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Engineering » Energy Engineering » "Thermoelectrics for Power Generation - A Look at Trends in the Technology", book edited by Sergey Skipidarov and Mikhail Nikitin, ISBN 978-953-51-2846-5, Print ISBN 978-953-51-2845-8, Published: December 21, 2016 under CC BY 3.0 license. © The Author(s).

# Nitrogen-Doped Carbon Nanotube/Polymer Nanocomposites Towards Thermoelectric Applications

DOI: 10.5772/65675

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## Overview

Figure 1. The approximate range of electrical conductivity covered by CPNs [1].

Figure 2. Major types of nitrogen bonding in N-CNTs [35].

Figure 3. Simplified diagram of the electronic band structure in the band theory, reproduced from [34].

Figure 4. Percolation curve of compression-molded CNT/polystyrene nanocomposite (a typical percolation curve of CPNs) [55].

Figure 5. Electrical conductivity of N-CNT/PVDF nanocomposites as a function of N-CNT content. N-CNTs were synthesized over Co, Fe, and Ni catalysts.

Figure 6. TEM micrographs of N-CNTs synthesized over Co, Fe, and Ni catalysts.

Figure 7. LM images of microtomed sections of 2.0 wt.% N-CNT/PVDF nanocomposites. N-CNTs were synthesized over Co, Fe, and Ni catalysts. The red squares represent areas employed for relative transparency quantifications.

Figure 8. TEM images of 2.0 wt.% N-CNT/PVDF nanocomposites with N-CNTs synthesized with different catalysts, illustrating nanodispersion state of N-CNTs.

Figure 9. Small amplitude oscillatory shear response at γ = 1 % and T = 240 °C for neat PVDF and N-CNT/PVDF nanocomposites with N-CNTs synthesized over different catalysts.

Figure 10. Oscillatory amplitude sweep response of neat PVDF and N-CNT/PVDF nanocomposites containing 3.5 wt.% N-CNTs synthesized over different catalysts for strain amplitudes of γ0 = 0.1–1000 % at an angular frequency of ω = 0.1 rad/s. The insets show the non-dimensionalized elastic Lissajous loops for strain amplitudes indicated by solid lines.

# Nitrogen-Doped Carbon Nanotube/Polymer Nanocomposites Towards Thermoelectric Applications

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## Abstract

This study investigates the impact of nitrogen doping on the performance of carbon nanotube (CNT)/polymer nanocomposites for thermoelectric applications; this was performed through measurement of conductivity of the generated nanocomposites. Three different catalysts (Co, Fe, and Ni) were used to synthesize nitrogen-doped CNTs (N-CNTs) by chemical vapor deposition technique. Synthesized N-CNTs were melt-mixed with a polyvinylidene fluoride (PVDF) matrix with a small-scale mixer at a broad range of loadings from 0.3 to 3.5 wt.% and then compression molded. Measurement of electrical conductivity of the generated nanocomposites showed superior properties in the following order of the synthesis catalyst: Co > Fe > Ni. We employed various characterization techniques to figure out the reasons behind dissimilar electrical conductivity of the generated nanocomposites, i.e., transmission electron microscopy, X-ray photoelectron spectroscopy, Raman spectroscopy, thermogravimetric analysis, light microscopy, and rheometry. It was found out, that the superior electrical conductivity of (N-CNT)Co nanocomposites was due to a combination of high synthesis yield, high aspect ratio, low nitrogen content, and high crystallinity of N-CNTs coupled with a good state of N-CNT dispersion. Moreover, it was revealed, that nitrogen doping had an adverse impact on electrical conductivity and, thus, on thermoelectric performance of CNT/polymer nanocomposites.

Keywords: carbon nanotube, nitrogen doping, polymer nanocomposites, electrical conductivity, thermoelectric

## 1. Introduction

### 1.1. Conductive filler/polymer nanocomposites

Conductive filler/polymer nanocomposites (CPNs) have recently drawn great interest to be employed in various applications due to their unique properties, such as tunable electrical conductivity, light weight, low cost, corrosion resistance, and processability [1, 2]. CPNs are generated by incorporating conductive filler into a polymer matrix. Conventional polymers, such as polycarbonate and polystyrene are insulative; however, incorporating conductive fillers to these polymer matrices can provide them with a broad range of conductivities through the formation of a two- or three-dimensional conductive network (Figure 1). Tunable electrical conductivity of CPNs entitles them to be used in a broad spectrum of applications, such as thermoelectric, charge storage, antistatic dissipation, electrostatic discharge (ESD) protection, and electromagnetic interference (EMI) shielding [38]. In fact, the level of electrical conductivity defines the applications in which CPNs can be employed. Charge storage and ESD protection are the major applications of CPNs necessitating low and medium electrical conductivity, respectively, whereas thermoelectric and EMI shielding require high electrical conductivity.

### Figure 1.

The approximate range of electrical conductivity covered by CPNs [1].

### 1.2. Conductive filler/polymer nanocomposites for thermoelectric applications

Thermoelectric devices provide an all solid-state means of heat to electricity conversion. These devices feature many advantages in heat pumps and electrical power generators, such as possessing no moving parts, generating zero noise, being easy to maintain, extended lifetime, and being highly reliable [9, 10]. However, their limited efficiency has restricted their usage to specialized applications, where cost and efficiency are of great importance. Thermoelectric efficiency is expressed in terms of dimensionless figure of merit (ZT),definedasZT=α2σTκ, in which α is Seebeck coefficient, σ is electrical conductivity, κ is thermal conductivity, and T is absolute temperature. The upper limit for thermoelectric energy conversion efficiency is Carnot limit. Several recently developed thermoelectric materials exhibit a ZT near unity, resulting in efficiency equal to 10 % of the Carnot efficiency limit. By reducing the physical dimensionality of the thermoelectric materials (quantum confinement) [11], it is possible to significantly increase Seebeck coefficient and decrease the thermal conductivity. This can lead to ZTs as high as 3 at 550 K in n-type PbSe0.98Te0.02/PbTe quantum-dot superlattices [12]. However, application of these heavy metal thermoelectric materials is associated with high cost of material and production processes, poor processability, and huge adverse environmental impacts [1315]. Accordingly, polymeric materials have drawn great interest to be used in thermoelectric applications. Improved processability, low cost, low thermal conductivity, and low density of polymeric materials are among key potential benefits stimulating the development of polymer-based thermoelectric materials [16, 17]. It is worth mentioning, that the figure of merit in polymer-based thermoelectric materials is ZT ~ 0(10− 3), which necessitates further investigations in this area [18].

Organic polymers exhibit poor electrical conductivity, which necessitates addition of conductive fillers in order to provide high electrical conductivity and reasonable thermoelectric performance. Recent studies suggested, that carbon-based nanofiller/polymer nanocomposites hold a significant promise in development of lightweight, low-cost thermoelectric materials [1923]. As an instance, Yu et al. [17] demonstrated, that by forming a segregated network of carbon nanotubes (CNTs), electrical conductivities as high as 48 S cm−1 were achievable, while Seebeck coefficient and thermal conductivity were marginally impacted by CNT presence. This resulted in a figure of merit of 0.006 at room temperature. These findings render CNT-based polymer nanocomposite as a basis for development of thermoelectric functional materials for future green energy applications.

### 1.3. Nitrogen-doped carbon nanotube/polymer nanocomposites for thermoelectric applications

Different types of conductive nanofillers have been employed to develop CPNs, viz., carbonaceous nanofillers and metallic nanowires [4, 5, 24], among which, CNT has appealed remarkable attention due to its large surface area and outstanding electrical, thermal, and mechanical properties [25, 26]. The luminous era of CNTs initiated in 1991 by their discovery from soot using an arc-discharge apparatus [27]. Like fullerene and graphene, CNTs consist of a sp2 network of carbon atoms. Among these, three carbonaceous poly-types, CNT is the only one produced in large industrial scale. There are two general types of CNTs: multi-walled CNT (MWCNT) and single-walled CNT (SWCNT). MWCNT consists of multiple rolled layers of graphite coaxially arranged around a central hollow core with van der Waals forces between contiguous layers, while SWCNT is made of a single rolled graphene [28]. The global market for CNT primary grades was $158.6 million in 2014 and is anticipated to reach$670.6 million in 2019. CNT/polymer nanocomposites represent, by far, the largest segment in the overall market of CNTs [29].

Manipulating the electronic energy gap of CNTs could lead to their superior performance. Since CNTs are sp2 carbon systems, theoretical [30] and experimental [31] studies showed, that substituting carbon atoms with heteroatoms can result in adjustment of electronic and structural patterns of carbon nanotubes. Nitrogen is the best choice for heteroatom substitution owing to its size proximity to carbon [32]. Bearing in mind, that nitrogen comprises one additional electron as compared to carbon, doping CNTs with nitrogen has emerged as an attractive research topic to improve the electronic properties of CNTs.

Essentially, there are three common nitrogen bonding configurations in nitrogen-doped CNTs (N-CNTs), viz., quaternary, pyridinic, and pyrrolic. As depicted in Figure 2, quaternary nitrogen is directly replaced for C atom in the hexagonal network, is sp2 hybridized, and creates electron-donor state. The pyridinic nitrogen is a part of sixfold ring structure and is sp2 hybridized, and two of its five electrons are localized sole pair. Pyrrolic nitrogen is a portion of a five-membered ring structure, is sp3 hybridized, and gives its remaining two electrons to a π orbital, integrating the aromatic ring [33]. Whereas the quaternary and pyridinic nitrogen lead to side-wall defects, the pyrrolic nitrogen is believed to form internal cappings, generating bamboo-like sections [34]. Besides these three types, there is also possibility for N2 molecules to get trapped inside the tube axis or intercalated into the graphitic layers of N-CNTs.

### Figure 2.

Major types of nitrogen bonding in N-CNTs [35].

Basically, as yet, most of the research studies have investigated the influence of nitrogen doping on electronic properties of CNTs via factors, such as density of states (DOS) and Fermi level; nevertheless, inspecting the electrical properties of polymer nanocomposites containing N-CNTs is still at its infancy [3538]. Hence, the current study aims to research the impact of nitrogen doping on the performance of N-CNT/polymer nanocomposites for thermoelectric applications by studying its influence on electrical conductivity of N-CNT/polymer nanocomposites. N-CNTs were synthesized with different types of catalyst (Co, Fe, and Ni) to obtain diverse nitrogen contents. Afterward, synthesized N-CNTs were melt mixed into polyvinylidene fluoride (PVDF) and compression molded. We evaluated electrical conductivity of the nanocomposites at different loadings and scrutinized the underlying causes behind dissimilar conductivities of the generated nanocomposites. As high electrical conductivity of CPNs is an important factor on their performance as thermoelectric materials, this study goes significantly beyond the state of the art and gives new insight on the role of nitrogen doping on conductivity and therefore performance of CNT nanocomposites for thermoelectric applications.

## 2. Experimental

### 2.1. Materials synthesis

We employed the incipient wetness impregnation technique to produce catalyst precursors. The catalyst precursors were dissolved in water and then impregnated onto aluminum oxide support (Sasol Catalox Sba-200). Thereafter, the developed materials were dried, calcinated, and reduced. Having high solubility and diffusion rate in carbon, Co, Fe, and Ni were chosen as the catalysts [39, 40]. Accordingly, we employed cobalt nitrate hexahydrate, iron (III) nitrate nonahydrate, and nickel (II) sulfate hexahydrate as the catalyst precursors. We set the metal loading at 20 wt.%. The catalyst calcination, reduction, and N-CNT synthesis were performed in CVD setup, detailed in a former study [41]. CVD setup comprised a quartz tubular reactor with an inner diameter of 4.5 cm encapsulated within a furnace. The following steps were implemented for the preparation of the catalysts: first, the catalysts were calcinated under air atmosphere with a flow rate of 100 sccm at 350 °C for 4 h. In this stage, metallic salts were translated into metal oxides. Thereafter, we used a mortar and pestle to achieve a fine powder. Hydrogen gas at a flow rate of 100 sccm at 400 °C for 1 h was utilized to obtain alumina-supported metal catalysts. Afterward, we conveyed a combination of ethane (50 sccm), ammonia (50 sccm), and argon (50 sccm) over the synthesized catalysts. Ethane played the role of carbon source, whereas ammonia and argon were nitrogen source and inert gas carrier, respectively. The synthesis temperature, synthesis time, and catalyst mass were set at 750 °C, 2 h, and 0.6 g, respectively. Catalyst preparation process is elucidated further elsewhere [39].

The polymer matrix utilized for the nanocomposite preparation was semicrystalline PVDF 11008/0001, purchased from 3M Canada, with an average density of 1.78 g/cm3 and melting point of 160 °C. PVDF was opted as the polymer matrix owing to its ferroelectricity, high dielectric strength (~13 kV mm−1), corrosion resistance, good mechanical properties, thermal stability, good chemical resistance (excellent with acid and alkali), and robust interaction of electrophilic fluorine groups with CNTs [4244]. The mixing of synthesized N-CNTs with PVDF matrix was carried out with Alberta Polymer Asymmetric Minimixer (APAM) at 240 °C and 235 rpm. PVDF matrix was first masticated within the mixing cup for 3 min, and then N-CNTs were inserted and mixed for an additional 14 min. For each catalyst, the nanocomposites with different N-CNT concentrations, i.e., 0.3, 0.5, 1.0, 2.0, 2.7, and 3.5 wt.%, were prepared. The nanocomposites were molded into circular cavities with 0.5 mm thickness using Carver compression molder (Carver Inc.) at 220 °C under 38 MPa pressure for 10 min. The molded samples were used for electrical, morphological, and rheological characterizations.

### 2.2.1.1. Transmission electron microscopy of N-CNTs

High-resolution transmission electron microscopy (HRTEM) was used to inspect the morphology of synthesized N-CNTs. HRTEM was conducted on Tecnai TF20 G2 FEG-TEM (FEI) at 200 kV acceleration voltage with a standard single-tilt holder. The images were taken with Gatan UltraScan 4000 CCD camera at 2048 × 2048 pixels. For HRTEM, around 1.0 mg of N-CNT powder was dispersed in 10 mL ethanol and bath sonicated for 15 min. A drop of the dispersion was mounted on the carbon side of a standard TEM grid covered with a ~40 nm holey carbon film (EMS). Measurement of the geometrical dimensions of N-CNTs was conducted for over 100 individual ones utilizing MeasureIT software (Olympus Soft Imaging Solutions GmbH).

### 2.2.1.2. X-ray photoelectron spectroscopy, Raman spectroscopy and thermogravimetric analysis

PHI VersaProbe 5000-XPS was used to obtain X-ray photoelectron spectra. The spectra were achieved employing monochromatic Al source at 1486.6 eV and 49.3 W with a beam diameter of 200.0 μm. The structural defects of N-CNTs were inspected using Raman spectroscopy. Renishaw inVia Raman microscope was used to obtain Raman spectra. Excitation was provided by the radiation of an argon-ion laser beam with 514 nm wavelength. A 5× objective was used to get Raman spectra. The yield of the synthesis process was inspected with Thermogravimetric Analyzer (TA instruments, Model: Q500). The samples were heated under air atmosphere (Praxair AI INDK) from ambient temperature to 950 °C at a rate of 10 °C/min. The samples were kept at 950 °C for 10 min before cooling.

### 2.2.2.1. Light microscopy

The microdispersion state of the nanofillers within PVDF matrix was enumerated using light transmission microscopy (LM) on thin cuts (5 μm thickness) of the compression-molded samples, prepared with Leica Microtome RM 2265 (Leica Microsystems GmbH). Olympus BH2 optical microscope (Olympus Deutschland GmbH) equipped with CCD camera DP71 was used to capture images with dimensions of 600 μm × 800 μm from different cut sections. The software Stream Motion (Olympus) was used to analyze the images. The agglomerate area ratio (in %) was defined by dividing the spotted area of non-dispersed nanofillers (with equivalent circle diameter > 5 μm, area > 19.6 μm2) over the whole sample area (15 cuts, ca. 7.2 mm2). Mean value and standard deviation, demonstrating the differences between the cuts and thus heterogeneity, were reckoned. The relative transparency of the cuts provided added information about the amount of dispersed nanofillers in the samples. The relative transparency was quantified by dividing the transparency of the cut over the transparency of the glass slide/cover glass assembly. Ten various areas per sample were used to obtain mean values and standard deviations. Further information on employing LM to evaluate microdispersion state of nanofillers within nanocomposites is presented elsewhere [45, 46].

### 2.2.2.2. TEM

Ultrathin sections of the samples were cut using ultramicrotome EM UC6/FC6 (Leica) setup with an ultrasonic diamond knife at ambient temperature. The sections were floated off water and thereafter transferred on carbon-filmed TEM copper grids. TEM characterizations were carried out employing TEM LIBRA 120 (Carl Zeiss SMT) with an acceleration voltage of 120 kV.

### 2.2.2.3. Rheology

Rheological measurements were performed using Anton-Paar MCR 302 rheometer at 240 °C using 25 mm cone-plate geometry with a cone angle of 1° and truncation of 47 μm. The thermal stability of the prepared samples was validated by conducting small-amplitude oscillatory shear measurements prior to and following the long-time exposure of the samples to elevated temperatures. Various rheological properties were measured at 240 °C to characterize the linear and nonlinear response for the neat and nanocomposite samples.

### 2.2.2.4. Electrical conductivity

Two conductivity meters with 90 V as the applied voltage were employed to measure the electrical conductivity of the generated materials. For nanocomposites with an electrical conductivity higher than 10−2 S m−1, the measurements were conducted according to ASTM 257-75 standards employing Loresta GP resistivity meter (MCP-T610 model, Mitsubishi Chemical Co.). An ESP probe was used to avert the effect of contact resistance. For electrical conductivities less than 10−2 S m−1, the measurements were carried out with Keithley 6517A electrometer connected to Keithley 8009 test fixture (Keithley Instruments).

## 3. Results and discussion

### 3.1. General background

#### 3.1.1. Mechanisms of electrical conductivity

Electrical conductivity derives from ordered movement of charge carriers (electric current). In the absence of an electric field, the conduction electrons are scattered freely in a solid owing to their thermal energy. If an electric field, E, is applied, the force on an electron, e, is –eE, and the electron is accelerated in the opposite direction to the electric field because of its negative charge. Accordingly, there is a net velocity and the current density is presented by [47]:

 J=Ne·e·μ·E (1)

where J is the current density, Ne is concentration of electrons, e is charge of electron, μ is the electron mobility, and E is the applied electric field. The applied electric field equals to the applied voltage over the thickness of a sample. Hence, the electrical conductivity can be determined as:

 σ=JE (2)

where σ is electrical conductivity and its SI unit is Siemens per meter (S m−1). Electrical conductivity of materials is an intrinsic property, which spans a very wide range. The conductivity of insulators is typically less than 10−10 S m−1, that of semiconductive materials covers the range 10−10 to around 10−2 S m−1, and for semimetals and metals is more than 10−2 S m−1.

Electrical conductivity of materials can be elucidated employing the band theory [48]. In the band theory, the energy level of each electron is reflected as a horizontal line. As any solid possesses a large number of electrons with various energy levels, the sets of energy levels form two continuous energy bands, named valence band and conduction band. The energy gap between the two bands signifies the forbidden region for electrons. Electrons restrained to individual atoms or interatomic bonds are, in the band theory, said to be in valence band. Those electrons, that can move freely in substance upon applying electric field lie in conduction band. Figure 3 depicts a schematic of the bands in a solid identifying three main types of materials: insulators, semiconductors, and metals. Valence and conduction bands in metals overlap each other; therefore, metals indicate very high conductivity. In intrinsic semiconductors, the valence-conduction band gap is adequately small, so that, electrons in valence band can be excited to conduction band by thermal energy. Among the three types of materials illustrated in Figure 3, insulators show the largest valence-conduction band gap, and, therefore, fewer electrons can be excited to their conduction band by thermal energy. This results in a very low conductivity in insulators.

### Figure 3.

Simplified diagram of the electronic band structure in the band theory, reproduced from [34].

#### 3.1.2. Electrical conductivity in CPNs

High electrical conductivity, i.e., conductive network formation, at very low filler contents has made CPNs distinctive materials for industrial applications [25, 26]. Conductive network formation in CPNs is better understood with the concept of percolation threshold [49, 50]. Percolation means, that at least one conductive pathway forms to allow electrical current to pass across CPNs, thereby transforming CPNs from insulative to conductive. Percolation happens at a narrow filler concentration range, where the electrical conductivity of CPNs drastically increases by several orders of magnitude. Low electrical percolation threshold in CPNs leads to the production of cost-effective composites.

Many statistical, geometric, thermodynamic, and structure-based models have been introduced to anticipate the percolation threshold and electrical conductivity of CPNs [49, 51]. Although the percolation theory is just valid at conductive filler concentrations above the percolation threshold, it is the most acceptable one. Statistical percolation theory estimates the percolation threshold of CPNs as:

 σ=σ0∙V-Vct (3)

where σ is electrical conductivity of CPN, σ0 is electrical conductivity of conductive filler, V is dimensionless volume content of conductive filler, and Vc and t are percolation threshold and critical exponent, respectively [49]. The equation is valid for filler concentration above the percolation threshold, i.e., V > Vc. Higher t value and lower percolation threshold correspond to well-dispersed, high-aspect-ratio fillers [5254].

Figure 4 illustrates a typical percolation curve of CPNs [55]. In general, percolation curve of CPNs can be divided into three regions: (1) region far below the percolation threshold (insulative region), (2) region where percolation occurs (percolation region), and (3) region far above the percolation threshold (conductive region). In the insulative region, the conductive filler loading is very low with the fillers far from each other; thus, polymer matrix dominates the charge transfer. As a matter of fact, at low filler concentrations, the insulating gaps are very large and the chance, that nomadic charge carriers are transferred between conductive fillers is very low.

By enhancing filler loading, the gaps between conductive fillers decrease, and a drastic increase in electrical conductivity is observed over a narrow concentration range (percolation region). In this region, hopping and direct-contact mechanisms become significant. When the mean particle-particle distance reaches below 1.8 nm, the dominant electron transfer mechanism become hopping mechanism [5658]. It is reported, that the presence of large conductive agglomerates in CPN results in a very high secondary internal electric field between the conductive islands [57, 59]. This high field strength assists free electrons in conductive filler having adequate energy to hop over the insulative gaps. Nevertheless, hopping takes place when an electron receives sufficient energy to pass over distance to nearest free site with lower energy to alter its lattice site. In the percolation region, due to proximity or direct contact of conductive fillers, the nomadic charge carriers in conductive fillers play the dominant role in conduction mechanism. Since these free charge carriers belong to the conduction band, the conductivity of the nanocomposite rises by several orders of magnitude in the percolation region. Next, by adding more filler loading, a well-developed, 3D conductive network initiates to form, but the electrical conductivity increases only marginally. This is due to substantial current dissipation at the contact spots between conductive fillers, i.e., the constriction resistance, leading to a plateau in the percolation curve [26].

### Figure 4.

Percolation curve of compression-molded CNT/polystyrene nanocomposite (a typical percolation curve of CPNs) [55].

### 3.2. Electrical conductivity of N-CNT/PVDF nanocomposites

The percolation curves of N-CNT/PVDF nanocomposites are depicted in Figure 5. It was observed, that (N-CNT)Co/PVDF nanocomposites presented the lowest percolation threshold (1.5 wt.%) and highest electrical conductivity (3 S m−1 at 3.5 wt.%). However, it was revealed, that Ni-based nanocomposites were insulative up to 2.7 wt.% and experienced a slight increase in electrical conductivity at 3.5 wt.%. The Fe-based nanocomposites presented an increase in electrical conductivity from 1.0 wt.% to 3.5 wt.% with a mild slope.

### Figure 5.

Electrical conductivity of N-CNT/PVDF nanocomposites as a function of N-CNT content. N-CNTs were synthesized over Co, Fe, and Ni catalysts.

There are many factors impacting the electrical conductivity of CPNs, such as loading, intrinsic conductivity, size, and aspect ratio of conductive filler, inherent properties of polymer medium, interfacial properties of CPN constituents, dispersion and distribution of filler, blending method, and crystalline structure of the matrix. The impacts of the aforementioned parameters on electrical conductivity of CNT/polymer nanocomposites have been well reviewed in the literature [1, 6062]. Accordingly, in succeeding section, we scrutinize structural and morphological features of N-CNTs and their nanocomposites to figure out the reasons behind different electrical behaviors of the generated nanocomposites.

### 3.3. Morphological and structural characterization of N-CNTs

The morphology and graphitic structure of N-CNTs were analyzed using TEM images. Figure 6 indicates, that the type of synthesis catalyst played a leading role in creating the final morphology of N-CNTs. As depicted in Figure 6, we observed an open-channel morphology for (N-CNT)Co and a bamboo-like morphology for (N-CNT)Fe and (N-CNT)Ni. Surface roughness is observed in bamboo-like N-CNTs, deriving from defected bonding of bamboo-like sections. The flawed parts in the wall of N-CNTs are attributed to replacement of nitrogen atoms [63, 64]. Since an open-channel structure was formed for (N-CNT)Co, we can say, that other factors, than nitrogen bonding, are involved in creation of bamboo-like morphology, such as type of catalyst.

### Figure 6.

TEM micrographs of N-CNTs synthesized over Co, Fe, and Ni catalysts.

Table 1 tabulates average length and diameter, nitrogen content, Raman feature, and synthesis yield of synthesized N-CNTs. We perceived, that N-CNTs synthesized over Fe catalyst had the largest diameter, over double that of (N-CNT)Ni. Statistical analysis of particle size of the catalysts revealed a good correlation between diameter of N-CNTs and size of catalyst particles. Discrepancies in original size of the catalyst particles and also dissimilar tendencies of the catalyst particles to sinter at synthesis temperatures are among significant parameters affecting the variation in diameter of N-CNTs. It should be noted, that metallic nanoparticles with sizes below 10 nm experience a drastic drop in melting point [65]. High synthesis temperature range (600–1000 °C) coupled with exothermic thermal decomposition of the precursor molecules results in higher temperature than nominal reaction temperature, contributing to metal liquefaction and coalescence of catalyst particles [66, 67].

CoFeNi
Length (μm)2.62.61.2
Diameter (nm)254620
Nitrogen content (at.%)2.22.23.3
ID/IG0.790.730.81
Synthesis yield %89.585.163.9

### Table 1.

Physical and structural features of N-CNTs synthesized over Co, Fe, and Ni catalysts.

Table 1 shows, that (N-CNT)Co and (N-CNT)Fe had average length about 2.6 μm, while (N-CNT)Ni exhibited considerably lower average length about 1.2 μm. The shorter length of (N-CNT)Ni can be attributed to either inferior activity of Ni catalyst, as will be shown by TGA, or the presence of larger amount of nitrogen in their structure, as will be exhibited by X-ray photoelectron spectroscopy (XPS) analysis. The presence of nitrogen can be envisaged as an important factor to bend, close, and cap N-CNTs. It is worth noting, that average length and diameter of N-CNTs are of high significance for electrical applications, since CNTs with high aspect ratio provide CPNs with superior electrical performance [68, 69].

The amount of nitrogen content can have a weighty effect on morphological, physical, and electronic properties of N-CNTs. The achieved data revealed, that atomic content of nitrogen incorporated into (N-CNT)Ni was 3.3 at.%, whereas (N-CNT)Co and (N-CNT)Fe had considerably lower nitrogen content, i.e., 2.2 at.%. As nitrogen could have the effect of closing the tube structures and thereby developing more disordered, bent, and capped structures, the larger nitrogen content of N-CNTNi could be a contributing factor to its lower length.

In Raman spectra of CNTs, tangential mode (G band) and defect-active mode (D band) offer valuable information about physical and electronic structure of CNTs [70, 71]. Hence, Raman spectroscopy was used to inspect the influence of nitrogen doping on physical and morphological features of N-CNTs. G band (~1600 cm−1) derives from the stretching of C─C bond in graphitic materials and is mutual to all sp2 carbon forms. D band (~1400 cm−1) is double-Raman scattering process, which requires lattice distortion to break the basic symmetry of the graphitic structure [72]. Therefore, the presence of structural defects stimulates D band feature. Accordingly, the ratio of D and G band intensities is often used as indicative tool to validate the structural perfection of CNTs [73]. We observed, that (N-CNT)Ni had the uppermost ID/IG ratio, signifying the poorest crystallinity. These results are in line with TEM images of (N-CNT)Ni, indicating poorer crystalline morphology than the other forms of N-CNTs. Moreover, Villalpando-Paez et al. [74] and Ibrahim et al. [75] reported good correlation between nitrogen concentration and ID/IG ratio. This is in agreement with our study and shows the opposing influence of nitrogen doping on the crystalline structure of N-CNTs. We also observed, that (N-CNT)Ni went through more breakage during the melt mixing process, ascribed to its poorer crystallinity.

TGA analysis helped investigate the synthesis yield. We obtained residues of 11.5 %, 14.9 %, and 36.1 %, relative to original mass, for (N-CNT)Co, (N-CNT)Fe, and (N-CNT)Ni, respectively. The residue consists of metallic oxide particles and alumina substrate [41, 76]. The higher the yield of the synthesis process, the lower is the amount of the remaining residue. Thus, we can claim, that Ni catalyst had an inferior performance compared to Co and Fe catalysts. The catalyst particles contained 80 wt.% alumina and 20 wt.% metallic particles. Alumina is insulative and metallic particles have much less surface area than synthesized N-CNTs, and their surface area even further reduced due to sintering phenomenon. This justifies the significance of synthesis yield on electrical properties.

### 3.4. Morphological characterization of N-CNT/PVDF nanocomposites

The dispersion state of conductive filler within polymer matrix is intensely influential on electrical properties. Hence, we inspected the dispersion state at three various scales. Microdispersion state of N-CNTs within the polymer medium was investigated via LM. LM talks about the portion of fillers, that appears as big agglomerates and is not disentangled well, and was enumerated as the agglomerate area ratio in our study. Moreover, gray appearance of LM samples helps us quantify the agglomerates with sizes equal to or slightly larger than the wavelength of visible light, ca. 400–700 nm, but smaller than visually identifiable agglomerates. Darker background denotes more nanotubes dispersed in this range. We also employed TEM to obtain information about nanodispersion state of carbon nanotubes, i.e., how well carbon nanotubes disentangle individually.

Figure 7 portrays examples for LM images of three different nanocomposites, corresponding to different synthesis catalysts, with 2.0 wt.% N-CNT content. Quantification of the agglomerate area ratio, as shown in Table 2, illustrates the lowest agglomerate area ratio for samples containing Fe-based N-CNTs, followed by Co and Ni. The corresponding relative transparency values indicate the lowest value for Co-based N-CNTs, followed by Fe-based and Ni-based.

### Figure 7.

LM images of microtomed sections of 2.0 wt.% N-CNT/PVDF nanocomposites. N-CNTs were synthesized over Co, Fe, and Ni catalysts. The red squares represent areas employed for relative transparency quantifications.

CoFeNi
Agglomerate area ratio %2.31.82.8
Relative transparency %375386

### Table 2.

LM microdispersion parameters of microtomed N-CNT/PVDF nanocomposites with N-CNTs synthesized over different catalysts.

TEM images look into nanodispersion state of N-CNTs in PVDF medium (Figure 8). The images clearly show, that (N-CNT)Ni had the worst dispersion state. TEM image of (N-CNT)Ni/PVDF nanocomposite shows a few individual nanotubes beside fairly large agglomerates. (N-CNT)Co presented the best state of nanodispersion, while (N-CNT)Fe/PVDF nanocomposites held small agglomerates with sizes around 500 nm. In conclusion, microscopy images showed, that (N-CNT)Co and (N-CNT)Fe had better both microdispersion and nanodispersion than their Ni-based counterpart. Co-based and Fe-based N-CNTs indicated only marginal discrepancies in their dispersion state.

### Figure 8.

TEM images of 2.0 wt.% N-CNT/PVDF nanocomposites with N-CNTs synthesized with different catalysts, illustrating nanodispersion state of N-CNTs.

### 3.5. Linear and nonlinear melt-state rheological response of N-CNT/PVDF nanocomposites

Figure 9 depicts storage modulus (G′) and loss modulus (G″) of N-CNT/PVDF nanocomposites as a function of frequency under small-amplitude oscillatory shear (γ = 1 %) for a frequency range from 0.1 rad/s to 625 rad/s at 240 °C.

As shown in Figure 9, G′ in low frequency region (ω ~ 0.1 rad/s) is significantly larger than G″ (a damping factor smaller than unity) for (N-CNT)Co/PVDF and (N-CNT)Ni/PVDF nanocomposites at concentrations as low as 0.5 wt.%. (N-CNT)Fe/PVDF nanocomposite samples, however, exhibited an elastic dominant response (G′ > G″) only at very high nanofiller concentrations (~2.0 wt.%). It is noticeable, that all N-CNT/PVDF nanocomposites, regardless of synthesis catalyst, showed a signature for the existence of an ultraslow relaxation process (a near-zero slope for G′ in low frequency region) at concentrations as low as 0.5 wt.%. This indicates, that linear rheological response was affected by the presence of N-CNTs at concentrations, that no significant enhancement in electrical conductivity was observable in N-CNT nanocomposites.

The linear melt-state rheological response is mainly controlled by several factors, such as inter-tube van der Waals interactions, micro- and nanodispersion states, individual CNT stiffness, and CNT network stiffness [7779]. Individual CNT stiffness is mainly controlled by intra-wall C─C bond strength and graphitic interlayer load transfer [80]. This suggests, that structural imperfections and defects in CNT graphitic walls can deteriorate their elastic properties. Moreover, stiffness of the network structure formed by CNT bundles is mainly determined by the load transfer across CNT/polymer and CNT-CNT interface [79]. In this context, it could be mentioned, that a scenario entirely based on individual CNT stiffness may not be able to fully describe the observations for elastic response of N-CNT nanocomposites in low frequency region. As can be seen in Figure 9, Co-, Fe-, and Ni-based N-CNT nanocomposites reached a storage modulus of 9640 Pa, 2290 Pa, and 3860 Pa at 0.1 rad, respectively. The presence of higher amount of structural imperfections may not be responsible for lower elasticity observed for Fe-based N-CNT nanocomposite as (N-CNT)Fe showed the lowest ID/IG. Therefore, the main contributing factors to linear melt-state rheological observations could be considered as the dispersion state and load transfer across the interfacial region.

### Figure 9.

Small amplitude oscillatory shear response at γ = 1 % and T = 240 °C for neat PVDF and N-CNT/PVDF nanocomposites with N-CNTs synthesized over different catalysts.

Figure 10 depicts oscillatory amplitude sweep response of neat PVDF and N-CNT/PVDF nanocomposites containing 3.5 wt.% N-CNTs synthesized over different catalysts over a range of applied strain amplitudes from 0.1 to 1000.0 % at an angular frequency of 0.1 rad/s. The responses observed for neat PVDF and N-CNT/PVDF nanocomposite samples demonstrated a transition from a linear regime to a nonlinear regime and also a drop in G′ as strain amplitude increases. It is noticeable, that in low-strain region, all nanocomposite samples exhibited elastic dominant response (G′ > G″). These results also feature a crossover strain amplitude γx (G′ = G″), which is a measure of N-CNT network sensitivity to deformation-induced microstructural changes. As shown in Figure 10, Co-, Fe-, and Ni-based N-CNT nanocomposites exhibited crossover strain amplitudes of 32.0 %, 5.8 %, and 5.6 %, respectively. This implies, that Co-based N-CNTs featured a very resilient behavior toward the applied deformation field and the stress-bearing backbone of the fractal clusters survived up to strain amplitudes one order of magnitude larger than N-CNTs synthesized over Fe and Ni. It is noticeable, that Ni-based N-CNT nanocomposite showed a multistep transition into a nonlinear regime as G′ dropped to an intermediate plateau and then significantly decreased. Moreover, the first step decrease in G′ in (N-CNT)Ni nanocomposite was accompanied by a dissipation process signified by a weak local peak in G″.

### Figure 10.

Oscillatory amplitude sweep response of neat PVDF and N-CNT/PVDF nanocomposites containing 3.5 wt.% N-CNTs synthesized over different catalysts for strain amplitudes of γ0 = 0.1–1000 % at an angular frequency of ω = 0.1 rad/s. The insets show the non-dimensionalized elastic Lissajous loops for strain amplitudes indicated by solid lines.

Insets in Figure 10 depict non-dimensionalized elastic Lissajous loops [8183], in which normalized torque Mnorm. is plotted as a function of normalized deflection angle φnorm.. At small strain amplitudes (γ ~ 1.0 %), for neat PVDF and nanocomposites, elastic Lissajous loops were elliptical, corresponding to a linear viscoelastic response. The area enclosed by elastic Lissajous loops is significantly smaller in N-CNT nanocomposites than neat PVDF, indicating an elastic dominant response in this region. As strain amplitude increased, Lissajous loops in N-CNT nanocomposite samples became distorted, indicative of thixotropy and a yielding process in nanocomposite samples. Furthermore, observed patterns for nanocomposites suggest that N-CNT at-rest microstructure partially survived in both weakly (γ ~ γx) and strongly (γ > γx) nonlinear regimes as the area enclosed by Lissajous loops is relatively smaller, than that observed for neat PVDF. The area enclosed by elastic Lissajous loops in weakly and strongly nonlinear regimes for N-CNT nanocomposites showed the following order: Ni > Fe ~ Co.

As demonstrated by LM observations, Ni-based N-CNT nanocomposite presented the highest agglomerate area and relative transparency, indicating a poor dispersion quality of N-CNTs within PVDF matrix. This could be responsible for observing a multistep transition into a nonlinear regime and strongly nonlinear response at intermediate strain amplitudes in Ni-based N-CNT samples. As explained in the preceding section, N-CNTs synthesized over different catalysts demonstrated fundamentally different dispersion states at different scales. The presence of densely aggregated N-CNT structures in Ni-based N-CNT nanocomposite led to poor load transfer across polymer-aggregate interface, resulting in deformation-induced microstructural changes initiated from aggregate-aggregate boundaries at intermediate strain amplitudes (γ ~ γx). This was followed by widespread disintegration of (N-CNT)Ni aggregates, marked by multistep transition into a nonlinear regime. However, in Co-based and Fe-based N-CNT nanocomposites, the transition into nonlinear regime occurred by stochastic erosion [84, 85] of network structures formed by individually dispersed N-CNTs bound polymer chains and polymer matrix entanglement network [86, 87].

In this context, it could be added, that Fe-based N-CNT nanocomposites demonstrated a dual nature in a sense, that it showed an almost one-step transition into a nonlinear regime; however, the crossover point occurred at fairly small strain amplitude (γx = 5.8 %). This dual behavior can be explained in conjunction with poorer load transfer across interfacial region than Co-based N-CNT nanocomposite as a result of denser N-CNT clusters present in Fe-based N-CNT nanocomposite. Moreover, one-step transition to a nonlinear regime compared to the multistep transition observed for Ni-based N-CNT nanocomposite can be attributed to better nanodispersion state achieved in Fe-based N-CNT nanocomposite (see TEM images in Figure 8 and relative transparency values in Table 2). Overall, it can be expressed, that no direct link between individual N-CNT structural features and rheological response was detectable, and thus N-CNT dispersion state played the main role in determining the melt-state rheological response.

## 4. Conclusions

In brief, this study revealed, that electrical conductivity of N-CNT/PVDF nanocomposites is highly dependent on N-CNT synthesis catalyst. Measuring electrical conductivity of the generated nanocomposites showed superior electrical conductivity and, thus, thermoelectric performance in the following order of the synthesis catalyst: Co > Fe > Ni. It was observed, that a combination of high synthesis yield, high aspect ratio, low structural defects, enhanced network formation, and good state of N-CNT dispersion can provide N-CNT/PVDF nanocomposites with superior electrical conductivity. Moreover, it was revealed, that nitrogen doping had an adverse impact on electrical conductivity of CNT/polymer nanocomposites and, therefore, their performance as thermoelectric materials.

## Acknowledgements

Financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) is highly appreciated. We would like to thank Prof. Uttandaraman Sundararaj for his supervision to perform this project. We are grateful to Dr. Lars Laurentius for his assistance with Raman spectroscopy. In addition, we thank Dr. Petra Pötschke and Ms. Uta Reuter from IPF Dresden for LM and TEM investigations. Dr. Mohammad Arjmand thanks IPF Dresden for granting a research stay.

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minimal polynomial calculator

Get the free "Max/Min Finder" widget for your website, blog, Wordpress, Blogger, or iGoogle. If we had two such polynomials, they must both have the same degree and the same leading coeﬃcient 1, and so their diﬀerence is a polynomial of smaller degree which still gives 0 when applied to A. In field theory, a branch of mathematics, the minimal polynomial of a value α is, roughly speaking, the polynomial of lowest degree having coefficients of a specified type, such that α is a root of the polynomial. The Minimal Polynomial By the Cayley-Hamilton theorem, there is a nonzero monic polynomial that kills a linear operator A: its characteristic polynomial.2 De nition 4.1. The #1 tool for creating Demonstrations and anything technical. Walk through homework problems step-by-step from beginning to end. A simple online degree and leading coefficient calculator which is a user-friendly tool that calculates the degree, leading coefficient and leading term of a given polynomial in a simple manner. It can also be said as the roots of the polynomial equation. Characteristic Polynomial Calculator. By using this website, you agree to our Cookie Policy. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Free polynomial equation calculator - Solve polynomials equations step-by-step This website uses cookies to ensure you get the best experience. This is the main site of WIMS (WWW Interactive Multipurpose Server): interactive exercises, online calculators and plotters, mathematical recreation and games The generator polynomial has the following three important properties [15,17–19,22,24–26]: 1. The minimal polynomial of an algebraic number is the unique irreducible monic polynomial of smallest degree with rational coefficients such that and whose leading coefficient is 1. If the minimal polynomial of α exists, it is unique. Dividing Polynomials Calculator is a free online tool that displays the result for the division of two polynomials. Hints help you try the next step on your own. BYJU’S online dividing polynomials calculator tool makes the calculation faster, and it displays the quotient in a fraction of seconds. De nition: Minimal Polynomial Let F be a nite eld of characteristic p, and let a2F. The zeros of a polynomial equation are the solutions of the function f(x) = 0. polynomial with rational coe cients. Explore anything with the first computational knowledge engine. Hence the JCF must be B 2, the non-diagonal one. SEE: Algebraic Number Minimal Polynomial, Extension Field Minimal Polynomial, Matrix Minimal Polynomial. Thus, we could have determined that this was the minimal polynomial for them without calculation. Find the zeros of an equation using this calculator. Wolfram Web Resources. 4. 4, 2008 An algorithm for the calculation of the minimal polynomial S. BIAŁAS1∗and M. BIAŁAS2 1 The School of Banking and Management, 4 Armii Krajowej St., 30-150 Kraków, Poland 2 Faculty of Management, AGH University of Science and Technology, 30 Mickiewicza St., 30-059 Kraków, Poland The generator polynomial and parity-check polynomial are factors of x n − 1. Join the initiative for modernizing math education. Interpolation Calculator. Minimal Polynomial. Minimal Polynomial Let V be a vector space over some eld k, and let : V -V be a linear map Any other polynomial … This online calculator calculates coefficients of characteristic polynomial of a square matrix using Faddeev–LeVerrier algorithm person_outline Timur schedule 2019-06-17 05:34:04 In linear algebra, the characteristic polynomial of a n×n square matrix A is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. Compute the minimal polynomials m s(x)where sruns over the set of representatives of cyclotomic cosets. Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. 2. Polynomial calculator - Parity Evaluator ( odd, even or none ) Polynomial calculator - Roots finder A quadratic equation is a second degree polynomial having the general form ax^2 + bx + c = 0, where a, b, and c Read More High School Math Solutions – Quadratic Equations Calculator, Part 2 Form a polynomial f(x) with real coefficients having the given degree and zeros. Recall that irreducible polynomials are required to be monic, and therefore a minimal polynomial m(x) for an element ais always a monic polynomial. KEEP LEARNING. To find the coefficients of the minimal polynomial of A, call minpoly with one argument. Minimal Polynomial and Jordan Form Tom Leinster The idea of these notes is to provide a summary of some of the results you need for this course, ... and it’s easy to calculate that (t 1) (t k) is an annihilating polynomial for . The properties and the applications of the minimal poly- A value of x that makes the equation equal to 0 is termed as zeros. 2 Minimal Polynomials and Jordan Canonical Forms Theorem 2.3. The minimal polynomial m A of A is the (unique) nonzero monic3 polynomial m of lowest possible degree, such that m(A) = 0 . ), with steps shown. The minimal polynomial can be computed using MinimalPolynomial [ zeta, var] in the Wolfram Language package AlgebraicNumberFields. Find The minimal polynomials for (at least some of) the k’s? The generator polynomial of an (n,k) cyclic code is unique (usually proved by contradiction). Then is an eigenvalue of T if and only if m( ) = 0. But: k+1 2k+1 = n (because k > 0) would contradict that x^n is the minimal polynomial of T. txt) or read online for free. For Polynomials of degree less than or equal to 4, the exact value of any roots (zeros) of the polynomial are returned. The monic polynomial ψ(λ) of least degree for which ψ(A) = 0 ∈ Mn is called the minimal polynomial of the matrix A∈ Mn. Three of these, s 3, s 4, and s 24, are really special, so specify Field Minimal Polynomial, Matrix Minimal In linear algebra, the minimal polynomial μ A of an n × n matrix A over a field F is the monic polynomial P over F of least degree such that P(A) = 0.Any other polynomial Q with Q(A) = 0 is a (polynomial) multiple of μ A.. Since A is numeric, minpoly returns coefficients as double-precision values: to Wolfram Alpha will compute the minimal polynomial of $\sqrt[5]{2}\left(1-\exp\left(\frac{2\pi i}{5}\right)\right)$. Matrix Theory: We apply the minimal polynomial to matrix computations. MINIMAL POLYNOMIAL IN HINDI. MinimalPolynomial[s, x] gives the minimal polynomial in x for which the algebraic number s is a root. Proof. KEEP WATCHING. What this means for a … Examples = i ) x2 + 1 = 0 = 4 p 3 p 9 + 7 3 p 3) x6 + 504 519 x4 2048 519 This polynomial, M ... algebraic number (norm calculator). Wolfram Alpha does not, alas, have a minimal polynomial calculator built-in, so you cannot get this from their search bar (it will give you the characteristic polynomial instead.) This Polynomial Calculator return the polynomials representing the sum and the difference of the two polynomials P1 and P2. High School Math Solutions – Quadratic Equations Calculator, Part 1. Finite field calculator This tool allows you to carry out algebraic operations on elements of a finite field. Moreover, I want to compute a minimal polynomial of linear combinations of algebraic numbers. By using this website, you agree to our Cookie Policy. Polynomial graphing calculator This page help you to explore polynomials of degrees up to 4. Free matrix Characteristic Polynomial calculator - find the Characteristic Polynomial of a matrix step-by-step This website uses cookies to ensure you get the best experience. However, I do not know how to compute a minimal polynomial of an algebraic number that is not expressible in radicals. But this would contradict the minimality of m. 56, No. Show Instructions. This website uses cookies to ensure you get the best experience. The calculator will try to factor any polynomial (binomial, trinomial, quadratic, etc. MINIMAL POLYNOMIAL EXAMPLES. In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. A finite field K = 𝔽 q is a field with q = p n elements, where p is a prime number. The minimal polynomial of A is unique. Minimal Polynomial and Jordan Form Tom Leinster The idea of these notes is to provide a summary of some of the results you need for this course, as well as a di erent perspective from the lectures. EXAMPLE 3 Consider the eld Z It can calculate and graph the roots (x-intercepts), signs , Local Maxima and Minima , Increasing and Decreasing Intervals , Points of Inflection and Concave Up/Down intervals . Knowledge-based programming for everyone. The minimal polynomial Michael H. Mertens October 22, 2015 Introduction In these short notes we explain some of the important features of the minimal polynomial of a square matrix Aand recall some basic techniques to nd roots of polynomials of small degree which may be useful. A quadratic equation is a second degree polynomial having the general form ax^2 + bx + c = 0, where a, b, and c... High School Math Solutions – Quadratic Equations Calculator, Part 2. The following methods are used: factoring monomials (common factor), factoring quadratics, grouping and regrouping, square of sum/difference, cube of sum/difference, difference of squares, sum/difference of cubes, the rational zeros theorem. Related Calculators. computes elements in a finite field. Any multiple of the generator polynomial is a codeword polynomial. Input the polynomials here: P1 = P2 = How to input. Able to display the work process and the detailed explanation. The minimal polynomial of a vector is a factor of the minimal polynomial of a linear transform Hot Network Questions Is this mathematical answer to randomness question philosophically satisfying? Polynomial calculator - Division and multiplication. For the minimal polynomial of an algebraic element of a field, see Minimal polynomial (field theory). Let d =the lowest possible degree mentioned above. Indeed, it must do since this is the characteristic polynomial of A and a result in the lectures tells us that this must be the case - the new information here is that the characteristic polynomial and the minimal polynomial coincide. Find more Mathematics widgets in Wolfram|Alpha. So this is the minimal polynomial. Cayley-Hamilton =)degm A n for n n matrices A. Uniqueness. Compute Coefficients of Minimal Polynomial. Interpolation Calculator.Input the set of points, choose one of the following interpolation methods (Linear interpolation, Lagrange interpolation or Cubic Spline interpolation) and click "Interpolate".The interpolation calculator will return the function that best approximates the given points according to the method chosen. The next question is to nd a matrix P such that P 1AP = B 2. Let mbe the minimal polynomial of Tand 2F. But what if the quadratic equation... To create your new password, just click the link in the email we sent you. This online calculator finds the roots of given polynomial. BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES Vol. The Problem The Motivation The SolutionEnd CM Points on the Shimura Curve There is a collection of \special" points fs kgˆS 6 called complex multiplication (CM) points. Enter a Polynomial Equation (Ex:5x^7+2x^5+4x^8+x^2+1) Eric Errthum Finding Minimal Polynomials with a Norm Calculator. Polynomial Factorization Calculator - Factor polynomials step-by-step This website uses cookies to ensure you get the best experience. 3. The nonzero monic polynomial in F[T] that kills Aand has least degree is called the minimal polynomial of Ain F[T]. +a1λ+a0, where an 6= 0 , is called monic if its leading coeﬃcient an = 1. Solving quadratics by factorizing (link to previous post) usually works just fine. A minimial polynomial for ais an irreducible polynomial m(x) 2Z p[x] such that m(a) = 0. Unlimited random practice problems and answers with built-in Step-by-step solutions. Polynomial. As another observation from this example, recall that the minimal polynomials are irreducible over GF(2) and are divisors of … primitive) polynomial x3 + x2 + 1, since it is the reciprocal of the original polynomial. In linear algebra, the minimal polynomial μA of an n × n matrix A over a field F is the monic polynomial P over F of least degree such that P(A) = 0. The characteristic polynomial of a matrix m may be computed in the Wolfram Language as. The calculator will show you the work and detailed explanation. SEE: Algebraic Number Minimal Polynomial, Extension Eric Errthum Finding Minimal Polynomials with a Norm Calculator. The calculator will find the characteristic polynomial of the given matrix, with steps shown. tice that x2 3x = x(x 3) divides the characteristic polynomial f A(x) = x2(x 3) for this A . DEFINITION 3. Note: the polynomial m s(x) = Q m s i=0 (x ˘ sqi), where to open up the brackets we perform operations in F qm We obtain the factorization in the form xn 1 = Q s m s(x) Example: Suppose we want to factor x9 1 over F 2:We get m= 6 since However, you can open a WA workbook and copy the code from Matrix Minimal Polynomial and execute it online: Practice online or make a printable study sheet. The following three statements are equivalent: λ is a root of μ A,; λ is a root of the characteristic polynomial χ A of A,; λ is an eigenvalue of matrix A. A common method for constructing minimal polynomials for numbers that are expressible over ℚ is “backwards ”: The number can be set equal to x, and the equation can be algebraically manipulated until a monic polynomial in ℚ  [x] is equal to 0. At least some of ) the k ’ s online Dividing polynomials calculator is a free online tool that the! Hints help you try the next step on your own has the following three minimal polynomial calculator properties [ 15,17–19,22,24–26 ] 1. Anything TECHNICAL your new password, just click the link in the Wolfram package! N for n n matrices A. Uniqueness eric Errthum Finding minimal polynomials with a Norm.!  5 * x  MinimalPolynomial [ zeta, var ] in the Wolfram Language package . Polynomial, Extension Field minimal polynomial display the work and detailed explanation eigenvalue T. High School Math solutions – quadratic Equations calculator, Part 1 contradict the minimality m...., I want to compute a minimal polynomial minimal polynomial can be computed in the Wolfram package. Problems and answers with built-in step-by-step solutions ) where sruns over the set of representatives of cyclotomic cosets characteristic... This website uses cookies to ensure you get the best experience process and the of! 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[ s, x ] gives the minimal polynomial polynomials with a Norm calculator ! Two polynomials P1 and P2 α exists, it is unique 2 minimal polynomials with a calculator... This polynomial calculator return the polynomials representing the sum and the detailed.... The next question is to nd a matrix P such that P 1AP = B 2, non-diagonal! A free online tool that displays the quotient in a fraction of seconds 2 minimal polynomials m s x. Call minpoly with one argument following three important properties [ 15,17–19,22,24–26 ]: 1 must be B,... A finite Field calculator this tool allows you to carry out algebraic operations elements... ) Interpolation calculator that is not expressible in radicals usually proved by contradiction ) means for a Dividing. Sciences Vol calculator tool makes the equation equal to 0 is termed as.! N, k ) cyclic code is unique enter minimal polynomial calculator polynomial equation are the solutions of minimal. 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Create your new password, just click the link in the Wolfram Language as representing the sum and detailed! Generator polynomial and parity-check polynomial are factors of x n − 1 Number minimal polynomial such. Also be said as the roots of the original polynomial high School Math solutions – quadratic calculator! Unlimited random practice problems and answers with built-in step-by-step solutions for ( at least some of ) the k s... Steps shown the zeros of a finite Field calculator this tool allows you to carry out algebraic on... P such that P 1AP = B 2 sent you and answers with built-in step-by-step.... With one argument polynomial can be computed in the Wolfram Language as )... Of given polynomial is not expressible in radicals an 6= 0, is called monic its...: 1 the minimality of m. the generator polynomial has the following three properties... For the division of two polynomials calculator - Factor polynomials step-by-step this website, you can skip the sign. The function f ( x ) = 0 = how to compute a minimal polynomial can be computed using [... To compute a minimal polynomial of an algebraic Number that is not expressible in radicals that displays the for. Display the work and detailed minimal polynomial calculator the roots of given polynomial usually proved by contradiction ) without calculation =.! Polynomial has the following three important properties [ 15,17–19,22,24–26 ]: 1 can be computed using MinimalPolynomial s. Polynomials and Jordan Canonical Forms Theorem 2.3 your own a, call minpoly with one argument through problems! Norm calculator n − 1 x ] gives the minimal polynomial for them without calculation ( usually proved by )...: 1 Ex:5x^7+2x^5+4x^8+x^2+1 ) Interpolation calculator MinimalPolynomial [ s, x ] gives the minimal polynomial can be computed the! What if the minimal polynomial, Extension Field minimal polynomial, matrix polynomial... Quadratic, etc Number minimal polynomial = B 2 matrix, with steps shown detailed.! 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## EmilyJernigan Group Title What is the inverse of the given relation? y = 3x + 9 2 years ago 2 years ago

1. y2o2 Group Title

$x = {{y-9} \over 3} = {1 \over3}y - 3$

2. alexwee123 Group Title

to find the inverse you would switch the x and y and then solve for y so.. y = 3x + 9 x=3y+9 3y=x-9 y=1/3*x-3

3. EmilyJernigan Group Title

Okay,, i had it right! Thank you

4. y2o2 Group Title

you're welcome


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## automotive intake question

If I can somehow increase and decrease the size of my intake tube with respect to engine's rpm (high rpm= bigger intake, low rpm=small intake to increase air velocity, about how much hp would I gain. Better question is would I gain any at all[?]
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 Although forced air intake systems (such as superchargers, turbochargers and ramjets) do increase air velocity the increase in power does not come directly from the increased air velocity, but the increased air flow. Think of a garden hose. If you put your thumb over the end the water may come out faster, but less water comes out of the smaller opening. You want more air to be taken in, not less air taken in faster.

Recognitions:

## automotive intake question

There is actually a nice resonance effect that you can take advantage of if you're clever enough by changing the lengths of the intake runners. BMW is I believe the only company which has a fully variable intake manifold. Look it up for more info.

Answering how much you'd gain from this is an impossible question. It depends so much on the size of the engine and how it is tuned. This technology is really more for broadening the torque curve rather than improving peak numbers very much though.

 Originally posted by one_raven ....Increasing it more than necessary makes no difference at all since it only takes in as much as the plunger action sucks in....
Not quite true. If your intake is too large, this slows down the gas velocity and reduces the cylinder efficiency (I think due to the lack of swirl on a slow moving mass of gas).

I race motorcycles, and engines that are 'over-carbed' are often less rideable than ones with smaller intakes.

As a general rule (if their is such a thing with tuning two-strokes...) smaller intakes increase the engines pick-up and acceleration, and bigger carbs increase top speed... but at the expense of narrowing your power band. The big carbs work better at peak power and revs, as by then, the intake velocity is sufficient to get the cylinders working properly.

As for exhaust length and intake tract length, these can make a massive difference. Changing your exhaust pipe length by only a few mm will noticeably change the engine characteristics. Also, with two-strokes, you want your inlet tract to be as short as possible.

 Originally posted by Adrian Baker Not quite true. If your intake is too large, this slows down the gas velocity and reduces the cylinder efficiency (I think due to the lack of swirl on a slow moving mass of gas). I race motorcycles, and engines that are 'over-carbed' are often less rideable than ones with smaller intakes. As a general rule (if their is such a thing with tuning two-strokes...) smaller intakes increase the engines pick-up and acceleration, and bigger carbs increase top speed... but at the expense of narrowing your power band. The big carbs work better at peak power and revs, as by then, the intake velocity is sufficient to get the cylinders working properly. As for exhaust length and intake tract length, these can make a massive difference. Changing your exhaust pipe length by only a few mm will noticeably change the engine characteristics. Also, with two-strokes, you want your inlet tract to be as short as possible.

So yes it will make more torque and hp if I can find a why to varies the size of my intake with respect to engine rpm?

 Originally posted by david90 So yes it will make more torque and hp if I can find a why to varies the size of my intake with respect to engine rpm?
Possibly.... but all my experience is with very peaky, high revving two-strokes. Your best bet is to make up various size manifolds and book some Dyno time at a local tuning shop......
 Recognitions: Homework Help Science Advisor I'm not an automotive engineer, or a fluid dyamics specialist, but resonant gas behavior can probably explain a lot of the issues. (This means that what I say might be completely incorrect.) Although it's convenient to think of the the air-flow in an engine as continous, it's actually pulsed. Since air is compressible, it also acts as a spring, and it has momentum. Effectively, an exhaust pipe/manifold acts like a pulse jet engine. When the cylinder is vented, there is a high-pressure shockwave that travels along the exhaust, and pushes enough air out that the interior of the exhaust pipe is at less than atmospheric pressure. As a result there is a second front of air that comes back into the exhaust pipe and can slam into a cylinder that is trying to vent. Since the cylinder must now work harder to vent there is a decrease in efficiency. Choosing an appropriate exhaust length can actually create a situation where the reverberation from the previous venting provides extra suction which can improve both horsepower and torque - although I have no clue to what degree. This effect is probably most pronounced in situations where there is a straight pipe running from the engine out, and where the equipment is usually in a narrow RPM band. A well tuned exhaust should also make less noise. I expect that the characteristic bop-bop-bop that cars make when they are at low RPMs is the result of cylinders firing at frequencies too low for the exhaust system to be efficient. There is potential for a similar situation at the intake manifold -- whenever air is taken in a shockwave travels through the intake manifold. If the timing is such that the pressure peak hits when the next cylinder is taking in air there will probably be an increase in compression which also leads to more power and more torque. Assuming that the shock waves travel at approximately the speed of sound (which we all know varies with temperature and pressure), leads t'o the conlusion that an exhaust pipe that has length equal to the time between firings multiplied by the speed of sound or longer. That way the when the second cylinder vents there should be maximal underpressure from the previous cylinder's shockwave. It's also possible to use other odd fractions of the that length, but it's much more sensitive to length changes since even fractions of this length are especially bad. If we assume that the speed of sound is 280 m/s the calculation comes out to: $$\frac{Strokes*16800}{RPM * Cylinders} meters.$$ (or approximately three times that many feet) You should probably err on the side of length and use your use your lowest operating RPM in the formula. Also remember that this formula is for the number of cylinders per manifold. I recommend that you experiment, and ask someone who has more experience than me, but I would guess that the optimal length for the exhaust pipe on a 2 stroke 2 cylinder engine at 8000 RPM would be somewhere around 3 feet. At lower RPM the pipes would probably want to be longer. By comparison, a 12 Cylinder 4 stroke engine at 8000 RPM wants a minimum length of only 1/2 foot or so. If you do experiment, I would be most interested to hear your results.
 An excellent reply Nate TG. It certainly is resonant gas behaviour. My racing bike has a power band from only 8,000 to 10,300 rpm. At anything below 7500rpm it barely runs. Opening up the throttle at lower revs just ends up spraying jets of fuel out of the carbs. The whole back of the bike is a real mess after a race. However, as it approaches 8,000rpm, the exhaust note changes, the induction note changes and it suddenly flies forwards like a rocket. In the power band, the pulses of air in the carbs rams the fuel air mix in to the cylinder (sort of...) and the return wave from the exhaust in the expansion pipe pushes unburnt mix back up the exhaust pipe into the cylinder. At any revs above or below the power band, everything is out of sync. Here is a rather pathetic video of the carbs when the engine is being revved in the pits... Revving Suzuki As you can see, masses of fuel mix are thrown OUT of the carbs towards the rear of the bike. When the revs are correct and the gas pulses in sync, then this is rammed into the cylinders instead! My bike is a Classic Racer (1967) and so is not allowed modern developments that increase the power band width. Modern two strokes have variable length exhaust pipes that change as the revs rise (they use a valve that opens or shuts a chamber in the exhaust port) Also reed valves in the inlet tract reduce blow back and helps the power band spread a little more. Incidentally, my 250cc bike averages about 15 miles per English gallon of petrol. (4.54 litres)

 Originally posted by Adrian Baker Modern two strokes have variable length exhaust pipes that change as the revs rise (they use a valve that opens or shuts a chamber in the exhaust port) Also reed valves in the inlet tract reduce blow back and helps the power band spread a little more.
So, in essense, what they are is woodwind instruments.
http://primates.ximian.com/~federico...t/2001-sax.jpg

-Chris
 Buy an Induction Kit and an Exhaust kit and then have them fitted by someone who knows what they are doing, this will increase the performance of your car. But you have to get both, either one on its own will have a negative effect.
 The following page shows a nice animation of how a tuned exhaust two-stroke works (2nd pic from top). This doesn't apply directly to four-strokes. http://www.coolskunk.com/two_stroke_engine.htm It takes a minute or two for the animation to load.
 Nice site - the animation does take ages to load but it is an excellent illustration of the two-stroke principle. Thanks.


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# Why is the peak-to-peak separation of cyclic voltammetry is 57 mV?

I'm studying CV, and I wonder if I can calculate the peak-to-peak separation of CV for a reversible redox system.

I found that the separation is 57 mV on my textbook, and my teacher taught me that is can be calculated theoretically using Nernst equation, but I think it is more complicated.

Is it easy enough (for sophomore) to calculate the value? And if it is, I want to know the way.

(Sorry for my poor English)

• The 57 mv is empirical and not related to the similar value from the Nernst equation. It arises from the numerical solution of the equations in the oxidation/reduction cycle. See Nicholson & Shain Anal. Chem. v36, p 706 1964 for the details of the calculation: not an easy read. In introduction is given by Mabbot J. Chem. Educ. v60, p697 1983 and Benschoten et al. ibid. v60 ,p772, 1983 – porphyrin Jul 26 '19 at 8:03


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Document de travail

# Portfolio Choices and Asset Prices: The Comparative Statics of Ambiguity Aversion

## Résumé

We investigate the comparative statics of "more ambiguity aversion" as defined by Klibanoff, Marinacci and Mukerji (2005) in the context of the static two-asset portfolio problem. It is not true in general that more ambiguity aversion reduces the demand for the uncertain asset. We exhibit some sufficient conditions to guarantee that, ceteris paribus, an increase in ambiguity aversion reduces the demand for the ambiguous asset, and raises the equity premium. For example, this is the case when the set of plausible distributions of returns can be ranked according to the monotone likelihood ratio order. We also show how ambiguity aversion distorts the price kernel in the alternative portfolio problem with complete markets for contingent claims.

## Référence

Christian Gollier, , TSE Working Paper, n° 09-068, juillet 2009.

## Publié dans

TSE Working Paper, n° 09-068, juillet 2009