetric, using a finite-element based computational method developed based on Nernst-Planck and Poisson equations as governing equations.

In the study [24] authors investigated the effect of the EDL on the voltammetric performance of nanometer single electrodes by developing a finite-element. For the electrochemical processes, a detailed description of the electrode reaction and mass transport can be found for single nanometer electrodes. For electrochemical electrodes, once they come into contact with an electrolytic solution, an EDL structure will form at the electrode surface due to the electrostatic interaction between electrons in the metal and ions in the solution [25]. For the electrical field and mass transport, an electrostatic problem governed by the Poisson equation can be solved in the compact layer and a combined Nernst-Planck and electrostatic problem can be solved in the electrolytic domain outside the compact layer [25]. Detailed EDL analysis was also provided in articles [19, 20, 21, 22, 23].

2.2 Modeling consideration

In electrostatic, under static conditions the electric potential, $V$ , is defined by the relationship:

$\displaystyle E=-\nabla V$

Combining this equation with the constitutive relationship

$\displaystyle D=\varepsilon E+P$

between the electric displacement $D$ and the electric field $E$ , it is possible to represent Gauss’ law as the following equation:

$\displaystyle-\nabla\cdot(\varepsilon\nabla V-P)=\rho$

In this equation, the physical constant, $\varepsilon$ (SI unit: F/m) is the permittivity of used dielectric material, $P$ (SI unit: C/m ${}^{2}$ ) is the electric polarization vector, and $\rho$ (SI unit: C/m ${}^{3}$ ) is a space charge density. This equation describes the electrostatic field in dielectric materials.

Figure 2.

Part of numerical model of MWCNTs “carpet” created using COMSOL multiphysics software.

2.3 Model parameters

In this study, a computational 3D models were developed using commercial finite element analysis package COMSOL Multiphysics to simulate supercapacitance.

Part of numerical model of supercapacitor electrode is presented at Fig. 2. Model was created for MWCNTs synthesized in temperature 1070 K. Based at described model, numerical calculations were performed.

Authors use Helmholtz model EDL, who is comparable to the classical problem of a parallel-plate capacitor i.e. EDL is capable of storing electric charge. The parameters of the Helmholtz layers were based, inter alia, on the basis of other researchers work [23, 19, 20].

In described case, the assumed value of electrical double layer (EDL) thickness is 0.6 nm, and density of electrical charges on this layer 8 $\mu$ C/cm ${}^{2}$ . Applied electrical potential is $\pm$ 0.25 V. The diameter of a single nanotube is 15 nm, and the distance between the nanotubes corresponds to their diameter. The capacitor filling is Sodium Chloride (NaCl) electrolyte – a water 2.2 M solution. Simulations were carried out for two dielectrics, pure water (as reference) and water with NaCl solution described above. Dielectric constant of distilled water is $\varepsilon_{r}=$ 80, water with NaCl $\varepsilon_{r}=$ 45. Correspondingly, electric conductivity is 5e-6 S/m and 5.5 S/m. For layer of ions dielectric constant is lower: $\varepsilon_{r}$ (Na+) $=$ 9.74 and $\varepsilon_{r}$ (Cl-) $=$ 1.99 [23].

Below at Fig. 3 MWCNTs model EDL and created mesh are shown.

Figure 3.

Mesh of MWCNTs (a) with EDL attached model (b).

Figure 4.

a) Capacitor plates configuration; b) Electric voltage potential distribution between plates.

Figure 5.

a) Voltage potential distribution – pure water used as dielectric, b) NaCl used as electrolyte.

Parameters of model shown at Fig. 3 are as follows:

degrees of freedom solved for over 4 million number,

number of boundary elements: over 8 thousand,

number of vertex elements: about 400 elements.

2.4 Calculations results

Figure 4a and b shows capacitor plates configuration and electric potential distribution inside the model.

Figures below show voltage potential distribution in the half of MWCNTs height cross section. Figure 5a nad b applies to pure water and NaCl solution used as electrolyte accordingly.

Specific capacitances calculated for described models are as follows:

distillated water used as dielectric – 0.04 F/g

NaCl solution (2.2 M concentration) – 28 F/g

Also specific capacitance dependence main diameter were calculated – Fig. 6.

Figure 6.

Specific capacitance CNTs’ diameter dependence.

The above calculations were made for room temperature $-$ 297.15 K. For the temperature of 330.15 K, considering Debye-Hückel theory of electrolyte solutions [29], specific capacitances values higher by about 40% were obtained $-$ 0.057 F/g for water and 39, 5, 5 F/g respectively for NaCl solution.

3. Discussion

Obtained results placing analyzed model’s parameters among high performance capacitors created and tested by other researchers [13]. Having high energy-storage capability and high power density MWCNTs based supercapacitor is device with broad area of applications, such as consumer electronics or light electric vehicles eg. electric bikes.

Used LSCVD synthesis technique not only gives possibility of altering supercapacitor’s parameters by precise tailoring MWCNTs parameters, but allowing continuous production of carbon nanotubes on moving substrate, making electrodes mass production possible.

The model used for calculation does not account for the dependence of the measured capacity on potential or electrolyte concentration. In the future, authors want simulate the fully capacitive effect of the EDL with electrochemical domain. A comparison with experimental data will also be required.

The first pilot measurements have already been made in the NanoLab lab of the Institute of Mechatronics and Information Systems.

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