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This chapter discusses several post-processing approaches applied to as-spun nanofibers to change their structure and/or enhance certain properties. Section 8.1 describes carbonization, sol-gel transformation and calcination, as well as metal-plating, used to make stiff, hollow or thermally and electrically conducting fibers. Sections 8.2 and 8.3 are devoted to cross-linking of solution-blown soy protein/nylon 6 nanofibers. The collected fiber mats can be bonded both chemically (using aldehydes and ionic cross-linkers, as in Section 8.2), and physically (by means of wet and thermal treatment, as in Section 8.3) to increase the tensile strength and therefore widen the range of applications of these green nonwovens. Chemical cross-linkers bond different amino groups, primary amides and sulfhydryl groups in the protein structure, which is beneficial for the enhancement of tensile strength. It is shown that treatment with ionic cross-linkers results in nanofiber mats with a higher Young’s modulus. Covalent bonds formed by aldehyde groups have a smaller effect on the mat strength. As cross-linked nanofibers are exposed to heat, the bonds formed between amino groups in the fibers are broken and they became less aggregated. In addition, in Section 8.3 it is shown that wet conglutination of soy protein/nylon 6 nanofiber mats leads to partial physical cross-linking of nanofibers and, consequently, to an increase in Young’s modulus. An enhancement of the tensile strength of soy protein nanofiber mats, as well as a slight plasticizing effect, can also result from exposure to water.
This chapter describes the machinery, mechanism and significant experimental and theoretical aspects of melt- and solution blowing. Meltblowing is a popular method of producing polymer micro- and nanofibers en masse in the form of nonwovens via aerodynamic blowing of polymer melt jets (Section 4.1). Its physical aspects were revisited recently. The process involves a complex interplay of the aerodynamics of turbulent gas jets with strong elongational flows of polymer melts, only recently uncovered and explained.
The role of turbulent pulsations (produced by turbulent eddies in the gas jet) in meltblowing is discussed first in Section 4.2 in the framework of a model experimental situation where solid flexible sewing threadlines are used to probe a parallel high-speed gas jet. After that, in Section 4.3, the dynamics of bending and flapping of flexible threadlines in a gas jet is considered. In Section 4.4 the aerodynamically driven stretching of a straight polymer jet is considered. In Section 4.5 it is shown how a severe bending instability leading to strong stretching and thinning of polymer jets can arise. This is done in the framework of a linearized version of the governing equations in the case of small bending perturbations of a single threadline or polymer jet in meltblowing. Then, in Section 4.6 the fully nonlinear case of large-amplitude planar bending perturbations of a single polymer jet is discussed. Both isothermal and non-isothermal cases are considered. In particular, it is shown how the cooling of the surrounding gas jet results in cooling of the polymer jet inside, and in the arrest of the bending perturbation growth due to melt solidification. Section 4.7 is devoted to predictions of three-dimensional configurations of polymer jets in meltblowing from die exit to deposition screen. Not only a single polymer jet, but multiple polymer jets are modeled simultaneously, as well as deposition on a screen moving normally to the principal jet direction being accounted for. The results include prediction of the fiber deposition patterns in lay-down and fiber-size distributions in the resulting nonwovens. The angular distributions in lay-down nonwovens are also predicted. Comparisons with the experimental data suggest that the model captures main trends rather accurately.
The first chapter is devoted to the traditional methods of fiber forming, which are used to produce macroscopic fibers. Since the novel methods used to form micro- and nanofibers described in this monograph have branched from the traditional methods, an introduction into the history of manmade fibers is instructive and fully appropriate (Section 1.1). There is a brief discussion of such traditional extrusion methods of fiber forming as melt spinning (Section 1.2), dry spinning (Section 1.3), wet spinning (Section 1.4) and the integrated process of spunbonding, which is used to form nonwoven fiber webs (Section 1.5). Melt and dry spinning are closely related to the electrospinning used to produce nanofibers, so the discussion of these traditional methods allows a first glimpse of electrospinning, covered in Chapter 5. One of the key elements of spunbonding is pulling polymer filaments by fast co-flowing air, which is known as meltblowing. Meltblowing, and its offshoot solution blowing, are also used to form micro- and nanofibers, as detailed in Chapter 4. In a sense, Section 1.5 serves as an introduction to the nonwoven nanofiber mats discussed later. Section 1.2 also contains some elements of quasi-one-dimensional theory; namely, its application to the draw resonance instability of melt spinning. In its more involved form a similar quasi-one-dimensional approach is applied in Chapters 3–6 to describe processes characteristic of melt- and solution blowing and electrospinning used to form micro- and nanofibers.
History and outlook
The term fiber originates from the French word fibre, from Latin fibra “a fiber, filament,” of uncertain origin, perhaps related to Latin filum “thread,” or from the root findere “to split” (Online Etimology Dictionary 2013). For centuries, the use of fibers was limited to natural materials such as cotton and linen, which had inherent problems with wrinkling. Silk was difficult to produce and was often too delicate. Wool was strong and abundant, but would shrink and was irritating next to the skin, and would not last long, as it was a food source for moths.
Several physical concepts that are of the utmost importance in fiber-forming processes are described in this chapter. The basic physical model of a flexible polymer macromolecule as a random walk is outlined in Section 2.1. The elongational and shear rheometry of polymer solutions and melts, which elucidate the stress relation with strains and strain rate, as well as stress relaxation is described in Section 2.2. The phenomenological rheological constitutive equations appropriate for the description of viscoelastic polymer solutions and melts are introduced in Section 2.3. The micromechanical foundations of the entropic elasticity responsible for viscoelasticity of polymer solutions and melts are sketched out in Section 2.4. Solidification and crystallization are discussed in Sections 2.5 and 2.6, respectively.
A linear polymer macromolecule can be represented as a succession of identical rigid segments connected at arbitrary angles, i.e. freely jointed with each other (Flory 1969, de Gennes 1979, Doi and Edwards (1986). Such a macromolecule is comprised of N segments, each of length b. The total length of a fully stretched macromolecule is then L = Nb. The rigid segments are called Kuhn segments. A real macromolecular chain consisting of n monomers is idealized as a random walk of N Kuhn segments, which are not monomers, nor is N identical to the degree of polymerization n. If the number of Kuhn segments in a macromolecule is not large, i.e. N is close to 1, it is rather inflexible, almost rod-like. On the other hand, if N >> 1, the macromolecule is very flexible, and on length scales that are significant compared to b, but much smaller than L, it can be viewed as a flexible string. Persistence length is another length scale that characterizes the resistance of segments of macromolecular chains to bending. It is of the same order of magnitude as the length of the Kuhn segments.
Nanotechnology has profound applications in healthcare and has improved healthcare research to a large extent. The therapeutic benefits of nanotechnology in the field of medicine have resulted in new areas, such as nanomedicine, nanobiotechnology, etc. Researchers in the field are attempting to find an effective nanoformulation to deliver growth factors, supplements or drugs safely and in a sustained manner at the required site. Their task is to attempt a different drug nanoformulation of existing blockbuster drugs that brings improved efficacy and a therapeutic breakthrough. Thus the ultimate objective of these nanotechnological drug-delivery systems is to fine tune the normal profile of potent drug molecules in the body following their administration to significantly improve their efficacy and also minimize potential intrinsic severe adverse effects. For treatment of breast cancer and non-small-cell lung cancer, Abraxane® (paclitaxel) is employed as a nanoparticular formulation, which increases drug delivery up to 70% in comparison with solvent-based paclitaxel delivery. In this novel nanoformulation, Abraxis Bio Sciences have used Bristol-Meyers Squibb’s blockbuster drug paclitaxel (Taxol) and a very common globular protein bovine serum albumin (BSA). There are numerous nanotechnology-based drug-delivery systems such as nanocrystals, nanoemulsions, lipid or polymeric nanoparticles, liposomes and nanofibers. While nanoemulsions and liposomal formulations did not make significant advances, despite huge research spending, the polymeric nanoparticulate systems show more promise. Nanoparticles of a polymeric nature find application as drug-delivery systems and are advantageous due to their scalability, cost, controlled and targeted delivery, compatibility, degradability, etc. Natural biopolymers are even better than the synthetic polymers in terms of biocompatibility and biodegradability. Nanoparticulate drug formulations alter the pharmokinetic profile of the therapeutic entity and program the release of the drug in sustained or controlled manner. Thus, nanoparticle or nanoformulated drugs outperform conventional systemic delivery in terms of delivery of an encapsulated drug and its sustained release. Slowly and surely nanoformulated drugs are coming onto the market, surpassing systemic delivery, which is believed to be the only mode of administration for a wide range of active pharmaceutical ingredients. Nanofibrous drug-delivery systems are being developed as potential scaffolds in tissue regeneration, wound healing and cancer drug-delivery applications. In this chapter we are going to discuss two promising nanotechnology-based drug-delivery tools, namely electrospun micro- and nanofibers and electrosprayed micro- and nanoparticles, which have a common synthetic procedure mediated by an electrical potential difference.
This chapter introduces the fundamental general equations of the dynamics of free liquid jets in Sections 3.1 and 3.2. The applications of these equations encompass practically all fiber-forming processes from melt spinning, as was sketched out in Section 1.2 in Chapter 2 to melt- and solution blowing, and electrospinning of polymer micro- and nanofibers discussed in detail in Chapters 4 and 5, or drawing of optical fibers outlined in Section 6.5 in Chapter 6. They also form the framework for description of several types of instabilities characteristic of the hydrodynamics of free liquid jets in general and of fiber-forming processes in particular. These include capillary instability (Section 3.3), aerodynamically driven bending instability (Section 3.4) and buckling of liquid impinging onto a wall (Section 3.5).
Mass, momentum and moment-of-momentum balance equations
The dynamics of free liquid jets moving in air, which are characteristic of fiber-forming processes, involve growth of various perturbations. The most notable are driven by surface tension and the dynamic interaction with the surrounding air, as well as electrically driven effects. Theoretical/numerical description of the jet evolution in general, and of perturbed jets in particular, is hindered by the fact that such problems typically involve a three-dimensional, time-dependent evolution of flows with free surfaces, the locations in time of which should also be established. Solving such problems in the framework of the rigorous equations of fluid mechanics, say the Navier–Stokes equations, in most cases would be prohibitively time-consuming, even using super-computers. Additional complicating factors arise due to the rheological complexity of polymer solutions used in fiber forming, as well as the temperature-dependent variation of material properties in nonisothermal situations. Accounting for all these factors together in the framework of the rigorous equations of non-Newtonian fluid mechanics would be tremendously difficult. However, these difficulties can be relatively easily overcome in the framework of a quasi-one-dimensional description of liquid motion in the bending jets. In the works of Entov and Yarin (1980, 1984a) and Yarin (1983, 1993) the general quasi-one-dimensional equations of the straight and bending jets were derived from the integral balances of mass, momentum and moment of momentum, as well as by averaging the three-dimensional equations of hydrodynamics over the jet cross-section.
This chapter outlines several applications of electrospun and solution-blown nanofibers and their mats. In the case of filters and membranes (Section 9.1), industrial application has already begun, but a number of research questions are still open. Applications of nanofiber mats as fluffy electrodes beneficial for fuel cells and Li-ion batteries have recently attracted significant attention and are the focus of Section 9.2. Two recent approaches based on nanofibers were proposed in the field of cooling of high-heat flux microelectronics (Section 9.3) and nanofluidics (Section 9.4).
Filters and membranes
Filter materials are used for air, water and blood filtration, while membranes are used in separation processes, in particular, for bioseparation and pathogen removal for direct blood transfusion. Filters can remove particles, droplets, bacteria, viruses or even individual molecules from a carrier fluid flowing through them, or, in principle, possess advanced detection and response features that are practically absent in today’s products (see Chapter 10).
Chemical warfare differs from conventional warfare, such as explosives, in the sense that the toxicity of the chemicals used is very dangerous to the combatants in the battlefield, or to the civil population being affected during a war or as a result of a terrorist attack. The threat is amplified by concealment and low cost and relatively easy production of these agents, which renders them weapons of mass destruction. It becomes essential in such a situation that soldiers wear protective clothing, respirators, face masks, gloves, etc. Initially German troops unleashed chemical warfare agents, such as mustard agent, HD, during World War I in 1915, and this resulted in much more concern during World War II. Thus research on decontamination of chemical and biological warfare agents (CWAs) has been the subject of serious interest in many research laboratories since World War II. Mustard gas or bis(2-chloroethyl) sulfide is one of the highly cytotoxic, readily methylating CWAs that are regulated under the 1993 Chemical Weapons Convention (CWC). It covalently binds to DNA bases or forms disulfide bonds with the thiol groups. This readily leads to programmed cell death or the mutated DNA leads to cancer. The simultaneous danger associated with this toxic agent is its high skin penetration owing to its high lipophilicity, which results in immediate blister formation and huge uptake through skin within a short duration of exposure (Ivarsson et al. 1992).
Conventional modes of decontamination involve activated charcoal or other such heavy physical adsorbants of the contaminant. When compared with technologies employing conventional catalysts and reactive sorbents, which are specific and sometimes time-consuming, nanotechnology-based nanomaterials are highly reactive, nonselective and multifunctional in character. Current development of nanoparticles such as MgO, Al2O3, Fe2O3, ZnO and TiO2 and their incorporation into nanofibers impart enhanced catalytic, disinfection and sensing capabilities, photo-protection capability, and stain-resistance and self-cleaning properties. The testing of these materials in textiles and protective clothing against CWAs showed that they have potential as replacement technology in such applications.
Fiber-forming processes and the resulting fibers have become a key element in many modern technologies. Today, practically everyone is directly or indirectly using these fibers. Manmade macroscopic fibers are widely used in our garments and many other items of everyday life. On the other hand, much smaller microscopic and, especially, nanofibers are only beginning their path to prominence. The chemical, physical and technological aspects of manufacturing of such fibers are still weakly linked and not fully understood. Two main processes associated with formation of micro- and nanofibers are melt- or solution blowing and electrospinning. They require concerted interaction of synthetic chemistry, responsible for polymers used as raw materials, polymer physics, providing a link to their viscoelastic behavior, rheological characterization of flow properties, non-Newtonian fluid dynamics of polymer solutions and melts, aerodynamics, associated with gas blowing, and electrohydrodynamics, in the case of electrospinning. The key element of the fiber-forming processes is a thin jet of polymer solution or melt, which rapidly changes its three-dimensional configuration under the action of the aerodynamic or electric forces applied to its surface and the internal viscous and elastic stresses. There is a definite and imperative need to interpret and rationalize these phenomena, which requires acquisition of extensive experimental data and establishment of an appropriate theoretical framework as an essential element in the further technological design and optimization. In addition to the above-mentioned broad spectrum of disciplines, this involves different aspects associated with materials science, such as the methods developed in polymer crystallography, and elasticity and plasticity theory. Although many aspects of fiber-forming processes can today be considered as uncovered and well described, either experimentally or theoretically/numerically, numerous important details are still to be explored. The importance of this subject is attested by an exponential increase in scientific publications devoted to microscopic and nanofibers and a broad involvement of the industries associated with fiber media, nonwovens, nano-textured materials, novel biomedical and healthcare products and optical fibers, as well as defense applications.
Interactions between waves and mean flows play a crucial role in understanding the long-term aspects of atmospheric and oceanographic modelling. Indeed, our ability to predict climate change hinges on our ability to model waves accurately. This book gives a modern account of the nonlinear interactions between waves and mean flows, such as shear flows and vortices. A detailed account of the theory of linear dispersive waves in moving media is followed by a thorough introduction to classical wave-mean interaction theory. The author then extends the scope of the classical theory and lifts its restriction to zonally symmetric mean flows. It can be used as a fundamental reference, a course text, or by geophysicists and physicists needing a first introduction. This second edition includes brand new material, including a section on Langmuir circulations and the Craik–Leibovich instability. The author has also added exercises to aid students' learning.
Problems of wave interaction with a body with arbitrary shape floating or submerged in water are of immense importance in the literature on the linearized theory of water waves. Wave-free potentials are used to construct solutions to these problems involving bodies with circular geometry, such as a submerged or half-immersed long horizontal circular cylinder (in two dimensions) or sphere (in three dimensions). These are singular solutions of Laplace’s equation satisfying the free surface condition and decaying rapidly away from the point of singularity. Wave-free potentials in two and three dimensions for infinitely deep water as well as water of uniform finite depth with a free surface are known in the literature. The method of constructing wave-free potentials in three dimensions is presented here in a systematic manner, neglecting or taking into account the effect of surface tension at the free surface or for water with an ice cover modelled as a thin elastic plate floating on the water. The forms of the wave motion at the upper surface (free surface or ice-covered surface) related to these wave-free potentials are depicted graphically in a number of figures for all the cases considered.
This work focuses on finding optimal dividend payment and capital injection policies to maximize the present value of the difference between the cumulative dividend payment and the possible capital injections with delays. Starting from the classical Cramér–Lundberg process, using the dynamic programming approach, the value function obeys a quasi-variational inequality. With delays in capital injections, the company will be exposed to the risk of financial ruin during the delay period. In addition, the optimal dividend payment and capital injection strategy should balance the expected cost of the possible capital injections and the time value of the delay period. In this paper, the closed-form solution of the value function and the corresponding optimal policies are obtained. Some limiting cases are also discussed. A numerical example is presented to illustrate properties of the solution. Some economic insights are also given.
It has been known for a long time that the equivariant $2+1$ wave map into the $2$-sphere blows up if the initial data are chosen appropriately. Here, we present numerical evidence for the stability of the blow-up phenomenon under explicit violations of equivariance.
There is much interest within the mathematical biology and statistical physics community in converting stochastic agent-based models for random walkers into a partial differential equation description for the average agent density. Here a collection of noninteracting biased random walkers on a one-dimensional lattice is considered. The usual master equation approach requires that two continuum limits, involving three parameters, namely step length, time step and the random walk bias, approach zero in a specific way. We are interested in the case where the two limits are not consistent. New results are obtained using a Fokker–Planck equation and the results are highly dependent on the simulation update schemes. The theoretical results are confirmed with examples. These findings provide insight into the importance of updating schemes to an accurate macroscopic description of stochastic local movement rules in agent-based models when the lattice spacing represents a physical object such as cell diameter.
A stochastic algorithm for bound-constrained global optimization is described. The method can be applied to objective functions that are nonsmooth or even discontinuous. The algorithm forms a partition on the search region using classification and regression trees (CART), which defines a region where the objective function is relatively low. Further points are drawn directly from the low region before a new partition is formed. Alternating between partition and sampling phases provides an effective method for nonsmooth global optimization. The sequence of iterates generated by the algorithm is shown to converge to an essential global minimizer with probability one under mild conditions. Nonprobabilistic results are also given when random sampling is replaced with points taken from the Halton sequence. Numerical results are presented for both smooth and nonsmooth problems and show that the method is effective and competitive in practice.