Induced Surface Charge Non Continuous Potential

Induced Charge

Even if the charge induced on a mineral surface using different charging materials is of the same sign, the amount of the charge transfer between two materials is dependent on the differences between the Fermi levels of the two materials.

From: Wills' Mineral Processing Technology (Eighth Edition) , 2016

Liquid crystalline derivatives of heterocyclic radicals

Piotr Kaszyński Szymon Kapuściński Sylwia Ciastek-Iskrzycka , in Advances in Heterocyclic Chemistry, 2019

3.3.3 Semiconductive properties

Transport of photo-induced charges in liquid crystalline phases was investigated by the time-of-flight (TOF) methods for several derivatives. Results for 61[8]a, 61[8]b, and 61[10]d derivatives demonstrated weak photocurrent (about 10^{−   13}  S   cm^{−   1}) (2014JMCC319) and the hole mobility in a range 1.5–3.3   ×   10^{−   3}  cm²  V^{−   1}  s^{−   1} in columnar phases (2012CC7064, 2012JA2465). These results are comparable with those for closed-shell discotic compounds. On the other hand, bent-core derivative 62[16]a exhibits ambipolar charge mobility in a similar range (2014JA14658).

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UHVDC Transmission Lines

In UHV Transmission Technology, 2018

18.3.4.2 Corona Requirements

For UHVDC lines, the induced charge on the ground wire is large enough that the electric field strength on the surface of the ground wire is high. If the electric field strength exceeds the corona inception field strength, the ground wire will suffer corona loss, radio interference, audible noise, and other problems. As such, the above electric field strength must be limited.

Currently, it is only possible to calculate the electric field strength on the surface of the ground wire when no corona occurs to the conductor. As the conductors are frequently subjected to corona, the electric field strength on the surface of the ground wire tends to increase (similar to the field strength of the conductor), but the magnitude of the increase is still under study. Therefore, the electric field strength on the surface of the ground wire of ordinary DC lines is calculated assuming that no corona occurs to the conductor, with a certain margin reserved. The electric field strength on the surface of the ground wire for ±800-kV YunnanGuangdong DC lines is recommended to be not larger than 12   kV/cm based on the actual conditions of the project. The ratio of the electric field strength on the surface of the ground wire to the corona inception field strength is around 0.63, as calculated based on the type and arrangement of the conductor and ground wire.

The corona inception field strength of the conductor of ordinary DC lines is taken as 15   kV/cm and a margin should be reserved for ground wires. As the conductors are frequently subjected to corona, the electric field strength on the surface of the ground wire tends to increase (similar to the field strength of the conductor), but the magnitude of the increase is still under study. Taking into account the effect of high altitude, it is temporarily suggested that the electric field strength on the surface of the ground wire not exceed 12   kV/cm and the diameter be around 18.0   mm, as shown in Table 18.14.

Table 18.14. Relation Between the Surface Electric Field Strength and Diameter of the Ground Wire of the XiangjiabaShanghai Project

Conductor Type	6×ACSR-720/50
Diameter (mm)	15	6	17.5	18	19
Electric field strength on the surface of ground wire (kV/cm)	13.3937	12.6573	11.7026	11.4179	10.8912

Note: Assume that no corona occurs to the conductor.

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Colloidal Crystals

Junpei Yamanaka , ... Akiko Toyotama , in Pattern Formations and Oscillatory Phenomena, 2013

Crystallization Due to Base Diffusion

On the basis of the charge-induced crystallization of silica colloids described in Section 5.3.1, Yamanaka and colleagues examined the unidirectional crystallization of colloidal silica due to diffusion of a base (Yamanaka et al., 2004; Murai et al., 2007). They used the weak base pyridine (Py). In aqueous solutions, Py molecules dissociate only slightly to provide basic species (py   +   H₂O   ↔   pyH⁺  +   OH⁻, where py and pyH⁺ denote undissociated and dissociated Py molecules, respectively). When Py is added to silica, negative surface charges are generated on the silica surfaces by the reaction SiOH   +   py → SiO⁻ + pyH⁺. For a strong base such as NaOH, unidirectional crystallization was not observed because the NaOH molecules reacted with the silanols almost completely, so the concentration of diffusing species was negligibly small. On the other hand, undissociated Py molecules in the medium were mobile and diffused in the silica colloids.

Figure 5.13(a) illustrates the experimental setup for unidirectional crystallization. Py molecules diffused into colloidal silica from a reservoir of an aqueous Py solution through a semipermeable membrane (sample cell size   =   1   ×   1   ×   4   cm³, reservoir volume   =   500   ml). Photographs of the crystallization process are also shown in Figure 5.13(b) (particle diameter   =   110   nm, salt-free, ϕ = 0.034, Py concentration in the reservoir [Py]₀  =   100   mM). Columnar crystals having lengths of a few centimeters were formed within one day. The crystal region showed bright Bragg diffraction of visible light. The crystals were larger at a slower growth rate and reached 1   ×   1   ×   3   cm³ at the largest. Figure 5.14 shows the crystal growth curves for three values of [Py]₀.

Figure 5.13. (a) Illustration and (b) images of typical crystal growth process of colloidal silica due to diffusion of Py.

Figure 5.14. Crystal growth curves for colloidal silica due to diffusion of Py at three values of [Py]₀. Solid and dotted curves are theoretical growth curves for salt-free conditions and in the presence of 5   μM of salt, respectively.

The unidirectional crystallization previously described can be regarded as a combination of three processes: (1) diffusion of a base accompanied by the reaction with silica particles, (2) an increase in the Z value of silica, and (3) crystallization of the colloidal silica due to the increase in Z. On the basis of this model, the crystal growth curves were obtained as follows.

The reaction between silica particles and Py is regarded as electrostatic adsorption of Py onto the silica surface. Hereafter, we denote the adsorption amount per particle as S and the concentration of free Py as C. Note that Z_a is proportional to S. The relationship between S and C was determined by performing separate titration experiments. The crystallization phase diagram of the silica   +   Py system, defined by S and [NaCl], is shown in Figure 5.12 (filled symbols). Because the strong base NaOH reacts almost completely with the silica, we can assume that [NaOH]   = S. The phase diagram shows good agreement with that obtained for the silica   +   NaOH system when the data were plotted using the S values.

We assume instantaneous equilibrium between S and C upon diffusion of Py. Then the adsorption–diffusion equation is given by

(5-23) $\partial C/\partial t=\left[D/\left(1+\partial S/\partial C\right)\right]{\partial }^{2}C/\partial x^2$

where x is the distance from the reservoir, and D is the diffusion coefficient of Py. We numerically solved Eq. (5-23) and obtained C(x, t) and S(x, t). Figures 5.15(a) and (b) show the profiles of C(x, t) and S(x, t), respectively, for [Py]₀  =   1   mM. From the crystallization phase diagram (Figure 5.12), we estimated the S value at crystallization, S*   =   20   μM. On the basis of the time evolution of S, we calculated the crystal growth curve as the x–t curve that satisfies S(x, t)   = S*. The solid curves in Figure 5.14 are the theoretical growth curves for the salt-free condition, which agree well with the observations. The slightly slower growth at low [Py] might be due to the presence of ionic impurities. The case of C _s   =   5   μM is shown by dotted curves, which are in rather good agreements with the experiments.

Figure 5.15. Time evolution of profiles of (a) C and (b) S at [Py]₀  =   1   mM. Values of t shown in (a) also apply to (b). Solid vertical lines in (a) and (b) indicate the membrane–colloid boundary (x  =   0.5   cm). In (b), the location of the crystal front (x  = x*) is shown when S*   =   20   μM at t  =   10   h.

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Volume 2

Shuichi Shoji , ... Roland Zengerle , in Comprehensive Microsystems, 2008

2.09.3.1.2 Traveling wave micropumps

In traveling wave EHD pump, the free charges induced in the bulk liquid were utilized to drive the fluid. In general, application of the electric field in the presence of the gradient in permittivity or conductivity induces free charge in the volume of a material. The electrical conductivity of slightly conducting liquids depends on the temperature. Therefore, the gradient of the electrical conductivity can be achieved by developing the temperature gradient in the liquid. Figure 19 illustrates an EHD pump using a temperature-induced conductivity gradient (Bart et al. 1990). The temperature and the conductivity gradient allow free charges to develop in the fluid volume. These charges are subjected to electric field and drag the liquid molecules around them. As a result, pumping force is induced. Figure 20 shows the schematic diagram of the traveling wave EHD pump. The planer electrodes are formed on the bottom of the pump channel. The electric field waves are produced by applying the phase-shifted rectangular pulses on the electrodes. The temperature gradient is produced by the traveling wave itself and not by additional heating (Fuhr et al. 1992). In contrast to the EHD injection pump, relatively conductive liquids such as water solutions could be pumped.

Figure 19. An electrohydrodynamic (EHD) pump employing a temperature-induced conductivity gradient. (Source: Bart S F, Tavrow L S, Mehran M, Lang J H 1990 Microfabricated electrohyrdrodynamic pumps. Sens. Actuators A21–A23, 193–7.)

Figure 20. Schematic drawing of the traveling wave electrohydrodynamic (EHD) micropump. (Source: Fuhr G, Hagedorn R, Muller T, Benecke W, Wagner B 1992 Microfabricated electrohydrodynamic (EHD) pumps for liquids of higher conductivity. J. Microelectromech. Syst. 1(3), 141–6.)

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TiO2-Based Dye-Sensitized Solar Cell

Shogo Mori , Shozo Yanagida , in Nanostructured Materials for Solar Energy Conversion, 2006

9. CONCLUSION

In the TiO₂ -based DSCs, photo-induced charge separation occurs by electron transferring at the TiO ₂/dye/electrolyte interfaces, and the charges travel in TiO₂ and an electrolyte separately. In order to achieve high efficiency of the solar cells, these charge transfer and transport rates should be controlled so that all electrons are extracted for external load without charge recombination. The transport and transfer rates are interdependent, and thus simultaneous control of the rates is important to achieve high efficiency. Challenging of the DSCs is that the development of the solar cell covers the wide range of chemistry and physics, and all components including organic and inorganic materials should be designed properly in view of kinetics of electron transport and transfer, whose mechanism is governed by nanosized structure. It is also an interesting aspect where collaboration among different fields of scientists is important to develop the solar cells and to elucidate the mechanism of solar cells' working principle. Further increase of the efficiency is gained by the interdisciplinary research.

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Equipment

Ian Sutton , in Plant Design and Operations, 2015

Floating Roof Tanks

The basic idea of a floating roof tank is that the roof is in contact with the liquid surface. It is attached to the sides of the tank with rollers. A seal between the moving roof and the tank wall prevents process vapors from leaking into the space above the roof. As liquid is added, the roof moves up; as liquid is removed, the roof moves down. At no time is there a vapor space directly above the liquid. Hence there is no need to vent the tank when it is being filled and there is no need to add inert gas when it is being emptied.

Many floating roof tanks do not have a fixed roof. Hence, when looking down on a large tank farm from an airplane, it is possible to see the status of the roof positions by looking down on them. This type of tank is illustrated in Figure 12.2. Some floating roof tanks have a fixed roof above the floating roof as shown in Figure 12.3.

Many tank fires are caused by lightning (either induced charges or direct strikes). The following actions minimize the chance of a fire:

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The roof seals should be maintained in a good condition.

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Floating roofs should be bonded to the tank walls by the use of shunts spaced at least 3 meters apart. The shunts should be in contact with the wall of the tank.

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The tank roof should be kept clean.

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All openings such as manways and inspection hatches should have sealed covers.

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Pontoons need to be inspected at least annually for presence of liquid or flammable vapors.

The potential for contamination of the atmosphere above the internal floating roof is great. Entry onto the roof of an internal floating roof tank constitutes entry into a confined space, so confined space entry procedures must be followed. Provisions for personnel support and distribution of weight must be provided when personnel are on the internal floating roof.

The internal floating roof in a covered tank may be a steel pontoon roof, a steel pan roof, an aluminum floating cover with pontoons or floats, or a fiberglass polyester skin panel deck. Aluminum and polyester roofs have a greater fall through potential than steel roofs. In addition, mechanical damage, corrosion, or other defects may not be readily apparent.

The following guidance applies to work carried out on the roof of a floating roof tank:

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The level of the floating roof should be within about 8 feet of the top of the tank, and no product movement should occur into or out of the tank for 24 hours prior to entry.

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The roof should be essentially horizontal with no evidence of tipping or "hanging up."

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The vent shown in Figure 12.3 should also be a vacuum breaker for use when the tank roof is falling.

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Workers should not enter the floating roof space to determine if liquids are present.

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No entry should be allowed if significant liquid hydrocarbon is present or is suspected to be present on the floating roof.

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Suitable personnel protective equipment should be used.

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Adequate roof support should be assured before anyone walks on the roof.

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Electrical equipment, including lighting, that is used during inspection/work on the internal floating roof should be explosion proof.

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On large-diameter tanks, more than one entry point should be provided to lessen the distance to be covered from the entry point and to minimize the possibility of air and tag (rescue) lines becoming entangled around columns.

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Hot work should not be permitted inside the space of an in-service internal floating roof tank.

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Chemical Functionalization and Photo-Induced Charge Transport

L. Gundlach , in Encyclopedia of Interfacial Chemistry, 2018

Sensitization with Nanoparticles Forming Hetero Interfaces

A limiting case for chemical functionalization of interfaces for light-induced charge transfer is the sensitization of a wide band gap semiconductor with small band gap semiconducting nanoparticles that form heterojunctions. These systems are similar to conventional semiconductor/semiconductor interfaces. The two types of heterojunctions that can lead to photo-induced charge transfer are type I where the smaller band gap lies inside the gap of the wide gap semiconductor and type II where both gaps are offset.

The combination of different semiconducting nanoparticles gives the advantage of generating a large effective surface area and of the stability of solid state materials. Several examples for such structures have been presented mostly based on ZnO as the wide gap semiconductor: Cu₂O/ZnO p–n heterojunction, ⁶² n-CaO-decorated n-ZnO nanorod structure, ⁶³ and hierarchical tree-like ZnO/CdSSe nanocomposite with CdSSe branches grown on ZnO nanowires. ⁶⁴ The latter form a type-II heterojunction and show fast charge transfer from CdSSe branches to ZnO stems. Band-to-band PL following charge transfer suggests that this material combination can be used for Z-scheme charge transfer ( Fig. 8 ).

Fig. 8. False color SEM image of A CdSSe (yellow) ZnO (blue) nanotree. Scale bar 200   nm.

From Li, Z.; Nieto-Pescador, J.; Carson, A.; Blake, J.; Gundlach, L. Efficient Z-Scheme Charge Separation in Novel Vertically-Aligned ZnO/CdSSe Nanotrees. Nanotechnology 2016, 27, 135401.

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Carbazole Polymers

Johannes Karl Fink , in High Performance Polymers (Second Edition), 2014

1.4.3.2 Holograms

Erasable holograms can be produced utilizing the photorefractive effect by forming a light-induced charge redistribution in a NLO material. Local changes in the index of refraction are produced so that dynamic, erasable holograms, which diffract visible light, can be formed. The photorefractive effect is achieved by exposing the material to an optical intensity pattern consisting of bright and dark regions, such as those formed by interfering two coherent laser-writing beams. Mobile charge generated in the material migrates to the appropriate region to form internal space-charge electric fields, which diffract the light during readout with a reading in accordance with the electro-optic effect [181].

The effect was recognized to be useful for storing volume phase holograms. The mechanism of the photorefractive effect may be explained as follows: crystals are illuminated with a light pattern, e.g., the interference pattern of two laser beams. As a result, charges are excited in the bright areas of the defects in the conduction band or valence band, redistributed, and recaptured preferentially in the darker areas. Space-charge patterns, which modulate the refractive index via the electro-optic effect, are formed. Charge sources and charge traps often may be transition metal ions, which occur in different valence states. Diffusion, the volume photovoltaic effect, and drift in external fields, in space-charge fields, and in pyroelectric fields are known as drive mechanisms for the charge transfer.

Photorefractive liquid crystals were first reported in 1994. Since then, the performance of these materials has dramatically improved.

Full-field, retroreflective holographic imaging through turbid media has been achieved using a photorefractive polymer composite as a coherence gate [182]. The photorefractive devices used, are based on PVK and TNFDM which is doped with the chromophore 1-(2′-ethylhexyloxy)-2,5-dimethyl-4-(4′-nitrophenylazo)benzene. There is certain evidence that TNFDM interacts with chromophores by complexation [183].

A recording technique of holograms and the non-destructive readout in a photorefractive polymer utilizes two-photon absorption. The holograms are formed through the photorefractive effect. The technique uses the excitation of the electroactive chromophore with femtosecond pulses, followed by charge injection into a PVK matrix. The holograms can be fully erased with a pulsed laser beam. However, they are insensitive to continuous-wave laser beams with the same wavelength [184].

N,N-Diphenyl-7-(2-(4-pyridinyl)-ethenyl)-9,9-di-n-decyl-9H-fluorene-2-amine (AF-50), c.f. Figure 1.17, has been studied as a chromophore in a PVK matrix for read/write applications [185]. Information was written onto a sample using the $325\text{nm}$ line of a continuous wave He-Cd laser. The PVK/AF-50 undergoes a chemical change upon laser irradiation causing the blue-shift in the PL spectrum. IR investigations suggest the formation of a new conjugated system, such as a ketimine. The chemical change appears to be irreversible.

Figure 1.17. N,N-Diphenyl-7-(2-(4-pyridinyl)-ethenyl)-9,9-di-n-decyl-9H-fluorene-2-amine.

A photorefractive polymer consisting of DMNPAA, TNF, ECZ, and PVK has been used for erasable/rewritable three-dimensional bit optical data storage under two-photon excitation [186]. A three-dimensional bit density of 5   Gbits/cm³ is achieved by pulsed beam illumination at an infrared wavelength of $800\text{nm}$ in the recording process.

Complete erasing of the recording information can be completely erased by ultraviolet illumination. Dual-use chromophore molecules allows one to write both irreversible photochromic and erasable photorefractive holographic gratings into the same storage volume [187].

At $675\text{nm}$ , the chromophore undergoes a photochemical reaction in creating irreversible holographic gratings. Later, at longer wavelengths, the storage of erasable photorefractive holograms in the same location can be achieved.

The photochemical gratings have a diffusion-limited dark half-life of about two weeks, depending on the glass transition temperature of the composite. The composites consist of PVK. As plasticizers, butyl benzyl phthalate or tricresyl phosphate is used. The sensitizer and charge generator consist of fullerene C-60 or TNF. The chromophores consist of moieties with the basic structure of 2-(5,5-dimethyl-3-styryl-cyclohex-2-enylidene)-malononitrile, c.f. Figure 1.18.

Figure 1.18. 2-(5,5-Dimethyl-3-styryl-cyclohex-2-enylidene)-malononitrile and related chromophores [187].

The chromophores serve for the formation of efficient photorefractive gratings and they are photochemically active, probably by 2   +   2 photochemical reactions, when triplet sensitized. The photorefractive external diffraction efficiency is highly dependent on the glass transition temperature of the composites. A low glass transition temperature favors the efficiency.

The following seems to be a general rule: The response times of photorefractive polymer composites are strongly dependent on both the glass transition temperature and the electro-optical chromophore [188].

Composites with a glass transition temperature below the temperature of measurement, with varying chromophore content, respond in comparable response times of $200''–''500\text{ms}$ .

However, significant differences occur in composites with a glass transition temperature above the temperature of measurement. In this case, the composites with the highest chromophore content show the best steady-state performance. However, their response time is much slower than that for those containing lower chromophore content. Reversible photorefractive grating and irreversible local photoinduced aggregation grating could be established in a low glass transition temperature polymer composite of PVK, TNF, ECZ, and N-(4-nitrophenyl)-1-prolinol [189].

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The development of piezoelectric materials and the new perspective

K. Uchino , in Advanced Piezoelectric Materials, 2010

Abstract:

Certain materials produce electric charges on their surfaces as a consequence of applying mechanical stress. The induced charges are proportional to the mechanical stress. This is called the direct piezoelectric effect and was discovered in quartz by Pierre and Jacques Curie in 1880. Materials showing have a geometric strain proportional to an applied electric field. This is the converse piezoelectric effect, discovered by Gabriel Lippmann in 1881. This article first reviews the historical episodes of piezoelectric materials in the sequence of quartz, Rochelle salt, barium titanate, PZT, lithium niobate/tantalate, relaxor ferroelectrics, PVDF, Pb-free piezoelectrics, and composites. Then, the detailed performances are described in the following section, which serves as the introduction to each chapter in this book. Third, since piezoelectricity is utilized extensively in the fabrication of various devices such as transducers, sensors, actuators, surface acoustic wave devices, frequency control, etc., applications of piezoelectric materials are introduced briefly in conjunction with materials. The author hopes that the reader can 'learn the history aiming at creating a new perspective for the future of piezoelectric materials'.

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Interfacial and Nanoscale Fracture

Y. Huang , Z.L. Wang , in Comprehensive Structural Integrity, 2003

(iii) The fundamental resonance frequency and nonlinear effect

As demonstrated in Figure 13 , a carbon nanotube can be charged by an externally applied voltage. The induced charge is distributed mostly at the tip of the carbon nanotube and the electrostatic force results in the deflection of the nanotube. Alternatively, if an applied voltage is an oscillating one, the charge on the tip of the nanotube also oscillates, and so does the force. Once the applied frequency matches the natural resonance frequency of the nanotube, mechanical resonance is induced. By tuning the applied frequency, the first and the second harmonic resonances can be observed ( Figure 14).

Figure 14. A selected carbon nanotube at (a) stationary, (b) the first harmonic resonance (ν ₁=0.53   MHz), and (c) the second harmonic resonance (ν ₂=3.01   MHz). The right-hand side shows the shape predicted based on elasticity theory for a uniform macroscopic beam.

Analysis of information obtained from the resonance experiments relies on the choice of theoretical models. It is unclear whether the classical continuum mechanics theory (e.g., elasticity), which works well for large size objects, is still applicable to carbon nanotubes. It may be necessary to use molecular dynamics, or the new continuum mechanics theory established from the interatomic potential in the next section. The validity of applying the classical elasticity theory or the interatomic-potential-based continuum mechanics theory to the experimental data analysis needs to be examined.

We have compared the following three characteristics between the elasticity theory and the experimental results shown in Figure 14. First, the theoretical mode for the second harmonic resonance occurs at 0.8L, and the experiment showed ∼0.76L. Second, the frequency ratio between the second to the first mode predicted by the theory is ν ₂/ν ₁=6.27, while the experimentally observed ratio is ν ₂/ν ₁=5.7. The agreement is reasonable considering the assumptions made in the theoretical model that the nanotube is a uniform and homogeneous beam, and the root of the clamping side is rigid. The latter assumption may not hold in experiments. Finally, the experimentally observed shape of the nanotube during resonance agrees very well with the shape calculated from the elasticity theory. It seems that the classical elasticity theory can still be used for the experimental data analysis.

If the nanotube is approximated as a uniform solid bar with one end fixed on a substrate, the classical elasticity theory gives the resonance frequency as (Meirovich, 1986)

(5) ${\nu }_i=\frac{{\beta }_i^2}{4\pi }\frac{1}{L^2}\sqrt{\frac{\left(R^2+R_i^2\right)E_{\text{b}}}{\rho }}$

where R and R _i are the outer and inner radii of the nanotube, respectively, L is the length, ρ is the mass density, E _b is the bending modulus, and β _i are constants for each mode. It is observed that the above resonance frequency depends strongly on the property and size of the nanotube.

The correlation between the applied frequency and the resonance frequency of the nanotube is not trivial. We know from Figure 13 that there are some electrostatic charges built on the tip of the carbon nanotube. With consideration of the difference between the surface work functions of the carbon nanotube and the counter electrode (Au), a static charge exists even when the applied voltage is withdrawn. Therefore, under an applied field the induced charge on the carbon nanotube can be represented by Q=Q ₀+αV ₀  cos ωt, where Q ₀ represents the charge on the tip to balance the difference in surface work functions, α is a geometrical factor, V ₀ is the amplitude of the applied voltage, and ω is the vibration frequency. The force acting on the carbon nanotube is

(6) $\begin{array}{c}F=\beta (Q_0+\alpha V_0\text{cos}\omega t)V_0\text{cos}\omega t\\ =\frac{1}{2}\alpha \beta V_0^2+Q_0\beta V_0\text{cos}\omega t+\frac{1}{2}\alpha \beta V_0^2\text{cos}2\omega t\end{array}$

where γ is a proportionality constant. Thus, resonance can be induced at ω or 2ω with vibration amplitudes proportional to V ₀ and V ₀ ², respectively. The former is a linear term in which the resonance frequency equals the applied frequency, while the latter is a nonlinear term and the resonance frequency is twice the applied frequency. The linear and nonlinear terms can be distinguished in experiments by observing the dependence of the vibration amplitude on the magnitude of the voltage V ₀. This observation is important to ensure the detection of the linear term.

Another factor that needs to be considered is the identification of the true fundamental resonance frequency. The frequency ratio between the first two modes is determined from Equation (5) as 6.27. If resonance occurs at ω, it could also occur at 2ω in practice, which is the double harmonic. Figure 15 shows the resonance of a bent nanotube in such a case. In order to identify the true fundamental frequency, one needs to examine the resonance at a frequency that is half or close to half of the observed resonance frequency; if no resonance occurs, the observed frequency is then the true fundamental frequency.

Figure 15. Resonance of a bent carbon nanotube at (a) ν ₍₁₎=1.87   MHz, V ₀=2   V and (b) ν ₍₂₎=3.93   MHz, V ₀=5   V, showing the multiple harmonic effect.

The diameters of the nanotube can be determined directly from TEM images to a high accuracy. The two-dimensional (2D) projection effect of the tube must be considered in the determination of tube length. It is essential to tilt the tube and catch its maximum length in TEM, which is likely to be the true length. This requires a TEM that gives a tilting angle as large as ±60°. It is also important to control the operation voltage of the TEM in order to minimize radiation damage. The 100   kV TEM used in our experiments has shown almost no detectable damage to a carbon nanotube, while 200   kV electrons could quickly damage a nanotube. The threshold for radiation damage of carbon nanotubes is ∼150   kV.

A carbon nanotube has been resonanced for more than 30   min in order to trace the sensitivity of resonance frequency on beam illumination and radiation damage at 100   kV. The resonance frequency has shown an increase of ∼1.4% over the entire period of experiment (Figure 16), but no dependence on the electron dose is found. The resonance peak is measured to be Δν/ν=0.6% in a vacuum of 10⁻⁴--10⁻⁵  torr. Slight increase in the resonance frequency could be related to the change of carbon structure under the electron beam, but this has a negligible effect on the measurement of the bending modulus.

Figure 16. Time dependence of the resonance frequency of a carbon nanotube, being illuminated by 100   kV electrons, showing the stability of the nanotube resonance frequency as a function of time.

We consider a 1D harmonic oscillator with an intrinsic resonance frequency ν ₁ in order to explore the intrinsic meaning of the measured Δν/ν ₁ value. If a viscosity (or friction) force acts on the particle and the force is proportional to the instantaneous particle speed, damping of the vibration amplitude is given by exp(−t/τ ₀), where τ ₀ is the life decay constant of the oscillator. For Δν/ν ₁≪1, this decay constant is related to Δν/ν ₁ by

(7) ${\tau }_0={\left[\left(\Delta \nu /{\nu }_1\right)\pi {\nu }_1/1.732\right]}^{-1}$

For Δν/ν ₁=0.65%, ν ₁=1.0   MHz, the life decay constant is τ ₀=85   μs. The viscosity/friction coefficient is η=2M/τ ₀, where M is the mass of the particle. Therefore, the time decay constant depends mainly on the viscosity coefficient of the nanotube in vacuum (10⁻⁴  torr) under which the measurement is made, and it is almost independent of the intrinsic structure of the carbon nanotube. This agrees with our experimental observation. Equation (7) can also be used to explain the larger value of Δν/ν ₁ obtained in air than that in vacuum, since the atmosphere should have a higher viscosity (friction) coefficient.

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