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Realization and characterization of carbon black based fractional order element
Microelectronics Journal 82 (2018) 22–28
Contents lists available at ScienceDirect
Microelectronics Journal journal homepage: www.elsevier.com/locate/mejo
Realization and characterization of carbon black based fractional order element☆ ∗
K. Biswas a , R. Caponetto b, , G. Di Pasquale c , S. Graziani b , A. Pollicino d , E. Murgano b a
IIT Karagpur, India DIEEI University of Catania Italy, Italy c DCS University of Catania Italy, Italy d DICAR University of Catania Italy, Italy b
A R T I C L E
I N F O
A B S T R A C T
Keywords: Fractional order element Structured material Frequency analysis Model identification
In this papers the possibility of realizing fractional capacitors by using carbon black nanostructured dielectrics is investigated. Capacitors have been realized by varying the percentage of distributed carbon black. The frequency analysis of the capacitors has been, therefore, performed. The Bode diagrams outline that this class of devices shows a non integer order behavior. Moreover, a dependance between the curing temperature and the fractional order has been shown.
When 𝛼 is an integer, it holds that:
1. Introduction Fractional calculus is a generalization of integration and differentiation to non-integer order fundamental operator a D𝛼t , where a and t are the limits of the operation and 𝛼 ∈ R. The continuous integro-differential operator is defined as: ⎧ d𝛼 ⎪ dt 𝛼 ⎪ 𝛼 D = ⎨1 a t ⎪ t −𝛼 ⎪∫ (d𝜏) ⎩ a
𝛼
h (t ) =
(1)
∶ 𝛼 < 0.
(2)
with 𝛼 ∈ ℝ+ and n − 1 < 𝛼 < n, where n 𝜖 ℕ. In the above definition, Γ(𝛼) is the factorial function given by the following expression [1]. ∞
Γ(𝛼) =
☆
∗
∫0
e−u u𝛼−1 du
h 1 ∑ lim 𝛼 (−1)k (𝛼k )h(t − kh) a Dt h(t ) = h→0 h k=0
(5)
where t is the time, a and t are the integration limits, and (𝛼k ) =
t
dn (t − 𝜏)n−𝛼−1 h(𝜏)d𝜏 Γ(n − 𝛼) dt n ∫a 1
h(t) may be any function for which the integral in (2) exists. The GL definition for the general fractional differintegral is given by: 𝛼
The three most frequently definitions used for the general fractional differintegral are the Riemann-Liouville (RL), the Grunwald-Letnikov (GL), and the Caputo definitions respectively [1,2]. The RL derivative is given by: a Dt
(4)
[ t −a ]
∶ 𝛼 > 0, ∶ 𝛼 = 0,
Γ(𝛼 + 1) = 𝛼!
(3)
Γ(𝛼+1) . Γ(k+1)Γ(𝛼−k+1)
Finally, the Caputo definition is: 𝛼 f (t )
a Dt
=
t f (n) (𝜏) 1 d𝜏, Γ(𝛼 − n) ∫a (t − 𝜏)𝛼−n+1
(6)
for (n − 1 < 𝛼 < n). The characteristic memory effect of Fractional Order Systems (FOSs) is outlined by (2) and (5), where the convolution integral in (2) and the infinite series in (5), represent the unlimited memory property. Such a memory is valuable when modeling hereditary and memory properties in physical systems and materials. This memory effect can be found in many physical processes. Some of them can found in: the
This paper has been support by the EU COST Action CA15225 - Fractional-order systems analysis, synthesis and their importance for future design. Corresponding author. E-mail address: [email protected] (R. Caponetto).
https://doi.org/10.1016/j.mejo.2018.10.008 Received 10 September 2018; Received in revised form 18 October 2018; Accepted 19 October 2018 Available online 25 October 2018 0026-2692/© 2018 Elsevier Ltd. All rights reserved.
K. Biswas et al.
Microelectronics Journal 82 (2018) 22–28
Fig. 3. Devices under test.
Table 1 List of the realized CB-FOEs.
Fig. 1. CB based FOE dielectric structure.
Name
Carbon Black %
Curing Temperature
Curing Time
C-125 C-140A C-140B C-150A C-150B C-160
8% 8% 8% 8% 8% 8%
125 ◦ C 140 ◦ C 140 ◦ C 150 ◦ C 150 ◦ C 160 ◦ C
38 min 32 min 32 min 28 min 28 min 24 min
order impedance (FOI) characterization [24,25], and in the studies of the fractional elements that exhibit a constant phase behavior, referred as a constant phase element (CPE), [26–30]. There is a consensus that FOE devices need to be fabricated and characterized ahead meaningful applications can be developed. In Ref. [31], it has been experimentally found out that the MWCNT (Multiwall Carbon Nanotubes) can be used to make fractional capacitors, (0 < 𝛼 < 1). The paper describes a process to fabricate such a components. The components have been realized using two copper plates as electrodes of dimension 1.5 cm by 1.5 cm and in between a dielectric made up of a mixture of MWCNT and epoxy to separate the electrodes. Inside the electrode, another copper plate of dimension 1 cm by 1 cm is placed. The middle plate has been coated with PMMA (Poly(methyl methacrylate)) polymer. The device can be considered to be a series combination of parallel R-C circuits, which provides distribution of time constant. It is assumed that a lossy capacitor is formed by two MWCNTs with polymer or PMMA in between. Different percentage of MWCNT loading results in different values of 𝛼 . In Ref. [32], some of the authors have already demonstrated the feasibility of fabricating FOEs by using Carbon Black (CB), dispersed in a polymeric matrix and proposed low pass filter as possible applications. Devices fabricated in Ref. [32] were realized using Sylgard, as polymeric matrix, and CB as dispersed filler, so that nanostructured devices were obtained. Also, the possibility of controlling the nature of the FOEs characteristics, by changing some fabrication process parameters, was suggested. Here, this aspect is further investigated and the dependance of the fractional order of cylindrical FOEs on the Curing Temperature (CT) is investigated. The frequency analysis of the Device Under Test (DUTs) has been performed and the corresponding experimental Bode diagrams have been estimated. Fractional order transfer functions have been fitted on the Bode diagrams. The value of 𝛼 has been used for characterizing the fractal nature of the DUTs. Obtained results show that, in
Fig. 2. Structure of the CB-FOEs under investigation.
charging discharging pattern of super capacitor [3], the relaxation of viscoelastic material [4], the chemical kinetic [5], and also large electrical networks [6]. Typically, in chemical kinetics, the non integer value of 𝛼 is used to represent anomalous diffusion. In case of viscoelastic materials, the stress response is known to be characterized by an exponential decay which is modelled using integer order differential equation. However, in case of microscopic models of polymer dynamics, which are also viscoelastic, the stress relaxation is algebraic decay [4], rather than an exponential decay. Such dynamics can easily be modelled using fractional derivatives. Typical applications of FOSs can be found in control [7–12], modelling [13–17], chaos [19,20], supercapacitor modelling [18], oscillators modeling [21], fractional order element (FOE) [22,23], fractional 23
K. Biswas et al.
Microelectronics Journal 82 (2018) 22–28
Table 2 Parameters of the transfer function obtained by using the Bode diagram fitting. Sample
𝛼
𝜏
K
Constant phase frequency range
C-125 C-140A C-140B C-150A C-150B C-160B
0.724 0.912 0.955 0.854 0.9262 0.898
2.7 · 10− 3 4081 · 10− 3 133918 · 10− 3 97.8 · 10− 3 4887 · 10− 3 745845 · 10− 3
0.163 · 106 3512 · 106 60117 · 106 19.5 · 106 1979.5 · 106 141251 · 106
(104 − 107 ) Hz (102 − 107 ) Hz (102 − 104 ) Hz (102 − 107 ) Hz (101 − 107 ) Hz (101 − 107 ) Hz
Fig. 4. Values of 𝛼 vs CT.
ites, with application, e.g., such as thermistors, deformation sensors [33], pressure sensors [34], tactile sensors, and gas sensors [35]. Here, the characteristics of CB based nano composites as FOEs are investigated. Sylgard184 has been used to realize the capacitors. It was purchased from DowCorning as a two part liquid elastomer kit. Part A (consisting in the vinyl-terminated PDMS prepolymer) was mixed with Part B (the crosslinking curing agent, consisting in a mixture of methylhydrosiloxane copolymer chains with a Pt catalyst and an inhibitor) in a 10:1 wt ratio as suggested by the manufacturer and
the investigated temperature range, the FOEs can be clustered into two classes.
2. Materials and device realization It is widely known in the literature, that, by adding a conductive filler to an insulating matrix (generally a polymeric matrix) composites can be fabricated, whose electrical and mechanical properties change dramatically with respect to the parent matrix. Such phenomenon has been widely exploited for realizing polymer conducting compos-
Fig. 5. Clustering of 𝛼 values vs CT. Results reported in Ref. [32] are also reported. 24
K. Biswas et al.
Microelectronics Journal 82 (2018) 22–28
Fig. 6. Comparison among Bode diagrams and the experimental based estimation. Samples 125, 140A and 140B.
25
K. Biswas et al.
Microelectronics Journal 82 (2018) 22–28
Fig. 7. Comparison among Bode diagrams and the experimental based estimation. Samples 150A, 150B and 160B.
26
K. Biswas et al.
Microelectronics Journal 82 (2018) 22–28
cured at room temperature for 48 h. Curing times at higher temperatures were chosen taking into account the manufacturer recommendation and the heat propagation through the mold, which inevitably results in a stabilization period required for the temperature of the curing PDMS approaching the required CT. Then curing times were fixed at 125 ◦ C 38 min, 140 ◦ C 32 min at 150 ◦ C 28 min and 160 ◦ C 24 min. CB (acetylene, 100% compressed, 99.9+%, specific area 75m2∕g, bulk density 170–230 g∕L, average particle size 0.042 μm) was purchased from AlfaAesar and used as received. The described, nano material CB based composite, is in fact the dielectric of the CB-FOEs under investigation. The resulting structure of the CB-FOE is given in Fig. 1. Cylindrical FOEs have been realized, whose geometry is shown in Fig. 2. More specifically, capacitances have copper-based shells with height h = 8 cm, internal diameter a = 0.6 cm and external diameter b = 1.2 cm, so that the dielectric is a hollow circular section of thickness 0.3 cm (see Fig. 3).
experimental module and phase values. For each Bode diagram, the corresponding parameters have been estimated. The mean values have been therefore computed. The obtained mean values of the parameters are given in Table 2, while Fig. 4 shows the dispersion of the 𝛼 values estimation for the four S1 − S4 measurements. Results reported in Fig. 4 show that, notwithstanding the data dispersion, the values of 𝛼 can be clustered into two classes. The first class contains the device obtained by using a CT equal to 125 ◦ C, while the second one contains all others devices. This suggests the possibility of modeling the fractional order of the CB-based FOEs by controlling the CT during the fabrication phase. For matter of comparison, in Fig. 5 the values of 𝛼 shown in 4 are compared with results already reported in Ref. [32] and labeled with a ∗. Though in Ref. [32] the investigation was performed in a different frequency range, it is still possible clustering all the devices, according to the CT. Such classes are indicated as Class A and Class B in the figure. The transition temperature interval looks to be between 125 ◦ C and 140 ◦ C. The values reported in Table 2 have been used along with (7), to estimate the Bode diagram of each device. The comparison among such Bode diagrams and the corresponding experimental based estimations are reported in Figs. 6(a) and 7(c).
3. FOE under test and experimental setup The CB-FOE samples have been prepared by mixing the PDMS and the crosslinking agent in a ratio of 1:10 in a Teflon crucible. The mixture was mixed for 10 min. The amount of CB required to achieve 8% concentration in mass has been added. The mixture was stirred for further 10 min, to enhance the dispersion of the CB. The mixture was poured into the devices and cured overnight. Since the focus of the paper is the dependance of the FOE order on the CT, used during the devices fabrication, devices with fixed geometry were realized at different CTs. More specifically, CT values in the range 125 ◦ C–160 ◦ C, have been investigated. Devices wit the characteristics reported in Table 1 have been fabricated for successive investigations. In order to find out the impedance of the fabricated FOEs at different frequencies, Novocontrol Alpha-A high performance frequency analyzer was used. Data were collected from 0.01 Hz to 4 MHz and 10 frequency points were taken in a decade. Measurements were repeated for 5 times at an interval of 24 h.
5. Conclusions In the paper, a class of carbon black-Sylgard nano composite fractional order elements has been investigated. The investigated devices were characterized by the same carbon black mass concentration, while differed for the curing temperature, used in the fabrication phase. Such a choice allowed for investigating the effect of the curing temperature on the device fractional order. Experimental results confirmed the possibility of clustering devices into two classes, according to the curing temperature. The transition between the two classes occurring in the interval 125 ◦ C and 140 ◦ C. Further investigations are needed in order to better resolve the transition region. Nevertheless, according to experimental results reported in the paper, it can be argued that the fractional order 𝛼 increases with the curing temperature up to the value of about 135 ◦ C. A kind of saturation looks to occur at higher temperatures. Finally, the 𝛼 value looks limited to values larger than 0.7. Furthermore, the present production procedure allows for realizing bulky and rigid devices. Investigation are required in order to establish if the production procedure can be modified in order to obtain smaller devices, characterized by design, by larger variation interval for the value of 𝛼 . Such improved production procedure could be a viable technology for realizing devices for analog filters implementation.
4. Estimation of the system order In order to determine the order 𝛼 for each CB-FOE, the acquired data have been elaborated to identify the DUTs transfer function in the form: Z (s ) =
K
(1 + 𝜏 s )𝛼
(7)
where K is the gain, 𝜏 is the pole of the system and 𝛼 ∈ ℝ, the fractional order. More specifically, the Bode module and phase experimental diagrams have been estimated by using the acquired data. The unknown parameters K, 𝛼 and 𝜏 , which rule the Bode diagrams, have been determined by using a standard genetic algorithm [36], with the following parameters: population size = 4000, generations number = 150, generation gap 0.9, and number of bit per variable equal to 8. The cost function, that has been minimized, takes into account the errors on both the magnitude and phase, and has been implemented in the following form:
References [1] I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications, Academic Press, 1999. [2] R. Caponetto, G. Dongola, L. Fortuna, I. Petras, Fractional Order Systems: Modelling and Control Applications, World Scientific Book, World Scientific Series on Nonlinear Science Series A, vol. 72, 2010. [3] S. Das, Theory of capacitors with a new formulation of charge storage concept and observed relaxation current as per curie-von schweidler law determining its rate histogram function as Zipfers distribution, 2018. [4] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing, 2000. [5] R. De Levie, On porous electrodes in electrolyte solutions: I. capacitance effects, Electrochim. Acta 8 (10) (1963) 751–780. [6] S. Das, Functional Fractional Calculus, Springer Science & Business Media, 2011. [7] I. Podlubny, Fractional-order systems and PI 𝜆 D𝜇 controllers, IEEE Trans. Automat. Contr. 44 (1) (1999) 208–214. [8] R. Caponetto, G. Dongola, F. Pappalardo, V. Tomasello, Auto-Tuning and fractional order controller implementation on hardware in the loop system, J. Optim. Theor. Appl. 156 (1) (2013) 141–152.
Erromod = sum(abs(20 ∗ log10(A) − 20 ∗ log10(moduleM ))) Errophase = sum(abs(rad2deg (Fi) − phaseM ))
(8)
Errore = Errophase + Erromod where A and Fi are respectively the module and the phase of the simulated transfer function (7), while the moduleM and phaseM are the 27
K. Biswas et al.
Microelectronics Journal 82 (2018) 22–28 [25] Bohannan G. Electrical component with fractional order impedance US patent n.20060267595 2006. [26] K. Biswas, S. Sen, K. Dutta, A constant phase element sensor for monitoring microbial growth, Sensor. Actuator. B 191 (2006) 186–191. [27] K. Biswas, S. Sen, K. Dutta, Realization of a constant phase element and its performance study in a differentiator circuit, IEEE Trans. Circ. Syst.-II: Expr. Brieff 53 (9) (2006) 802–806. [28] S. Das, M. Sivaramakrishna, K. Biswas, B. Goswami, Performance study of a constant phase angle based impedance sensor to detect milk adulteration, Sensor. Actuator. 167 (2011) 273–278. [29] M. Elshurafa, N. Almadhoun, K. Salama, H. Alshareef, Microscale electrostatic fractional capacitors using reduced graphene oxide percolated polymer composites, Appl. Phys. Lett. 102 (23) (2013). [30] R. Caponetto, S. Graziani, L. Pappalardo, F. Sapuppo, Experimental characterization of ionic polymer metal composite as a novel fractional order element, Adv. Math. Phys. (2013) 953695. [31] D. John, S. Banerjee, G. Bohannan, K. Biswas, Solid-state fractional capacitor using mwcnt-epoxy nanocomposite, Appl. Phys. Lett. 110 (16) (2017). [32] A. Buscarino, R. Caponetto, G. Di Pasquale, L. Fortuna, S. Graziani, A. Pollicinoc, Carbon Black based capacitive Fractional Order Element towards a new electronic device, Int. J. Electron. Commun. 84 (2018) 307–312. [33] P. Giannone, S. Graziani, E. Umana, Investigation of carbon black loaded natural rubber piezoresistivity, proc, in: IEEE I2MTC 2015, Pisa, May 11-14, 2015, pp. 1477–1481. [34] M. Ding, L. Wang, P. Wang, Changes in electrical resistance of carbon-black-filled silicone rubber composite during compression, J. Polym. Sci. B Polym. Phys. 45 (19) (2007) 2700–2706. [35] P. Wang, T. Ding, Creep of electrical resistance under uniaxial pressures for carbon black-silicone rubber composite, J. Mater. Sci. (45) (2010) 3595–3601. [36] D. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley Professional, 1989.
[9] R. De Keyser, C. Muresan, C. Ionescu, A novel auto-tuning method for fractional order PI/PD controllers, ISA (Instrum. Soc. Am.) Trans. 62 (2016) 268–275. [10] G. Bohannan, Analog fractional order controller in temperature and motor control applications, J. Vib. Contr. 14 (9) (2008) 1487–1498. [11] R. Caponetto, V. Tomasello, P. Lino, G. Maione, Design and efficient implementation of digital non-integer order controllers for electro-mechanical systems, J. Vib. Contr. 22 (9) (2015) 2196–2210. [12] H. Malek, S. Dadras, Y.Q. Chen, Performance analysis of fractional order extremum seeking control, ISA (Instrum. Soc. Am.) Trans. 63 (2016) 281–287. [13] W. Zheng, Y. Luo, Y. Chen, Fractional-order modeling of permanent magnet synchronous motor speed servo system, J. Vib. Contr. 22 (9) (2016). [14] G. Navarro, G. Tang, Fractional order model reference adaptive control for anesthesia, Int. J. Adapt. Contr. Signal Process. 31 (9) (2017) 1350–1360. [15] R. Magin, Fractional Calculus in Bioengineering, BeggelHouse, 2006. [16] R. Caponetto, G. Dongola, L. Fortuna, S. Graziani, S. Strazzeri, A fractional model for IPMC actuators, in: IEEE International Instrumentation and Measurement Technology Conference, I2MTC, 2008, pp. 2103–2107. [17] V. De Santis, V. Martynyuk, A. Lampasi, M. Fedula, M. Ortigueira, Fractional-order circuit models of the human body impedance for compliance tests against contact currents, Int. J. Electron. Commun. 78 (2017) 238–244. [18] C. Liu, M. Liu, F. Li, Cheng, Frequency response characteristic of single-walled carbon nanotubes as supercapacitor electrode material, J. Appl. Phys. Lett. 92 (14) (2008). [19] R. Caponetto, S. Fazzino, An application of adomian decomposition for analysis of fractional-order chaotic systems, Int. J. Bifurc. Chaos 23 (3) (2013). [20] I. Petras, Fractional-order Nonlinear Systems: Modeling, Analysis and Simulation (Nonlinear Physical Science), Springer, 2011. [21] G. Bohannan, B. Knauber, A physical experimental study of the fractional harmonic oscillator, IEEE Int. Sym. Circ. Syst. (2015) 2341–2344. [22] D. Mondal, K. Biswas, Packaging of single-component fractional order element, IEEE Trans. Device Mater. Reliab. 13 (2013) 73–80. [23] I. Jesus, T. Machado, Development of fractional order capacitors based on electrolyte processes, Nonlinear Dynam. 56 (1) (2009) 45–55. [24] S. Krishna, S. Das, K. Biswas, B. Goswami, Fabrication of a fractional order capacitor with desired specifications: a study on process identification and characterization, IEEE Trans. Electron. Dev. 58 (11) (2011) 4067–4072.
28
Contents lists available at ScienceDirect
Microelectronics Journal journal homepage: www.elsevier.com/locate/mejo
Realization and characterization of carbon black based fractional order element☆ ∗
K. Biswas a , R. Caponetto b, , G. Di Pasquale c , S. Graziani b , A. Pollicino d , E. Murgano b a
IIT Karagpur, India DIEEI University of Catania Italy, Italy c DCS University of Catania Italy, Italy d DICAR University of Catania Italy, Italy b
A R T I C L E
I N F O
A B S T R A C T
Keywords: Fractional order element Structured material Frequency analysis Model identification
In this papers the possibility of realizing fractional capacitors by using carbon black nanostructured dielectrics is investigated. Capacitors have been realized by varying the percentage of distributed carbon black. The frequency analysis of the capacitors has been, therefore, performed. The Bode diagrams outline that this class of devices shows a non integer order behavior. Moreover, a dependance between the curing temperature and the fractional order has been shown.
When 𝛼 is an integer, it holds that:
1. Introduction Fractional calculus is a generalization of integration and differentiation to non-integer order fundamental operator a D𝛼t , where a and t are the limits of the operation and 𝛼 ∈ R. The continuous integro-differential operator is defined as: ⎧ d𝛼 ⎪ dt 𝛼 ⎪ 𝛼 D = ⎨1 a t ⎪ t −𝛼 ⎪∫ (d𝜏) ⎩ a
𝛼
h (t ) =
(1)
∶ 𝛼 < 0.
(2)
with 𝛼 ∈ ℝ+ and n − 1 < 𝛼 < n, where n 𝜖 ℕ. In the above definition, Γ(𝛼) is the factorial function given by the following expression [1]. ∞
Γ(𝛼) =
☆
∗
∫0
e−u u𝛼−1 du
h 1 ∑ lim 𝛼 (−1)k (𝛼k )h(t − kh) a Dt h(t ) = h→0 h k=0
(5)
where t is the time, a and t are the integration limits, and (𝛼k ) =
t
dn (t − 𝜏)n−𝛼−1 h(𝜏)d𝜏 Γ(n − 𝛼) dt n ∫a 1
h(t) may be any function for which the integral in (2) exists. The GL definition for the general fractional differintegral is given by: 𝛼
The three most frequently definitions used for the general fractional differintegral are the Riemann-Liouville (RL), the Grunwald-Letnikov (GL), and the Caputo definitions respectively [1,2]. The RL derivative is given by: a Dt
(4)
[ t −a ]
∶ 𝛼 > 0, ∶ 𝛼 = 0,
Γ(𝛼 + 1) = 𝛼!
(3)
Γ(𝛼+1) . Γ(k+1)Γ(𝛼−k+1)
Finally, the Caputo definition is: 𝛼 f (t )
a Dt
=
t f (n) (𝜏) 1 d𝜏, Γ(𝛼 − n) ∫a (t − 𝜏)𝛼−n+1
(6)
for (n − 1 < 𝛼 < n). The characteristic memory effect of Fractional Order Systems (FOSs) is outlined by (2) and (5), where the convolution integral in (2) and the infinite series in (5), represent the unlimited memory property. Such a memory is valuable when modeling hereditary and memory properties in physical systems and materials. This memory effect can be found in many physical processes. Some of them can found in: the
This paper has been support by the EU COST Action CA15225 - Fractional-order systems analysis, synthesis and their importance for future design. Corresponding author. E-mail address: [email protected] (R. Caponetto).
https://doi.org/10.1016/j.mejo.2018.10.008 Received 10 September 2018; Received in revised form 18 October 2018; Accepted 19 October 2018 Available online 25 October 2018 0026-2692/© 2018 Elsevier Ltd. All rights reserved.
K. Biswas et al.
Microelectronics Journal 82 (2018) 22–28
Fig. 3. Devices under test.
Table 1 List of the realized CB-FOEs.
Fig. 1. CB based FOE dielectric structure.
Name
Carbon Black %
Curing Temperature
Curing Time
C-125 C-140A C-140B C-150A C-150B C-160
8% 8% 8% 8% 8% 8%
125 ◦ C 140 ◦ C 140 ◦ C 150 ◦ C 150 ◦ C 160 ◦ C
38 min 32 min 32 min 28 min 28 min 24 min
order impedance (FOI) characterization [24,25], and in the studies of the fractional elements that exhibit a constant phase behavior, referred as a constant phase element (CPE), [26–30]. There is a consensus that FOE devices need to be fabricated and characterized ahead meaningful applications can be developed. In Ref. [31], it has been experimentally found out that the MWCNT (Multiwall Carbon Nanotubes) can be used to make fractional capacitors, (0 < 𝛼 < 1). The paper describes a process to fabricate such a components. The components have been realized using two copper plates as electrodes of dimension 1.5 cm by 1.5 cm and in between a dielectric made up of a mixture of MWCNT and epoxy to separate the electrodes. Inside the electrode, another copper plate of dimension 1 cm by 1 cm is placed. The middle plate has been coated with PMMA (Poly(methyl methacrylate)) polymer. The device can be considered to be a series combination of parallel R-C circuits, which provides distribution of time constant. It is assumed that a lossy capacitor is formed by two MWCNTs with polymer or PMMA in between. Different percentage of MWCNT loading results in different values of 𝛼 . In Ref. [32], some of the authors have already demonstrated the feasibility of fabricating FOEs by using Carbon Black (CB), dispersed in a polymeric matrix and proposed low pass filter as possible applications. Devices fabricated in Ref. [32] were realized using Sylgard, as polymeric matrix, and CB as dispersed filler, so that nanostructured devices were obtained. Also, the possibility of controlling the nature of the FOEs characteristics, by changing some fabrication process parameters, was suggested. Here, this aspect is further investigated and the dependance of the fractional order of cylindrical FOEs on the Curing Temperature (CT) is investigated. The frequency analysis of the Device Under Test (DUTs) has been performed and the corresponding experimental Bode diagrams have been estimated. Fractional order transfer functions have been fitted on the Bode diagrams. The value of 𝛼 has been used for characterizing the fractal nature of the DUTs. Obtained results show that, in
Fig. 2. Structure of the CB-FOEs under investigation.
charging discharging pattern of super capacitor [3], the relaxation of viscoelastic material [4], the chemical kinetic [5], and also large electrical networks [6]. Typically, in chemical kinetics, the non integer value of 𝛼 is used to represent anomalous diffusion. In case of viscoelastic materials, the stress response is known to be characterized by an exponential decay which is modelled using integer order differential equation. However, in case of microscopic models of polymer dynamics, which are also viscoelastic, the stress relaxation is algebraic decay [4], rather than an exponential decay. Such dynamics can easily be modelled using fractional derivatives. Typical applications of FOSs can be found in control [7–12], modelling [13–17], chaos [19,20], supercapacitor modelling [18], oscillators modeling [21], fractional order element (FOE) [22,23], fractional 23
K. Biswas et al.
Microelectronics Journal 82 (2018) 22–28
Table 2 Parameters of the transfer function obtained by using the Bode diagram fitting. Sample
𝛼
𝜏
K
Constant phase frequency range
C-125 C-140A C-140B C-150A C-150B C-160B
0.724 0.912 0.955 0.854 0.9262 0.898
2.7 · 10− 3 4081 · 10− 3 133918 · 10− 3 97.8 · 10− 3 4887 · 10− 3 745845 · 10− 3
0.163 · 106 3512 · 106 60117 · 106 19.5 · 106 1979.5 · 106 141251 · 106
(104 − 107 ) Hz (102 − 107 ) Hz (102 − 104 ) Hz (102 − 107 ) Hz (101 − 107 ) Hz (101 − 107 ) Hz
Fig. 4. Values of 𝛼 vs CT.
ites, with application, e.g., such as thermistors, deformation sensors [33], pressure sensors [34], tactile sensors, and gas sensors [35]. Here, the characteristics of CB based nano composites as FOEs are investigated. Sylgard184 has been used to realize the capacitors. It was purchased from DowCorning as a two part liquid elastomer kit. Part A (consisting in the vinyl-terminated PDMS prepolymer) was mixed with Part B (the crosslinking curing agent, consisting in a mixture of methylhydrosiloxane copolymer chains with a Pt catalyst and an inhibitor) in a 10:1 wt ratio as suggested by the manufacturer and
the investigated temperature range, the FOEs can be clustered into two classes.
2. Materials and device realization It is widely known in the literature, that, by adding a conductive filler to an insulating matrix (generally a polymeric matrix) composites can be fabricated, whose electrical and mechanical properties change dramatically with respect to the parent matrix. Such phenomenon has been widely exploited for realizing polymer conducting compos-
Fig. 5. Clustering of 𝛼 values vs CT. Results reported in Ref. [32] are also reported. 24
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Fig. 6. Comparison among Bode diagrams and the experimental based estimation. Samples 125, 140A and 140B.
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Fig. 7. Comparison among Bode diagrams and the experimental based estimation. Samples 150A, 150B and 160B.
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cured at room temperature for 48 h. Curing times at higher temperatures were chosen taking into account the manufacturer recommendation and the heat propagation through the mold, which inevitably results in a stabilization period required for the temperature of the curing PDMS approaching the required CT. Then curing times were fixed at 125 ◦ C 38 min, 140 ◦ C 32 min at 150 ◦ C 28 min and 160 ◦ C 24 min. CB (acetylene, 100% compressed, 99.9+%, specific area 75m2∕g, bulk density 170–230 g∕L, average particle size 0.042 μm) was purchased from AlfaAesar and used as received. The described, nano material CB based composite, is in fact the dielectric of the CB-FOEs under investigation. The resulting structure of the CB-FOE is given in Fig. 1. Cylindrical FOEs have been realized, whose geometry is shown in Fig. 2. More specifically, capacitances have copper-based shells with height h = 8 cm, internal diameter a = 0.6 cm and external diameter b = 1.2 cm, so that the dielectric is a hollow circular section of thickness 0.3 cm (see Fig. 3).
experimental module and phase values. For each Bode diagram, the corresponding parameters have been estimated. The mean values have been therefore computed. The obtained mean values of the parameters are given in Table 2, while Fig. 4 shows the dispersion of the 𝛼 values estimation for the four S1 − S4 measurements. Results reported in Fig. 4 show that, notwithstanding the data dispersion, the values of 𝛼 can be clustered into two classes. The first class contains the device obtained by using a CT equal to 125 ◦ C, while the second one contains all others devices. This suggests the possibility of modeling the fractional order of the CB-based FOEs by controlling the CT during the fabrication phase. For matter of comparison, in Fig. 5 the values of 𝛼 shown in 4 are compared with results already reported in Ref. [32] and labeled with a ∗. Though in Ref. [32] the investigation was performed in a different frequency range, it is still possible clustering all the devices, according to the CT. Such classes are indicated as Class A and Class B in the figure. The transition temperature interval looks to be between 125 ◦ C and 140 ◦ C. The values reported in Table 2 have been used along with (7), to estimate the Bode diagram of each device. The comparison among such Bode diagrams and the corresponding experimental based estimations are reported in Figs. 6(a) and 7(c).
3. FOE under test and experimental setup The CB-FOE samples have been prepared by mixing the PDMS and the crosslinking agent in a ratio of 1:10 in a Teflon crucible. The mixture was mixed for 10 min. The amount of CB required to achieve 8% concentration in mass has been added. The mixture was stirred for further 10 min, to enhance the dispersion of the CB. The mixture was poured into the devices and cured overnight. Since the focus of the paper is the dependance of the FOE order on the CT, used during the devices fabrication, devices with fixed geometry were realized at different CTs. More specifically, CT values in the range 125 ◦ C–160 ◦ C, have been investigated. Devices wit the characteristics reported in Table 1 have been fabricated for successive investigations. In order to find out the impedance of the fabricated FOEs at different frequencies, Novocontrol Alpha-A high performance frequency analyzer was used. Data were collected from 0.01 Hz to 4 MHz and 10 frequency points were taken in a decade. Measurements were repeated for 5 times at an interval of 24 h.
5. Conclusions In the paper, a class of carbon black-Sylgard nano composite fractional order elements has been investigated. The investigated devices were characterized by the same carbon black mass concentration, while differed for the curing temperature, used in the fabrication phase. Such a choice allowed for investigating the effect of the curing temperature on the device fractional order. Experimental results confirmed the possibility of clustering devices into two classes, according to the curing temperature. The transition between the two classes occurring in the interval 125 ◦ C and 140 ◦ C. Further investigations are needed in order to better resolve the transition region. Nevertheless, according to experimental results reported in the paper, it can be argued that the fractional order 𝛼 increases with the curing temperature up to the value of about 135 ◦ C. A kind of saturation looks to occur at higher temperatures. Finally, the 𝛼 value looks limited to values larger than 0.7. Furthermore, the present production procedure allows for realizing bulky and rigid devices. Investigation are required in order to establish if the production procedure can be modified in order to obtain smaller devices, characterized by design, by larger variation interval for the value of 𝛼 . Such improved production procedure could be a viable technology for realizing devices for analog filters implementation.
4. Estimation of the system order In order to determine the order 𝛼 for each CB-FOE, the acquired data have been elaborated to identify the DUTs transfer function in the form: Z (s ) =
K
(1 + 𝜏 s )𝛼
(7)
where K is the gain, 𝜏 is the pole of the system and 𝛼 ∈ ℝ, the fractional order. More specifically, the Bode module and phase experimental diagrams have been estimated by using the acquired data. The unknown parameters K, 𝛼 and 𝜏 , which rule the Bode diagrams, have been determined by using a standard genetic algorithm [36], with the following parameters: population size = 4000, generations number = 150, generation gap 0.9, and number of bit per variable equal to 8. The cost function, that has been minimized, takes into account the errors on both the magnitude and phase, and has been implemented in the following form:
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Erromod = sum(abs(20 ∗ log10(A) − 20 ∗ log10(moduleM ))) Errophase = sum(abs(rad2deg (Fi) − phaseM ))
(8)
Errore = Errophase + Erromod where A and Fi are respectively the module and the phase of the simulated transfer function (7), while the moduleM and phaseM are the 27
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