Disintegration of Water Drops in an Electric Field
28 Jul 1964-Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences (The Royal Society)-Vol. 280, Iss: 1382, pp 383-397
TL;DR: In this article, it was shown that a conical interface between two fluids can exist in equilibrium in an electric field, but only when the cone has a semi-vertical angle 49.3$^\circ$.
Abstract: The disintegration of drops in strong electric fields is believed to play an important part in the formation of thunderstorms, at least in those parts of them where no ice crystals are present. Zeleny showed experimentally that disintegration begins as a hydrodynamical instability, but his ideas about the mechanics of the situation rest on the implicit assumption that instability occurs when the internal pressure is the same as that outside the drop. It is shown that this assumption is false and that instability of an elongated drop would not occur unless a pressure difference existed. When this error is corrected it is found that a drop, elongated by an electric field, becomes unstable when its length is 1.9 times its equatorial diameter, and the calculated critical electric field agrees with laboratory experiments to within 1%. When the drop becomes unstable the ends develop obtuse-angled conical points from which axial jets are projected but the stability calculations give no indication of the mechanics of this process. It is shown theoretically that a conical interface between two fluids can exist in equilibrium in an electric field, but only when the cone has a semi-vertical angle 49.3$^\circ$. Apparatus was constructed for producing the necessary field, and photographs show that conical oil/water interfaces and soap films can be produced at the caloulated voltage and that their semi-vertical angles are very close to 49.3$^\circ$. The photographs give an indication of how the axial jets are produced but no complete analytical description of the process is attempted.
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TL;DR: In this article, a comprehensive review is presented on the researches and developments related to electrospun polymer nanofibers including processing, structure and property characterization, applications, and modeling and simulations.
6,987 citations
Cites background from "Disintegration of Water Drops in an..."
...Jet initiation The basic principles for dealing with the electrospinning jet fluids were developed by [146–148]....
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TL;DR: Electrospinning is a highly versatile method to process solutions or melts, mainly of polymers, into continuous fibers with diameters ranging from a few micrometers to a few nanometers, applicable to virtually every soluble or fusible polymer.
Abstract: Electrospinning is a highly versatile method to process solutions or melts, mainly of polymers, into continuous fibers with diameters ranging from a few micrometers to a few nanometers. This technique is applicable to virtually every soluble or fusible polymer. The polymers can be chemically modified and can also be tailored with additives ranging from simple carbon-black particles to complex species such as enzymes, viruses, and bacteria. Electrospinning appears to be straightforward, but is a rather intricate process that depends on a multitude of molecular, process, and technical parameters. The method provides access to entirely new materials, which may have complex chemical structures. Electrospinning is not only a focus of intense academic investigation; the technique is already being applied in many technological areas.
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TL;DR: More than 20 polymers, including polyethylene oxide, nylon, polyimide, DNA, polyaramid, and polyaniline, have been electrospun in this paper.
Abstract: Electrospinning uses electrical forces to produce polymer fibres with nanometre-scale diameters. Electrospinning occurs when the electrical forces at the surface of a polymer solution or melt overcome the surface tension and cause an electrically charged jet to be ejected. When the jet dries or solidifies, an electrically charged fibre remains. This charged fibre can be directed or accelerated by electrical forces and then collected in sheets or other useful geometrical forms. More than 20 polymers, including polyethylene oxide, nylon, polyimide, DNA, polyaramid, and polyaniline, have been electrospun in our laboratory. Most were spun from solution, although spinning from the melt in vacuum and air was also demonstrated. Electrospinning from polymer melts in a vacuum is advantageous because higher fields and higher temperatures can be used than in air.
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TL;DR: In this paper, the effects of two of the most important processing parameters, spinning voltage and solution concentration, on the morphology of the fibers formed were evaluated systematically, and it was found that spinning voltage is strongly correlated with the formation of bead defects in the fibers, and that current measurements may be used to signal the onset of the processing voltage at which the bead defect density increases substantially.
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TL;DR: In this paper, the authors analyzed and explained the reasons for the instability of a viscous jet of polymer solution at a pendent droplet, showing that the longitudinal stress caused by the external electric field acting on the charge carried by the jet stabilized the straight jet for some distance.
Abstract: Nanofibers of polymers were electrospun by creating an electrically charged jet of polymer solution at a pendent droplet. After the jet flowed away from the droplet in a nearly straight line, it bent into a complex path and other changes in shape occurred, during which electrical forces stretched and thinned it by very large ratios. After the solvent evaporated, birefringent nanofibers were left. In this article the reasons for the instability are analyzed and explained using a mathematical model. The rheological complexity of the polymer solution is included, which allows consideration of viscoelastic jets. It is shown that the longitudinal stress caused by the external electric field acting on the charge carried by the jet stabilized the straight jet for some distance. Then a lateral perturbation grew in response to the repulsive forces between adjacent elements of charge carried by the jet. The motion of segments of the jet grew rapidly into an electrically driven bending instability. The three-dimensional paths of continuous jets were calculated, both in the nearly straight region where the instability grew slowly and in the region where the bending dominated the path of the jet. The mathematical model provides a reasonable representation of the experimental data, particularly of the jet paths determined from high speed videographic observations.
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"Disintegration of Water Drops in an..." refers background in this paper
...Zeleny (1917) photographed drops held at the end of capillary tubes and raised to a high potential....
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TL;DR: In this article, the authors studied the behavior of the Legendre functions of non-integral order (P„(cos 6), where n and m are restricted to integral values.
Abstract: Introduction. In the theory of the propagation of spherical waves in free space the angular wave functions are Legendre polynomials, P„(cos 6), or associated Legendre polynomials, PTM (cos 6), where n and m are restricted to integral values. These functions are polynomials in cos 8, their properties have been widely studied, numerical values have been tabulated, and in general they may be regarded as known functions. In more recent years, however, Legendre functions of non-integral order, which we shall denote by PJcos 6), have also occurred in physical problems. Thus, for wave propagation inside a circular horn of given angle, the boundary conditions introduce a characteristic equation which is actually an equation in the parameter v. It has been customary to simplify the problem by choosing horn angles corresponding to integral values of v, but a complete solution should include a study of the behavior of P„(cos 6) as a function of v. Similarly, in the mode theory of antennas developed by Schelkunoff the appropriate angular wave functions in the antenna region are Legendre functions of order n + 120/K, where n is an integer and K is the characteristic impedance of the biconical antenna to the principal wave. For thin cones K is large and the order of the Legendre functions is nearly, but not quite, integral. Further, when the cone angle is large, v may have quite general real values. Another application has appeared early this year, when P. Grivetf used Legendre functions of fractional order in the approximate solution of an electron lens problem, with particular emphasis on small values of v. Thus it appears that the properties of Legendre functions of non-integral order are of quite general interest, and it may be worth while to put on record some formulas that were developed a few years ago in connection with Schelkunoff's antenna theory. At that time the formulas were used to compute values of P„(cos 6), 0 5S d < x, for values of v between 0 and 2 at intervals of 0.1, and curves based on these computations have already been published.** Those curves show P„(cos 6) as a function of 6 for the fractional values of v; in this memorandum we include a table of numerical values (Appendix, Table I), and also a new set of curves (Figure 1) showing P„(cos 6) as a function of v for values of 6 between 0° and 175°. We have confined our computations to real values of v, but it might be worth noting that the approximate formulas, and in particular the fundamental series expansions (3) and (17), are also valid for complex values of v, in all regions in which they converge. The function P„(cos 6) has a logarithmic singularity at 6 = v for all non-integral values of v, and it may be expressed in closed form at 6 = ir/2 for all values of v; hence
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