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Thus, 1 nanometre = 1 nm = 10–9 metre = 10–9 m. Etymology. The Greek word ..... (2.4) where p is the internal pressure of the fluid, and τ is the viscous force.
Electrospun Nanofibres and Their Applications

Ji-Huan He, Yong Liu, Lu-Feng Mo, Yu-Qin Wan and Lan Xu

Smithers Rapra Update

Electrospun Nanofibres and Their Applications Ji-Huan He, Yong Liu, Lu-Feng Mo, Yu-Qin Wan and Lan Xu

iSmithers A wholly owned subsidiary of The Smithers Group Shawbury, Shrewsbury, Shropshire, SY4 4NR, United Kingdom Telephone: +44 (0)1939 250383 Fax: +44 (0)1939 251118 http://www.rapra.net

Published 2008

iSmithers Shawbury, Shrewsbury, Shropshire, SY4 4NR, UK

©2008, iSmithers

All rights reserved. Except as permitted under current legislation no part of this publication may be photocopied, reproduced or distributed in any form or by any means or stored in a database or retrieval system, without the prior permission from the copyright holder.

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Cover micrograph reproduced with permission from Dennis Kunkel Microscopy, Inc.

ISBN: 978-1-84735-145-6

Typeset by iSmithers Printed and bound by Lightning Source Inc.

C

ontents

1.

Introduction ......................................................................1 1.1

What is Nanotechnology? ........................................1

1.2

What is Electrospinning? .........................................6

1.3

What Affects Electrospinning? ...............................10

1.4

Applications ..........................................................12

1.5

Global Interest in the Field of Electrospinning .......13

References .......................................................................16 2.

Mathematical Models for the Electrospinning Process.....17 2.1

One-Dimensional Model .......................................17

2.2

Spivak-Dzenis Model .............................................18

2.3

Wan-Guo-Pan Model .............................................19

2.4

Modified One-Dimensional Model ........................20

2.5

Modified Conservation of Charge Model ..............22

2.6

Reneker’s Model ....................................................28

2.7

E-Infinity Theory ...................................................34

References .......................................................................39 3.

Allometric Scaling in Electrospinning ..............................41 3.1

Allometric Scaling in Nature ..................................42

3.2

Allometric Scaling Laws in Electrospinning ...........45

3.2.1

Relationship Between Radius r of Jet and Axial Distance z.............................. 45

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Electrospun Nanofibres and Their Applications

3.2.2

Allometric Scaling Relationship Between Current and Voltage ................ 53

3.2.3

Allometric Scaling Relationship Between Solution Flow Rate and Current ........................................... 56

3.2.4

Effect of Concentration on Electrospun PAN Nanofibres ................. 59

3.2.5

Allometric Scaling Law Between Average Polymer Molecular Weight and Electrospun Nanofibre Diameter .... 63

3.2.6

Effect of Voltage on Morphology and Diameter of Electrospun Nanofibres...... 66

3.2.7 3.3

Improving Electrospinnability Using Non-ionic Surfactants............................ 72 Allometric Scaling Law for Static Friction of Fibrous Materials ..................................................80

3.3.1

Solid–Solid Friction ............................... 81

3.3.2

Viscous Friction for Newtonian Flow .... 82

3.3.3

Friction for Soft Materials ..................... 82

3.3.4 Fibre–Fibre Friction ............................... 82 Allometric Scaling in Biology .................................84 References .......................................................................87 3.4

4.

ii

Application of Vibration Technology to Electrospinning ...93 4.1

Effect of Viscosity on Diameter of Electrospun Fibre ...................................................93

4.2

Effect of Vibration on Viscosity .............................94

4.3

Application of Vibration Technology to Polymer Electrospinning ........................................95

4.4

Effect of Solution Viscosity on Mechanical Characteristics of Electrospun Fibres ...................103

Contents

4.5 Carbon-Nanotube-Reinforced Polyacrylonitrile Nanofibres by Vibration Electrospinning .............105 References .....................................................................112 5.

Magneto-Electrospinning: Control of the Instability......115 5.1

Critical Length of Straight Jet in Electrospinning ...116

5.2

Controlling Stability by Magnetic Field ...............119

5.3

Controlling Stability by Temperature ...................123

5.4

Siro-electrospinning .............................................127

References .....................................................................129 6.

Bubble Electrospinning: Biomimic Fabrication of Electrospun Nanofibres with High Throughput.............131 6.1

Spider Spinning ....................................................131

6.1.1

Intelligent Spider Fibre ........................ 132

6.1.2

Mathematical Model for SpiderSpun Fibres.......................................... 134

6.2

Electrospinning of Silk Fibroin Nanofibres ..........135

6.3

Solving the Mystery of the Spider Spinning Process ..................................................136

6.4

Bubble Electrospinning ........................................143

References .....................................................................154 7.

Controlling Numbers and Sizes of Beads in Electrospun Nanofibres .................................................157 7.1

Experimental Observations ..................................157

7.2

Effects of Different Solvents .................................158

7.3

Effect of Polymer Concentration ..........................163

7.4

Effect of Salt Additive ..........................................165

References .....................................................................167

iii

Electrospun Nanofibres and Their Applications

8.

Electrospun Nanoporous Microspheres for Nanotechnology ............................................................169 8.1

Electrospun Nanoporous Spheres with a Traditional Chinese Drug.....................................170

8.2

Electrospinning Dilation ......................................180

8.3

Single Nanoporous Fibres by Electrospinning ......183

8.4

Microspheres with Nanoporosity ........................187

References .....................................................................190 9.

A Hierarchy of Motion in the Electrospinning Process and E-Infinity Nanotechnology .........................191 9.1

E-Infinity Nanotechnology ...................................191

9.2

Application of E-Infinity Theory to Electrospinning ....................................................193

9.3

9.2.1

Hausdorff Dimension for the Hierarchy of Motion ........................... 194

9.2.2

Experimental Verification .................... 195

Super Carbon Nanotubes: An E-Infinity Approach .............................................................199

References .....................................................................202 10. Mechanics in Nanotextile Science..................................203 10.1 Jet Vortex Spinning and Cyclone Model ..............203 10.2 Two-Phase Flow of Yarn Motion in High-Speed Air and Micropolar Model ...................................205 10.3 Mathematical Model for Yarn Motion in a Tube..................................................................209 10.4 Nanohydrodynamics ...........................................212 10.5 A New Resistance Formulation for Carbon Nanotubes and Nerve Fibres ...............................218

iv

Contents

10.6 Differential–Difference Model for Nanotechnology ..................................................220 References .....................................................................222 11. Nonlinear Dynamics in Sirofil/SiroSpun Yarn Spinning ...225 11.1 Convergence Point ...............................................227 11.2 Linear Dynamic Model ........................................229 11.3 Nonlinear Dynamic Model ..................................236 11.4 Stable Working Condition for Three-Strand Yarn Spinning ......................................................241 11.5 Nano-Sirospinning ...............................................246 References .....................................................................247 Acknowledgement ........................................................248 Abbreviations ........................................................................249 Index .....................................................................................251

v

Electrospun Nanofibres and Their Applications

vi

1

Introduction

1.1 What is Nanotechnology? The importance of nanotechnology as an emerging technology has been recognised in the USA, where a National Nanotechnology Initiative (see http://www.nano.gov) was launched, with an investment of over $1 billion in nanotechnology research over the past few years. Nanotechnology has attracted much attention recently, and it can be applied to all aspects of science and engineering, as well as to life. But what is nanotechnology? There are many definitions of the term, but here we adopt El-Naschie’s definition [1]. The naive and direct answer to the frequently posed question what exactly is Nanotechnology is to say that it is a technology concerning processes which are relevant to physics, chemistry and biology taking place at a length scale of one divided by 100 million of a metre. Thus, 1 nanometre = 1 nm = 10–9 metre = 10–9 m.

Etymology The Greek word nanos or nannos means ‘little old man’ or ‘dwarf’, from nannas, meaning ‘uncle’. The metric prefix nanomeans one billionth of a unit or 10–9. A single human hair is around 80,000 nanometres in width.

1

Electrospun Nanofibres and Their Applications An obvious phenomenon is the remarkably large surface-to-volume ratio of nanomaterials. Consider a fibre with radius of 1 mm and length of 10 mm. Its surface area is:

Now we divide the fibre into nanofibres with radius of 10 nm. The number of nanofibres can be calculated as:

So the total surface of the nanofibres is:

The volume remains unchanged, while the surface area increases remarkably by the factor:

Maybe a little bit more enlightening although equally naive is to say, according to El Naschie [1], that nanotechnology is the art of producing little devices, machines and systems that have novel properties. These include extremely small electronic devices and circuits built from individual atoms and molecules, DNA computers, microelectromechanical systems, motors, nanosensors, nanowires, nano-satellite missions, and others, somewhat at atomic, molecular, or macromolecular scales. Figure 1.1 gives an example of the possibilities. Atoms are roughly angstroms (Å) in size: a hydrogen atom is about 1 Å in diameter, a carbon atom is about 2 Å in diameter, and the diameter of an oxygen atom is about 1.75 Å. One angstrom (1 Å) is one tenbillionth of a metre or one-tenth of a nanometre. Thus 1 nm = 10 Å. On the molecular or angstrom scale, quantum-like phenomena occur. In a scientific sense, El Naschie [1] defined nanotechnology as a 2

Introduction

Figure 1.1 Aqua technology on the angstrom scale, shaping the world atom by atom.

technology applied in the grey area between classical mechanics and quantum mechanics. Classical mechanics is the mechanics governing the motion of all the objects we can see with our naked eye. This is mechanics that obeys deterministic laws (Newton’s laws) and that we can control to a very great extent. By contrast, quantum mechanics, which is the mechanics controlling the motion of things like the electron, proton, neutron and the like, is completely probabilistic. We know nothing about the motion of the electron except that there is a probability that the electron may be here or there. Even crazier than this, if we know the exact location of an electron, it is impossible to know its speed; and if we know the exact speed of the electron, it is impossible to know its exact location. Such a relationship is called the Heisenberg uncertainty principle. 3

Electrospun Nanofibres and Their Applications Nanotechnology links to both deterministic classical mechanics and chaotic quantum mechanics [1]. There should be a law controlling the change from a classical object like a stone to a quantum object like an electron. Somewhere between these two scales these changes happen, but this does not happen suddenly. There is a grey area between these two scales, which is neither classical nor quantum [1]. To model quantum processes, we can use the deterministic chaotic geometry, which is used in El Naschie’s E-infinity theory [2-4]. Consider an extremely simple quadratic equation in iterated form [5]:

where λ is a parameter. The incredible complexity displayed by the bifurcation diagram of this equation as λ varies was one of the most amazing discoveries in this field (see Figure 1.2). The main application of E-infinity theory shows miraculous scientific exactness, especially in determining theoretically the coupling constants and the mass spectrum of the standard model of elementary

Figure 1.2 From deterministic path to chaotic path.

4

Introduction particles. For example, the absolute zero temperature can be derived using E-infinity theory [1–5] as:

or the mass of an expectation proton [1–5] is:

where

is the golden mean.

On the nanoscale, some new properties occur, and in many instances the origins of the new properties are, at present, not fully understood. Scale is of utter importance in physics as well as nanotechnology. Different scales lead to different laws (or theories) and thus result in different dimensions, as illustrated in the following formula: dimensions = 3 + 1 + φ 3 = 4.236… 3 dimensions

Newton: three-dimensional absolute space

3 + 1 dimensions Einstein: four-dimensional continuous space–time 4.236 dimensions El-Naschie: infinite-dimensional discontinuous space–time. In view of El-Naschie’s E-infinity theory, nanoscale systems may possess entirely new physical and chemical characteristics that result in properties that are well described neither by those of a single molecule of the substance nor by those of the bulk material. A similar phenomenon is observed on the quantum scale (see Figure 1.3). On such a scale, Einstein’s space–time resembles a stormy ocean and his original Riemannian smooth manifold is only an approximation [6]. This is the very reason for Einstein’s failure to unify gravity with electromagnetism. 5

Electrospun Nanofibres and Their Applications

Figure 1.3 Nanomechanics – is it chaotic or deterministic?

1.2 What is Electrospinning? There are various approaches to producing nanofibres. For example, the following can be used: drawing technology for producing micro/nanofibres using a micropipette with a diameter of a few micrometres; template synthesis of carbon nanotubes, nanofibre arrays and electronically conductive polymer nanostructures; and thermally induced phase separation method for producing nanoporous nanofibres. Electrospinning is the cheapest and the most straightforward way to produce nanomaterials. Electrospun nanofibres are of indispensable importance for the scientific and economic revival of developing countries. Structured polymer fibres with diameters in the range from several micrometres down to tens of nanometres are of considerable interest for various kinds of applications. It is now possible to produce a low-cost, high-value, high-strength fibre from a biodegradable and renewable waste product for easing environmental concerns. For instance, a pore structured electrospun nanofibrous membrane used as a wound dressing can promote the exudation of fluid from the wound, so as to prevent either build-up under the covering or wound desiccation. The electrospun nanofibrous membrane shows controlled liquid evaporation, excellent oxygen permeability and 6

Introduction promoted fluid drainage capacity, while still inhibiting exogenous microorganism invasion because its ultrafine pores. Other examples include thin fibres for filtration application, bone tissue engineering, drug delivery, catalyst supports, fibre mats serving as reinforcing component in composite systems, and fibre templates for the preparation of functional nanotubes. In 1934, a process was patented by Formhals [7] entitled ‘Process and apparatus for preparing artificial threads’, wherein an experimental set-up was outlined for the production of polymer filaments using electrostatic force (Figure 1.4). When used to spin fibres this way, the process is termed electrospinning.

Figure 1.4 Formhals’ electrospinning set-up.

Electrospinning is a novel process for producing superfine fibres by forcing a viscous polymer, composite, sol–gel solution or melt through a spinneret with an electric field to a droplet of the solution, most often at a metallic needle tip (Figure 1.5). The electric field draws this droplet into a structure called a Taylor cone [8]. If the viscosity and surface tension of the solution are appropriately tuned, varicose break-up is avoided (if there is varicose break-up, then electrospray occurs) and a stable jet is formed.

7

Electrospun Nanofibres and Their Applications

Figure 1.5 The most frequently used electrospinning set-up.

Taylor cone A Taylor cone [8] is caused by equilibrium between the electronic force of the charged surface and the surface tension. A higher applied voltage leads to an elongated cone; when it exceeds its threshold voltage, a jet is emanated.

Electrospinning traces its roots to electrostatic spraying. Electrospinning now represents an attractive approach for polymer biomaterials processing, with the opportunity for control over morphology, porosity and composition using simple equipment. Because electrospinning is one of the few techniques to prepare long fibres of nano- to micro-metre diameter (Figure 1.6), great progress has been made in recent years. 8

Introduction

Figure 1.6 SEM photographs of electrospun fibres.

9

Electrospun Nanofibres and Their Applications Nanofibre is defined in this book as a slender, elongated thread-like object or structure on the nanoscale, from several hundred to several thousand nanometres. Nanofibre is an emerging, interdisciplinary area of research, with important commercial applications, and will, most assuredly, be a dominant technology in new-world economies. Materials in nanofibre form have an exceptionally high specific surface area, which enables a high proportion of atoms to be on the fibre surface. This will result in quantum efficiency, nanoscale effect of unusually high surface energy, surface reactivity, high thermal and electrical conductivity, and high strength. Electrospun fibres can be used in the following applications: nonwoven fabrics, reinforced fibres, support for enzymes, drug delivery systems, fuel cells, conducting polymers and composites, photonics, sensorics, medicine, pharmacy, wound dressings, filtration, tissue engineering, catalyst supports, fibre mats serving as reinforcing component in composite systems, and fibre templates for the preparation of functional nanotubes, to name just a few.

1.3 What Affects Electrospinning? We have difficulty in precisely controlling the diameter, morphology and porosity of electrospun fibres, which means that we should develop a new theory linked to classical mechanics and quantum mechanics. As a first step towards the new theory, we should define the dimensions needed for theoretical analysis on a suitable scale, certainly different from our three-dimensional space or the four dimensions of space–time. Thus El-Naschie’s E-infinity theory is needed. Note that Einstein’s special theory of relativity forbids discontinuous space, leading to the failure to unify gravity with electromagnetism. El Naschie set out to resolve the contradiction and the result was his famous E-infinity theory [1]. According to El Naschie, space and time are discontinuous. The main conceptual idea of E-infinity theory is in fact a sweeping generalisation of what Einstein did in 10

Introduction

(a)

(b) Figure 1.7 SEM photographs of polybutylene succinate (PBS) electrospun fibres under the same conditions, showing the different morphologies formed: (a) beads; and (b) enlarged beads with surface porosity. The concentration and voltage are 12 wt% and 10 kV. The diameter of the inner needle orifice is 0.5 mm. 11

Electrospun Nanofibres and Their Applications his general theory of relativity, namely introducing a new geometry for space–time which differs considerably from the space–time of our sensual experience. El Naschie [1] points out that on extremely small scales, at very high observational resolution equivalent to a very high energy, space–time resembles a stormy ocean. We would like to emphasise that E-infinity theory stresses the fact that everything we see or measure in nature is resolution-dependent. On the macroscale, pure water is definitely continuous, with three dimensions of space. As we said above, one angstrom (1 Å) is one ten-billionth of a metre or one-tenth of a nanometre, and atoms are roughly angstroms in size. Thus, on the angstrom scale, continuum water becomes discontinuous. On the angstrom scale, the fractal dimension reads:

At such a resolution, the world of water must look like a universe that is full of empty space. On the nanoscale, quantum-like phenomena can be observed. In our experiments, we find uncertainty: under almost the same conditions, we cannot obtain exactly the same nanofibres, beads or microspheres (see Figure 1.7). This is similar to Heisenberg’s uncertainty principle in quantum mechanics. Of course, deterministic characteristics also exist on the nanoscale. The following parameters affect the electrospinning process: molecular weight, viscosity, conductivity, surface tension, applied electric potential, flow rate, solvent, concentration, and distance between the capillary and collection screen. A complete mathematical analysis is available in the forthcoming chapters.

1.4 Applications An important characteristic of electrospinning is the ability to make fibres with diameters in the range of nanometres to a few 12

Introduction micrometres. Consequently, these fibres have a large surface area per unit mass. Thus, nanowoven fabrics of these nanofibres collected on a screen can be used, for example, for filtration of submicrometre particles in the separation industries and biomedical applications, wound dressings, tissue engineering scaffolds, artificial blood vessels, photonics, sensorics, pharmacy, drug delivery, catalyst supports, fibre mats serving as reinforcing components in composite systems, fibre templates for the preparation of functional nanotubes, and others. The use of electrospun fibres in ‘invisibility devices’ (e.g., ‘stealth’ planes) and at critical places in advanced composites to improve crack resistance is also promising.

1.5 Global Interest in the Field of Electrospinning According to the database of the Web of Science the number of publications on electrospinning is rocketing, as shown in Figures 1.8–1.12 and Table 1.1.

Figure 1.8 Increase of publications in electrospinning. 13

Electrospun Nanofibres and Their Applications

Figure 1.9 Global interest in electrospinning.

Figure 1.10 Top 10 electrospinning institutes in the world.

Figure 1.11 Top 10 electrospinning institutes in the Peoples’ Republic of China.

14

Introduction

Figure 1.12 Top 10 scientists in the field of electrospinning.

Table 1.1 Increase of investment in electrospinning in the USA, for financial years 2000–2004 Federal Department or Agency

Actual ($ million) 2000

2001

2002

2003

2004

National Science Foundation

97

150

204

221

249

Department of Defense

70

125

224

243

222

Department of Energy

58

88

89

133

197

National Institutes of Health

32

40

59

65

70

National Institute of Standards and Technology (NIST)

8

33

77

69

62

National Aeronautics and Space Administration (NASA)

5

22

35

33

31

Environmental Protection Agency



6

6

6

5

Homeland Security (TSA)





2

2

2

Department of Agriculture



1.5

0

1

10

Department of Justice (DOJ) Total



1.4

1

1

1

270

465

697

774

849

15

Electrospun Nanofibres and Their Applications

References 1.

M.S. El Naschie, Chaos, Solitons & Fractals, 2006, 30, 4, 769.

2.

M.S. El Naschie, Chaos, Solitons & Fractals, 2004, 19, 1, 209.

3.

M.S. El Naschie, International Journal of Nonlinear Sciences and Numerical Simulation, 2005, 6, 3, 331.

4.

M.S. El Naschie, International Journal of Nonlinear Sciences and Numerical Simulation, 2006, 7, 2, 119.

5.

M.S. El Naschie, Chaos, Solitons & Fractals, 2006, 30, 3, 579.

6.

J-H. He, International Journal of Nonlinear Sciences and Numerical Simulation, 2005, 6, 2, 93.

7.

A. Formhals, inventor; no assignee; US Patent 1,975,504, 1934.

8.

G. Taylor, Proceedings of the Royal Society of London Series A, 1964, 280, 383.

16

2

Mathematical Models for the Electrospinning Process

This chapter deals with modelling the electrospinning process. Theoretical models offer in-depth insights into the physical understanding of many complex phenomena that cannot be fully explained experimentally. A simple model (e.g., allometric approach, see Chapter 3) might be very useful to shed light on the contributing factors. Theoretical models are a powerful tool to realise some control over morphology, porosity and physical characteristics. Many basic properties and some special properties (such as biocompatibility, degradation) are tunable by adjusting the electrospinning parameters, such as voltage, flow rate and others. There are two main ways to model the process: one is the deterministic approach used in classical mechanics, including the Euler approach (Sections 2.1–2.5) and the Lagrange approach (Section 2.6); and the other is the probabilistic approach, where E-infinity theory [1] can be powerfully applied and quantum-like properties can be fully elucidated.

2.1 One-Dimensional Model Details of this model can be found in [2–6]. A steady-state flow of an infinite viscous jet pulled from a capillary orifice and accelerated by a constant external electric field is considered. (1) Conservation of mass gives: (2.1) where Q is the volume flow rate, r is the radius of the jet, ρ is the liquid density, and u is the velocity.

17

Electrospun Nanofibres and Their Applications (2) Letting the surface charge be σ, conservation of charge gives: (2.2) where k is the dimensionless conductivity of the fluid, E is the applied electric field, and I is the current passing through the jet. (3) Force balance gives: (2.3) For steady-state flow, we have:

(2.4) where p is the internal pressure of the fluid, and τ is the viscous force.

2.2 Spivak-Dzenis Model Spivak and co-workers [7, 8] established a model of a steady-state jet in the electrospinning process. (1) Equation of mass balance gives: (2.5) (2) Linear momentum balance is: (2.6) (3) Electric charge balance reads: (2.7)

18

Mathematical Models for the Electrospinning Process The right-hand side of Equation (2.6) is the sum of viscous and electric forces.

2.3 Wan-Guo-Pan Model This model [9] considers the coupled thermal, electrical and hydrodynamic effects. The modified Navier–Stokes equations governing heat and fluid flow under the influence of an electric field, and the constitutive equations describing the behaviour of the fluid are: (2.8)

(2.9)

(2.10) This set of conservation laws can constitute a closed system when it is supplemented by appropriate constitutive equations for the field variables, such as polarisation. The most general theory of constitutive equations determining the polarisation, electrical conduction current, heat flux and Cauchy stress tensor has been developed by Eringen and Maugin [10, 11]: (2.11) (2.12) (2.13) (2.14) Here, the coefficients εp, k, σ, σT, κ, κE, η are material properties and depend only on temperature in the case of an incompressible 19

Electrospun Nanofibres and Their Applications fluid. For the physical importance of these properties, see the review paper by Ko and Dulikravich [12]. Equation (2.14) is valid only for Newtonian flows; of course, it can be extended to more complex nonlinear constitutive equations.

2.4 Modified One-Dimensional Model An unsteady flow of an infinite viscous jet pulled from a capillary orifice and accelerated by a constant external electric field is considered in this section. Further details may be found in [9]. (1) The conservation of mass equation gives: (2.15) where r is the radius of the jet at axial coordinate z, and u is the axial velocity. (2) Conservation of charge reads:

(2.16) where σ is the surface charge density, and E is the electric field in the axial direction. The current is composed of three parts: (i) ohmic bulk conduction current:

(ii) surface convection current:

(iii) current caused by temperature gradient:

20

Mathematical Models for the Electrospinning Process (3) The Navier–Stokes equations become: (2.17)

(2.18) where p is the internal pressure of the fluid expressed as: (2.19) Here κ is twice the mean curvature of the interface, κ = 1/R1 + 1/R2, with R1 and R2 the principal radii of curvature, ε is the dielectric constant of the fluid, and is the dielectric constant of air. The rheological behaviour of many polymer fluids can be described by a power-law constitutive equation in the form: (2.20) In addition to conducting bodies, there are also dielectrics. In dielectrics, the charges are not completely free to move, but the positive and negative charges that compose the body may be displaced in relation to one another when a field is applied. The body is said to be polarised. The polarisation is given in terms of a dipole moment per unit volume P, called the polarisation vector. The bound charge or polarisation charge in the dielectric is given by: (2.21)

21

Electrospun Nanofibres and Their Applications In an isotropic linear dielectric case, the polarisation is assumed to be proportional to the field that causes it, thus: (2.22) where εp is the electrical susceptibility.

2.5 Modified Conservation of Charge Model Details of this model may be found in [13]. The current in the electrospinning jet is composed of two parts, the bulk conduction current:

and surface convection current:

Conservation of charge reads:

(2.23) We know from Ohm’s law that current flows down a voltage gradient in proportion to the resistance in the circuit. The bulk current is therefore expressed as I = E/R. In all the open literature [2–6], the calculation of resistance is still based on the traditional formulation for metal resistors, which can be written as: (2.24) where Rc is the resistance of a conductor, A is the cross-sectional area, and k is the resistance coefficient. If the bulk resistance in the electrospinning jet obeys the same law as illustrated above, then Equation (2.23) reduces to Equation (2.2). 22

Mathematical Models for the Electrospinning Process Actually, Equation (2.24) is valid only for metal conductors where there are plenty of electrons in the conductor. However, in an electrospinning jet, the current is not caused by electrons, so Equation (2.24) should be modified in order to accurately describe the jet conduction [14–16]. The resistance of a conductive metal scales as: (2.25) For an insulator, the resistance is independent of its cross-sectional area. In scaling form, we write: (2.26) or (2.27) For an electronic fluid, we assume that: (2.28) where α is constant relative to the conductive character of the fluid. The conductive behaviour of a conductive fluid lies between that for a metal and that for an insulator, so the value of α lies between 0 and 1: (2.29) In ancient China, there were many interpolation formulae [17]. Among others, here we will use He Chengtian’s interpolation [18] to fix approximately the value of α. Consider the inequality: 23

Electrospun Nanofibres and Their Applications

(2.30) where a, b, c and d are integers. According to He Chengtian’s interpolation, the value of x can be approximated as: (2.31) where m and n are weighting factors. Proof of He Chengtian’s interpolation can be found in [18], and application of this ancient mathematical tool can be found in [14–17].

He Chengtian In an ancient history book is written the following: He Chengtian uses 26/49 as the strong, and 9/17 as the weak. Among the strong and the weak, Chengtian tries to find a more accurate denominator of the fractional day of the Moon. Chengtian obtains 752 as the denominator by using the 15 and 1 as weighting factors, respectively, for the strong and the weak. No other calendar can reach such a high accuracy after Chengtian, who uses heuristically the strong and weak weighting factors. The statement is rather cryptic. In modern mathematical terms, the statement can be explained as follows. According to the observation data, He Chengtian (369?–447AD) finds that:

Using the weighting factors (15 and 1), He Chengtian obtains

so:

24

Mathematical Models for the Electrospinning Process He Chengtian actually uses the following inequality:

where a, b, c and d are real numbers. Then:

and x is approximated by:

where m and n are weighting factors. Applying He Chengtian’s interpolation, the value of α in Equation (2.28) can be written in the form: (2.32) where m and n are integers, such that δ = n/m. So for a non-metal material, we have: (2.33) where δ is a constant relative to the charge concentration in the section. In the case δ → ∞, the exponent α tends to 1, and it becomes a metal-like conductor. When δ = 1, α tends to 1/2. Such a case arises in surface convection currents. See Figures 2.1–2.3 for some illustrations of the possibilities for exponent α. 25

Electrospun Nanofibres and Their Applications

Figure 2.1 Resistance of an ideal electronic jet: R ~ 1/r2, δ → ∞, i.e., there are plenty of charged particles in the section just like electrons in a metal conductor. The exponent ‘α = 2’ can be explained as the fractal dimension of the ‘charged’ section.

Figure 2.2 Resistance of surface convection: R ~ 1/r1, δ = 1. The exponent ‘α = 1’ can be explained as the fractal dimension of the ‘charged’ perimeter.

26

Mathematical Models for the Electrospinning Process

Figure 2.3 Resistance of an electronic jet: R ~ 1/rα, α = 2δ / (δ+1). The exponent ‘α ’ can be explained as the fractal dimension of the ‘charged’ section.

So Equation (2.2) should be modified as: (2.34a) or in a more general form as: (2.34b) where α and β can be explained as the fractal dimensions of the ‘charged’ perimeter and ‘charged’ section, respectively; h and k are constants. For a conductive textile we have:

If we want to design an electrochemical cell constructed of two electrodes, which are made of knitted, woven or non-woven 27

Electrospun Nanofibres and Their Applications conductive textile material, the allometry leads to the following formulation:

where k is a constant, L is the distance between the electrodes, A is the surface area of the electrodes, and c is the concentration of the electrolyte solution.

2.6 Reneker’s Model Reneker and his colleagues [19] suggested a mathematical model to analyse the reasons for the instability in electrospinning. The rheological complexity of the polymer solution is included, which allows consideration of viscoelastic jets. In this model, the jet is modelled as a system of beads possessing charge e and mass m connected by viscoelastic elements as shown in Figure 2.4.

Figure 2.4 Bending electrospun jet model.

28

Mathematical Models for the Electrospinning Process Now we consider a segment of the jet (i – 1, i). Let the position of bead i – 1 be fixed. The Coulomb repulsive force acting on bead i is . The force applied to i due to the external electrical field is . The stress σi–1,i, which pulls i back to i – 1 is given by: (2.35) where t is time, G and µ are the elastic modulus and viscosity, respectively, and li–1,i is the segment length, which is given by: (2.36) where xi, yi, zi, … are the Cartesian coordinates of the beads. Therefore, the viscoelastic force acting on bead i is , where a is the cross-sectional radius of the segment. The total number of beads, N, increases over time as new electrically charged beads are inserted at the top of Figure 2.4 to represent the flow of solution into the jet. The net Coulomb force acting on the ith bead from all the other beads is given by: (2.37)

where ri and rj are the positions of bead i and bead j, and Rij is the distance between bead i and bead j. The electric force imposed on the ith bead by the electric field is: (2.38) where V0 is the electrical voltage, and h is distance from the pendent drop to the collector. 29

Electrospun Nanofibres and Their Applications The viscoelastic force acting on the ith bead of the jet is:

(2.39)

where ai–1,i and ai,i+1 are the filament radii. The surface tension force acting on the ith bead, and tending to restore the equilibrium shape of the bending part of the jet, is given by: (2.40) where α is the surface tension coefficient, ki is the curvature of jet segment (i–1, i+1), and the meaning of ‘sign’ is: sign(x) = 1,

if x > 0

sign(x) = –1,

if x < 0

sign(x) = 0,

if x = 0

(2.41)

Therefore, the momentum equation for the motion of the beads is: (2.42) where FC, FE, Fve and FB are the forces described previously. For the first bead, i = 1, and N, the total number of beads, is also 1. As more beads are added, N becomes larger and the first bead i = 1 remains at the bottom end of the growing jet. For this bead, all the parameters with subscript i–1,i should be set equal to zero. Zeng and co-workers applied the model to the simulation of the instability of electrospinning [20]. 30

Mathematical Models for the Electrospinning Process Given initial perturbations: (2.43) , with a0 the where L is defined as the length scale, initial cross-sectional radius at t = 0, the numerical simulation is illustrated in Figures 2.5–2.7 [20].

(a) t = 1

(b) t = 2 Figure 2.5 Perturbations develop into a bending instability.

31

Electrospun Nanofibres and Their Applications

(c) t = 3

(d) t = 4 Figure 2.5 Continued

Carroll and Joo [21] studied the electrospinning of viscoelastic Boger fluids, and presented a theoretical model for electrospun Newtonian and viscoelastic jets. In particular, the effect of electrical conductivity and viscoelasticity on the jet profile during the initial stage of electrospinning was examined. In the theoretical study, the fluid is described as a leaky dielectric with charges only on the jet surface. Viscoelastic models for polymer solutions such as Oldroyd-B 32

Mathematical Models for the Electrospinning Process

Figure 2.6 The dimensionless jet diameter d/d0 as a function of time.

Figure 2.7 The dimensionless jet diameter d/d0 as a function of the distance from pendent drop to collector.

and FENE-P are fully coupled with the fluid momentum equations and Gauss’s law. A theoretical model for the jet is derived using a thin filament approximation. 33

Electrospun Nanofibres and Their Applications

2.7 E-Infinity Theory In recent years there has been a flurry of original papers published on the foundation and application of El Naschie’s E-infinity Cantorian space–time theory [1]. The main application of E-infinity theory [22] shows miraculous scientific exactness, especially in determining the theoretical coupling constants and the mass spectrum of the standard model of elementary particles. In this section we will show the possible application of E-infinity theory to nanofibres. We start by explaining what we consider to be possibly an epochmaking theory, namely the infinite-dimensional Cantorian space–time E-infinity proposal. First let us make a careful inspection of Einstein’s field equation: (2.44) where Rij is the Ricci tensor, Tij is the energy–momentum tensor, K is a coupling constant and gij is the metric tensor. On the right-hand side of Equation (2.44) we have the mass tensor. According to Einstein’s famous formula E = mc2, matter and energy are equivalent by virtue of the special theory of relativity. However, energy on a fundamental level obeys Planck’s quantum equation, so, at the quantum scale, the right-hand side of Equation (2.44) becomes quantised or discrete, while the left-hand side of the equation is still continuous. Thus, at the quantum scale, Einstein’s space–time must become discrete in the sense of quantum mechanics, resembling a stormy ocean due to quantum fluctuation or the equation would be extremely limited. The problem in Einstein’s field equation can be eliminated using El Naschie’s E-infinity theory [22], which regards discontinuities of space and time in a transfinite way. Introducing a new Cantorian space–time, El Naschie admitted formally an infinite-dimensional ‘real’ space–time, which is hierarchical in a strict mathematical way. Let us consider first the classical triadic Cantor set in n-dimensional space, and write down the following Hausdorff capacity dimensions 34

Mathematical Models for the Electrospinning Process using the bijection formula Cantor set. In that way we find:

in the case of an orderly

Note that for n = 6 we have , the dimension of the conventional super string theory, while D(6) = 6 is the dimension of the compactified sector. It is also obvious that, at low resolution or equivalently at low energy, we have only when n = 4. That indicates that Cantorian space-time mimics the appearance of our fourdimensional space–time manifold. We also have when n > 4. It is only when n = 4 that we have a quasi-ergodic behaviour for which . We can say that for n < 4 our ‘world’ set is stable; while for n > 4 the set is totally unstable, in fact, chaotic, and makes the transition from classical mechanics to turbulence and special forms of statistical mechanics that we call quantum mechanics. Thus space-time is inherently a Cantorian structure at the small scale, or equivalently at high energy resolution. We note that in the case of a random Cantor set , we find . 35

Electrospun Nanofibres and Their Applications

Setting dimension to be:

we find the familiar E-infinity

The four dimensions of classical space–time upon which Einstein’s theory is based is only an approximation of the true geometry of the Universe. Electrospinning provides a simple approach to fabricating nanofibres and assemblies with controllable hierarchical structures [23, 24]. Figure 2.8 illustrates a schematic representing a section of a nanofibre where the large circle and the medium circles refer to aggregated macromolecules, and the small circles on the fibre surface represent atoms. This will result in quantum-like properties of unusually high surface energy, surface reactivity, and high thermal and electrical conductivity. Figure 2.8 is very similar to the fractal kissing problem (see Figure 2.9), which is inherently related to E-infinity theory. E-infinity theory is a powerful tool for dealing with hierarchical structures, fascinating applications of E-infinity theory to turbulence (see Figure 2.10) [25] and biology (see Figure 2.11) [26]. At the nano-scale, nano-effects arise similarly to those in the quantum world. For example, unusual current conduction properties arise when the size of wires is reduced below a certain critical thickness (nanoscale): smaller wires may conduct more current. This nano-effect is very similar to Arnold diffusion: the higher the dimensionality, the stronger Arnold diffusion is [1]. A review on mathematical models for electrospinning is available in [27]. 36

Mathematical Models for the Electrospinning Process

Figure 2.8 A section of a nanofibre, showing aggregated macromolecules in the centre and a high proportion of atoms on the fibre surface.

Figure 2.9 The fractal kissing problem [1].

37

Electrospun Nanofibres and Their Applications

Figure 2.10 Hierarchical structure in E-infinity turbulence model [25].

Figure 2.11 Hierarchical structure of the blood-vessel system where E-infinity theory can be powerfully applied [26].

38

Mathematical Models for the Electrospinning Process

References 1.

M.S. El Naschie, Chaos, Solitons & Fractals, 2006, 30, 3, 579.

2.

J.J. Feng, Journal of Non-Newtonian Fluid Mechanics, 2003, 116, 1, 55.

3.

A.M. Ganan-Calvo, Physical Review Letters, 1997, 79, 2, 217.

4.

A.M. Ganan-Calvo, Journal of Fluid Mechanics, 1997, 335, 165.

5.

A.M. Ganan-Calvo, Journal of Aerosol Science, 1999, 30, 7, 863.

6.

A.M. Ganan-Calvo, J. Davila and A. Barrero, Journal of Aerosol Science, 1997, 28, 2, 249.

7.

A.F. Spivak and Y.A. Dzenis, Applied Physics Letters, 1998, 73, 21, 3067.

8.

A.F. Spivak, Y.A. Dzenis and D.H. Reneker, Mechanics Research Communications, 2000, 27, 1, 37.

9.

Y-Q. Wan, Q. Guo and N. Pan, International Journal of Nonlinear Sciences and Numerical Simulation, 2004, 5, 1, 5.

10. A.C. Eringen and G.A. Maugin, Electrodynamics of Continua 1: Foundations and Solid Media, Springer-Verlag, New York, NY, USA, 1990. 11. A.C. Eringen and G.A. Maugin, Electrodynamics of Continua 2: Fluids and Complex Media, Springer-Verlag, New York, NY, USA 1990. 12. H.J. Ko and G.S. Dulikravich, International Journal of Nonlinear Sciences and Numerical Simulation, 2000, 1, 4, 247. 13. J-H. He and Y.Q. Wan, Polymer, 2004, 45, 19, 6731. 39

Electrospun Nanofibres and Their Applications 14. J-H. He, Neuroscience Letters, 2005, 373, 1, 48. 15. J-H. He, Polymer, 2004, 45, 26, 9067. 16. J-H. He, Chaos, Solitons & Fractals, 2006, 29, 2, 303. 17. J-H. He, International Journal of Modern Physics B, 2006, 20, 10, 1141. 18. J-H. He, Applied Mathematics and Computation, 2004, 151, 3, 887. 19. D.H. Reneker, A.L. Yarin, H. Fong and S. Koombhongse, Journal of Applied Physics, 2000, 87, 9, 4531. 20. Y.C. Zeng, Y. Wu, Z.G. Pei and C.W. Yu, International Journal of Nonlinear Sciences and Numerical Simulation, 2006, 7, 4, 385. 21. C.P. Carroll and Y.L. Joo, Physics of Fluids, 2006, 18, 5, 053102. 22. M.S. El Naschie, Chaos, Solitons & Fractals, 2004, 19, 1, 209. 23. R. Ostermann, D. Li, Y.D. Yin, J.T. McCann and Y. Xia, Nano Letters, 2006, 6, 6, 1297. 24. D. Li, Y.L. Wang and Y.N. Xia, Advanced Materials, 2004, 16, 4, 361. 25. J-H. He, Chaos, Solitons & Fractals, 2006, 28, 2, 285. 26. J-H. He, Chaos, Solitons & Fractals, 2006, 30, 2, 506. 27. J-H. He, L. Xu, Y. Wu and Y. Liu, Polymer International, 2006, 56, 11, 1323.

40

3

Allometric Scaling in Electrospinning

In this chapter, we will theoretically study the allometric scaling laws in electrospinning. Generally, the form of an allometric relation can be expressed as: (3.1) where X and Y are two parameters describing an event, such as applied voltage, average radius of the electrospun fibres, flow rate and others in the process of electrospinning, and b is an allometric exponent. When b = 1 the relationship is isometric, and when b ≠ 1 the relationship is allometric. The exponent is critically important. Generally, it is relevant to the space dimension, so Equation (3.1) can be rewritten in the form: (3.2) or (3.3) where D is dimension of the discussed problem, and k is an integer. In this chapter we will apply the allometric approach to the search for various relationships between the main controlling parameters in the process of electrospinning [1–10].

41

Electrospun Nanofibres and Their Applications

3.1 Allometric Scaling in Nature Allometry exists everywhere in our daily life and scientific activity. Allometric law offers in-depth scientific understanding. Consider first a simple pendulum. We want to obtain a relationship between the two main parameters: the length of the pendulum and its period. The pendulum moves approximately in one-dimensional space, D = 1, and we have: (3.4) where R is the length of the pendulum, and T is the period. Now we recapitulate the well-known Kepler’s third law, which says that the squares of the periods of revolution around the Sun are proportional to the cubes of the distances. In allometric form, Kepler’s third law can be expressed, with D = 1 and k = 3, in the form: (3.5) where R is the distance from the planet to the Sun, and T is the period. A planet moves around the Sun in a circle or an ellipse in one topological dimension, D = 1, so the exponent should be 1/2 or a multiple of 1/2. Allometry is also widely applied in biology, the most fruitful achievement being the allometric scaling relationship relating metabolic rate (B) to an organism’s mass (M): (3.6) where D is the dimension of the organism’s construction, for example D = 2 for a leaf [11], D = 3 for an animal [12], and D = 4 for a human brain [13]. Now consider a particle’s motion in turbulence (see Figure 3.1). 42

Allometric Scaling in Electrospinning

Figure 3.1 An approximate one-dimensional flow.

The velocity varies along the path line. The motion is an approximate one-dimensional one (D = 1), so we predict that: (3.7) where V is the velocity of the particle, and Re is the Reynolds number. The prediction from Equation (3.7) is same as that in [14]. For a fully developed turbulence, it is of three-dimensional construction (D = 3), so we have: (3.8) The scaling exponent b in Equation (3.8) characteristically takes a limited number of values, all of which are simple multiples of 1/4. For example, the Kolmogorov length, η, scales as [14, 15]: (3.9) Allometry is very effective in quantitative analysis. Now we consider the relation between the viscosity of a macromolecular solution and molecular weight. 43

Electrospun Nanofibres and Their Applications Viscosity is caused by chemical forces acting on macromolecules; among others, the weak electric force of surface charge is dominant. We know that the surface charge is distributed on the outside of the system of macromolecules, so D = 2, leading to the result: (3.10) where η is the viscosity, and Mw is molecular weight. Tacx and co-workers [16] have obtained the following Mark– Houwink relationship for polyvinyl alcohol in water: (3.11) The prediction, Equation (3.10) agrees very well with Equation (3.11). Conductive textiles are widely used as a new kind of intelligent material. The classical Ohm’s law is not valid for calculating the resistance of an intelligent textile. For a non-metal material, a modified Ohm’s law is suggested [17, 18]: (3.12) Here, d is the fractal dimension of the longitudinal length, and D can be considered as the fractal dimension of the ‘charged’ section: D = 2 for the case when the moving charges are distributed on a section (metal-like resistance), and D = 1 for the case when the moving charges are distributed only on its surface (see Figures 2.1–2.3). When D = 2 and d = 1, Equation (3.12) turns out to correspond to metal resistance.

44

Allometric Scaling in Electrospinning

Fourth Dimension of Life The scaling relationship between metabolic rate, B, and body mass, M, can be generally expressed as B ∝ M3/4. The height of an animal scales as L ∝ M1/4 ∝ V1/4, where V is the total volume of the animal, implying a fourth dimension of life [19]. (See also section 3.4.)

3.2 Allometric Scaling Laws in Electrospinning 3.2.1 Relationship Between Radius r of Jet and Axial Distance z The relationship between the radius r of the jet and the axial distance z from the nozzle has been the subject of regular investigation [6], since the electrospinning process was first patented by Formhals in 1934 [20]. The different stages in electrospinning obey different scaling laws. Understanding the regulation of allometry in electrospinning would have broad implications on furthering our knowledge of the process and on controlling the diameter of the electrospun fibres. Spivak and Dzenis [21] obtained the relationship:

(3.13)

where R is the dimensionless jet radius, Z is the dimensionless axial coordinate, and NW, NE and NR are the Weber number, Euler number and effective Reynolds number, respectively. 45

Electrospun Nanofibres and Their Applications Spivak and co-workers [22] obtained a power-law asymptote with an exponent –1/4 for the jet radius, for example: (3.14) Shin and co-workers [23] reported an experimental investigation of the electrically forced jet, and the data revealed that the radius decreased as z increased. Fridrikh and co-workers [24] gave a simple analytical model for the forces that determine the jet diameter during electrospinning as a function of surface tension, flow rate and electric current in the jet. Rutledge’s group at the National Textile Center also suggested the scaling law [25]:

where

(3.15)

Ganan-Calvo and his group [26–28] suggested some asymptotic scaling laws in electrospraying. In this subsection, we shall consider the steady-state flow of an infinite viscous jet pulled from a capillary orifice and accelerated by a constant external electric field (Figure 3.2) is considered. In the absence of an electric field, a meniscus is formed at the exit of the capillary. The meniscus is pulled out into a cone (called the Taylor cone) when the electric force is applied. It was shown that a conducting fluid can exist in equilibrium in the form of a cone under the action of an electric field but only when the semi-vertical angle is 49.3° [29]. When the electric force surpasses a threshold value, the electric force exceeds the surface tension and a fine charged jet is pulled out and is accelerated. Under the condition of high voltage, calculations and experiment indicated that the spinning velocity probably reaches and perhaps exceeds the velocity of sound in air. When the jet is accelerated by the electrical force, the viscous resistance, πr2dτ/dz, becomes higher and higher, and the jet becomes unstable when the value of the viscous resistance almost reaches or 46

Allometric Scaling in Electrospinning

Figure 3.2 Different stages in electrospinning.

surpasses that of the electrical force, 2πrσE. Under such a condition, a slight perturbation by air might lead to oscillation (see Figure 3.3). When the moving fibre moves to the boundary M, the velocity in the x-direction becomes zero. There are three main forces acting on the fibre: viscous force, electronic force and inertia force. The direction of the viscous force is in the opposite direction to its motion, i.e., in the direction of MQ in Figure 3.3. The direction of the resultant of the electronic force and the inertia force is in the z-direction. By the parallelogram law, it moves in the direction of its diagonal, i.e., the direction of MO. At point M, it has the maximal acceleration, and when it reaches point O, the velocity in the x-direction becomes maximum while its acceleration becomes zero; due to inertia, it moves to another boundary N. The instability motion in electrospinning is analogous to pendulum motion. 47

Electrospun Nanofibres and Their Applications

Figure 3.3 Instability analysis.

When z → ∞ (i.e., close to jet break-up), surface charge advection is dominant and the velocity in the z-direction remains unchanged. The process is just like a parachute jump. Initially it accelerates due to the gravity of the parachutist and the velocity becomes higher and higher; as the velocity increases, the air resistance also increases while its acceleration decreases until it becomes zero. Many experiments show the scaling relationship between r and z, which can be expressed as an allometric equation of the form: (3.16) where b is the power exponent. Assume that the volume flow rate (Q) and the current (I) remain unchanged during the electrospinning procedure. We have the 48

Allometric Scaling in Electrospinning following scaling relationships: Q ~ r0 and I ~ r0. From Equation (2.1), we have the following scaling relationships: (3.17) From Equation (2.2), we have: (3.18) and (3.19) From Equations (3.18) and (3.19), we obtain: (3.20) and (3.21)

Initial Stage for Steady Jet At the initial stage of the electrospinning, the electrical force is dominant over the other forces acting on the jet. Under such a case, Equation (2.4) can be approximately expressed as:

(3.22) Substituting Equation (3.17), (3.20) and (3.21) into Equation (3.22), we have: (3.23)

49

Electrospun Nanofibres and Their Applications which leads to the following scaling: (3.24) The prediction Equation (3.24) is valid only for the case when the electrical force acting on the jet dominates over the body force, the viscous force and the internal pressure of the fluid, i.e., the steady jet in Figure 3.2.

Experimental Verification The apparatus used in this work is designed to ensure operation at a uniform voltage and flow rate. We use polyhydroxybutyrate-co-valerate (PHBV) and cellulose as the solutions. Table 3.1 illustrates the parameters applied in the experiment. Figure 3.4 shows that our prediction r ~ z–0.5 at the initial stage is in good agreement with the experimental observation, r ~ z–0.44, for PHBV and r ~ z–0.54 for cellulose.

Table 3.1 Experimental parameters (d0 and z0 are the diameter and coordinate, respectively, at the point where instability occurs) PHBV

Cellulose

Voltage (kV)

30

30

Current (nA)

500

35

Flow rate (ml/h)

2

2

d0 (µm)

80

105

z0 (mm)

60

2

Instability of Viscous Jet Under some circumstances a jet was observed that appeared to rise steadily for a short distance and then to disappear suddenly. Instability

50

Allometric Scaling in Electrospinning

Figure 3.4 The dimensionless jet diameter d/d0 versus the dimensionless axial coordinate z/z0: d0 = d(0) and z = z0 is the instability point (i.e., the point A in Figure 3.3).

occurs when the resultant of the electric force and viscous force approximately vanishes: (3.25) We assume that the pressure gradient remains unchanged during this stage: (3.26)

51

Electrospun Nanofibres and Their Applications Equation (2.4), under the instability condition, reduces to:

(3.27) In view of the scaling relationship Equation (3.17), we have: (3.28) from which we obtain the following scaling law: (3.29) The result is the same as that in the open literature – see Figure 3.5.

Figure 3.5 Shin and co-workers’ experimental data [23]. The dashed line AB corresponds to the slope of a line that scales as r ~ z–0.53, while BC scales as r ~ z–0.25 and CD scales as r ~ z0.

52

Allometric Scaling in Electrospinning

Terminal State When z → ∞, the acceleration in the z-direction vanishes completely: (3.30) which leads to the scaling law: (3.31)

3.2.2 Allometric Scaling Relationship Between Current and Voltage Suppose that the allometric relationship between current I and voltage e can be expressed as [1, 8]: (3.32) where b is the power exponent. When b = 1 the relationship is isometric and corresponds to Ohm’s law: Assume that the volume flow rate (Q) remains unchanged during the electrospinning procedure, i.e. Q ∝ r0. Assume that: (3.33) From Equation (2.2), we have: (3.34) and

(3.35)

From Equation (3.33)-(3.35), we have: (3.36) and

(3.37) 53

Electrospun Nanofibres and Their Applications The unknown constant a varies among different polymer solutions, and can be determined by experiment. Theron and co-workers experimental data [30] show experimentally the scaling law between current and voltage for 10 wt% polycaprolactone (PCL) in methylene chloride:dimethylformamide (DMF; 75:25) in the form I ∝ E3. Demir and co-workers [31] obtained a similar result I ∝ E2.7 for polyurethane (PU) solutions.

Experimental Verification A variable high-voltage power supply was used for the electrospinning. It was used to produce voltages ranging from 10 to 50 kV. Polyacrylonitrile (PAN) solution was poured into a syringe attached with a capillary tip of 0.5 mm diameter. The applied voltage and flow rate were changed separately, and the distance between the capillary tip and the collector is constant, 8 cm. A voltmeter was used to measure the voltage across the resistance; the voltages measured were then converted to currents.

Materials PAN with a molecular weight of 70,000 was supplied by Sinopec Shanghai Petrochemical Co. Ltd and DMF was purchased from Shanghai Chemical Co.

Solution Preparation PAN (12 wt% and 18 wt%) was added separately to DMF and the solution was stirred magnetically for one hour at 80 °C.

Results and Discussion The PAN/DMF (12 wt% and 18 wt%) solutions were spun separately under different flow rates. The experimental results are shown in 54

Allometric Scaling in Electrospinning

Figure 3.6 The scaling relationship between voltage and current: (a) 12 wt% PAN in pure DMF; (b) 18 wt% PAN in pure DMF.

Figure 3.6 for the two solutions. Figure 3.6 reveals that the scaling exponent is approximately equal to 3 for both 12 wt% and 18 wt% PAN in pure DMF. 55

Electrospun Nanofibres and Their Applications

3.2.3 Allometric Scaling Relationship Between Solution Flow Rate and Current Assume that the allometric relation between flow rate Q and current I can be expressed as [5]: (3.38) where b is the allometric exponent. Assume that the voltage (E) remains unchanged during the electrospinning procedure: (3.39) From Equation (2.1), we have the following scaling relationship: (3.40) From Equation (2.2), we have: (3.41) (3.42) Assume that: (3.43) Considering that E ∝ r0, from the above scaling relations, we can easily obtain: (3.44) Therefore, we immediately obtain the following allometric scaling relationship between I and Q: (3.45) The value of a can be determined through experiment. 56

Allometric Scaling in Electrospinning Shin and co-workers [23] obtained results showing that the solution flow rate and current follow a linear relationship, which is also proved by Theron and co-workers research [30]. Therefore, the value of the exponent 2/(3 – a) of Equation (3.45) should be 1, so a = 1. The relationship between current and solution flow rate is in the form: (3.46)

Experimental Verification A variable high-voltage power supply was used for the electrospinning. It was used to produce voltages ranging from 10 to 50 kV. PAN solution was poured into a syringe attached with a capillary tip of 0.5 mm diameter. The applied voltage and flow rate were changed separately, and the distance between the capillary tip and the collector is constant, 8 cm. A voltmeter was used to measure the voltage across the resistance; voltages measured were then converted to currents.

Materials PAN with a molecular weight of 70,000 was supplied by Sinopec Shanghai Petrochemical Co. Ltd and DMF was purchased from Shanghai Chemical Co.

Solution Preparation PAN (12 wt% and 18 wt%) was added separately to DMF and the solutions were stirred magnetically for one hour at 80 °C.

Results and Discussion The PAN/DMF (12 wt% and 18 wt%) solutions were spun separately under different voltages. The experimental results are shown in Figure 3.7 for the two solutions. Figure 3.7 reveals that, when the voltage is 57

Electrospun Nanofibres and Their Applications

Figure 3.7 The relationship between solution flow rate and current: (a) 12 wt% PAN; (b) 18 wt% PAN.

58

Allometric Scaling in Electrospinning constant, the current is linear with solution flow rate, i.e., the scaling exponent equals 1, which agrees well with our above-mentioned results. For different solutions, the linear relation is different. For the same solution, when it flows in different rate ranges, the relationship will also be different – see Figures 3.7(a) and 3.7(b).

3.2.4 Effect of Concentration on Electrospun PAN Nanofibres See [10] for more details. Experimental data [32, 33] and theoretical analysis [6, 7] show that viscosity greatly affects the diameter of electrospun fibres. Baumgarten [32] pointed out that, as the viscosity increased, the spinning drop changed from approximately hemispherical to conical, and the length of the jet increased as well. Fibre diameter also increased with solution viscosity and was approximately proportional to jet length. The jet length was measured from the tip of the spinning drop to the onset of waves in the fibre: (3.47) where d is the diameter of the electrospun fibre, η is the viscosity and α is the scaling exponent. The exponent value might differ between different polymers. For acrylic solution, Baumgarten [32] found that the fibre diameter increased with solution viscosity in the form: (3.48) Many experimental observations [34] and theoretical analysis [6, 7] also show that viscosity has a power relationship with concentration: (3.49) where C is the solution concentration, and β is the scaling exponent. A simple analysis of Kenawy and co-workers experimental data [34] on Brookfield viscosity data for polyethylene-co-vinyl acetate 59

Electrospun Nanofibres and Their Applications and polylactic acid (PLA) in chloroform shows that the relationship between viscosity and concentration can be expressed in the form: (3.50) Combining the scaling relationships (Equation (3.47) and Equation (3.49)), we obtain a new allometric law between the diameter of the electrospun fibres and the solution concentration: (3.51) where δ is the scaling exponent, which differs greatly between different polymers, and for the same polymer with different molecules or the same molecules with different properties.

Experimental Verification A high-voltage power supply is used for electrospinning. The voltages range from 0 to 50 kV; the voltage used in the experiment is about 20 kV. PAN solution was poured into a syringe attached with a capillary tip of 0.7 mm diameter.

Material PAN with a molecular weight of 70,000 was supplied by Sinopec Shanghai Petrochemical Co. Ltd and DMF was purchased from Shanghai Chemical Co.

Results and Discussion Our experimental results on PAN in DMF are shown in Figures 3.8 and 3.9. A scanning electron microscope (SEM) microphotograph of electrospun PAN fibres is illustrated in Figure 3.10. 60

Allometric Scaling in Electrospinning

Figure 3.8 Relationship between diameter of electrospun polyacrylonitrile PAN nanofibre and its concentration.

Figure 3.9 Relationship between viscosity and concentration, η ∝ C5.98. 61

Electrospun Nanofibres and Their Applications

Figure 3.10 SEM microphotograph of electrospun PAN fibres.

It was found that fibre diameter is linear with solution concentration, and solution viscosity is allometric with concentration, in the form: (3.52) and (3.53) Demir and co-workers [31] showed experimentally the dependence of the average fibre diameter on solution concentration in a power-law relationship for electrospun PU fibres: d ∝ C3. So the scaling exponent in Equation (3.51) differs greatly between different polymers, and for the same polymer with different molecules or the same molecules with different properties. 62

Allometric Scaling in Electrospinning

3.2.5 Allometric Scaling Law Between Average Polymer Molecular Weight and Electrospun Nanofibre Diameter This is covered in more detail in [35]. Viscosity is due to friction between the molecules. We consider an ideal case as illustrated in Figure 3.11, where all the macromolecules are almost parallel. In such a case, the viscosity, η, scales linearly with the average chain length of macromolecules, L: (3.54) By a simple Euclidean analysis, we have (3.55) where M is the solution weight per volume, which scales linearly as the polymer’s average molecular weight, Mw: (3.56)

Figure 3.11 An ideal polymer solution or melt as a system of almost parallel macromolecules.

63

Electrospun Nanofibres and Their Applications

Figure 3.12 A concentrated and entangled polymer solution or melt as a system of macromolecules.

From Equations (3.54)–(3.56), we can easily obtain the following scaling relation for the ideal case of parallel macromolecules: (3.57) Now we consider another extreme case as illustrated in Figure 3.12. Macromolecules in the polymer solution are concentrated and entangled with each other. In such a case, fluid friction depends upon not only the friction between two neighbouring molecules, but also their topological structure. Fluid friction scales approximately as the solution volume, so the solution viscosity scales linearly with the solution weight per volume: (3.58) In view of Equation (3.58), we have the following scaling relationship for a concentrated and entangled polymer solution: (3.59)

64

Allometric Scaling in Electrospinning Generally the scaling relationship between molecular weight and solution viscosity can be expressed in the following Mark–Houwink form [16, 36, 37] for a real polymer solution: (3.60) where α is the scaling exponent. The value of α lies between 1/3 and 1: (3.61) Applying He Chengtian’s interpolation, in view of Equation (3.61), the value of α can be written in the form: (3.62) So the scaling relationship between molecular weight and solution viscosity can be expressed in the following form: (3.63) or (3.64) where δ = m/n. There are two free parameters, m and n, in Equation (3.63). By a suitable choice of m and n, the value of α can change smoothly from 1/3 to 1. It is obvious that: and

(3.65)

and

(3.66)

Generally, the exponent α is specific to polymer chemistry, configuration, molar mass distribution, solvent and temperature;

65

Electrospun Nanofibres and Their Applications α = 0 for spherical molecules and 0.5 for unperturbed coils [33]. Infinitely thin, rigid rods have α values of 2.0, whereas worm-like polymer chains exhibit α values that range from 0.764 to 2.0 [36]. Our theory is valid for α values from 1/3 to 1: (3.67) The scaling law Equation (3.67) agrees well with Zeng and coworkers’ experiment [33].

3.2.6 Effect of Voltage on Morphology and Diameter of Electrospun Nanofibres If the electrical force is zero or weak, the pendent droplet of the polymer solution at the capillary tip is deformed into a conical shape, where the surface tension is dominant. If the voltage surpasses a threshold value, the electrostatic force overcomes the surface tension and a fine charged jet is ejected. In the initial case, we assume that the acceleration of the charged jet is constant, and then we have: (3.68) where u0 is the initial velocity, and a is the acceleration. In the initial stage, the viscous force is ignored, and thus the acceleration depends upon the electrostatic force: (3.69) From Equation (2.2), we have: (3.70) We therefore obtain: (3.71)

66

Allometric Scaling in Electrospinning We assume that the acceleration in the initial stage is constant, and thus we obtain: (3.72) Now we assume that the allometric scaling relationship between current and voltage has the form: (3.73) where b is a scaling exponent, which differs among different polymer solutions. In view of Equation (3.73) we have: (3.74) Considering that the volume flow rate is constant during electrospinning, from Equation (2.1), we have: (3.75)

Experimental Verification Via Electrospun Biodegradable (PBS) Fibres Biodegradable polymers are designed to degrade upon disposal by the action of living organisms. Extraordinary progress has been made in the development of practical processes and products from polymers such as starch, cellulose and lactic acid [37-40]. Recently, fabrication of biodegradable ultrafine fibres by electrospinning has been given much attention [41-46] due to their small diameter, low density, high specific surface area and excellent surface properties. The utilisation of biodegradable polymers is expected to decrease the ecological pollution problems that currently occur all over the world due to waste plastics, particularly those for outdoor purposes. The most important family of biodegradable polymers currently developed consists of aliphatic polyesters such as PCL, poly(L-lactic 67

Electrospun Nanofibres and Their Applications acid), poly(3-hydroxybutyrate) and polybutylene succinate (PBS). Among these aliphatic polyesters, PBS is a typical two-component aliphatic polyester that is considered to be one of the most accessible biodegradable polymers in terms of ease of synthesis [42], as a result of the combination of the electrospinning process and the excellent biodegradability of PBS in a natural environment. The enhanced properties of these polymers are required for applications in various fields, such as catalysis, filtration, nanoelectromechanical systems, medical implants, cell supports, nanocomposites, nanofibrous structures, tissue scaffolds, drug delivery systems, protective textiles, and storage cells for hydrogen fuel cells [48].

Materials To prepare electrospun PBS fibres, PBS with a molecular weight of 200,000 to 300,000 g/mol (provided by Shanghai Institute of Organic Chemistry, Chinese Academy of Science) was used. The solvent system studied was chloroform (CHCl3) (supplied by Shanghai Chemical Reagent Co. Ltd) with a molecular weight of 119.38 g/mol and density of 1.471–1.484 g/cm3. The polymer, PBS, was dissolved in CHCl3 solvent at room temperature with two hours of stirring in an electromagnetic stirrer (Angel Electronic Equipment (Shanghai) Co. Ltd) to prepare a polymer solution with a concentration of 14 wt%.

Electrospinning Experiment The polymer solution was placed into a 20 ml syringe attached to a syringe pump. Electrospinning was carried out under room temperature in a vertical spinning configuration using a 0.5 mm inner diameter needle with spinning distance of 5–10 cm. The applied voltages were in the range of 25–50 kV connected to the needle by a DC high-voltage power supply (F180-L, Shanghai Fudan High School) via an alligator clip. 68

Allometric Scaling in Electrospinning For the investigation of the morphology and diameter of the electrospun nanofibres, PBS/CHCl3 films were determined by a scanning electron microscope (SEM, JSM-5610). To obtain SEM images the fibres were collected on an SEM disc and coated with gold. SEM micrographs are illustrated in Figures 3.13–3.16. The average diameter varies from 700 nm to 1200 nm and can be adjusted by the applied voltage. The relationship between current and applied voltage is illustrated in Figure 3.17, from which we know that: (3.76) According to Equation (3.75) we have: (3.77)

Figure 3.13 SEM photograph of PBS/CHCl3 electrospun fibres. The concentration and voltage are 14 wt% and 30 kV. The average fibre diameter is about 1087 nm. 69

Electrospun Nanofibres and Their Applications

Figure 3.14 SEM photograph of PBS/CHCl3 electrospun fibres. The concentration and voltage are 14 wt% and 35 kV. The average fibre diameter is about 994 nm.

Figure 3.15 SEM photograph of PBS/CHCl3 electrospun fibres. The concentration and voltage are 14 wt% and 40 kV. The average fibre diameter is about 856 nm.

70

Allometric Scaling in Electrospinning

Figure 3.16 SEM photograph of PBS/CHCl3 electrospun fibres. The concentration and voltage are 14 wt% and 50 kV. The average fibre diameter is about 760 nm.

Figure 3.17 The relationship between applied voltage and current: points, experimental data; full line, theory I ∝ E0.29.

71

Electrospun Nanofibres and Their Applications

Figure 3.18 Average diameter versus applied voltage: points, experimental data; full line, theory r ∝ E–0.645.

The prediction Equation (3.77) agrees very well with the experimental data as illustrated in Figure 3.18.

3.2.7 Improving Electrospinnability Using Non-ionic Surfactants Surface tension is a major factor affecting the morphology of nanofibres and their electrospinnability [49]. The smaller the surface tension, the lower the threshold voltage needed to overcome the surface tension of the electrospun solution. Surfactants can reduce surface tension remarkably, and as a result electrospinnability can be improved. In this section, we add a non-ionic surfactant, Triton X-100 is added, to polyvinylpyrrolidone solution to study the effects of surfactant on electrospinning [49]. 72

Allometric Scaling in Electrospinning

Materials Polyvinylpyrrolidone (PVP) with the serial number of K-30, Triton X-100 and absolute alcohol were purchased from Shanghai Chemical Reagent Co. Ltd, China. Deionised water was supplied by the College of Chemistry, Donghua University. A mixture of deionised water and absolute alcohol with the weight ratio 1:5 was used as solvent. All materials were used without any further purification.

Instrumentation The electrospinning set-up consisted of a syringe, a needle, a grounded collected plate, a flow meter and a variable DC high-voltage power generator (0–100 kV, F180-L, Shanghai Fudan High School). Fibre diameters and morphology images of the obtained nanofibres were analysed using a scanning electron microscope. A contact angle–surface tension machine (DCA-322) was used to test the surface tension of the solution.

Electrospinning Process All concentration measurements were done as weight by weight (w/w). The mixture of deionised water and absolute alcohol with the weight ratio 1:5 was used as solvent. PVP with a concentration of 35% was dissolved in the above solvent. Triton X-100 was added to the obtained solution in ratios of 2, 5, 8, 11, and 14 wt%. The prepared solution was magnetically stirred at 40 °C. The morphology of the electrospun PVP fibres was investigated using a scanning electron microscope (SEM, JSM-5610). The fibre mat was collected on an SEM disc and coated with gold before photographing. SEM micrographs are illustrated in Figure 3.19. Surface tension was remarkably reduced when a surfactant, Triton X-100, was added to the solution – see Figure 3.20. Figure 3.21 shows that the average diameter of the fibres decreased while the concentration of Triton X-100 increased. 73

Electrospun Nanofibres and Their Applications

(a)

(b)

(c)

(d)

Figure 3.19 Scanning electron micrographs of 35 wt% PVP electrospun fibres at 1.5 kV/cm with different Triton X-100 concentrations (wt%): (a) 0, (b) 2, (c) 5, (d) 8, (e) 11, (f) 14.

74

Allometric Scaling in Electrospinning

(e)

(f) Figure 3.19 Continued

Figure 3.20 Surface tension of 35 wt% PVP solution with different Triton X-100 concentrations.

75

Electrospun Nanofibres and Their Applications

Figure 3.21 Average diameter of 35 wt% PVP nanofibres with different Triton X-100 concentrations.

When a surfactant is added, the surface tension becomes smaller. Consider the initial stage of the ejection of the charge jet. According to Newton’s second law, we have: (3.78) where FE is the electric force acting on the charged surface of the Taylor cone, and FS is the surface tension of the Taylor cone. Addition of the surfactant results in smaller surface tension, FS. As a result, a higher acceleration of the charge jet is estimated, and accordingly a higher velocity of the charged jet is anticipated. According to the conservation of mass during electrospinning, we have: (3.79) where u is the velocity of the jet, Q is the flow rate (which remains unchanged during the electrospinning), ρ is the density, and r is the radius of the nanofibre. We predict that: (3.80) 76

Allometric Scaling in Electrospinning This means that the addition of the surfactant results in smaller nanofibres. In our experiment, the average diameter of the fibres with 14% Triton X-100 is about 380 nm while the average diameter of the fibres without any surfactant is 1155 nm. From Equation (3.80), we obtain: (3.81) or (3.82) We can approximately write Equation (3.82) in the form: (3.83) As the distance between the tip and the collector is fixed during the electrospinning, so the time for a jet from the tip to the collector is assumed to be unchanged. We therefore have: (3.84) In view of Newton’s second law, Equation (3.78), we finally obtain the following relationship: (3.85) where k is a constant. For our experiment, we obtain approximately (see Figure 3.22): or

(3.86)

where d is the diameter in nm and FS is the surface tension in mN/m. 77

Electrospun Nanofibres and Their Applications

Figure 3.22 Effect of the surface tension on the diameter of the electrospun nanofibres.

Effect of the Surfactant on Electrospinnability Our experiment showed that, when the concentration of PVP exceeded a critical value, 48%, the solution could not be electrospun (Figure 3.23). However, this was changed when Triton X-100 was added to the PVP solution. It was shown that, even when the concentration of PVP reached 48%, the solution can be electrospun into nanofibres with a diameter of about 780 nm. The experimental results show that the electrospinnability of a polymer solution depends on its concentration. The diameter of an electrospun fibre depends greatly on the surface tension. The surfactant, Triton X-100, added to the solution can dramatically decrease fibre diameter and improve electrospinnability as well. 78

Allometric Scaling in Electrospinning

(a)

(b) Figure 3.23 Effect of surfactant, Triton X-100, on the electrospinnability of 48 wt% PVP at 3 kV/cm: (a) no surfactant is added; (b) Triton X-100 concentration 6 wt%. 79

Electrospun Nanofibres and Their Applications We also give a very simple theoretical prediction of the average diameter of the nanofibres under different surfactant concentrations. Our theoretical prediction agrees very well with our experimental observation.

3.3 Allometric Scaling Law for Static Friction of Fibrous Materials Friction is a vitally important factor in the application of fibrous materials to engineering [50]. Fibres are assembled into various textile products and structures largely through friction mechanisms. However, due to the complex nature of inter-fibre friction, our understanding of this fundamental phenomenon is scarce and primitive. As is well known, the static friction can be written in the form: (3.87) where F is the frictional force, µ is the friction coefficient, and N is the normal force. This frictional behaviour was probably understood by Leonardo da Vinci, who found that the frictional force is independent of the area of contact between the two faces and is proportional to the normal force between them. The phenomena were rediscovered by Amontons in 1699 and verified by Coulomb in 1781. He also pointed out the distinction between static friction and kinetic friction. Amontons’ law for a yarn passing round a guide can be expressed in the form: (3.88) where T1 and T2 denote incoming tension and leaving tension, respectively, µ is the coefficient of friction and θ is the angle of contact. 80

Allometric Scaling in Electrospinning Many experimental data show that the ratio of frictional force f to normal load N for fibres is found to decrease as the load is increased. In other words, Coulomb’s law or Amontons’ law is not obeyed. The study of fibrous friction between soft materials or between soft material and metal has largely been via experimental observation of departures from these laws, the reasons for such departures and their consequences. Among the various mathematical relations that have been used to fit the experimental data are the following [51]: (3.89) (3.90) (3.91) (3.92) where S is the area of contact, and µ, α, A, B, a, b and c are constants determined from experiment data. Here, we will apply the allometric approach to the discussed problem. The technique has emerged as an interesting and fascinating mathematical tool in the search for the scaling relations of various real-life problems.

3.3.1 Solid–Solid Friction Coulomb’s law is valid for metal–metal friction, for which the scaling relation between frictional force F and normal force N can be written in the form: (3.93) The frictional metal–metal force is independent of the area (A) of the contact between the two surfaces: (3.94) 81

Electrospun Nanofibres and Their Applications These two scaling relationships Equations (3.93) and (3.94) are not valid for non-metals.

3.3.2 Viscous Friction for Newtonian Flow We consider the viscous friction of Newtonian flow. The viscous friction, F, is proportional to its area: (3.95) The relationships Equations (3.94) and (3.95) are valid for two extreme conditions, respectively: the former for stable contact and the latter for unstable contact.

3.3.3 Friction for Soft Materials The frictional force between soft materials, e.g., cotton flow, scales as: (3.96) where the scaling exponent α represents the mechanical character for the frictional partners. When α = 0, the friction is of solid–solid type, and when α = 1, it turns out to be viscous friction. Soft material behaves in the manner of neither Newtonian flow nor solid contact, so the value of α lies between 0 and 1. According to He Chengtian’s interpolation (see Section 2.5), the value of α can be approximated as: (3.97) where m and n are integers. The most successful relationship is α(1,1) = 1/2.

3.3.4 Fibre–Fibre Friction Now we consider fibre–fibre (e.g., Nylon–Nylon, Nylon–acetate) friction. 82

Allometric Scaling in Electrospinning We have: (3.98) In the case β = 0, the frictional partners can be considered as rigid bodies; and β = α holds for flexible bodies. Most textile products have a fixed surface similar to a solid surface, but the fibrous flexibility leads to a similar phenomenon as flow, especially as cotton flow, so the value of β generally lies between 0 and 1/2: (3.99)

Table 3.2 Experimental and predicted values of δ Acetate

Nylon

Viscose rayon

Terylene polyester fibre

Wool

0.94

0.89

0.90

0.86

0.92

δ(1,11) = 0.93

δ(1,6) = 0.88

δ(1,7) = 0.90

δ(1,4) = 0.85

δ(1,10) = 0.92

0.86

0.81

δ(1,4) = 0.85

δ(1,2) = 0.80

Viscose rayon

0.89

0.88

0.91

0.88

0.87

δ(1,6) = 0.88

δ(1,6) = 0.88

δ(1,10) = 0.92

δ(1,6) = 0.88

δ(1,6) = 0.88

Terylene polyester fibre

0.88

Wool

0.88

0.86

0.92

0.86

0.90

δ(1,6) = 0.88

δ(1,4) = 0.85

δ(1,10) = 0.92

δ(1,4) = 0.85

δ(1,7) = 0.90

Acetate

Nylon

δ(1,6) = 0.88

83

Electrospun Nanofibres and Their Applications By He Chengtian’s interpolation, we have:

If we choose β(1,1) = 1/3, then we have the scaling relation F ~ A1/3. The relationship, Equation (3.93) is modified as: (3.100) For fibre–fibre material friction, F ~ ND/(D+1), D = 2, so F ~ N2/3 and

Some typical values for fibre–fibre friction are listed in Table 3.2.

3.4 Allometric Scaling in Biology Allometric scaling laws in biology have attracted considerable attention after the insightful works by West and co-workers [19, 52]. Now we consider a multicellular organ and assume that there are n basal cells with characteristic (or typical) radius r. The metabolic rate b of the organ scales linearly with respect to its total surface area, for example: (3.101) where A is the total surface area of all the cells in the organ. The total surface area of cell boundaries is of fractal construction. Therefore, we have: (3.102) where D is the fractal dimension of the total cell surface area. Owing to the smallness of the cells, we can assume that cells are space-filling (thus, for example, nutrition can reach each cell in the organism). 84

Allometric Scaling in Electrospinning Consequently, the fractal dimension, D, tends to 3, i.e., D = 3 for most three-dimensional organs. Combination of Equations (3.101) and (3.102) leads to the following scaling relationship: (3.103) The mass M of the organ scales linearly with respect to its total volume of cells: (3.104) Note that Equation (3.104) is not valid for bone tissue. From the scaling relationships, Equations (3.101), (3.103) and (3.104), we obtain the well-known Kleiber’s 3/4 allometric scaling law, which reads: (3.105) In 1977, Blum [53] suggested that the 3/4 law can be understood by a four-dimensional approach. In D-dimensional space, the ‘area’ A of the hypersurface enclosing a D-dimensional hypervolume scales like A~V(D–1)/D. When D = 4, we have A~V3/4, a four-dimensional construction: (3.106) In view of El Naschie’s E-infinity theory [54], we modify the scaling law, Equation (3.106) in the form: (3.107) where DH is the expectation value of the Hausdorff dimension of ε(∞) [54]: (3.108) is the golden mean, which is the building block where of El Naschie’s ε(∞) network [54]. 85

Electrospun Nanofibres and Their Applications Note that D = 4 is the dimension of classical space-time upon which Einstein’s theory is based. This is only an approximation of the true geometry of the Universe in the large. Similarly D = 4 is used in fourdimensional life, and we argue that El Naschie’s theory might lead to an accurate prediction of metabolic rate. The main application of E-infinity theory shows miraculous scientific exactness, especially in determining theoretically the coupling constants and the mass spectrum of the standard model of elementary particles. Now we consider a leaf of a plant or a hepatic cellular plate. For an approximate two-dimensional (2D) organ, the fractal dimension d in Equation (3.102) tends to 2, i.e. D = 2. As a result, we have nr2 = A~r2, leading to the result: (3.109) We therefore obtain Rubner’s 2/3 law for 2D organs: (3.110) It is obvious that Rubner’s 2/3 law is valid for 2D lives and Kleiber’s 3/4 law for 3-dimensional (3D) lives, as predicted by He and Chen [55]. If the exponent in Equation (3.110) is linked to the golden mean, then Rubner’s 2/3 law can be modified as: (3.111) for two-dimensional organs. It is interesting to note the scaling relationship n~r (see Equation (3.103)), which is valid for 3D organs. Remember that r is a space dimension, so the number of cells in an organ endows another life dimension [56]. If a cell is isolated from the heart of a rat, or heart cells are cultivated on a plane (D = 2), no life dimension is endowed. Therefore, these cells have no life functional characteristic. However, if a sufficient number of heart cells are cultivated and accumulated together in three dimensions, the isolated cardiac cell begins to beat [57, 58] and the life dimension is endowed. 86

Allometric Scaling in Electrospinning We have also illustrated that n~r0 (see (3.109)) for 2D lives. This is really very interesting and gives much insight into the explanation of the transplantation of 2D tissues, for example, most plants, liver and cornea. The life characteristic of 2D organs does not seriously depend upon the number of cells involved. This is the reason why a damaged leaf can still photosynthesise and can replace dead cells, whereas the adult human brain cannot replace lost neurons [59]. In adult rodents, however, one region of the brain, the subventricular zone, generates thousands of neurons every day [60]. There could be important implications for future developments in neuroregenerative therapy by constraining brain cells in a niche of unique organisation in the adult human brain to be of 2D tissue. Considering the effect of temperature, we suggest a modified scaling relationship in the form: (3.112) The prediction Equation (3.112) is valid within the limited range of ‘biologically relevant’ temperatures between approximately 0 °C and 40 °C. This is the range within which organisms commonly operate under natural conditions. Near 0 °C, metabolic reactions cease due to the phase transition associated with freezing water and above approximately 40 °C, metabolic reaction rates are reduced by the increasing influence of catabolism. Lifespan therefore is predicted as: (3.113) Increase in temperature leads to increase in metabolic rate, and thus to decrease in lifespan.

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Allometric Scaling in Electrospinning 45. H. Yoshimoto, Y.M. Shin, H. Terai and J.P. Vacanti, Biomaterials, 2003, 24, 12, 2077. 46. N. Dharmaraj, H.C. Park, C.H. Kim, K.P. Viswanathamurthi and H.Y. Kim, Materials Research Bulletin, 2006, 41, 3, 612. 47. S. Thandavamoorthy, G.S. Bhat, R.W. Tock, S. Parameswaran and S.S. Ramkumar, Journal of Applied Polymer Science, 2005, 96, 2, 557. 48. E. Harel, J. Granwehr, J.A. Seeley and A. Pines, Nature Materials, 2006, 5, 4, 321 49. S-Q. Wang, J-H. He and L. Xu, Polymer International, 2008, 57, 9, 1079. 50. Y. Wu, Y.M. Zhao, J.Y. Yu and J-H. He in Computational and Information Science (Lecture Notes in Computer Science, 3314), Springer-Verlag, Berlin, Germany, 2004, p.465. 51. W.E. Morton and J.W.S. Hearle, Physical Properties of Textile Fibres, The Textile Institute and Heinemann, London, UK, 1975. 52. G.B. West, J.H. Brown and B.J. Enquist, Science, 1997, 276, 5309, 122. 53. J.J. Blum, Journal of Theoretical Biology, 1997, 64, 3, 599. 54. M.S. El Naschie, International Journal of Nonlinear Sciences and Numerical Simulation, 2007, 8, 1, 11. 55. J-H. He and H. Chen, International Journal of Nonlinear Sciences and Numerical Simulation, 2003, 4, 4, 429. 56. J-H. He and J. Zhang, Cell Biology International, 2004, 28, 11, 809. 57. N. Bursac, M. Papadaki, R.J. Cohen, F.J. Schoen, S.R. Eisenberg, R.L. Carrier, G. Vunjak-Novakovic and L.E. Freed, American Journal of Physiology – Heart Circulation and Physiology, 1999, 277, 2, H433. 91

Electrospun Nanofibres and Their Applications 58. R.L. Carrier, M. Papadaki, M. Rupnik, F.J. Schoen, N. Bursac, R. Langer, L.E. Freed and G. Vunjak-Novakovic, Biotechnology and Bioengineering, 1999, 64, 5, 580. 59. P. Rakic, Nature, 2004, 427, 6976, 685. 60. N. Sanai, A.D. Tramontin, A. Qulnones-Hinojosa, N.M. Barbaro, N. Gupta, S. Kunwar, M.T. Lawton, M.W. McDermott, A.T. Parsa, J.M.G. Verdugo, M.S. Berger and A. Alvarez-Buylla, Nature, 2004, 427, 6976, 740. 61. Y. Liu, J-H. He and J.Y. Yu, Fibres and Textiles in Eastern Europe, 2007, 15, 4, 30.

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4

Application of Vibration Technology to Electrospinning

A facile strategy for preparing electrospun nanofibres by vibration technology is suggested in this chapter. Vibration is applied to the polymer solutions or melts, leading to a dramatic reduction in viscosity. As a result, only a moderate voltage is needed to produce fine nanofibres, which commonly require an elevated voltage during conventional electrospinning procedures. This novel strategy produces finer nanofibres than those obtained without vibration, and expands the range of this technique by making fibres from macromolecules that cannot be electrospun in traditional ways.

4.1 Effect of Viscosity on Diameter of Electrospun Fibre For further details, see [1–3]. Under very high voltage, not every solution can be electrospun. Instead, in the case of low-viscosity liquids of small molecules, the jet might break up into droplets as a result of surface tension (known as electrospraying). For highviscosity liquids of macromolecules, the jet does not break up, but travels as a jet to the grounded target (known as electrospinning). For very highly viscous solutions or melts, the process can be used in electro-extrusion in the plastics industry [4]. Experimental data and theoretical analysis [1] show that solution viscosity greatly affects the diameter of electrospun fibres, initiating droplet shape and jet instability. Increasing the solution viscosity has been associated with the production of larger-diameter fibres, and it was shown that fibre diameter d depends allometrically on solution viscosity d in the form [1]:

93

Electrospun Nanofibres and Their Applications (4.1) where d is the average diameter of the electrospun fibre, η is the solution viscosity, and α is the scaling exponent. The value of the exponent might differ between different polymers. For acrylic solution, Baumgarten [5] found that the scaling exponent is about 1/2.

4.2 Effect of Vibration on Viscosity Vibration technology [6–8] was introduced into polymer processing many years ago. Initially, it was only applied in research on polymer melt viscosity measurement. Subsequently, the principle of melt vibration for reduction of viscosity was introduced into practical applications including injection moulding, extrusion and compression moulding/thermoforming to lower the processing temperature and pressure, and thus eliminate melt defects and weld lines and enhance mechanical properties by modification of the amorphous and semicrystalline texture and orientational state. Ibar [9] observed the effect of low-frequency vibration during the processing of polymethyl methacrylate (PMMA) (Figure 4.1). The result showed that increasing frequency makes the solution viscosity decrease. The lower the frequency, the bigger the viscosity differences were among various temperatures. In the low-frequency regime, the solution viscosity decreased quickly. In the regime of frequency 100–500 rad/s, the increase in frequency did not work obviously. However, frequencies higher than 200 rad/s would decrease the viscosity gradient caused by temperature. Ibar explained the effect of vibration frequency and amplitude on melt viscosity in terms of shear-thinning criteria. When a vibrating force is applied to a concentrated and entangled polymer solution or melt, the weak van der Waals forces connecting macromolecules becomes weaker, and entanglement is relaxed, so that the viscous force between the macromolecules decreases dramatically, resulting in the reduction of viscosity. 94

Application of Vibration Technology to Electrospinning

Figure 4.1 Solution viscosity (η) of PMMA versus vibration frequency (ω). (Redrawn from experimental data in [9].)

According to the experimental observations, we have the following allometric scaling: (4.2) where β is a scaling exponent that varies with the polymer’s characteristics. For PMMA solution at 239 °C, Ibar’s experiment [9] showed that η ∝ ω–2/5. Our experiment revealed that η ∝ ω–7/10 for polyacrylonitrile in dimethylformamide (PAN/DMF) solution.

4.3 Application of Vibration Technology to Polymer Electrospinning When an additional vibrating force is applied to conducting polymer solutions or melts, dramatic reduction in viscosity occurs. The voltage 95

Electrospun Nanofibres and Their Applications

Figure 4.2 The effect of an electric field acting on entangled macromolecules: (a) without an electric field; (b) with an electric field plus a vibrating force, which leads to high molecular orientation due to the electrical force, and as a result higher mechanical properties are expected.

is applied to polarise dielectrics, where the charges are not completely free to move, but the positive and negative charges that compose the body may be displaced in relation to one another when a vibrating force is applied – see Figure 4.2. Owing to its lower viscosity, we need a relatively lower voltage to eject a jet from the spinneret. From relationships (4.1) and (4.2), we have the following power law [1]: (4.3) where δ is the scaling exponent, which varies among different polymers. The main advantages of vibration electrospinning are as follows: 96

Application of Vibration Technology to Electrospinning (1) There is a dramatic reduction of viscosity, so that the jet length can be controlled by changing the frequency. (2) Finer fibres can be produced than those obtained by traditional electrospinning apparatus under the same conditions. The patented apparatus (Figure 4.3) can even spin such fibres that cannot be spun by traditional apparatus, or can spin fibres with the same diameter at much lower voltage. (3) The patented electrospinning process (Figure 4.3), as that in polymer extrusion, can enhance the mechanical properties of electrospun fibres by modification of the amorphous and semicrystalline texture and orientational state.

Figure 4.3 Schematic of patented vibration technology for polymer electrospinning. (This apparatus is patented [10]. To use this principle to prepare electrospun fibres, a transfer agreement must be made.)

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Electrospun Nanofibres and Their Applications

Experimental Verification 1 In this method [2], an ultrasonic generator is used to generate an ultrasonic vibration. A high-voltage power supply is used for electrospinning. The voltage ranges from 0 to 50 kV, and the voltage used in the experiment is about 20 kV. PAN with a molecular weight of 70 000 was supplied by Sinopec Shanghai Petrochemical Co. Ltd, and DMF was purchased from Shanghai Chemical Co. PAN solution was poured into a syringe attached with a capillary tip of 0.7 mm diameter. The diameter of the capillary is wider than those usually used to ensure that the solution can flow through the capillary with no initial flow rate. When no vibration is applied, the solution can flow through the capillary at the beginning and the fibres are drawn out. But the solution quickly solidified at the tip of the capillary. When ultrasonic vibration was applied, the solution flowed through the capillary during the spinning process. Figure 4.4 reveals the vibro-rheological effect on viscosity. It is clear that the increase in frequency leads to a dramatic reduction in polymer viscosity. Figure 4.5(a) shows a scanning electron microscope (SEM) photograph of fibres spun without vibration and Figure 4.5(b) shows fibres spun while vibration was applied. To check the effect of vibration on polymer molecule structure, X-ray diffraction was carried out without and with vibration (Figures 4.6 and 4.7). It is obvious that the crystal peak was weakened by vibration.

Experimental Verification 2 In this method [3], polyethylene oxide (PEO) with a molecular weight of 3000 000 was supplied by Shanghai Lian Sheng Chemical Co. Ltd, and pure alcohol was purchased from Shanghai Chemical Co. An ultrasonic generator SFSA-1 (Shenbo Ultrasonic Device Co. Ltd) was used to apply ultrasonic vibration to the solution at 400 kHz. 98

Application of Vibration Technology to Electrospinning

(a)

(b) Figure 4.4 Vibro-rheological effects on viscosity.

99

Electrospun Nanofibres and Their Applications

(a)

(b)

Figure 4.5 SEM photographs of fibres spun (a) without vibration and (b) with vibration. The average diameter of the electrospun fibres in (a) is about 1000 nm, while in (b) it is about 700 nm.

Figure 4.6 Result of X-ray diffraction of the electrospun fibre without vibration: counts per second versus angle of incidence.

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Figure 4.7 Result of X-ray diffraction of the electrospun fibre with vibration: counts per second versus angle of incidence.

PEO was added to pure alcohol at a concentration of 10 wt% and stirred magnetically for one hour at 30 °C. The solution obtained was divided into two parts (solution A and solution B). Solution A was reserved under normal conditions to act as a control; solution B was taken to spin under ultrasonic vibration. The electrospun fibre diameter and morphology were analysed using a JSM-5600 LV SEM. Fifty measurements on random fibres for each electrospinning condition were performed and average fibre diameters are reported. Solution B was placed in a 20 ml syringe. The positive lead of a high-voltage power supply (F180-L; Shanghai Fudan Middle School Affiliated Factory) was connected to the 7-gauge syringe needle via an alligator clip. A grounded metal target was placed 8 cm from the needle tip. The polymer solution flowed by gravity without an initial 101

Electrospun Nanofibres and Their Applications

(a)

(b)

(c)

Figure 4.8 The 10% PEO solution after various treatments: (a) 10% PEO solution (gel); (b) coagulated solution A; (c) solution under ultrasonic vibration. Vibration improves electrospinnability.

flow rate, and the voltage was maintained at 15 kV. Under the same conditions without vibration, solution A cannot be electrospun. Pure alcohol was added to PEO, and stirred magnetically at 30 °C. The solution became a gel, and the solution (gel) obtained was too viscous to flow – see Figure 4.8(a). As solution (gel) A became cooler, PEO separated from alcohol and coagulated – see Figure 4.8(b). Applying ultrasonic vibration to the separated solution A, the coagulated PEO dissolved again – see Figure 4.8(c). The solution (gel) B obtained was placed in the syringe, and ultrasonic vibration is applied to the solution (gel), which became dilute and flowing. The vibration technology can therefore be successfully applied to electrospinning. Compared with the traditional electrospinning, vibration electrospinning has at least two main merits: (1) moderate voltage is needed; (2) finer fibre is obtained. 102

Application of Vibration Technology to Electrospinning

Figure 4.9 SEM micrograph of 10% PEO electrospun nanofibres. The average diameter of the nanofibres is about 100 nm.

We also show in this volume that vibration electrospinning produces nanofibres (Figure 4.9), which cannot be obtained by traditional electrospinning.

4.4 Effect of Solution Viscosity on Mechanical Characteristics of Electrospun Fibres It is found that the viscosity is most influential on the fibre morphology, size and mechanical characteristics [11]. Solution viscosity had a much greater effect on fibre diameter than did applied voltage. The inescapable conclusion is that electrospun fibres generally have low molecular orientation and poor mechanical properties. Lee and co-workers [11] studied the mechanical behaviour of electrospun fibre mats of polyvinyl chloride/polyurethane (PVC/PU) blends, revealing t ∝ η1.4, where τ is the tensile strength – see Figure 4.10. 103

Electrospun Nanofibres and Their Applications

Figure 4.10 Tensile strength of electrospun PVC/PU blend nonwoven mats versus solution viscosity. (Redrawn from Lee and co-workers data [11].)

In order to obtain an extremely tough electrospun fibre, high viscosity is best. But high viscosity leads to a large diameter of electrospun fibres and high applied voltage. On nanoscales, however, the mechanical properties mainly depend upon the diameter of the nanofibres due to the nano-effect [12]: τ ∝ ηb

d > 100 nm

τ ∝ d–1/2 and τ >> τ0

d < 100 nm

The threshold voltage is proportional to the viscosity of the electrospun solutions or melts: Ethreshold ∝ ηa 104

Application of Vibration Technology to Electrospinning

4.5 Carbon-Nanotube-Reinforced Polyacrylonitrile Nanofibres by Vibration Electrospinning Carbon nanotubes (CNT) are thought to be the perfect material for composites. In this section, multi-walled carbon nanotubes (MWCNT) were directly electrospun intoPAN nanofibres via both traditional electrospinning and vibration electrospinning. The fibres obtained were examined by SEM and X-ray diffraction. CNT were aggregated heavily in the fibres obtained by traditional electrospinning, while CNT were well distributed and aligned in PAN fibres by vibration electrospinning. The invention of the CNT [13] brought a revolution not only in materials but also in biology, electronics and even life. Owing to its perfect structure, the CNT has unique mechanical, electronic and other properties, such as high surface-to-volume ratio, small size, high strength and stiffness, low density, high conductivity, high flexibility and extensibility, ability to withstand cross-sectional and twisting distortions, and ability to withstand compression without fracture. As a result of these properties, CNT are expected to have a great potential in forming intelligent composite materials. Ko’s group has successfully electrospun CNT into nanofibres [14], and the reinforcement and rupture behaviour of CNT polymer nanofibres has also been studied in detail in [15, 16]. Salalha and co-workers. [17] embedded single-walled carbon nanotubes (SWCNT) in a PEO matrix by an electrospinning process. Reneker and co-workers [18] observed for the first time that the orientation of carbon nanotubes within nanofibres was much higher than that of the PAN polymer crystal matrix. The reinforcement depends mainly on the dispersion and alignment of CNT in the polymer solution. In order to enhance the dispersion of CNT in polymer solution, the CNT used were chemically modified. Modification methods included purification, polymerisation, ‘cutting’ or ‘disentangling’ and other activation treatments [19]. Such modifications require a strong acid (such as concentrated sulfuric 105

Electrospun Nanofibres and Their Applications acid and nitric acid) and high temperature, which are very dangerous for researchers during preparation; furthermore the preparation is time-consuming. Our aim is to apply the vibration technology to ensure good distribution of CNT in the polymer solution, so that no further preprocessing is needed [20]. As a result, costs can be reduced dramatically. In this section, we will show how to electrospin MWCNT-reinforced PAN nanofibres directly by vibration electrospinning.

Experimental Materials PAN with a molecular weight of 70,000 g/mol was supplied by Sinopec Shanghai Petrochemical Co. Ltd. DMF was purchased from Shanghai Chemical Co. and used as solvent without any further purification. MWCNT with a diameter of 40–60 µm were supplied by Shenzhen Nanotech Port Co. Ltd.

Instrumentation The scheme of the electrospinning set-up was shown in Figure 1.5, and the scheme of the vibration electrospinning set-up was shown in Figure 4.3. The needle tip was connected to a DC high-voltage generator (F180-L, Shanghai Fudan High School) via an alligator clip. An ultrasonic generator SFSA-1 (Shenbo Ultrasonic Device Co. Ltd) was used to provide the vibration in the vibration electrospinning set-up. Fibre morphology images of MWCNT-reinforced PAN fibres were analysed using a SEM. The crystal structures of PAN fibres and PAN/MWNT fibres were analysed using an X-ray diffractometer (XRD), and molecular chain structures were analysed using a Fourier transform infrared (FTIR) spectrometer.

Electrospinning Process All concentration measurements were done as weight by weight (w/w). PAN/DMF solution at 18% polymer concentration was 106

Application of Vibration Technology to Electrospinning prepared by dissolving PAN grains in DMF and stirring for 1 hour at 70 °C in an electromagnetic stirrer, at 1200 rev/min, then cooling to room temperature. Then 2% MWCNT were poured into the PAN/DMF solution and stirred for an hour. The solution was placed in a 20 ml syringe. The positive lead of a high-voltage power supply was connected to the 7-gauge syringe needle via an alligator clip. A grounded metal target was placed 8 cm below the needle tip. In the vibration electrospinning process, an ultrasonic generator SFSA-1 (Shenbo Ultrasonic Device Co. Ltd) was used to provide vibration at 20 kHz. The polymer solution flowed by gravity without an initial flow rate, and the voltage applied was maintained at 35 kV.

Results and Discussion The fibres obtained have an average diameter of about 1,000 nm. The surface of PAN fibres was smooth. The PAN/MWCNT fibres obtained by the traditional electrospinning process were of rough, cobblestone-like surface morphology, similar to that reported in an atomic force microscope (AFM) study of ~300 nm PAN/SWNT fibres [14] and that reported in a transmission electron microscope (TEM) study of polylactic acid /SWNT fibres [2]. In contrast to the traditional electrospinning, a smoother surface was detected by SEM for the PAN/SWNT fibres electrospun by the vibration electrospinning process (Figure 4.11). TEM observation of PAN/MWNT fibre mats obtained by the vibration electrospinning process (Figure 4.12) showed that most MWNT maintained their straight shape and were parallel to the axial direction of the PAN fibre after vibration electrospinning, indicating that a better alignment of MWNT was achieved in PAN fibres when vibration was applied in the electrospinning process. A good distribution was also obtained by vibration electrospinning. 107

Electrospun Nanofibres and Their Applications

(a)

(b)

(c)

(d)

(e) Figure 4.11 SEM micrographs of electrospun fibres: (a) traditionally electrospun neat PAN fibres; (b, c) traditionally electrospun PAN/MWCNT; (d, e) vibration electrospun PAN/MWCNT.

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Application of Vibration Technology to Electrospinning

Figure 4.12 TEM micrograph of vibration electrospun MWCNTreinforced PAN fibre. MWCNT were parallel to the axial direction of the PAN fibre and well distributed in the fibre. The welldistributed MWCNT could be considered as ‘artificial atoms’ which behaved with quantum-like properties.

With MWCNT added, the XRD crystal peaks of PAN/MWNT fibre mats were sharpened (Figure 4.13). This served as a direct confirmation of the successful filling of the fibrils with MWCNT. The peaks of PAN/MWNT fibres obtained by vibration electrospinning were strongest among the three XRD scans, which proved that the crystal structures of PAN/MWCNT fibres were greatly improved by means of an applied vibration. This result shows the promise of vibration electrospinning. Regarding the FTIR results (Figure 4.14), a wavelength in the 1600–1750 cm–1 range could be seen in neat PAN fibres but almost disappeared in the PAN/MWCNT fibres. The wavelength in the 500–750 cm–1 range was strengthened in PAN/MWCNT fibres. These two changes could also serve as a confirmation of the filling of MWCNT in the electrospun fibres. 109

Electrospun Nanofibres and Their Applications

Figure 4.13 XRD of PAN fibre mats and MWCNT-reinforced PAN fibre mats.

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Application of Vibration Technology to Electrospinning

Figure 4.14 FTIR of PAN and MWCNT-reinforced PAN fibres: (a) PAN neat fibres; (b) traditionally electrospun MWCNT-reinforced PAN fibres; (c) vibration electrospun MWCNT-reinforced PAN fibres.

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J-H. He, Y.Q. Wan and J.Y. Yu, International Journal of Nonlinear Sciences and Numerical Simulation, 2004, 5, 3, 253.

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Y.Q. Wan, J-H. He, Y. Wu and J.Y. Yu, Materials Letters, 2006, 60, 27, 3296.

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Y.Q. Wan, J-H. He, J.Y. Yu and Y. Wu, Journal of Applied Polymer Science, 2007, 103, 6, 3840.

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J. Zhang, Y.Q. Wan, J-H. He and Y.S. Zhu, Chinese Patent ZL200410015685.3.

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P.K. Baumgarten, Journal of Colloid Interface Science, 1971, 36, 1, 71.

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H. Wu, S. Guo, G. Chen, J. Lin, W. Chen and H. Wang, Journal of Applied Polymer Science, 2003, 90, 7, 1873.

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A.I. Isayev, C.M. Wong and X. Zeng, Advances in Polymer Technology, 1990, 10, 1, 31.

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B.H. Bersted, Journal of Applied Polymer Science, 2003, 28, 9, 2777.

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J.P. Ibar, Polymer Engineering and Science, 1998, 38, 1, 1.

10. Y.Q. Wan, J. Zhang, J-H. He and J.Y. Yu, Chinese Patent ZL200420020596.3. 11. D.Y. Lee, B.Y. Kim, S.J. Lee, M.H. Lee, Y.S. Song and J.Y. Lee, Journal of the Korean Physical Society, 2006, 48, 6, 1686. 12. J-H. He, Y.Q. Wan and L. Xu, Chaos, Solitons & Fractals, 2007, 33, 1, 26. 13. H.W. Kroto, J.R. Heath, S.C. O’Brien, R.F. Curl and R.E. Smalley, Nature, 1985, 318, 6042, 162. 112

Application of Vibration Technology to Electrospinning 14. F. Ko, Y. Gogotsi, A. Ali, N. Naguib, H. Ye, G.L. Yang, C. Li and P. Willis, Advanced Materials, 2003, 15, 14, 1161. 15. H.H. Ye, H. Lam, N. Titchenal, Y. Gogotsi and F. Ko, Applied Physics Letters, 2004, 85, 12, 1775. 16. J. Ayutsede, M. Gandhi, S. Sukigara, H.H. Ye, C.M. Hsu, Y. Gogotsi and F. Ko, Biomacromolecules, 2006, 7, 1, 208. 17. W. Salalha, Y. Dror, R. Khalfin, Y. Cohen, A.L. Yarin and E. Zussman, Langmuir, 2003, 19, 17, 7012. 18. J.J. Ge, H. Hou, Q. Li, M.J. Graham, D.H. Reneker, F.W. Harris and S.Z.D. Cheng, Journal of the American Chemical Society, 2004, 126, 48, 15754. 19. M.R. Anderson, B.R. Mattes, H. Reiss and R.B. Kaner, Science, 1991, 252, 5011, 1412. 20. Y.Q. Wan, J-H. He and J.Y. Yu, Polymer International, 2007, 56, 11, 1367.

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5

Magneto-Electrospinning: Control of the Instability

It is well known that, during the electrospinning process, the charged jet is intrinsically unstable. This leads to the construction of uneven nanofibres and a significant waste of energy (Figure 5.1), which otherwise could be used to further stretch the jet into even smaller fibres. The instability also leads to low molecular orientation, as a result of which low mechanical properties arise. This chapter suggests a completely novel approach to controlling the instability in electrospinning – the technology is termed magnetoelectrospinning [1, 2]. We conclude that the magnetic approach is the most effective and economical way to control the instability and to improve the mechanical properties as well.

Figure 5.1 The instability in electrospinning. Much energy is wasted due to the instability.

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5.1 Critical Length of Straight Jet in Electrospinning This section will provide a rational theory that can predict simply the length of the straight jet in electrospinning (Figure 5.2) [1]. See [2, 3] for further information. For the sake of clarity and to avoid unnecessary complexity, we consider the steady-state flow of an infinite viscous jet pulled from a capillary orifice and accelerated by a constant external electric field.

Figure 5.2 Critical straight length (AB) in electrospinning.

We rewrite the force balance equation for the initial stage as:

(5.1) Using Cauchy’s inequality, and in view of Equation (2.2), from Equation (5.1) we obtain the following inequality:

(5.2)

116

Magneto-Electrospinning: Control of the Instability The critical value occurs when: (5.3) In view of Equation (2.1), we have u = Q/πr2ρ). Substituting this into Equation (5.3), we obtain: (5.4) We limit ourselves to the initial stage, i.e., AB in Figure 5.2. In this stage, the electrical force is dominant over other forces acting on the jet, so the inequality Eqaution (5.2), by a simple operation, reduces to: (5.5) from which we can immediately obtain: (5.6) This inequality can be rewritten in the form: (5.7) where β is the minimal radius number defined as: (5.8) We modify inequality Equation (5.7), in the following form in order to describe the actual electrospinning: (5.9)

117

Electrospun Nanofibres and Their Applications which corresponds to r ~ z–1/2 for the initial stage. Here α is an unknown function of the viscosity of the polymer solution, α = α(µ), which can be determined experimentally or theoretically. The minimal value of r in the initial stage reads: (5.10)

where L is the length of the straight length (AB in Figure 5.2) of the electrospun fibre, and we call Rcr the critical radius number. In view of Equation (5.4), we have the relationship:

(5.11) From this we can obtain the critical straight length (AB in Figure 5.2) from the capillary orifice to the point where instability occurs:

(5.12) where:

Instability occurs when the viscous force is almost the same as the electrical force. As the viscosity increases, the viscous force increases as well, and as a result the length of the jet increases. 118

Magneto-Electrospinning: Control of the Instability The fibre diameter also increases with solution viscosity and is approximately proportional to the jet length. The jet length is measured from the tip of the spinning drop to the onset of waves in the fibre.

5.2 Controlling Stability by Magnetic Field If we apply a magnetic field in the electrospinning process, as illustrated in Figure 5.3, an Ampere force is generated due to the current in the polymer jet:

where I is the current flowing inside the conductor, B is the value of the magnetic field induction, and L is the conductor length. The resultant of the electrical force and the viscous force of the jet flow increases the radius of the whipping circle. If we apply a magnetic field in electrospinning, the problem can be completely overcome. The current in the jet, under the magnetic field, produces a centripetal force, i.e., the direction of the Ampere force is always towards the initial equilibrium point (see Figure 5.3), leading to the reduction in the radius of the whipping circle. As a result, the stability condition is enormously improved – see Figure 5.4. The magnetic force is perpendicular to the velocity, so it contributes no work for the moving jet. The reduced circle size means less energy waste in the instability process, and the energy saved is used to increase the kinetic energy of the moving jet. According to Equation (2.1), r2~1/u, the radius becomes much smaller than that without magnetic field – see Figure 5.5. The magnetic force produced can ameliorate the inner structure of the macromolecules, resulting in remarkable amelioration of the nanofibre’s strength (Figure 5.6). 119

Electrospun Nanofibres and Their Applications

(a)

(b) Figure 5.3 Effect of a magnetic field in electrospinning: (a) application of a magnetic field to electrospinning; (b) the Ampere force in the electronic jet induced by the magnetic field.

120

Magneto-Electrospinning: Control of the Instability

(a)

(b) Figure 5.4 The whipping circle is reduced using a magnetic field: (a) without a magnetic field; (b) with a magnetic field.

121

Electrospun Nanofibres and Their Applications

(a)

(b) Figure 5.5 Effect of the magnetic field on the diameters of the nanofibres: (a) without a magnetic field; (b) with a magnetic field.

122

Magneto-Electrospinning: Control of the Instability

Figure 5.6 Macromolecules under the coupled action of electrical force and magnetic force.

5.3 Controlling Stability by Temperature In order to control solvent evaporation, a furnace is added, in which the temperature field can be adjusted. The temperature near the orifice is lower than that of the melt, and higher than that near the collector. Owing to the temperature gradient, cold air flows up along the furnace wall, and down along the fibre. The low-velocity gas further impacts the polymer jet, as illustrated in Figure 5.7. Convection currents occur in the upper half of the furnace and were enhanced by the injection of comparatively cool feedstock down the axis. We believe that the gas moved upwards adjacent to the furnace walls and downwards in the centre of the tube. As a result, the polymer jet is further attenuated and jet length can be increased, so that the instability can be controlled [2, 3]. 123

Electrospun Nanofibres and Their Applications

Figure 5.7 Schematic of the electrospinning process with a controllable temperature field.

It would be more effective if we could devise an experimental setup that takes full account of these controllable temperatures (see Figure 5.8). High temperature can also dramatically reduce the viscosity, and the electrospinning of polymer melts can offer an advantage over solution electrospinning. Figure 5.9 illustrates the apparatus for a melt electrospinning system. Some melts have a high viscosity. We can devise an apparatus for vibration melt electrospinning, as illustrated in Figure 5.10. Hot air is applied in aero-electrospinning – see Figure 5.11. Hot air further attenuates the filaments, making the electrospun fibres ever smaller. Melts are extruded by electrostatic force through a small orifice into convergent streams of hot air that rapidly attenuate the 124

Magneto-Electrospinning: Control of the Instability

Figure 5.8 Schematic illustration of the set-up for temperaturecontrolled melt electrospinning.

Figure 5.9 Melt electrospinning does not require a solvent. The temperature of melts can be controlled by an electrical heater.

125

Electrospun Nanofibres and Their Applications

Figure 5.10 Experimental set-up for vibration melt electrospinning.

Figure 5.11 Aero-electrospinning.

126

Magneto-Electrospinning: Control of the Instability charged jet into small diameter fibres. The air streams can also control the instability in the electrospinning process, and transport fibres to a collector where they bond at fibre–fibre contact points to produce a cohesive non-woven web.

5.4 Siro-electrospinning Two-strand spun or Sirofil yarns [4, 5] have now been widely used in the worsted industry (see Figure 5.12). The strands are texturised to improve the bulk of the resultant yarns, which have been demonstrated to possess more desirable properties. For example, the weavability of the fabric formed by Sirofil yarns is significantly improved over its counterpart yarns.

Figure 5.12 Sirofil technology to improve the mechanical properties of the mother fibres.

127

Electrospun Nanofibres and Their Applications

Figure 5.13 Siro-electrospinning.

Dragline silk is made of many nanofibres. To mimic spider spinning, Siro-electrospinning has been created. An apparatus has been devised in which a twist is added (see Figure 5.13), causing the fibres to have a tendency to twist and crimp. The fabrics so produced can improve their mechanical characteristics; furthermore, strength, toughness, softness and feel can be improved dramatically. Li and co-workers have patented a conjugate electrospinning apparatus [6–8], as illustrated in Figure 5.14. Two or more fibres are combined together by weak electrostatic forces.

Figure 5.14 The schematic set-up used for conjugate electrospinning experiments [6–8].

128

Magneto-Electrospinning: Control of the Instability The present technology can also be taken in conjunction with other ones. For example, vibration technology can be added. Using this technique, it is now possible to spin conductive polymer fibres under ambient conditions by depositing nanometre metal particles in the polymer solutions or melts. If we add some titanium dioxide to the solutions or melts, the electrospun fibres are self-cleaning.

References 1.

Y. Wu, J.Y. Yu, J-H. He and Y.Q. Wan, Chaos, Solitons & Fractals, 2007, 32, 1, 5.

2.

J-H. He, Y.Q. Wan and L. Xu, Chaos, Solitons & Fractals, 2007, 33, 1, 26.

3.

J-H. He, Y. Wu and W.W. Zuo, Polymer, 2005, 46, 26, 12637.

4.

J-H. He, Y.P. Yu, J.Y. Yu, W.R. Li, S.Y. Wang and N. Pan, Textile Research Journal, 2005, 75, 2, 181.

5.

J-H. He, Y.P. Yu, N. Pan, X.C. Cai, J.Y. Yu and S.Y. Wang, Mechanics Research Communications, 2005, 32, 2, 197.

6.

X. Li, C. Yao, F. Sun and T. Song, ACS Polymeric Materials: Science and Engineering, 2006, 94, 1, 26.

7.

X. Li, Y. Chen and S. Tangyin, inventors; Chinese Patent ZL200510038571.5.

8.

X. Li, Y. Chen and S. Fuqian, inventors; Chinese Patent ZL200510095384.0.

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130

6

Bubble Electrospinning: Biomimic Fabrication of Electrospun Nanofibres with High Throughput

Spider-spun fibre is of extraordinary strength and toughness compared to those of electrospun fibres. To produce the latter from the water-soluble protein ‘soup’ produced by a spider requires a very high voltage (from several thousand volts to several tens of thousands of volts) applied to the ‘soup’. Furthermore, the mechanical strength of such fibres is dramatically decreased compared to spider silk. A possible mechanism in the spider spinning process is discussed in this chapter. The distinct character in spider spinning is that the spider’s spinneret consists of millions of nanoscale tubes, and a bubble can be produced at the apex of each nanotube. The surface tension of each bubble is extremely small, such that it can be spun into a nanofibre with an awfully small force, either by the spider’s body weight or by tension created by the rear legs. We mimic the spider spinning process in electrospinning using an aerated solution, which leads to various small bubbles on the surface with very small surface tension. As a result the bubble can easily be electrospun into nanofibres with low applied voltage. This fabrication process possesses the features of high productivity and versatility; in addition, the minimum diameter of nanofibres produced by this process can reach as small as 50 nm.

6.1 Spider Spinning After 400 million years of evolution, nature has endowed spiders with the genius of spinning flexible, lightweight fibres that are the

131

Electrospun Nanofibres and Their Applications strongest in the world, at least five times stronger by weight than steel, and with remarkable toughness and elasticity [1–3]. Even in modern times, it is difficult to synthesise a material having the advantages of strength and toughness, except for carbon nanotube fibres, which are spun from solution at very high temperature or pressure [4], while spider silks are produced at room temperature and from aqueous solutions. Spider silk is the only natural material that combines properties unmatched by any known synthetic highperformance fibre. Spider silks are protein-based ‘biopolymer’ filaments or threads secreted by specialised epithelial cells as concentrated soluble precursors of highly repetitive primary sequences [5]. Many experiments have been conducted, and much research has focused on gene sequencing of the spider, and developed a biomimicry technology [6–8]. However, no theoretical analysis has yet dealt with this, and our understanding of the mechanism of spider spinning is scarce and primitive. If the mystery in its mechanism can be solved, then we could apply the mechanism to synthetic high-performance fibres, such as electrospun fibres, to achieve the combination of great strength and stretch. The method devised must be very economical, so this could be the beginning of a new materials revolution.

6.1.1 Intelligent Spider Fibre When a spider is in the centre of its web, the spider will not respond when the web is acted on by natural forces, such as wind and rain. But when a mosquito struggles to get free from its web, the spider will find its prey immediately, no matter whether it is in the centre or on the fringe of the web. The spider can distinguish various prey with ease by their fluttering frequency. So the spider can feel oscillations just like people can hear sound, which is also an oscillation. Our eardrum is oscillated when the ear is subjected to an oscillating sound, and information is sent to the brain through nerves. But spider silk in the web has no life, so the 132

Bubble Electrospinning … oscillation information cannot be sent in a similar way to ion pumps in biology. Instead, the oscillation information is transferred through the electrical behaviour of the web caused by the vibration. The spider can only ‘hear’ some oscillations with defined frequency. It cannot ‘hear’ the oscillation caused by the wind or rain. Some species of orb builders do not stay on the web itself, but attach a dragline to the sheet of webbing, then sit in a nearby bush holding the line taut. The web diaphragm vibrates, the message travels down the non-sticky radial threads, and the spider instantly ‘hears’ the vibration and knows in which direction to head for its meal. It seems that a spider web and dragline have an intelligent character. When acted on by a vibrating force with definite frequency, the spider web and dragline may produce an electric field. Or do they oscillate when subjected to an electric field? This still needs experimental verification. However, this intelligent character differs from that of piezoelectric materials (Figure 6.1), which have been widely used in engineering. Piezoelectricity was first discovered by Pierre and Jacques Curie in 1880, when they found that certain crystals may, when stressed, produce an electric field, or when subjected to an electric field, deform.

Figure 6.1 Spider silk has maximal strength and has a special smart character different from traditional piezoelectricity.

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Electrospun Nanofibres and Their Applications

6.1.2 Mathematical Model for Spider-Spun Fibres At present, and despite decades of concentrated effort, we do not have a rational theory that can explain the intelligent character of spider dragline, let alone the mystery of the spider spinning process. Here we consider the governing equations for linear spider dragline silk [9]. (1) Equilibrium equations: (6.1) in which σij,j is the symmetric stress tensor, σij,j = ∂σij/∂xj. (2) Constitutive equations: (6.2) (6.3) in which γij is the symmetric strain tensor, Di is the electric displacement vector, Ei is the electric field vector, ωi is the frequency, and ki is a constant. The elastic moduli aijkl measured at constant (zero) electric field, and the piezoelectric moduli emij and the dielectric permittivity εij have the following symmetry properties, respectively: aijkl = ajikl = aijlk = aklij emij = emji

and

εij = εji

When ki = 0 or ωi = 0, our model turns out to be the traditional one in piezoelectricity. (3) Strain–displacement relationship: (6.4) where ui is the elastic displacement vector. 134

Bubble Electrospinning … (4) Quasi-static Maxwell’s equations: (6.5) We know that changes in hydrogen bonding and the electrostatics of the environment affect the vibrations of the peptide bonds in the fibroin molecule. Vibration gives rise to a sensitivity of the amide normal modes to the secondary structure of the protein. In addition, subtle changes in side-chain vibrations can also be correlated with alterations in conformation or solvent. So vibration affects the folding and crystallisation of the main protein constituents, and its viscosity as well. Vibration technology in spider spinning represents an exciting scientific opportunity and a technological challenge. However, the technology has been under-utilised primarily because of the most rudimentary understanding of how the highly complex fibroin molecules self-organise to form a high-strength polymer, and how they do so under the mildest of conditions. It is intriguing how the spider is able to control different glands to spin different silks – the possible answer is vibration. It is also still an enigma how a spider produces a silk that a human needs ten thousand volts to do. One possible answer is that a spider has the power to generate an oscillation by environmental stimuli when it ‘spins’ silks. The generated oscillation leads to the reduction of viscosity, or equivalently surface tension, and to an electrical force that surpasses the surface tension. As a result, a non-sticky silk is ejected. The ejection stops when the oscillation vanishes. So the spider can control the ‘spinning’ process arbitrarily.

6.2 Electrospinning of Silk Fibroin Nanofibres If an aqueous solution is electrospun, we need a very high voltage under at room temperature. Recently Ohgo and co-workers [10] successfully prepared non-woven nanofibres of Bombyx mori and Samia cynthia ricini silk fibroins, and a recombinant hybrid fibre involving the crystalline domain of B. mori silk and the non-crystalline domain of S. cynthia ricini silk from hexafluoroacetone solution using an 135

Electrospun Nanofibres and Their Applications electrospinning method. In the electrospinning process, a high electric voltage of 15–30 kV is applied to a droplet of a silk solution at the tip of a plastic capillary in which a platinum wire is used as an electrode [10]. The electrospinning of B. mori cocoon silk and of Nephila clavipes dragline silks have also been reported by other authors [11, 12]. It is really very perplexing that natural silk fibres can be produced with minimal force, while man-made silk fibres from aqueous fibroin solution require very high pressure or voltage. Spiders have evolved far superior methods for processing polymers into materials, methods that allow nearly optimal utilisation of the quantum-like properties of the polymer chains. Imagine a silk-like material with the diameter of a pencil: this is so strong that it could stop a Boeing 747 in flight! The stress and strain of dragline threads can reach up to 1500 MPa and 500%, respectively. However, electrospun non-woven silk fibres have dramatically low stress and strain [10].

6.3 Solving the Mystery of the Spider Spinning Process How do we answer this enigma of how a spider produces a silk that needs ten thousand volts for us to make it. Another possible answer is that a spider efficiently applies the nano-effect of bubble dynamics [13]. The pressure difference between the inside and outside of a bubble depends upon the surface tension and the radius of the bubble. For a bubble with two surfaces providing tension, the net upward force on the top hemisphere of the bubble is just the pressure difference times the area of the equatorial circle: (6.6) The surface tension force downward around a circle is twice the surface tension times the circumference, since two surfaces contribute to the force: (6.7)

136

Bubble Electrospinning … This gives: (6.8) When an electric field is present, it induces charges into the surface of bubbles in clusters and on the solution surface. The coupling of surface charge and the external electric field creates a tangential stress, resulting in the deformation of the small bubble into a protuberanceinduced upward-directed re-entrant jet. Once the electric field exceeds the critical value needed to overcome the surface tension, a fluid jet ejects from the apex of the conical bubble. The threshold voltage needed to overcome the surface tension depends upon the size of the bubble and the inlet air pressure. Hendricks and co-workers [14] calculated the minimum spraying potential of a suspended, hemispherical, conducting drop in air as: (6.9) Buchko and co-workers [15] modified equation (6.9) in the form: (6.10)

(6.11) where Ethreshold is the critical voltage needed for the ejection of the jet from the Taylor cone, H is the separation distance between the needle and the collector, L is the length of the needle (or capillary), R is the radius of the needle and γ is the surface tension of the solution. From Figure 6.2, we can see clearly that dragline silk is made of many nanofibres, each with diameter of about 20 nm. We have reason to assume that a bubble can be produced at the apex of each tube with a diameter of about 20 nm. According to Equation (6.7), the surface tension of such bubbles is extremely small, and can easily be 137

Electrospun Nanofibres and Their Applications

Figure 6.2 Spider spinning process. The diameter of a single nanofibre is about 20 nm. The spinneret on the posterior portion of the spider’s abdomen consists of millions of nanoscale tubes, and a bubble can be produced at the apex of each nanotube. The surface tension of each bubble is extremely small, such that it can be spun into nanofibres with an awfully small force, either by the spider’s body weight or by tension created by the rear legs. (Reproduced with the permission of Dennis Kunkel Microscopy, Inc.)

overcome either by the spider’s body weight or by tension created by the rear legs. Most natural spider silk is only 2.5-4 µm in diameter, and the weight of most spiders varies from 90 to 1500 mg. Consider a dragline silk with a diameter of 3 × 10–6 m, which consists of many nanoscale fibrils with diameter of about 20 nm; (see Figure 6.2). We can estimate the number of nanofibres in the assembly, as follows: 138

Bubble Electrospinning …

(6.12) That means there are tens of thousands of nanofibres in the assembly! Similar to the Hall–Petch relationship, nanofibre strength depends upon fibre diameter in nanoscale (from a few nanometres to tens of nanometres): (6.13) where kτ are the fitting parameters (material constants), τ0 is the strength of the bulk material, and d is the fibre diameter. Comparing the strength of a single fibre with diameter of 3 × 10–6 m, τSF, with the strength of a dragline assembly consisting of 2 × 104 nanofibres with diameter of 20 × 10–9 m, τDA, we estimate: (6.14) Thus, the extraordinarily high strength of a dragline assembly is predicted compared with a single fibre with the same cross-sectional area and made of the same material. This finding shows that it is a challenge to developing technologies to be capable of preparing nanofibres within the 50 nm range. There are tens of thousands of nanotubes in a spider’s spinneret. We assume that a spider has weight of G = 10–4 kg. In the spinning process, the spider can use its weight to overcome the surface tension of all the bubbles produced at the apex of its nanotubes. This requires that: (6.15) 139

Electrospun Nanofibres and Their Applications from which we can determine the pressure difference between the inside and outside of a bubble, which reads: (6.16) An extremely small force indeed is needed in spider spinning! Although the maximal or minimal size of a bubble might depend upon the solution viscosity, the surface tension of the bubbles is independent of the properties of the spun solutions, such as viscosity, which is the main obstruction in traditional electrospinning. We can also use temperature to adjust the bubble sizes in practical applications of the new electrospinning process (described later). The key advantages of this electrospinning process used here, compared to the traditional electrospinning process, are that it can produce smaller nanofibres without the requirement for nozzles, which are the main shortcoming of the traditional electrospinning technology. Furthermore, in our method, with a relatively low voltage, millions of protruded bubbles can be easily produced, which can be electrospun into nanofibres simultaneously, and electrospinnability does not strongly depend upon the solution viscosity, overcoming completely the main shortcoming of traditional electrospinning. The average diameter of nanofibres can be easily controlled, and the minimum diameter in our experiment reached as small as 50 nm. The novel electrospinning method resulted in thinner nanofibres and potential application for mass production of nanofibres. Gu and co-workers [16] measured the Young’s modulus of a single electrospun polyacrylonitrile (PAN) fibre using an atomic force microscopy cantilever. We reanalysed the data, revealing that the nano-effect happens near 150 nm (see Figure 6.3). Generally, electrospun nanofibres are too large (hundreds to a few thousand nanometres) to exhibit a nano-effect. Thus, we have: 140

Bubble Electrospinning …

Figure 6.3 The E modulus of nanofibres. (Redrawn according to Gu and co-workers’ experimental data [16], shown by the dots.)

τ ≤ τ0 τ ∝ d–1/2

when d > 200 nm, and

τ >> τ0

when 0 < d < 100 nm,

where τ is the strength of a single electrospun fibre, τ0 is the strength of its bulk material, and d is the diameter of the fibre. Spiders can have as many as eight types of silk-making glands, but no spider has been found to possess all eight kinds. Most spiders, however, possess at least three silk glands: the aciniform glands, the pyriform glands, and the ampullaceal glands [17]. All the glands produce silk for different functions, and it seems that each gland synthesises and secretes different types of silk for various functions. Some produce silk for cocoons with moderate viscosity, whereas others produce sticky silk to trap prey. Non-sticky dragline threads, produced in the ampullate gland, are the strongest and form the radiating spokes of 141

Electrospun Nanofibres and Their Applications a web. This phenomenon cannot be explained by what is observed in electrospun fibres, where the inverse phenomenon arises: the higher the viscosity, the higher the strength of the electrospun fibres (see Figure 6.4). It is found that the viscosity has most influence on the fibre morphology, size and mechanical characteristics. Solution viscosity had a much greater effect on fibre diameter than did applied voltage. The inescapable conclusion is that electrospun fibres generally have low molecular orientation and poor mechanical properties. Such deterioration can be ameliorated by applying vibration when electrospinning. Spider spinning is controlled by changing pH (from 6.9 to 4.8) and viscosity (from high value to low one). Lee and his colleagues [18] studied the mechanical behaviour of electrospun fibre mats of polyvinyl chloride/polyurethane (PVC/PU) blends, revealing that τ ∝ η 1.4, where τ is the tensile strength – see Figure 6.4.

Figure 6.4 Tensile strength of electrospun polyblend non-wovens versus PVC/PU solution viscosity. (Redrawn according to Lee and co-workers’ data [18].) 142

Bubble Electrospinning … In order to obtain an extremely tough electrospun fibre, high viscosity is best. But high viscosity leads to large diameter of electrospun fibres and high applied voltage. In order to obtain highstrength nanofibre with high toughness, the most challenging way is to prepare nanofibres within the 100 nm range. Within such a scale, quantum-like properties and nano-effects can be fully used. For the first time, Maensiri and Nuansing [19] obtained sodium cobalt oxide (NaCo2O4) nanofibres with a diameter of about 20–200 nm by electrospinning a precursor mixture of sodium acetate/cobalt acetate/polyacrylonitrile, followed by calcination treatment of the electrospun composite nanofibres.

6.4 Bubble Electrospinning With these hints as to the mechanism of spider spinning, we designed a new approach to overcoming the bottleneck in present electrospinning technology, that is to minimise the surface tension of the electrospun solutions mimicking the spider spinning. Our system (see [20–23] for further details) consists of a vertical solution reservoir with a gas tube feeding from the bottom, with a metal electrode fixed along the centreline of the tube, and a grounded collector over the reservoir – see Figure 6.5. It has been found that many small bubbles with different sizes were produced on the solution surface. The mechanism of the new electrospinning process is deceptively simple. In the absence of an electric field, the aerated solution forms various bubbles on the surface. When an electric field is present, it induces charges into the bubble surface, which quickly relax to the solution surface. The coupling of surface charge and external electric field creates a tangential stress, resulting in the deformation of the small bubbles into protuberance-induced upward-directed reentrant jets. Once the electric field exceeds the critical value needed to overcome the surface tension, a fluid jet ejects from the apex of the conical bubble. The threshold voltage needed to overcome the 143

Electrospun Nanofibres and Their Applications surface tension depends upon the size of the bubble and the inlet air pressure. The most fascinating character of the surface tension of a bubble is independent of the properties of the electrospun solutions, such as viscosity. This new technology is of critical importance for the new generation of electrospinning, especially for specialists in the design, manufacture and use of nanofibres.

Experimental In order to verify our above idea, an experiment was carefully designed. The experimental set-up is illustrated in Figure 6.5.

Figure 6.5 The experimental set-up for aerated solution electrospinning. (This principle for preparing nanoproducts is patented [20]. To use this principle to prepare nanoporous products, a transfer agreement must be made).

144

Bubble Electrospinning … Polyacrylonitrile (PAN) was dissolved in N,N-dimethylformamide (DMF) solvent and the weight concentration was adjusted to 12 wt%. The polymer solution was poured into the reservoir. The height of the liquid surface was higher than that of the electrode and tube. The pneumatic pressure control valve was turned on slowly. Several bubbles were seen to be generated at the apex of the tube. The bubbles produced broke down into smaller ones on the solution surface. When the surface tension of the small bubbles reduces to the critical value that can be overcome by the applied electric field, nanojets eject from the apex of the bubbles. The diameters of the electrospun nanofibres (see Figure 6.6) depend mainly on the sizes of the bubbles produced on the solution surface, which can be adjusted by the inlet air pressure and the position of the apex of the air tube. The minimal fibre diameter produced in the experiment was as small as 50 nm (Figure 6.7). These materials with diameters less than 100 nm always display quantum-like properties

Figure 6.6 Scanning electron microscope (SEM) photograph of PAN electrospun nanofibres. 145

Electrospun Nanofibres and Their Applications

Figure 6.7 The minimum diameter of nanofibres was 50 nm.

and have many fascinating nano-effects, such as a remarkable increase in strength, high surface energy and surface reactivity, and thermal and electrical conductivity [21].

Visualisation of Aerated Solution Electrospinning The new method can produce nanofibres as small as 50 nm, which cannot be seen with the naked eye. What we could see in our experiment is actually a nanofibre aggregation, which consists of many smaller nanofibres. Figures 6.8 and 6.9 show photographs of the ejected jets and a nanofibre mat produced in this experiment. In order to observe clearly the ejected jets in the electrospinning process, we designed an experiment as follows. The experiment was performed at room temperature, which was about 28 °C. Polyethylene oxide (PEO) with a molecular weight of 500,000 at a weight concentration of 2%, was dissolved in a mixture 146

Bubble Electrospinning …

Figure 6.8 Ejected jets, as captured by a digital camera.

Figure 6.9 Nanofibre mat, as captured by a digital camera.

147

Electrospun Nanofibres and Their Applications

Figure 6.10 Visualised evolution of the aerated solution electrospinning process with an interval of 17 ms.

148

Bubble Electrospinning …

Figure 6.10 Continued

of around 10% distilled water and 90% ethanol. Such an electrospun solution can lead to a large diameter of the charged jet, but the jet cannot be solidified into a fibre. A high-speed motion camera was used to observe the motion of jets, as shown in Figure 6.10. The images were produced by the camera at 60 frames per second.

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Morphology of Nanofibres Two polymers, PEO Mw = 500,000 g/mol) and polyacrylonitrile (PAN, Mw = 70,000 g/mol), were purchased from Shanghai Lian Sheng Chemical Co. Ltd and Sinopec Shanghai Petrochemical Co. Ltd, respectively. Solvent, pure alcohol, distilled water and DMF were bought from Shanghai Chemical Co. Ltd. All the chemicals were used as received without further purification. The polymer, PEO, was dissolved in a mixed solvent, pure alcohol plus distilled water with the weight ratio of 9:1. The solution concentration was adjusted to 2 wt%. The other polymer, PAN, was dissolved in DMF solvent and the weight concentration was adjusted to 12 wt%.

Experimental Set-up and Electrospinning Process The system consists of a vertical solution reservoir with a gas tube feeding from the bottom, with a metal electrode fixed along the centreline of the tube, and a grounded collector over the reservoir as described in previous work [21]. The polymer solution was poured into the reservoir so that the liquid surface was higher than the tip of the positive electrode and the gas tube. When the pneumatic pressure control valve was turned on slowly, several bubbles were generated at the free surface of the polymer solution. After an electric field was applied and the voltage was above the threshold voltage, multiple jets were ejected from the bubbles to the collector. The morphology and diameter of the PAN nanofibres obtained by the bubble electrospinning process were determined by a scanning electron microscope (SEM, JSM-5610, Japan). An Olympus digital compact camera (FE-160, Japan) and a high-speed motion camera, Redlake’s MotionXtra system (HG-LE, Redlake, USA), were employed to capture the motion of the jets in the experiments. Figures 6.11–6.13 show the SEM photographs and diameter distributions of PAN electrospun nanofibres with different applied voltages observed in this experiment. The diameters of the electrospun nanofibres in the present experiment were in the range between 50 and 800 nm.

150

Bubble Electrospinning …

Figure 6.11 SEM photograph of PAN electrospun nanofibres and their diameter distribution, at an applied voltage of 10 kV.

151

Electrospun Nanofibres and Their Applications

Figure 6.12 SEM photograph of PAN electrospun nanofibres and their diameter distribution, at an applied voltage of 20 kV.

152

Bubble Electrospinning …

Figure 6.13 SEM photograph of PAN electrospun nanofibres and their diameter distribution, at an applied voltage of 35 kV.

153

Electrospun Nanofibres and Their Applications The distributions showed that the majority of the nanofibre diameters in this experiment were in the range 100–400 nm, and the average value was about 200 nm. The number average diameters increased from 185 nm to 251 nm with increasing applied voltage. The reason for the fine nanofibres obtained might be attributed to the thin polymer solution layer of the bubbles. There might be less macromolecular chain entanglement in the thin solution layer of the bubbles. Therefore, fine jets were ejected from the tip of the bubble. The travelling liquid jets were subject to a variety of forces and slenderised in stretching during this spinning process. Additionally, the electrostatic force increased as the electric field strength increased. The bigger electrostatic repulsion force of the charges in the bubble tends to pull out more and thicker jets with higher speed. The travel distance, however, was not changed. Thus, thicker nanofibres could be produced because the solidification time was shorter.

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J. Koover in Ecophysiology of Spiders, Ed., W. Nentwig, Springer, Berlin, Germany, 1987, p.160–188.

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A. Lazaris, S. Arcidiacono, Y. Huang, J.F. Zhou, F. Duguay, N. Chretien, E.A. Welsh, J.W. Soares and C.N. Karatzas, Science, 2002, 295, 5554, 472.

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J. Gatesy, C. Hayashi, D. Motriuk, J. Woods and R. Lewis, Science, 2001, 291, 5513, 2603.

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J-H. He and K.L. Xie, Facta Universitatis Series: Mechanics, Automatic Control and Robotics, 2006, 5, 1, 53.

10. K. Ohgo, C. Zhao, M. Kobayashi and T. Asakura, Polymer, 2003, 44, 3, 841. 11. S. Sukigara, M. Gandhi, J. Ayutsede, M. Micklus and F. Ko, Polymer, 2003, 44, 19, 5721. 12. B.M. Min, G. Lee, S.H. Kim, Y.S. Nam, T.S. Lee and W.H. Park, Biomaterials, 2004, 25, 7-8, 1289. 13. J-H. He, Journal of Animal and Veterinary Advances, 2008, 7, 2, 207. 14. C.D. Hendricks, Jr., R.S. Carson, J.J. Hogan and A.J. Schneider, AIAA Journal, 1964, 2, 5, 955. 15. C.J. Buchko, L.C. Chen, Y. Shen and D.C. Martin, Polymer, 1999, 40, 26, 7397. 16. S.Y. Gu, Q.L. Wu, J. Ren and G. J. Vancso, Macromolecular Rapid Communications, 2005, 26, 9, 716. 17. R.F. Foelix, Biology of Spiders, 2nd Edition, Oxford University Press, New York, NY, USA, 1996. 18. D.Y. Lee, B.Y. Kim, S.J. Lee, M.H. Lee, Y.S. Song and J.Y. Lee, Journal of the Korean Physical Society, 2006, 48, 6, 1686. 19. S. Maensiri and W. Nuansing, Materials Chemistry and Physics, 2006, 99, 1, 104. 20. Y. Liu, J-H. He, J.Y. Yu, L. Xu and L.F. Liu, inventors, Chinese Patent 200710036447.4. 21. Y. Liu and J-H. He, International Journal of Nonlinear Sciences and Numerical Simulation, 2007, 8, 3, 393. 155

Electrospun Nanofibres and Their Applications 22. J-H. He, Y. Liu, L. Xu, J.Y. Yu and G. Sun, Chaos, Solitons & Fractals, 2008, 37, 3, 643. 23. Y. Liu, J-H. He, L. Xu and J.Y. Yu, Journal of Polymer Engineering, 2008, 28, 1–2, 55.

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Controlling Numbers and Sizes of Beads in Electrospun Nanofibres

Beads are considered the main demerit of electrospun fibres. There are many factors affecting the occurrence of beads, such as the applied voltage, the viscoelasticity of the solution, charge density, and the surface tension of the solution, etc. In this chapter we will suggest three methods [1] to reduce the numbers and sizes of beads: adjusting the weight concentration, adding a salt additive, or choosing a suitable solvent for electrospun products.

7.1 Experimental Observations Materials Polybutylene succinate (PBS) pellets were supplied by Shanghai Institute of Organic Chemistry, Chinese Academy of Science. The molecular weight is about 2 × 105 g/mol. The solvents, chloroform, dichloromethane (DCM), 2-chloroethanol and isopropanol (IPA), were purchased from Shanghai Chemical Reagent Co. Ltd. Lithium chloride (LiCl) was purchased from Pinjiang Chemical Co. All the chemicals were used directly without further purification. The polymer pellets were dissolved in a single solvent or mixed solvents mentioned above. The weight concentration of the solutions were adjusted to 11–17 wt%. LiCl, as a salt additive, with a concentration of either 0.5 or 1.0 wt%, was added to the polymer solution at 14 wt%.

Electrospinning Process The electrospinning set-up, equipped with a variable DC high-voltage power generator (0–100 kV, F180-L, Shanghai Fudan High School),

157

Electrospun Nanofibres and Their Applications was used in this work. The polymer solution was transferred into a 20 ml plastic syringe vertically and delivered to the orifice of the stainless-steel needle by a syringe pump (AJ-5803, Shanghai Angel Electronic Equipment Co.) at a constant feeding rate. A voltage was applied to the needle by the high-voltage power generator via an alligator clip. A flat aluminium foil, as a collector, was connected to ground under the needle. The distance between the orifice and the collector was 10 cm. The diameter of the orifice was 0.9 mm. The polymer pellets were dissolved in a single solvent or mixture of solvents and stirred for about 2 hours at 40 °C. All the electrospinning processes were performed in air at room temperature.

Characterisation The morphology of the electrospun products was examined using a scanning electron microscope (SEM; JSM-5610, JEOL, Japan) after being coated with gold. The surface tensions of different polymer concentrations were measured with a ThermoCahn DCA322 Surface Dynamic Contact Angle Analyser.

7.2 Effects of Different Solvents PBS was chosen because it is soluble in common organic solvents such as CHCl3, DCM, chloroethanol, and so on. In order to investigate the morphology of beads in electrospun PBS nanofibres, the polymer was dissolved in a single solvent (CHCl3 or DCM), or a mixed solvent system at different weight ratios (w/w), such as CHCl3/DcM (7:3), CHCl3/IPA (8:2) or CHCl3/chloroethanol (7:3). The reason for the choice of the mixed solvents was that the electrospinnability and efficiency were better. All the process parameters were fixed as follows: applied voltage 10 kV; solution concentration 11 wt%; orifice-to-collector distance 10 cm; diameter of the orifice 0.9 mm. The solution concentrations were adjusted to a constant value of 11 wt%, because a large number of beads and microspheres were observed on the electrospun PBS products in SEM images at such a concentration in this study. 158

Controlling Numbers and Sizes of Beads in Electrospun Nanofibres SEM photographs of electrospun PBS products with different polymer concentrations are illustrated in Figures 7.1–7.5. When the solvent was 100% CF, the PBS electrospun products were like ‘beads on a string’ (see Figure 7.1). Microspheres were observed for the 100% DCM solvent (not shown). Bigger beads with plenty of microporosity appeared when the solvent was changed to the mixed solvent CHCl3/DCM (7:3). There are few single fibres but a lot of beads and microspheres under these solvent conditions. In Figure 7.2, the shape of the beads changed again and a number of bowl-shaped or spoon-shaped beads were produced as the solvent system changed. The fibres were in the majority in the electrospun products when the mixed solvent was CHCl3/chloroethanol (7:3) (see Figure 7.3), and the efficiency of the spinning was the best in our experiments. Furthermore, the products also showed plenty of beads or microspheres and few fibres for other mixed solvent systems such as CHCl3/IPA (9:1), DCM/IPA (9:1) and CHCl3/DCM/IPA (8:1:1) (not shown).

Figure 7.1 SEM photograph of PBS electrospun product. The solvent was CF at a concentration of 11 wt%, voltage 10 kV, diameter of needle orifice 0.9 mm and orifice-to-collector distance 10 cm. 159

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Figure 7.2 SEM photograph of PBS electrospun product. The solvent was the mixed solvent CHCl3/IPA (8:2) at a concentration of 11 wt%, voltage 10 kV, diameter of needle orifice 0.9 mm and orifice-to-collector distance 10 cm.

Figure 7.3 SEM photograph of PBS electrospun product. The solvent was the mixed solvent CHCl3/chloroethanol (7:3) at a concentration of 11 wt%, voltage 10 kV, diameter of needle orifice 0.9 mm and orifice-to-collector distance 10 cm.

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Figure 7.4 SEM photograph of PBS electrospun product. The concentration was 14 wt% in the mixed solvent CHCl3/ chloroethanol (7:3) at a voltage of 10 kV, diameter of needle orifice 0.9 mm and orifice-to-collector distance 10 cm.

Figure 7.5 SEM photograph of PBS electrospun product. The concentration was 17 wt% in the mixed solvent CHCl3/ chloroethanol (7:3) at a voltage of10 kV, diameter of needle orifice 0.9 mm and orifice-to-collector distance 10 cm. 161

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Table 7.1 Electrospun products in different solvents Polymer concentration (wt%)

Electrospun products

CHCl3

11

Beads and a few fibres

CHCl3/DCM (7:3)

11

Microspheres and a few fibres

CHCl3/IPA (8:2)

11

Spoon-shaped beads and fibres

CHCl3/chloroethanol (7:3)

11

Few beads and fibres

DCM/chloroethanol (5:5)*

15

Beaded fibres

CHCl3/1-CP (9:1)*

15

Beaded fibres

DCM/3-CP (9:1)*

15

Beaded fibres

Solvent (w/w)

DCM/1-CP (9:1)*

15

Beaded fibres

CHCl3/3-CP (9:1)*

15

Fibres

DCM/chloroethanol (7:3)*

15

Fibres

DCM/ chloroethanol (6:4)*

15

Fibres

*Jeong and co-workers experimental results [2] in different solvents for PBS (Number average molecular weight (Mn) = 75 000).

Our results and those of other researchers [2] in different solvents are listed in Table 7.1. From Table 7.1, we can see that the appropriate choice of solvent in the electrospinning process can produce fewer beads at the polymer concentration of 11 wt%. The electrospun PBS products were fibres with the three mixed solvent systems, CHCl3/3CP (9:1), DCM/chloroethanol (7:3) and DCM/chloroethanol (6:4), at the polymer concentration of 15 wt% in Jeong and co-workers experiments [2]. The possible reasons for the different morphologies with different solvent systems are complex. There are many possible factors influencing the morphologies in different solvents, such as 162

Controlling Numbers and Sizes of Beads in Electrospun Nanofibres volatilisation rate, solvent polarity, solution conductivity, surface tension coefficient, solution viscoelastic behaviour, molecular weight, chain entanglements, or ambient temperature [3–6]. This shows that the optimal solvent system could be chosen to decrease or even eliminate the by-products in the electrospinning process. Additionally, the solvent system was very important to the efficiency in the electrospinning process.

7.3 Effect of Polymer Concentration It is well known that polymer concentration is one of the most important parameters in the electrospinning process because the concentration influences the viscosity of the solution. On the other hand, the fabrication and the resulting fibre morphology are dependent on solution viscosity [7]. When the polymer concentration is low, the products are a lot of beads or microspheres because the process becomes electrospraying [7, 8]. Therefore, by increasing the polymer concentration it might be possible to decrease the numbers and sizes of beads and even to eliminate the beads as discussed previously. In order to confirm this guess, seven concentrations of PBS polymer, 11–17 wt%, were used in electrospinning under the same conditions. SEM images of the different concentrations are shown in Figures 7.3–7.5. The number of beads gradually diminished with higher polymer concentration (see Figure 7.4). There were no beads and microspheres in the electrospun products when the concentration equalled or exceeded 16 wt% (see Figure 7.5). The electrospun fibres produced were more uniform in these conditions. The reason for this might be that the beads are mainly caused by the surface tension, which minimises the surface area. In the case of no surface tension, the jets will break down into drops. Lower surface tension tends to form beads in the fibres. With the increase of polymer concentration, the surface tension becomes bigger, and the beads are fewer. The surface tensions of different polymer solutions and their products are listed in Table 7.2 and shown in Figure 7.6. 163

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Table 7.2 The surface tensions of different polymer solutions and their products Polymer concentration (wt%)

Surface tension (mN/m)

Electrospun products

13

32.51452

A lot of beads and fibres

14

32.92377

Beads and fibres

15

33.17413

Few beads and fibres

16

33.9906

Fibres

Figure 7.6 The relationship between solution concentration and surface tension.

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7.4 Effect of Salt Additive Fong and co-workers [5] thought that the net charge density carried by the electrospinning jet was another important factor that greatly influenced the morphology of the electrospun products besides the viscosity and the surface tension of the solution. Their results showed that the beads became smaller and the shape became more spindle-like as the net charge density increased. So we added a small amount of salt, LiCl, to the 14 wt% PBS/(CHCl3/chloroethanol = 7:3) solution. Other fixed parameters were mixed solvent CHCl3/chloroethanol (7:3), voltage 20 kV, diameter of needle orifice 0.7 mm, and orifice-to-collector distance 14 cm. The SEM images of these products were compared to those of the samples with no salt under the same parameters. From Figure 7.7 (solution with no salt), Figure 7.8 (solution with 0.5 wt% LiCl) and Figure 7.9 (solution with 1.0 wt% LiCl), we can see that the number of beads decreases with increasing amount of salt. The reason might be that the addition of salt led to more electric charges carried

Figure 7.7 SEM photograph of PBS electrospun product. The concentration was 14 wt% in the mixed solvent CHCl3/chloroethanol (7:3) at a voltage of 20 kV, diameter of needle orifice 0.7 mm and orifice-to-collector distance 14 cm.

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Figure 7.8 SEM photograph of PBS electrospun product. The concentration was 14 wt% with 0.5 wt% LiCl in the mixed solvent CHCl3/chloroethanol (7:3) at a voltage of 20 kV, diameter of needle orifice 0.7 mm and orifice-to-collector distance 14 cm.

Figure 7.9 SEM photograph of PBS electrospun product. The concentration was 14 wt% with 1 wt% LiCl in the mixed solvent CHCl3/chloroethanol (7:3) at a voltage of 20 kV, diameter of needle orifice 0.7 mm and orifice-to-collector distance 14 cm.

166

Controlling Numbers and Sizes of Beads in Electrospun Nanofibres by the electrospinning jet; thus higher elongation forces were imposed on the jet in the electric field [9]. As a result, the beads became fewer and more spindle-like as the charge density increased. The beads are mainly caused by the lower surface tension. With the higher surface tension, the size and number of beads in electrospun products are smaller and less numerous. The concentration of polymer solution and the inorganic additive have the typical effect of preventing the occurrence of beads in the electrospinning process. Another important parameter, the solvent, can affect the number and size of beads, and the morphology of the electrospun fibres as well.

References 1.

Y. Liu, J-H. He, J.Y. Yu and H.M. Zeng, Polymer International, 2008, 57, 4, 632.

2.

E.H. Jeong, S.S. Im and J.H. Youk, Polymer, 2005, 46, 23, 9538.

3.

W.W. Zuo, M.F. Zhu, W. Yang, H. Yu, Y.M. Chen and Y. Zhang, Polymer Engineering and Science, 2005, 45, 5, 704.

4.

T. Lin, H.X. Wang, H.M. Wang and X.G. Wang, Nanotechnology, 2004, 15, 9, 1375.

5.

H. Fong, I. Chun and D.H. Reneker, Polymer, 1999, 40, 16, 4585.

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S.L. Shenoy, W.D. Bates, H.L. Frisch and G.E. Wnek, Polymer, 2005, 46, 10, 3372.

7.

K.H. Lee, H.Y. Kim, H.J. Bang, Y.H. Jung and S.G. Lee, Polymer, 2003, 44, 14, 4029.

8.

C.M. Hsu and S. Shivkumar, Journal of Materials Science, 2004, 39, 9, 3003.

9.

X. Zong, K. Kim, D. Fang, S. Ran, B.S. Hsiao and B. Chu, Polymer, 2002, 43, 16, 4403. 167

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Electrospun Nanoporous Microspheres for Nanotechnology

Nanoporous microspheres were formed by electrospinning polybutylene succinate (PBS) solutions with an additive of Yunnan Baiyo, a traditional Chinese drug, and a mixed solvent of chloroform and isopropyl alcohol in a single processing step. The numbers and sizes of the electrospun nanoporous microspheres could be controlled by a tunable voltage and flow rate applied in the electrospinning process. The increase of voltage and the decrease of flow rate produced ever-increasing numbers and everdecreasing sizes of the nanoporous microspheres. The electrospun nanoporous microspheres offer the potential for direct fabrication of biologically based, high-surface-area porous materials without the use of multiple synthetic steps or post-processing surface treatments. In this chapter we will discuss a new approach to producing nanoporous microspheres by electrospinning. Because of their ultrahigh specific surface area, nanoporous structures, which are potentially of great technological interest for the development of electronic, catalytic and hydrogen-storage systems, invisibility devices, e.g., stealth plane, stealth clothes, and others, have attracted much attention recently. Pore structure and connectivity determine how microstructured materials perform in applications such as adsorption, separation, filtering, catalysis, fluid storage and transport, as electrode materials or as reactors. Farreaching implications are emerging for applications, including medical implants, cell supports, materials that can be used as instructive three-dimensional environments for tissue regeneration and others.

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8.1 Electrospun Nanoporous Spheres with a Traditional Chinese Drug In this section we use a kind of traditional Chinese drug called Yunnan Baiyo [1, 2] is used as an additive for possible application as a drug carrier. The chosen drug Yunnan Baiyo is a highly valued and important traditional Chinese drug. It is a kind of fawn powder mixture composed of several kinds of drugs. There are a wide variety of therapeutic uses of Yunnan Baiyo, including promotion of blood circulation, removal of blood stasis, anti-inflammatory action, haemostasis, induction of blood clotting, relief of swelling and alleviation of pain. It can also be used for the treatment of traumatic injury, spitting blood, haemoptysis, surgical bleeding, suppurative and pyogenic infections, soft tissue bruises, closed fracture and infective diseases on skin, etc. It has potential applications in wound dressings, i.e. haemostatic bandages. In the field of tissue engineering, Yunnan Baiyo can be added to scaffolding materials so that the material can restrain an inflammatory response. The base material for the electrospun fibres is PBS, a type of biodegradable material.

Experimental Materials PBS with a molecular weight of 20,000–30,000 g/mol was supplied by Shanghai Institute of Organic Chemistry, Chinese Academy of Science. Yunnan Baiyo, a traditional Chinese drug, was produced by Yunnan Baiyo Group Co. Ltd, and was used as an additive. Chloroform (CHCl3) and isopropyl alcohol were obtained from Shanghai Chemical Reagent Co. Ltd, China. A mixture of isopropyl alcohol and CHCl3 with the weight ratio 1:9 was used as the solvent. All materials were used without any further purification.

Instrumentation The electrospinning set-up consisted of a syringe, a needle, a grounded collecting plate, a flow meter and a variable DC high-voltage power supply (0–100 kV, F180-L, Shanghai Fudan High School). 170

Electrospun Nanoporous Microspheres for Nanotechnology Analysis of the chemical components was studied through infrared investigation. Fibre diameters, pore sizes and morphology images of films were analysed using a scanning electron microscope (SEM). Fibres for SEM analysis were collected on a steel mesh, mounted on a SEM disc and sputter-coated with platinum. Typical images were analysed under various conditions.

Electrospinning Process All concentration measurements were done as weight by weight (w/w). A control amount of PBS particles and Yunnan Baiyo powder were dissolved in the mixed solvent with the weight ratio 1:6:43 (Yunnan Baiyo:PBS:mixed solvent). The solution obtained was magnetically stirred at 25 °C for 4 hours in an electromagnetic stirrer (Angel Electronic Equipment (Shanghai) Co. Ltd) with a stirring speed of about 1000 rev/min. The solution was dropped into a 20 ml syringe, which was mounted in a syringe pump. A grounded metal mesh screen was placed vertically under the needle tip. The inner diameter of the needle orifice was 0.57 mm. The tip-to-collector distance was 10 cm. The applied voltages connected to the needle varied from 8 to 15 kV. The flow rate varied from 1.5 to 8 ml/h. All electrospinning processes were carried out at room temperature in a vertical spinning configuration.

IR Spectra Figure 8.1 displays the IR spectra for (a) pure PBS fibres and (b) the drug-containing electrospun fibres. In Figure 8.1(a), absorption peaks can be seen at about 2944, 1712, 1424 (1387), 1329 (1311), 1153, 1043, 985 (953) and 917 cm–1, assigned to the hydroxyl group (ν-oh), carboxyl group (ν-co) and benzene ring of pure PBS. In Figure 8.1(b), besides these peaks assigned to pure PBS, there are new peaks at about 1471 (1447), 1245 (1207) and 804 cm–1 corresponding to Yunnan Baiyo. 171

Electrospun Nanofibres and Their Applications

(a)

(b)

Figure 8.1 IR spectra for (a) pure PBS and (b) the drug-containing electrospun nanoporous microspheres and fibres. 172

Electrospun Nanoporous Microspheres for Nanotechnology

SEM Analysis Figures 8.2–8.4 show SEM pictures of the electrospun microspheres with different voltages applied in the electrospinning process.

Figure 8.2 SEM pictures of the electrospun nanoporous microspheres at a voltage of 8 kV.

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Figure 8.3 SEM pictures of the electrospun nanoporous microspheres at a voltage of 10 kV.

The flow rate was constant at 1.5 ml/h. It could be seen that the diameters of the microspheres ranged from 5 to 40 µm, and there appeared ever-increasing numbers and ever-decreasing sizes of the electrospun nanoporous microspheres with increase of voltage. 174

Electrospun Nanoporous Microspheres for Nanotechnology

Figure 8.4 SEM pictures of the electrospun nanoporous microspheres at a voltage of 15 kV.

Figures 8.5–8.8 show SEM images of the electrospun microspheres with different flow rates applied in the electrospinning process. The applied voltage was constant at 10 kV. It can be seen that the diameters of the microspheres ranged from 5 to 50 µm, and there 175

Electrospun Nanofibres and Their Applications appeared ever-decreasing numbers and ever-increasing sizes of the electrospun nanoporous microspheres with increase of flow rate.

Figure 8.5 SEM pictures of the electrospun nanoporous microspheres at a flow rate of 1.5 ml/h.

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Figure 8.6 SEM pictures of the electrospun nanoporous microspheres at a flow rate of 3 ml/h.

177

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Figure 8.7 SEM pictures of the electrospun nanoporous microspheres at a flow rate of 5 ml/h.

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Figure 8.8 SEM pictures of the electrospun nanoporous microspheres at a flow rate of 8 ml/h.

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Electrospun Nanofibres and Their Applications

8.2 Electrospinning Dilation During the electrospinning process, the charged jet is accelerated by a constant external electric field, and the spinning velocity probably reaches a maximum and perhaps exceeds the velocity of sound in air in a very short time before the spinning becomes unstable [3]. According to the mass conservation equation: (8.1) where r is the radius of the jet, u is the velocity, Q is the flow rate, and ρ is the density, the radius of the jet decreases with the increase of the velocity of the incompressible charged jet. Macromolecules of the polymers are compacted together tighter and tighter during the electrospinning process, as illustrated in Figure 8.9. There must exist a critical minimal radius rcr. For all electrospun jets r ≤ rcr for continuous ultrafine fibres, and the critical maximal velocity is: (8.2)

Figure 8.9 Macromolecular chains are compacted during the electrospinning process.

180

Electrospun Nanoporous Microspheres for Nanotechnology However, the velocity can exceed this critical value ucr if a higher voltage is applied and the distance between the needle and the collecting plate is infinitely long. When the radius of the jet reaches the critical value, r = rcr, and the jet speed exceeds its critical value, u > ucr, in order to satisfy the conservation of mass equation, the jet dilates by decreasing its density, leading to porosity of the electrospun fibres. This phenomenon is called ‘electrospinning dilation’ [1, 2, 8]. In the case of higher voltage, the charged jet can be more easily accelerated to the critical speed before the charged jet is collected. Higher voltage means higher value of the jet speed (u0) at r = rcr, and a more drastic electrospinning dilation process happens. This results in lower density ( ) of the dilated microspheres, smaller size (R) of the microspheres, and smaller pores as well – see Figure 8.10.

Figure 8.10 Effect of applied voltage on the diameter of the electrospun nanoporous microspheres.

181

Electrospun Nanofibres and Their Applications This phenomenon can be explained by the equation: (8.3) where is the density of the dilated microspheres, u0 is the velocity of the charged jet at r = rcr, R is the maximal radius of the microspheres, and umin is the minimal velocity. According to Equation (8.2), the higher the flow rate, the higher the critical speed. This means that electrospinning dilation occurs easily when the flow rate is relatively low – see Figure 8.11. In conclusion, we suggest a general strategy for the synthesis of microspheres with nanoporosity by electrospinning. Pores having uniform but tunable diameters can be achieved by controlling the

Figure 8.11 Effect of flow rate on the diameter of the electrospun nanoporous microspheres.

182

Electrospun Nanoporous Microspheres for Nanotechnology applied voltage or the flow rate in the electrospinning process. The flexibility and adaptation provided by the method have made it a strong candidate for producing nanoporous materials.

8.3 Single Nanoporous Fibres by Electrospinning A single nanoporous fibre with an ultrahigh specific surface area was formed by electrospinning PBS/CHCl3 solution in a single processing step [4, 5]. As the concentration of PBS/CHCl3 increased, the morphology evolved from beads only to ultrafine continuous fibres. At a suitable concentration, PBS/CHCl3 13%, uniform nanoporous threads of polymers are emitted from the jet in the electrospinning process. The electrospun nanoporous fibres were fabricated with an average pore diameter of about 200 nm. The electrospun PBS/CHCl3 fibres offer the potential for direct fabrication of biologically based, high-surface-area porous fibres without the use of multiple synthetic steps, complicated electrospinning designs, or post-processing surface treatments.

Experimental Materials PBS with a molecular weight of 20,000–30,000 g/mol was supplied by Shanghai Institute of Organic Chemistry, Chinese Academy of Science. CHCl3 was obtained from a commercial source and used as solvent without any further purification.

Instrumentation The scheme of the electrospinning process is described in Section 8.1. The needle tip was connected to a DC high-voltage supply (F180-L, Shanghai Fudan High School) via an alligator clip. Fibre diameter and morphology images of PBS/CHCl3 films were analysed using a scanning electron microscope. Fibres for SEM analysis were collected on a steel mesh, mounted on a SEM disc 183

Electrospun Nanofibres and Their Applications and sputter-coated with platinum. Typical images were analysed under various conditions

Electrospinning Process All the concentration measurements were done as weight by weight (w/w). PBS/CF solutions at various polymer concentrations (12%, 13% or 14%) were prepared by dissolving PBS grains in CHCl3 and stirring for 2 hours at 30 °C with an electromagnetic stirrer (Angel Electronic Equipment (Shanghai) Co. Ltd), at a stirring speed from 950 to 1200 rev/min, then cooling to room temperature. The PBS/ CHCl3 solution was placed in a 20 ml syringe, which was mounted in a syringe pump. A grounded metal mesh screen was placed vertically under the needle tip. The tip-to-collector plate distance varied from 5 to 10 cm. The voltage varied from 1.3 to 2.0 kV. In order to obtain continuous PBS fibres, various PBS/CHCl3 concentrations were tested. When the concentration is less than 8%, the solution is too dilute to be electrospun, and only beads can be obtained. When the concentration reaches 12%, continuous PBS fibres with partly porous fibres can be electrospun (see Figure 8.12). The optimal concentration for nanoporous fibres is 13% (see Figure 8.13), where the average pore diameter is about 200 nm. With increasing concentration, the porous morphology is observed more clearly and the diameters of the nanoscale holes are more uniform (see Figures 8.14 and 8.15, where the concentration is 14%). When the concentration exceeds 15%, the solvent evaporates too fast so the PBS solution suffers from gelation and the gel blocks the needle of the syringe.

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Figure 8.12 The porous morphology of PBS fibres at a concentration of 12%, voltage of 1.5 kV, and flow rate of 4.5 ml/h.

Figure 8.13 The porous morphology of PBS fibres at a concentration of 13%, voltage of 1.5 kV, and flow rate of 3 ml/h. The average pore diameter is about 200 nm.

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Figure 8.14 The porous morphology of PBS fibres at a concentration of 14%, and voltage of 2 kV; the flow rate is adjusted artificially.

Figure 8.15 The porous morphology of PBS fibres at a concentration of 14%, and voltage of 2 kV; the flow rate is adjusted artificially. The average pore diameter is about 300 nm.

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8.4 Microspheres with Nanoporosity Microspheres with nanoporosity [6] are prepared by PBS mixed with CHCl3.

Experimental Materials PBS (provided by Shanghai Institute of Organic Chemistry, Chinese Academy of Science) was used as received without further purification. The molecular weight is about 200,000–300,000 g/mol. The other solvent system in this study was CHCl3 (Shanghai Chemical Reagent Co. Ltd). The polymer was dissolved in CHCl3 solvent at 40 °C with 2 hours of stirring in an electromagnetic stirrer (Angel Electronic Equipment (Shanghai) Co. Ltd) and cooled to room temperature before electrospinning. The weight concentrations were adjusted to 8, 10 or 12 wt%.

Electrospinning Process The electrospinning set-up with a variable DC high-voltage power generator (0–100 kV, F180-L, Shanghai Fudan High School) was used in this study. The polymer solution was placed into a 20 ml syringe attached to a syringe pump. All fibre spinning was carried out at room temperature in a vertical spinning configuration. The orifices of the needles were 0.5, 0.7 and 0.9 mm, and the spinning distance between the orifice and the collector was 5 cm. The applied voltage was 10 kV connected to the needle from the high-voltage power supply via an alligator clip. SEM photographs of the resulting PBS electrospun fibres at concentrations of 10 and 8 wt% and a voltage of 10 kV are shown in Figures 8.16 and 8.17, respectively. 187

Electrospun Nanofibres and Their Applications

Figure 8.16 SEM photograph of PBS electrospun fibres. The concentration and voltage are 10 wt% and 10 kV. The diameter of the inner needle orifice is 0.5 mm. Spherical beads with nanoporosity are formed.

Figure 8.17 SEM photograph of PBS electrospun fibres. The concentration and voltage are 8 wt% and 10 kV. The diameter of the inner needle orifice is 0.5 mm. Microspheres with nanoporosity are formed.

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8.5 Microcomposite Fibres by Electrospinning This new method for producing microcomposite fibres using electrospinning is described in detail elsewhere [7]. Figure 8.18 shows the core–shell structured polyacrylonitrile/polyvinylpyrrolidone (PAN/PVP) fibres produced. These microcomposites show great potential applications as tissue engineering scaffolds, for wound dressings, in drug release systems and for sound absorption, and they will no doubt attract a good deal of attention in the future. The mathematical modelling and theoretical analysis of the method are worthy of further study.

Figure 8.18 Optical microscopy images of PAN/PVP composite coaxial microfibres.

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J-H. He, L. Xu, Y. Wu and Y. Liu, Polymer International, 2007, 56, 11, 1323.

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L. Xu, J-H. He and Y. Liu, International Journal of Nonlinear Sciences and Numerical Simulation, 2007, 8, 2, 199.

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J-H. He, Y.Q. Wan and L. Xu, Chaos, Solitons & Fractals, 2007, 33, 1, 26.

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Y. Wu, J.Y. Yu and C. Ma, Textile Research Journal, 2008, 78, 9, 812.

6.

J-H. He, Y. Liu, L. Xu and J.Y. Yu, Chaos, Solitons & Fractals, 2007, 32, 3, 1096.

7.

D-H. Shou and J-H. He in Proceedings of the 2007 International Symposium on Nonlinear Dynamics (2007 ISND), Shanghai, China Journal of Physics: Conference Series, 2008, 96, 012213.

8.

D-H. Shou and J-H. He, Journal of Polymer Engineering, 2008, 28, 1–2, 115.

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9

A Hierarchy of Motion in the Electrospinning Process and E-Infinity Nanotechnology

In this chapter we will show that nanotechnology could be seen as a link between deterministic classical mechanics and indeterministic quantum mechanics, and that E-infinity theory could help in understanding various nano-effects. We also predict that the Hausdorff dimension for the hierarchy of motion should be close to the average Hausdorff dimension of E-infinity space–time, and that the characteristics of electrospun nanofibres depend mainly upon the Hausdorff fractal dimension. An experiment is performed to verify our predictions. In view of E-infinity theory [1, 2], processes at the nanoscale may possess entirely new physical and chemical characteristics, which result in properties that are well described neither by those of a single elementary particle of the substance, nor by those of the bulk material. At the nanoscale, quantum-like phenomena occur [3].

9.1 E-Infinity Nanotechnology Nanotechnology links both deterministic classical mechanics and chaotic quantum mechanics. Of course, there ought to be a law controlling the change from a classical object like a stone to a quantum object like an electron. Somewhere between the two scales, these changes happen. However, the changes do not happen abruptly. There is a grey area between these two scales that is neither classical nor quantum. Since E-infinity theory is valid for all scales, it follows then that it can deal with this grey area. In recent years there has been a flurry of original papers published on the foundation and application of E-infinity Cantorian space–time

191

Electrospun Nanofibres and Their Applications theory. Various applications of this theory show impressive exactness, especially in predicting the theoretical coupling constants and the mass spectrum of the standard model of elementary particles. In this paper we will apply E-infinity theory to a problem of immense technical importance, namely, electrospun nanofibres. In the theory of n-dimensional space, what we mean by n dimensions is simply that we need n numbers to represent n coordinates to fix the position of a point in this space. In our classical space–time, these are the familiar triplet x, y, z; while in relativity, we have a fourth coordinate or dimension, namely the time t. The formal dimension in E-infinity theory [1, 2], however, is: (9.1) The physical interpretation of Equation (9.1) in nanofibres will be discussed later. The topological dimension in E-infinity theory [1, 2] is: (9.2) The dimension 3 + 1 = 4 means that, in a tangible visible sense, the charged jet in the electrospinning process appears to us as if it were topologically four-dimensional, and the Navier–Stokes equations can be approximately applied [4]. Next we write down something different, namely the average Hausdorff dimension in E-infinity theory [1, 2]: (9.3)

is the golden mean, which plays an important where role in E-infinity theory. For example, the mass of an expectation MeV, and the absolute zero temperature π-meson is . can be simply derived as 192

A Hierarchy of Motion in the Electrospinning Process …

9.2 Application of E-Infinity Theory to Electrospinning There is more about this topic in [5]. Using an electronic camera, Reneker and co-workers [6] recorded up to 2000 frames per second with exposure times as short as 0.0125 ms, and found a secondary instability of the electrospinning process – see illustrations in [6]. Our experiment showed a hierarchy of motion in the electrospinning process (see Figure 9.1). Hierarchy is one of the essential characteristics of E-infinity theory. Now we consider the first circle motion with unit diameter (see Figure 9.1). A second circle motion occurs during the electrospinning process, and then a third circle motion, a fourth circle motion, and so on – see Figure 9.1. The diameter of the second circle is , and the diameter of the nth circle is . An infinite number of circles could be produced if the electrospinning process could be infinite, i.e., the distance between the electrospinning nozzle and the collector of the electrospun nanofibres is infinitely long.

Figure 9.1 Hierarchy of motion in the electrospinning process. 193

Electrospun Nanofibres and Their Applications We sum an infinite number of Hausdorff fractal dimensions of an infinite number of one-dimensional Cantor sets, with Hausdorff dimensions , where n = 0, 1, 2, … . Consequently we have [1, 2]: (9.4)

Here we assume the diameter of the first circle to be equal to 1, and is the diameter of the nth circle. The expectation value of this sum in terms of is [1, 2]: (9.5) This formulation implies that DF = ∞ in the electrospinning process. This is because we have been summing over n = 0, 1, … , ∞ and this is the formal dimension of E-infinity theory.

9.2.1 Hausdorff Dimension for the Hierarchy of Motion Fluid mechanics is normally concerned with the behaviour of matter in the large on a macroscopic scale. If the diameter of the charged jet were large enough, say 10 mm or larger, the continuum hypothesis in fluid dynamics should still be valid, and the Navier–Stokes equations could be approximately applied [4], as if it were four-dimensional: for large scale.

(9.6)

However, we have to take into account the macromolecular structure of the charged jet in the electrospinning process when the jet is thin enough on the nanoscale. At such a small scale, the (3 + 1)-dimensional Navier–Stokes equations become invalid [4]. Modified Navier–Stokes equations in 4 + 1 Hausdorff dimensions are suggested for turbulence using E-infinity theory [4]. The smaller 194

A Hierarchy of Motion in the Electrospinning Process … the scale, the higher the Hausdorff dimension that is required. In the case when the scale tends to zero, we predict: for scale tending to zero.

(9.7)

This is the formal dimension of E-infinity [1, 2]. When the scale tends to several hundreds of nanometres, which is should very large compared with the quantum scale, the value of be smaller than the five dimensions needed in Kaluza–Klein theory. should be very close to the average We, therefore, predict that Hausdorff dimension in E-infinity theory: for nanoscale.

(9.8)

9.2.2 Experimental Verification In order to verify the previous prediction, was an experiment designed using polyacrylonitrile/dimethylformamide (PAN/DMF) solution. The polymer was dissolved in DMF solvent and the weight concentration was adjusted to 12 wt%. The electrospinning set-up with variable DC high-voltage power generator was used in this study. The electrospinning process was performed using the PAN/DMF solution. The solution was placed into a 20 ml plastic syringe with a stainless-steel needle. The spinning process was carried out at room temperature in a vertical spinning configuration. The orifice of the needle was 0.9 mm and the spinning distance between the orifice and the collector was about 6 cm. The applied voltage was 20 kV connected to the needle from the highvoltage power supply via an alligator clip. The flow rate of the syringe pump was constant, 2.0 ml/h. The morphology and diameter of the PAN nanofibres and their circle motions in this experiment, were determined by a scanning electron microscope (SEM) and an optical microscope, respectively. The 195

Electrospun Nanofibres and Their Applications samples were collected on an SEM disc and coated with gold before being observed through the SEM. An SEM photograph is illustrated in Figure 9.2; and optical microscope photographs for various circle motions in the collected products are values are calculated. given in Figure 9.3, where the approximate is According to these optical micrographs, the average value of about 4.3, which is very close to the average Hausdorff dimension of E-infinity theory [1–4], DH = 4.236 06, revealing the remarkable connection and accuracy of the theoretical prediction [1, 2]. The diameter of the primary circle is about 0.1 m, and the average diameter of the obtained nth circle motion in Figure 9.2 is about 16 µm. We, therefore, have:

Figure 9.2 SEM photograph of the circle motion in the electrospinning process.

196

A Hierarchy of Motion in the Electrospinning Process … According to E-infinity theory, we predict a Hausdorff dimension for the hierarchy of motion to be larger than four. The exact value of differs for various polymer solutions and various electrospinning parameters, e.g., applied voltage and flow rate, and it tends on average to 4 + φ3 = 4.23606. In the case n + 1 = 8, we obtain = 4.5; and if n + 1 = 9, we have = 4.25. Thus the nanofibres obtained in Figure 9.2 were the result of at least ninth circle motion. A high Hausdorff dimension larger than four is an indication of a complex process during electrospinning.

(a) Figure 9.3 Optical microscope photographs for various circle motions in electrospinning: (a) eighth circle motion, with average diameter about 0.024 µm, r9/r8 = 0.666, = 4.5; (b) seventh circle motion, with average diameter about 0.06 µm, r8/r7 = 0.4, = 4.166; (c) sixth circle motion, with average diameter about 0.10 µm, r7/r6 = 0.6, = 4.166; and (d) fifth circle motion, with average diameter about 0.15 µm, r6/r5 = 0.666, = 4.5. 197

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(b)

(c)

Figure 9.3 Continued

198

A Hierarchy of Motion in the Electrospinning Process …

(d) Figure 9.3 Continued

9.3 Super Carbon Nanotubes: An E-Infinity Approach Super carbon nanotubes have wide potential applications such as field-emission displays, high-strength composites, hydrogen storage, and nanometre-sized semiconductor devices [7–9]. These materials with hierarchical geometrical structures always display quantum-like properties and have many fascinating nano-effects, such as a remarkable increase in strength, high surface energy and surface reactivity, excellent thermal and electrical conductivity, and extraordinarily fast flow in nanotubes. However, the precise role of these intricate phenomena is unknown. Recently Coluci and co-workers [7, 8] proposed a new structure for the so-called super carbon nanotubes, which can be either metallic or semiconducting prototypes for electromechanical actuators, serve as hosts for large biomolecules, and have a wide range of potential 199

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Figure 9.4 The hierarchy of a super carbon nanotube. E-infinity theory could explain the nano-effects, including unusual strength, high surface energy, high surface reactivity, and high thermal as well as electrical conductivity, which depend strongly upon the hierarchical structure of the super carbon nanotubes and especially upon the ratio of rn+1/rn.

applications. Here we will show that the characteristics of super carbon nanotubes depend mainly on their hierarchical structure, as illustrated in Figure 9.4. Let us following [10], again sum up an infinite number of intersecting Hausdorff fractal dimensions of an infinite number of one-dimensional Cantor sets, with Hausdorff dimension , where n = 0, 1, 2, … . As stated earlier, we obtain (9.9) We assume that the radius of the mother cylinder in Figure 9.4 is equal is the radius of the nth child cylinder. As before, to one, and is: the expectation value of this sum in terms of 200

A Hierarchy of Motion in the Electrospinning Process …

(9.10) This formula already implies that DF = ∞ in super carbon nanotubes. This is because we have been summing over n = 0, 1, … , ∞, and this is the formal dimension of E-infinity theory. In the case of = 1/2, we have = 4. This means that, when = rn+1/rn = 1/2, where rn and rn+1 are the radii of the nth and (n+1) th carbon cylinders, respectively, the super carbon nanotube will behave like those in four-dimensional space–time. More explicitly, super carbon nanotubes with = 4, have the same characteristics as those of the single-walled carbon nanotube from which they are built. For > 1/2, we can easily find that the average Hausdorff dimension is larger than four; and for , we have a space–time filling structure with . Such a super carbon nanotube displays quantum-like properties and has many fascinating nano-effects, such as a remarkable increase in strength, higher surface energy and surface reactivity, as well as better thermal and electrical conductivity. We see that the preceding Cantor set based on a hierarchical geometrical picture could help in understanding these nano-effects. We have proposed an E-infinity nano-model for super carbon nanotubes that deals for the first time with the seemingly complex properties of critical importance in nanotechnology. Particularly, this will be of great practical importance for specialists in designing, manufacturing and using nanotubes. This E-infinity model is able to give a complete theoretical description of a complex dynamic process. Of course, the authors understand that, no matter how rigorous, further experimental verifications are needed to validate the model. We hope the present study will stimulate further experimental studies to verify our prediction that the main characteristics depend upon the Hausdorff dimension of the hierarchy structure, and E-infinity theory might turn out to have more important ramifications in nanotechnology than hitherto believed possible. 201

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References 1.

M.S. El Naschie, International Journal of Nonlinear Sciences and Numerical Simulation, 2007, 8, 1, 11.

2.

M.S. El Naschie, International Journal of Nonlinear Sciences and Numerical Simulation, 2007, 8, 1, 5.

3.

J-H. He, Y.Q. Wan and L. Xu, Chaos, Solitons & Fractals, 2007, 33, 1, 26.

4.

J-H. He, Chaos, Solitons & Fractals, 2006, 30, 2, 506.

5.

J-H. He and Y. Liu, Journal of Polymer Engineering, 2008, 28, 1–2, 101.

6.

D.H. Reneker, A.L. Yarin, H. Fong and S. Koombhongse, Journal of Applied Physics, 2000, 87, 9, 4531.

7.

R. Coluci, D.S. Galvao and A. Jorio, Nanotechnology, 2006, 17, 3, 617.

8.

V.R. Coluci, S.O. Dantas, A. Jorio and D.S. Galvão, Physical Review B, 2007, 75, 075417.

9.

M. Wang, X.M. Qiu and X. Zhang, Nanotechnology, 2007, 18, 075711.

10. J-H. He and L. Xu, International Journal of Electrospun Nanofibres and Applications, 2007, 1, 3, 161.

202

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Mechanics in Nanotextile Science

This chapter introduces some mathematical models arising in textile engineering, especially in nanotextile science [1].

10.1 Jet Vortex Spinning and Cyclone Model Air vortex spinning technology has been applied successfully to commercial wrapped (fascinated) yarn due to its high productivity (600 m/min, a productivity rate 20 times that of ring spinning and 2.5 times that of rotor spinning), low hairiness, and wide spinnability – see Figure 10.1.

Figure 10.1 Schematic diagram of the air vortex twisting chamber.

203

Electrospun Nanofibres and Their Applications We can use the cyclone model to improve its effectiveness. In weather terms, cyclones need warm tropical oceans, moisture and light winds above them. The mechanism is illustrated in Figure 10.2. We can use this mechanism to create a novel cyclone spinning for fibres, as illustrated in Figure 10.3.

Figure 10.2 Cyclone model.

Figure 10.3 Cyclone spinning. 204

Mechanics in Nanotextile Science The tangential velocity distribution vt in the radial direction is assumed to obey a law of the form: (10.1) where C is a constant, r is the radius and the exponent n depends on r. The equilibrium between centrifugal force and radial pressure gradient can be written in the form: (10.2)

10.2 Two-Phase Flow of Yarn Motion in High-Speed Air and Micropolar Model The dynamics of fibres in a fluid are important to understanding many modern problems arising in engineering, biology and other areas. Fibre motion in high-speed air differs dramatically from general two-phase flows. Considering the volume element illustrated in Figure 10.4, due to the work done by the fibre that is involved in

Figure 10.4 A volume element involving a moving fibre, such that the symmetry of the stress tension Navier–Stokes model is broken.

205

Electrospun Nanofibres and Their Applications the element, the symmetry of the stress tensor Navier–Stokes model is broken: (10.3) So the Navier–Stokes equations become invalid for textile fluid mechanics. We can, however, decompose each fibre into a series of discrete particles, as illustrated in Figure 10.5. We can therefore use the twophase model to describe the motion of fibres under the following constraints: (1) geometry constraint, i.e., lij = constant, where lij is the distance between two adjoining particles; and (2) force constraint, i.e., fij = –fji, which is obtained from Newton’s third law. The micropolar fluid model is an essential generalisation of the wellestablished Navier–Stokes model in the sense that it takes into account the microstructure of the fluid. Micropolar fluids [2, 3] are fluids with microstructure. They belong to a class of fluids with non-symmetric

Figure 10.5 Two-phase model for fibre motion in high-speed air. 206

Mechanics in Nanotextile Science stress tensor and are called polar fluids. The polar fluids, when the coupled stress is not taken into account, reduce to the well-known Navier–Stokes model of classical fluids. Physically, micropolar fluids represent such fluids consisting of rigid, randomly oriented (or spherical) particles suspended in a viscous medium, where the deformation of fluid particles is ignored. The general governing equations can be obtained by taking account of the balances of mass, momentum and angular momentum, as follows. (1) Equation of continuity: (10.4a) So it follows that: (10.4b) (2) Momentum equations: (10.5a) where tn is the normal stress, tn = n·T. By the transport theorem and Green’s theorem, we have: (10.5b) (3) Conservation of angular momentum: (10.6a) where r = xi + yj + zk. By Green’s theorem we have: 207

Electrospun Nanofibres and Their Applications

(10.6b) Equation (10.6a) reduces to the following equation: (10.6c) For ordinary fluids, Tij = Tji holds. In such a case, the right-hand side of Equation (10.6c) vanishes completely, leading to the well-known Navier–Stokes model. In the case of textile fluid mechanics, we have. By introducing microrotation ω and coupled stress tensor C, we can obtain the following equation [2]: (10.7) where I is the micro-inertia coefficient, g is the body torque per unit mass, and σjk = εijkTjk. For incompressible micropolar flow, the previous equations can be reduced to: (10.8) (10.9)

(10.10) where u = (u1, u2, u3) is the velocity field, P is the pressure, and ω = (ω1, ω2, ω3) is the microrotation field, interpreted as the angular velocity field of rotation of the particles. The fields f = (f1, f2, f3) and g = (g1, g2, g3) are external forces and moments, respectively. 208

Mechanics in Nanotextile Science The constants ν and νr are the usual Newtonian viscosity and microrotation viscosity, respectively, and the constants c0, cd, ca are viscosity coefficients related to microfluids. These parameters can describe well the effects of fibre orientation and fibre motion in flowing suspensions in air flow.

10.3 Mathematical Model for Yarn Motion in a Tube This topic is covered in more detail in [4]. The mathematical model of high-speed yarn transport systems has played and continues to play an important role in enhancing our understanding of the movement of yarn in an air-jet insertion, which is a complex motion, and is not a positively controlled process. Once the yarn is released from the clamp, there is little, if anything, that can be done to control its movement along the insertion length. In this section, we derive the following governing equations for air motion. (1) Continuity equation: (10.11) (2) Momentum equation: (10.12) where A is area of the tube, a is the area of the yarn, V is the air velocity, U is the yarn velocity, and τ is the shear stress. The area of the yarn in a cross-section depends upon the air pressure, and we suggest that: (10.13) where β is a constant. 209

Electrospun Nanofibres and Their Applications (3) Motion of yarn: (10.14)

(10.15) where ρ0 is the density of the yarn, and σ is the tension of the yarn. Now we suggest a yarn sliver flow model. Using high drafting on a sliver-to-yarn ring spinning frame requires some critical conditions. In order to understand what occurs during this process, a mathematical model is needed to determine the most influential parameters and their interactions. As a preliminary study of the problem, we consider the ideal case as illustrated in Figure 10.6, where yarn slivers enter into the funnel

Figure 10.6 Yarn slivers passing through a funnel condenser.

210

Mechanics in Nanotextile Science condenser with speed v0 and yarn stress P0, which become v1 and P1, respectively, after the funnel condenser. In order to calculate the force of resistance of the inner surface of the funnel condenser acting on the yarn sliver, we use the Euler approach in fluid mechanics to formulating the needed equations. We assume that the trajectories of the yarn slivers are pathlines of an imaginary flow passing through the funnel condenser. The imaginary flow is assumed to be a steady inviscid compressible homentropic one. Accordingly we have the following equations: (1) Continuity equation: (10.16) (2) Homentropic equation: (10.17) (3) Bernoulli equation: (10.18) Here P is the actual pressure acting on the yarn silver, ρ is an artificially introduced density, k is an artificial specific heat ratio, and B is the Bernoulli constant. The subscripts ‘0’ and ‘1’ denote before and after the funnel condenser, respectively. Substituting Equation (10.17) into Equation (10.18), we can identify the artificial specific heat ratio k from the following relationship: (10.19) where:

211

Electrospun Nanofibres and Their Applications After identification of the artificial specific heat ratio k, the Bernoulli constant B is determined. In view of Equation (10.17), we have the following equation for the pressure at the inner surface of the funnel condenser: (10.20) The total force of resistance of the inner surface of the funnel condenser can be easily calculated as: (10.21) where µ is the coefficient of resistance between the sliver and the funnel condenser, and L is the funnel length illustrated in Figure 10.6.

10.4 Nanohydrodynamics Nanotechnology is defined as a technology applied in the grey area between classical mechanics and quantum mechanics. Classical mechanics is the mechanics governing the motion of all the objects we can see with our naked eye. This is mechanics that obey deterministic laws (Newton’s laws) and that we can control to a very great extent. By contrast, quantum mechanics, which is the mechanics controlling the motion of things like the electron, the proton, the neutron and the like, is completely probabilistic. There is more about the subject matter of this section in [5, 6]. Nanotechnology links both deterministic classical mechanics and chaotic quantum mechanics. There ought to be a law controlling the change from a classical object like a stone to a quantum object like an electron. Somewhere between these two scales these changes happen, but this does not happen suddenly. There is a grey area between these two scales which is neither classical nor quantum. 212

Mechanics in Nanotextile Science Nanoscale structures that could mimic the selective transport and extraordinarily fast flow in biological cellular channels would have a wide range of potential applications. Majumder and co-workers. [7] found that liquid flow through a membrane composed of an array of aligned carbon nanotubes is 4–5 orders of magnitude faster than would be predicted from conventional fluid flow theory, and similar phenomena have been observed by other researchers. Why does the fluid in nanotubes flow extraordinarily fast? We will give a heuristic explanation. The governing equations for continuous media are well established by using the Gauss divergence theorems: (10.22)

(10.23)

(10.24) Mathematically, these Gauss divergence theorems are invalid for fractal media, such as porous media and weaves. So the governing equations for non-continuous media should be derived by a fractal approach, or by an allometric approach or fractional derivatives. Consider a flow whose density, ρ, varies in time and space, where the spatial variation is restricted to one spatial variable x. This situation is illustrated in Figure 10.7, where the flow moves in a long, thin porous tube with constant cross-sectional area A. (1) Mass conservation. In any fixed region R along the tube, mass conservation can be expressed as: (10.25)

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Electrospun Nanofibres and Their Applications

Figure 10.7 Conservation in a one-dimensional porous medium.

where a is the total pore area in the cross-section. The relationship between pore area a and the cross-sectional area A can be expressed as: (10.26) where D is the fractal dimension of pores in the section, and ξ is a function of D, satisfying ξ(2) = 1 and ξ(0) = 0. When D = 2, it turns out to be a continuous medium; and the case D = 0 arises when there are no pores in the cross-section so that no flow can pass through it. So the equation of mass conservation can be expressed as: (10.27) (2) Darcy’s law. If the porous medium has a structure that is statistically isotropic, so that a pressure gradient applied in different directions produces the same flux, we may write: (10.28) The above relationship is known as Darcy’s law. If we consider the fractal character of the section, Darcy’s law can be modified as follows: 214

Mechanics in Nanotextile Science

(10.29) where Di is the fractal dimension of each section. The modified Darcy’s law should be valid for all porous media. (3) Fourier’s equation for heat conduction reads: (10.30) where q is the heat flux, and T is the temperature. Its modification for a non-continuous medium can be expressed as: (10.31) (4) The reaction–diffusion equation: (10.32) can be modified as: (10.33) where c is the concentration of a chemical species, J is the flux, f is the reaction term, and A is the cross-sectional area. We write the conservation of mass in a continuous medium in the form: (10.34) where Q is the flow rate. For a continuous medium, we have: (10.35)

215

Electrospun Nanofibres and Their Applications For nanoscale hydrodynamics, for example, the flow in carbon nanotubes, the perimeter of a section is of fractal character: (10.36) where D is the fractal dimension of the perimeter. We rewrite Equation (10.34) in the form: (10.37) According to the fractal integral suggested in [8], Equation (10.37) can be approximately calculated as: (10.38) So the velocity in discontinuous carbon nanotubes can be expressed as: (10.39) Comparing Equation (10.39) with Equation (10.35), we find that: (10.40) When D = 1, this admits uE = u, so Equation (10.40) becomes: (10.41) Equation (10.41) is approximately valid. We write it as a scaling relationship: (10.42)

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Figure 10.8 Fractal boundary of a carbon nanotube.

Consider the case illustrated in Figure 10.8. The fractal dimension, D, can be calculated as: (10.43) Considering a nanotube with a radius of 10 nm, we have: (10.44) which agrees with Majumder and co-workers’ experimental observation [7]. Now we consider the Hagen–Poiseuille equation in a continuous medium, which reads: (10.45) where ∆P is the pressure difference between the two ends, and L is the length of the continuous tube. 217

Electrospun Nanofibres and Their Applications For nanohydrodynamics, we predict that: (10.46) where d is the fractal dimension of the longitudinal length. For carbon nanotubes (Figure 10.8), we consider the case: (10.47) We therefore predict that: (10.48) Experimental verification of our prediction is very much needed.

10.5 A New Resistance Formulation for Carbon Nanotubes and Nerve Fibres This topic is covered in more detail in [9]. Sundqvist and co-workers found that the resistance of carbon nanotubes does not follow the same behaviour as that of a metal conductor [10]. We know from Ohm’s law that current flows down a voltage gradient in proportion to the resistance in the circuit. Current is therefore expressed as: (10.49a) where I is the current, E is the voltage, R is the resistance, and g is the conductance. The resistance, R, in Equation (10.49a) is expressed in the form: (10.49b)

218

Mechanics in Nanotextile Science where A is the cross-sectional area of the conductor, L is its length, and k is a resistance parameter. Actually Equation (10.49b) is valid only for metal conductors where there are plenty of electrons in the conductor. For non-conductors, we suggest the following scaling relationship [11–14]: (10.50) As above, for carbon nanotubes, we consider the case: (10.51) and our prediction reads: (10.52)

Figure 10.9 Sundqvist and co-workers’ [10] experiment, showing resistance (kΩ) versus length (µm) for single-walled carbon nanotubes, and our prediction. 219

Electrospun Nanofibres and Their Applications It is obvious that R = 0 when L = 0. However, in Sundqvist and coworkers’ experiment [10] they found that R(0) = 50 kΩ. This is the error due to the contact resistance at the tip, so this initial error should be taken away from every obtained data point. The experimental data so obtained are illustrated in Figure 10.9. We find that our prediction agrees well with the experimental data.

10.6 Differential–Difference Model for Nanotechnology Further details about this are given in [15]. For a continuum, many governing equations can be expressed in the form: (10.53) This equation is invalid for nanotechnology, e.g., heat conduction, electric current and flow in carbon nanotubes. We modify Equation (10.53) in the following differential–difference form: (10.54) where the subscript n represents the nth lattice. Considering nonlinear effects, we have: (10.55) or, in a more general form: (10.56) where αk are constants. 220

Mechanics in Nanotextile Science Considering the inertia effect, we have:

(10.57) or, in a more general form: (10.58)

Considering the viscosity, we have: (10.59)

Some modifications of these equations are listed next: (10.60)

(10.61)

(10.62) (10.63)

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References 1.

J-H. He, International Journal of Modern Physics B, 2008, 20, 21.

2.

G. Lukaszewicz, Micropolar Fluids: Theory and Applications, Birkhäuser, Boston, MA, USA, 1999.

3.

J-H. He, Journal of Hydrodynamics B, 2003, 15, 3, 119.

4.

J-H. He in Proceedings of the Fourth International Conference on Fluid Mechanics, Eds., F. Zhuang and J. Li, Springer-Verlag, Berlin, Germany, 2004.

5.

J-H. He, International Journal of Electrospun Nanofibers and Applications, 2008, 2, 1, 23.

6.

J-H. He, Journal of Animal and Veterinary Advances, 2008, 7, 2, 207.

7.

M. Majumder, N. Chopra, R. Andrews and B.J. Hinds, Nature, 2005, 438, 7064, 44.

8.

J-H. He, Chaos, Solitons & Fractals, 2008, 36, 3, 542.

9.

J-H. He in Proceedings of the 2007 International Symposium on Nonlinear Dynamics (2007 ISND), Journal of Physics: Conference Series, 2008, 96, 012218.

10. P. Sundqvist, F.J. Garcia-Vidal, F. Flores, M. MorenoMoreno, C. Gomez-Navarro, J.S. Bunch and J. GomezHerreo, Nano Letters, 2007, 7, 9, 2568. 11. J-H. He, Neuroscience Letters, 2005, 373, 1, 48. 12. J-H. He and X.H. Wu, Neurocomputing, 2005, 64, 543. 13. J-H. He, Polymer, 2004, 45, 26, 9067. 222

Mechanics in Nanotextile Science 14. J-H. He, L. Xu, Y. Wu and Y. Lui, Polymer International, 2007, 56, 11, 1323. 15. J-H. He and S.D. Zhu in Proceedings of the 2007 International Symposium on Nonlinear Dynamics (2007 ISND), Journal of Physics: Conference Series, 2008, 96, 012189.

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Nonlinear Dynamics in Sirofil/SiroSpun Yarn Spinning

The Siro-spinning technology [1–6] is a very effective and economical method for producing composite yarns, and it has become a hot subject in recent years. Two-strand spun yarns have now been widely used in the worsted industry. The strands are texturised to improve the bulk of the resultant yarns, which have been demonstrated to possess more desirable properties. For example, the weavability of the fabric formed by SiroSpun yarns is significantly improved over those made with its counterpart yarns. Figure 11.1 illustrates twostrand yarn spinning in an asymmetric case, and Figure 11.2 is the actual experimental set-up.

Figure 11.1 Asymmetric two-strand yarn spinning.

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Electrospun Nanofibres and Their Applications

Figure 11.2 The experimental set-up.

The SiroSpun technique is especially suited to the production of lightweight trans-seasonal fabrics, and a significant proportion of the world’s worsted spinning installations have been converted to this cost-saving and innovative CSIRO technology. A recent study revealed that dragline silk is made of many nanofibres, similar to SiroSpun worsted yarns. Siro-electrospinning was suggested by He and co-workers [7]. The multi-strand spinning process can improve The properties of worsted nanofibres, especially the physical and chemical characteristics. Though much research has been conducted experimentally and theoretically, mathematical modelling, which is the key point for interpreting the spinning process, is still far from perfect. This is possibly due to the nonlinear phenomena and complex properties involved, and to the many factors affecting the procedure, such as the distance 226

Nonlinear Dynamics in Sirofil/SiroSpun Yarn Spinning between the rollers, the convergence point, the force on the yarns, the twist of the yarns, the densities of the two strands and the spun yarn, and the velocities of the two strands and the spun yarn. In this section a complete model is established which can be used to optimise the system parameters, and to reveal how to control its instability.

11.1 Convergence Point The system (Figures 11.1 and 11.2) must obey the basic laws of mechanics, including force balance, mass conservation and energy conservation, at the convergence point (Figure 11.3) [1].

Figure 11.3 Control volume for stable spinning process.

The governing equations for the system can be written as follows. (1) Force equations: (11.1) (11.2) 227

Electrospun Nanofibres and Their Applications (11.3) where F and M are the tension and the elastic torque in the twostrand yarn below the convergence point, respectively Fi and Mi (i = 1, 2), are the tension and the elastic torque in the two strands above the convergence point, respectively, and R1 and R2 are the radii of the two strands. (2) Momentum equations: (11.4) (11.5) where ρ1 and ρ2 are the densities of the above two strands, ρ is the density of the spun yarn, u1 and u2 are the velocities of the two strands, and u is the velocity of the spun yarn. (3) Mass conservation: (11.6) Solving the above equations, we find that: (11.7)

(11.8)

(11.9)

(11.10)

228

Nonlinear Dynamics in Sirofil/SiroSpun Yarn Spinning

(11.11) where:

Considering α1 = α2 = α, F1 = F2 = f, M1 = M2 = m, u1 = u2 = u0, ρ1 = ρ2 = ρ0, and R1 = R2 = r for a SiroSpun system, we have: (11.12)

(11.13)

(11.14)

(11.15)

11.2 Linear Dynamic Model More details about this model may be found in [2]. We first assume that the system is in a stable condition. By our quasistatic model [5], the convergence point can be determined with ease. Owing to some perturbations, the convergence point (equilibrium position O in Figure 11.4) moves to an instantaneous new position (O′). The distances x and y are measured from the equilibrium position. From Figure 11.4 it is easy to see that the projections in the x and y directions of the force F in the two-strand yarn below the convergence point, and of the forces F1 and F2 in the two strands 229

Electrospun Nanofibres and Their Applications

Figure 11.4 The dynamics of two-strand spun yarn.

above the convergence point are, respectively: 0, –F; F1 cos α, F1 sin α; and –F2 cos β, F2 sin β. The angles α and β are defined in Figure 11.4. Let the ends of the two strands above the convergence point be fixed at a distance 2L apart, and the equilibrium position be H below. The equations of motion in the x and y directions are: (11.16)

(11.17) Here M is the total mass of the fi xed control volume ABCD illustrated in Figure 11.5; the control volume is chosen in such a way that the centre of mass coincides with the convergence point (O) of the two strands. The mass, M, is then determined from the following relationship: 230

Nonlinear Dynamics in Sirofil/SiroSpun Yarn Spinning

Figure 11.5 The control volume and the centre of mass at the convergence point.

(11.18) where l1, l2 and ρ1, ρ2 are, the length and density per unit length of the two parent strands above the convergence point, respectively, h is the distance of the two-strand yarn below the convergence point, and ρ is the density per unit length of the resultant twostrand yarn. , applying the binomial If x and y are much smaller than theorem to expand the square-root terms, we have:

231

Electrospun Nanofibres and Their Applications

When the system is in equilibrium, we consider a simple symmetric case, i.e., α1 = α2 = α0 and F1 = F2 = f = constant. Substituting all the above into equations (11.16) and (11.17), we obtain: (11.19)

(11.20) where α0 as in Figure 11.4 is the convergence angle in equilibrium, i.e.,

Then we can rearrange Equations (11.19) and (11.20) in the forms: (11.21)

(11.22) where ωx and ωy are the oscillation frequencies in the x and y directions, respectively, defined as: 232

Nonlinear Dynamics in Sirofil/SiroSpun Yarn Spinning

(11.23)

(11.24) The oscillation periods in the x and y directions are, written in the forms: (11.25)

(11.26) The solutions for x and y can be expressed in the forms: (11.27) (11.28) where A and B are the amplitudes in the x and y directions, ) we have: respectively. In the case Tx = Ty (i.e., H = L, (11.29) where k = B/A. Equation (11.29) states that the convergence point (O′) always tends to its equilibrium position (O) only when the ratio y/x = k becomes a constant. Various trajectories of the convergence point (O′) can be obtained using our quasistatic model [5] by a suitable choice of the parameter H, as shown in Figure 11.6. The linear dynamic model reveals that the optimal equilibrium , and that, under such a condition, the convergence angle is system behaves like a mass–spring system. However, the dynamic system in general is of inherent nonlinearity, and we will now study the nonlinear case. 233

Electrospun Nanofibres and Their Applications

(a) (b)

(d) (c)

Figure 11.6 Trajectories of the convergence point under different conditions, namely, ωx:ωy = 1:1 (a), 1:2 (b), 1:3 (c), 1:4 (d), 2:1 (e), 3:1 (f), and 4:1 (g). 234

Nonlinear Dynamics in Sirofil/SiroSpun Yarn Spinning

(f)

(e)

(g)

Figure 11.6 Continued

235

Electrospun Nanofibres and Their Applications

11.3 Nonlinear Dynamic Model More details about this model may be found in [3]. As before, assume that the convergence point (equilibrium position) moves to an instantaneous position (see Figure 11.4), and the distances x and y are measured from the equilibrium position. The equations of motion in the x and y directions can again be expressed as: (11.30)

(11.31) Here M is the total mass of the fixed control volume ABCD illustrated in Figure 11.5, the control volume being chosen such that the centre of mass is located on the convergence point (O) of the two strands. We now consider the symmetric simple case, i.e., α1 = α2 and F1 = F2 = f, when the system is in equilibrium. Substituting these into Equations (11.16) and (11.17), similarly to before we now obtain: (11.32)

(11.33) where:

236

Nonlinear Dynamics in Sirofil/SiroSpun Yarn Spinning We use the homotopy perturbation method [8] to solve the system. To this end, we construct a homotopy system in the forms: (11.34)

(11.35) It is obvious that, when p = 0, the system of Equations (11.34) and (11.35) becomes linearised; and when p = 1, it turns out to be the original system of Equations (11.32) and (11.33). The embedding parameter p increases monotonically from zero to unity as the linear system (p = 0) is continuously deformed to the original system of Equations (11.32) and (11.33). So if we can construct an iteration formula for the system of Equations (11.34) and (11.35), the series of approximations comes along the solution path, by incrementing the imbedding parameter from zero to one. This continuously maps the initial solution of Equations (11.34) and (11.35) into the solution of the original system of Equations (11.32) and (11.33). According to the homotopy perturbation method [8], the solutions can be expressed in the following form: (11.36) (11.37) Substituting Equations (11.36) and (11.37) into Equations (11.34) and (11.35), and collecting terms of the same power of p, we obtain the following differential Equations for x0, x1 and y0, y1: (11.38)

237

Electrospun Nanofibres and Their Applications

(11.39)

(11.40)

(11.41) The solutions of Equations (11.38) and (11.40) are, Equations (11.42) and (11.43), respectively: (11.42) (11.43) where A and B are the amplitudes in the x and y directions, respectively. Using the initial conditions for x1 and y1, namely x1(0) = x′1(0) = 0 and y1(0) = y′1(0) = 0, we can easily obtain the solutions for x1 and y1. If the first-order approximations are enough, we set p = 1 in Equations (11.36) and (11.37) to yield the following approximate solutions:

(11.44)

(11.45)

238

Nonlinear Dynamics in Sirofil/SiroSpun Yarn Spinning If 2ωx = ωy (i.e. L = 2H or α0 = 26.56°), resonance occurs. The phenomenon should be completely avoided in textile applications. We can also apply the variational iteration method [1] to solve the system. According to the variational iteration method, we can construct the following iteration formulae: (11.46)

(11.47) We begin with the initial solutions: (11.48) (11.49) where Ωx and Ωy are the frequencies in the x and y directions, respectively. By the iteration of formulae (11.46) and (11.47), we have:

(11.50) 239

Electrospun Nanofibres and Their Applications

and

(11.51) If the first iteration is enough, then use the conditions: (11.52) We can determine the values of Ωx and Ωy. From Equations (11.50) and (11.51), we can obtain the resonance condition of the coupled oscillator Equations (11.34) and (11.35), which reads: (11.53) In the case where a is small, then Ωx ≈ ωx. In such a case, resonance occurs when: (11.54) i.e. L = 2H or the convergence angle 2α0 = 2 × 63.43°. 240

Nonlinear Dynamics in Sirofil/SiroSpun Yarn Spinning To conclude, we suggest a nonlinear dynamic model for two-strand spinning, which can be applied directly to the textile industry. The study reveals that the optimal convergence angle of the two strands in equilibrium is 90°, so that when the convergence angle is near 127°, resonance occurs.

11.4 Stable Working Condition for Three-Strand Yarn Spinning Traditionally, two-strand yarns have been used for weaving because they are stronger, and the twisting operation binds the surface fibres into the yarn structure so that it is smoother and more resistant to abrasion during weaving. As mentioned in the introduction to this chapter, texturised two-strand spun yarns are used in the worsted industry to improve the bulk of the resultant yarns and to impart more desirable properties, such as the weavability of the fabric formed by SiroSpun yarns. For multiple-strand yarn spinning, the stable working condition can be readily obtained as follows: (11.55) where ρk, uk and Rk are, the density, velocity and radius of the kth strand, respectively. Three-strand yarns (Figure 11.7) can be designed for smart fabrics and might have many advantages over two-strand yarns. Our group in Donghua University, Shanghai, China, is doing research on threestrand yarns. A major parameter of the spinning system is the convergence angle, which determines whether the composite yarns could have fine properties. Using force balance and the basic laws of fluid mechanics, e.g., mass conservation and momentum conservation, 241

Electrospun Nanofibres and Their Applications

Figure 11.7 Three-strand yarn spinning.

we can identify the convergence point with ease, while some other known models require experimental data to determine the convergence angle. A dynamic model is established by suitable choice of a control volume (Figure 11.8) where the centre of mass focuses on the convergence point. The variational iteration method is applied to solve analytically the nonlinear system. The results obtained show that there is an optimal condition for the system, and resonance arises for some special cases, which should be avoided in engineering applications. We design a three-strand yarn spinning system where one or two convergence points possibly exist depending on the spinning conditions. A description of stable working conditions for threestrand yarn is also given, and the conditions for one or two convergence points are obtained, which are essential to determine the dynamical properties of the spinning system. 242

Nonlinear Dynamics in Sirofil/SiroSpun Yarn Spinning

(a)

(b)

Figure 11.8 Control volumes for stable spinning of three-strand yarns.

243

Electrospun Nanofibres and Their Applications Using the control volume in Figure 11.8(a), we have the following governing equations, from momentum and mass conservation: (11.56) (11.57) (11.58) Using the control volume in Figure 11.8(b), we have the following governing equations: (11.59) (11.60) (11.61) (11.62)

(11.63) (11.64) From the above system, we obtain: (11.65) or (11.66)

244

Nonlinear Dynamics in Sirofil/SiroSpun Yarn Spinning

(11.67)

(11.68) where:

So the stable working condition for three-strand yarn spinning (see Figure 11.9 for the various angles and lengths) is: (11.69)

Figure 11.9 Stable working condition for three-strand yarn spinning.

245

Electrospun Nanofibres and Their Applications

11.5 Nano-Sirospinning A new direction in Sirospinning is nano-Sirospinning, i.e., the composition of a microcomposite from thousands of nanofibres, like spider spinning, to take full advantage of the nano-effect. Bubble electrospinning can be used for this purpose, as illustrated in Figures 11.10 and 11.11.

Figure 11.10 Nano-Sirospinning using bubble electrospinning.

246

Nonlinear Dynamics in Sirofil/SiroSpun Yarn Spinning

Figure 11.11 Multiple jets ejected from polyvinylpyrrolidone solution using bubble electrospinning.

References 1.

W-Y. Liu, Y-P. Yu, J-H. He and S-Y. Wang, Textile Research Journal, 2007, 77, 4, 195.

2.

W-Y. Liu, Y-P. Yu, J-H. He and S-Y. Wang, Textile Research Journal, 2007, 77, 4, 200.

3.

J-H. He, Fibres and Textiles in Eastern Europe, 2007, 60, 1, 31.

4.

J-H. He, Y-P. Yu, J-Y. Yu, W-R. Li, S-Y. Wang and N. Pan, Textile Research Journal, 2005, 75, 2, 181.

5.

J-H. He, Y-P. Yu, N. Pan, X-C. Cai, J.Y. Yu and S.Y. Wang, Mechanics Research Communications, 2005, 32, 2, 197. 247

Electrospun Nanofibres and Their Applications 6.

J-H. He, Y-P. Yu, J-Y. Yu, W-R. Li, S-Y. Wang and N. Pan, Textile Research Journal, 2005, 75, 1, 87.

7.

J-H. He, International Journal of Modern Physics B, 2006, 20, 10, 1141.

8.

J-H. He, Y-Q. Wan and L. Xu, Chaos, Solitons & Fractals, 2007, 33, 1, 26.

Acknowledgement This material is based on work supported by National Natural Science Foundation of China under the grand Nos. 10372021, 10772054 and 10572038, the 111 project under the grand No. B07024 and the Program for New Century Excellent Talents in University under grand No. NCET-05-0417. The first author should thank the other authors for their contributions. Wan mainly contributes her work on vibration-electrospinning ; Wu on magneto-electrospinning; Liu on bubble-electrospinning; and Xu on nanoporous materials.

248

A

bbreviations

1-CP

1-Chloropropanol

2D

Two-dimensional

3-CP

3-Chloropropanol

3D

Three-dimensional

CHCl3

Chloroform

CNT

Carbon nanotubes

CSIRO

Commenwealth Scientific and Industrial Research Organisation

DC

Direct current

DMF

Dimethylformamide

DNA

Deoxyribonucleic acid

DOJ

Department of Justice

FTIR

Fourier transform infrared

IPA

Isopropanol

LiCl

Lithium chloride

Mn

Number average molecular weight

MWCNT

Multi-walled carbon nanotube(s)

NASA

National Aeronautics and Space Administration

NIST

National Institute of Standards and Technology

PAN

Polyacrylonitrile

PBS

Polybutylene succinate

PCL

Polycaprolactone

PEO

Polyethylene oxide

249

Electrospun Nanofibres and Their Applications PHBV

Polyhydroxybutyrate-co-valerate

PMMA

Polymethylmethacrylate

PU

Polyurethane

PVC

Polyvinylchloride

PVP

Polyvinylpyrrolidone

SEM

Scanning electron microscope

SWCNT

Single-walled carbon nanotube(s)

TEM

Transmission electron microscope

TSA

Homeland Security

XRD

X-ray diffractometer

250

I

ndex

A absolute zero 5 aciniform glands 141 aero-electrospinning 126 air vortex spinning 203–204 allometric scaling 41 biological systems 84–87 electrospinning effect of concentration 59–62 non-ionic surfactants to improve electrospinning 72–80 relationship between average polymer molecular weight and nanofibre diameter 63–66 relationship between current and voltage 53–55 relationship between jet radius and axial distance 45–53 relationship between nanofibre diameter and voltage 66–72 relationship between solution flow rate and current 56–59 occurence in nature 42–45 static friction of fibrous materials 80–81 fibre–fibre friction 82–84 soft materials 82 solid–solid friction 81–82 viscous friction in Newtonian flow 82 Amontons’ law 80 Ampere force 119, 120 ampullaceal glands 141 artificial atoms 109 axial distance relationship with jet radius 45–49

251

Electrospun Nanofibres and Their Applications experimental verification 50 initial stage for steady jet 49–50 instability of viscous jet 50–52 terminal state 53

B beads, controlling numbers and sizes 157 experimental procedure characterisation 158 electrospinning process 157–158 materials 157 polymer concentration 163–164 salt additive 165–167 solvents 158–163 bubble electrospinning 131, 143–144 experimental procedure 144–146, 150–154 morphology of nanofibres 150 nano-Sirospinning 246–247 silk fibroin nanfibres 135–136 solving mystery of spider spinning 136–143 spider spinning 131–132 intelligent spider fibre 132–133 mathematical model 134–135 visualisation 146–149

C Cantor sets 34–35 carbon nanotubes (CNT) 105–111 resistance formulation 218–220 Cauchy inequality 116 Cauchy stress tensor 19 chaotic quantum methanics 4–5 charge balance 18 charge, conservation of 18, 20, 21, 22 concentration, effect on nanofibres 59–60 experimental verification 60–62 252

Index conservation of charge 18, 20, 21, 22 conservation of charge model, modified 22–28 conservation of mass 17, 20 convergence point 227–229 linear dynamic model 229–235 nonlinear dynamic model 236–241 coupling constant 34 critical maximal velocity 180–181 current ohmic bulk conduction 20, 22 relationship with solution flow rate 56–57 experimental verification 57–59 relationship with voltage 53–54 experimental verification 54–55 surface convection 20, 22 temperature gradient 20 cyclone model 204–205

D Darcy’s law 214–215 differential–difference model for nanotechnology 220–221 dimensions of space–time 6 fractal dimensions 12 dragline silk 128 drug delivery 169, 170

E E-infinity theory 4–5, 10–12, 37–38 hierarchical structure in blood-vessel system model 38 hierarchical structure in turbulence model 38 Einstein’s field equation 34 El Naschie see E-infinity theory electrical conduction current 19 electrical resistance 22, 44 electrospinning applications 12–13 253

Electrospun Nanofibres and Their Applications controlling parameters 10–12 definition 6–10 dilation 180–183 global interest 13–15 methods Formhals 7, 45 Taylor cone 7–8 non-ionic surfactants 72–80 stages 46–47 energy–momentum tensor 34

F fibres see also nanofibres diameter vibration technology, effect of 93–94 surface area 2 flow rate relationship with current 56–57 experimental verification 57–59 flow, steady-state 18 force balance 18 Formhals’s electrospinning method 7, 45 Fourier’s equation for heat conduction 215 fourth dimension of life 45 fractal dimensions 12, 217 fractal kissing problem 36, 37 friction 80–81 fibre–fibre friction 82–84 soft materials 82 solid–solid friction 81–82 viscous friction in Newtonian flow 82

G Gauss divergence theorems 213 golden mean 6 254

Index

H Hagen–Poiseuille equation 217 Hall–Patch relationship 139 Hausdorff dimension 85 He Chengtian 23–5 heat flux 19

I instability analysis 47–48 control of 115 critical jet length 116–119 magnetic fields 119–123 temperature 123–127

J jet length critical length 116–119 jet radius relationship with axial distance 45–49 experimental verification 50 initial stage for steady jet 49–50 instability of viscous jet 50–52 terminal state 53 jet vortex spinning 203

K Kepler’s third law 42 Kolmogorov length 43

L linear dynamics 229–235

M magneto-electrospinning 115 255

Electrospun Nanofibres and Their Applications critical jet length 116–119 instability, control of 119–123 Mark–Houwink relationship 44 mass, conservation of 17, 20, 180 mass balance 18 mathematical models 17 conservation of charge model, modified 22–28 E-infinity model 37–38 one-dimensional model 17–18 one-dimensional model, modified 20–22 Reneker’s model 28–33 Spivak–Dzenis model 18–19, 45–46 Wan–Guo–Pan model 19–20 mechanics 203 carbon nanotube resistance formulation 218–220 differential–difference model for nanotechnology 220–221 jet vortex spinning and cyclone model 203–205 nanohydrodynamics 212–218 two-phase flow 205–209 yarn motion in a tube 209–212 melt electrospinning 125 metabolic rate and body mass 42 metric tensor 34 microcomposite fibres 189 micropolar fluids 206–209 molecular weight relationship with nanofibre diameter 63–66 momentum balance 18 multi-walled carbon nanotubes (MWCNTs) 105–111

N nano-effect 104 nanofibres 10 see also fibres relationship between diameter and molecular weight 63–66 relationship between diameter and voltage 66–67 experimental verification 67–72 256

Index nanohydrodynamics 212–218 nanoporous microspheres 169 electrospinning 187–188 dilation 180–183 microcomposite fibres 189 single fibres 183 incorporating drugs 170 electrospinning process 171 instrumentation 170–171 IR spectra 171–172 materials 170 SEM analysis 173–179 nanotechnology, definition 1–6 Navier–Stokes equations 19, 21 Navier–Stokes model 205–207 Newton’s second law 76 non-ionic surfactants 72–80 nonlinear dynamics 236–241

O ohmic bulk conduction current 20, 22 Ohm’s law 44 one-dimensional flow 42–43 one-dimensional model of flow 17–18 one-dimensional model of flow, modified 20–22

P pendulum, motion of 42 polarisation 19, 22 polarisation charge 21 polyacrylonitrile (PAN) 59–62 polybutylene succinate (PBS) 67–72, 157 concentration 163–164 electrospinning 183–186 salt additive 165–167 solvents 158–163 257

Electrospun Nanofibres and Their Applications polyethylene oxide (PEO) 98–103 polymethyl methacrylate (PMMA) 94, 95 polyvinylpyrrolidone (PVP) 73–78 proton mass 5 pyriform glands 141

Q quantum-like phenomena 2–3

R reaction–diffusion equation 215 Reneker’s model 28–33 restistance, electrical 22, 44 Reynolds number 43 rheological behaviour 21 Ricci tensor 34 Rubner’s 2/3 law 86

S single-walled carbon nanotubes (SWCNTs) 105–111 Siro-electrospinning 127–129, 225–227 convergence point 227–229 nano-spinning 246–247 stable three-strand spinning 241–245 sodium cobalt oxide nanfibres 143 spiders 131 electrospinning of silk fibroin nanfibres 135–136 intelligent fibre 132–133 mathematical model for fibres 134–135 solving mystery of spinning 136–143 spinnerets 138, 141 spinning 131–132 Spivak–Dzenis model 18–19, 45–46 steady-state flow 18 surface area of fibres 2 surface convection current 20, 22 258

Index

T tangential velocity distribution 205 Taylor cone electrospinning method 7–8 temperature and control of instability 123–127 turbulence 42–43 two-phase flow 205–209

V van der Waals forces 94 vibration melt electrospinning 126 vibration technology 93 applications 95–97 experimental verification 98–103 carbon nanotubes (CNT) 105–111 effect of viscosity on fibre mechanical characteristics 103–104 fibre diameter, effect on 93–94 viscosity, effect on 94–95 viscosity 44, 64 effect on fibre mechanical characteristics 103–104 vibration technology, effect of 94–95 viscosity–shear rate plots 99 viscous resistance 46, 51 voltage effect on morphology and diameter of nanofibres 66–67 experimental verification 67–72 relationship with current 53–54 experimental verification 54–55

W Wan–Guo–Pan model 19–20 whipping circle 119, 121

X X-ray diffraction of electrospun fibre multi-walled carbon nanotubes (MWCNTs) 110 259

Electrospun Nanofibres and Their Applications no vibration 100 with vibration 101

Y yarn motion in a tube 209–212 yarn sliver flow 210–211 Yunnan Baiyo (traditional Chinese drug) 169 nanoporous microsphere delivery 170 electrospinning process 171 instrumentation 170–171 IR spectra 171–172 materials 170 SEM analysis 173–179

260

Published by iSmithers, 2008

This Update covers all aspects of electrospinning as used to produce nanofibres. It contains an array of colour diagrams, mathematical models, equations and detailed references. Electrospinning is the cheapest and the most straightforward way to produce nanomaterials. Electrospun nanofibres are very important for the scientific and economic revival of developing countries. It is now possible to produce a lowcost, high-value, high-strength fibre from a biodegradable and renewable waste product for easing environmental concerns. For example, electrospun nanofibres can be used in wound dressings, filtration applications, bone tissue engineering, catalyst supports, non-woven fabrics, reinforced fibres, support for enzymes, drug delivery systems and many other applications which are discussed in this Update. It will be invaluable to anyone who is interested in using this technique and also to those interested in finding out more about the subject.

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