Simulating the dynamical behaviour
of electrospinning processes
Rob van Vught
DC 2010.25
Traineeship report
Coach(es): dr. H.J.C. Huijberts
Supervisor: prof. dr. H. Nijmeijer
Eindhoven University of TechnologyDepartment of Mechanical EngineeringDynamics and Control Group
Eindhoven, March, 2010
Abstract
Electrospinning is a relatively new technique to produces nano scale fibres. These fibres canbe used for many applications, including the fabrication of scaffolds for tissue engineering.A drawback of the electrospinning process however, is the unstable behaviour of the liquidjet. This causes the fibres to be collected randomly. Controlling this process is thereforedesirable. Studying the dynamics of the jet becomes easier and faster if it can be simulated,rather than doing experiments.
This project focusses on simulating the electrospinning process. An existing mathe-matical model, which describes the jet as a mechanical system of masses that are intercon-nected with viscoelastic elements, was used to build a Matlab script on. The simulationfeatures the possibility to cope with the introduction of new, and the removal of old ele-ments in the system.
Simulations that were performed with the script yielded figures that do not perfectlymatch the results of previous experiments and simulations. The simulations could onlybe performed with a low amount of beads. This resulted in rough jets and also influencedthe reality of the simulation. The electrospinning process depends on a lot of differentparameters. Changing these parameters results in a difference in the dynamical behaviourof the jet. Increasing the elastic modulus resulted in less unstable behaviour of the jet.Increasing the voltage in more unstable behaviour as was the case by increasing the initialjet radius. The influence of the viscosity and surface tension coefficient seems to be small.
Furthermore, the possibilities for a dynamical analysis were studied. A mathematicalproof for the instability of the jet was searched. This could not be found.
Preface
Electrospinning is a technique to create one-dimensional structures in the form of nanofibre yarns. The potential of the electrospinning process is very large, since it is a relativelysimple and cost effective method to create these yarns. However, the instability of thefluid jet limits the applications of the yarns that are produced by this production method.Only when the unstable behaviour of this process can be controlled, the full potential ofthis process can be exploited. Therefore the electrospinning process may count on a lotof attention. Especially the study group of Reneker at the University of Akron, did a lot ofresearch on this topic and popularised it.
One of the other studies was performed in May 2009, by a group of masters studentsat the School of Engineering and Material Sciences of Queen Mary University of London(QMUL). They performed a Master of Engineering group project about the developmentof a controlled electrospinning system for anterior cruciate ligament tissue engineering.A part of this study focussed on simulating the electrospinning process. However thesesimulations did not cope with the flow of the material in and out of the process.
As part of my Masters programme in Mechanical engineering at the Eindhoven Uni-versity of Technology (TU/e), an abroad master traineeship was offered by the School ofEngineering and Material Sciences of QMUL, to improve the Matlab script that was madeby the above mentioned students. This script was based on a mathematical model thatwas found in one of the studies of Reneker et al.. More understanding of this model wasdesirable, since it can also be used for a further dynamical analysis.
This report shows the approach on how to gain more understanding of the electrospin-ning process. It shows that some routes to obtain correct simulations have been taken andmany of them seemed to be wrong. Furthermore, a start is given to do a dynamical analysison the behaviour of the electrospinning process.
I would like to thank Dr. Henri Huijberts for creating the opportunity to do my mastertraineeship at QMUL. Furthermore I would like to thank him for his great support. He didnot only provide crucial mathematical advice, but also the critical notes he made on thisreport were very welcome. Finally, I also would like to thank Prof. Dr. Henk Nijmeijer forthe time and effort he took to find a suitable assignment at a great location.
London, December 2009
Rob van Vught
Contents
1 Introduction 31.1 Electrospinning in scaffold fabrication . . . . . . . . . . . . . . . . . . . . . 31.2 Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Matlab program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Dynamical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Overview of the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Literature review 62.1 Tissue engineering and the anterior cruciate ligament . . . . . . . . . . . . 62.2 Electrospinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Development of the Matlab simulation 93.1 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1.1 Viscoelastic force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.2 Coulomb forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.1.3 Surface tension force . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.4 Electric force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1.5 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Transformation to Matlab script . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.1 Different ways of handling the state vector size . . . . . . . . . . . . 173.2.2 Transformation of differential equations to a state vector . . . . . . . 183.2.3 Handling disappearing and new beads . . . . . . . . . . . . . . . . . 193.2.4 Dimensionless parameters . . . . . . . . . . . . . . . . . . . . . . . 20
4 Results of electrospinning simulations 224.1 Simulating a straight jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Simulations with the first version of the script . . . . . . . . . . . . . . . . . 234.3 Simulations with the second version of the script . . . . . . . . . . . . . . . 26
5 Dynamical analysis 305.1 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.2 Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.3 Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1
6 Conclusions and recommendations 356.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
A Scaffolds for ACL tissue engineering 37A.0.1 Scaffold designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37A.0.2 Scaffold materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
B Electrospinning history 40
C Examples of previous electrospinning studies 42
D Simulation parameters 45
E Effects of changing parameters on simulation results 46E.1 Increasing the amount of beads . . . . . . . . . . . . . . . . . . . . . . . . . 46E.2 Influence of the viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46E.3 Influence of the surface tension . . . . . . . . . . . . . . . . . . . . . . . . . 47E.4 Influence of the elastic modulus . . . . . . . . . . . . . . . . . . . . . . . . 47E.5 Influence of the initial jet radius . . . . . . . . . . . . . . . . . . . . . . . . 50E.6 Influence of the applied voltage . . . . . . . . . . . . . . . . . . . . . . . . . 50E.7 Dimensionless parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
F Definition of dimensionless groups 55
G Radius derivation for curvature 58
H Coordinate transformation 60
I Derivation of Liouville’s theorem for nonlinear differential equations 62
2
Chapter 1
Introduction
1.1 Electrospinning in scaffold fabrication
Many people suffer from tissue loss or end-stage organ failure. To treat these disorders,transplantation of organs, surgical reconstruction or mechanical devices are commonlyused by physicians. It is understandable that these treatments are imperfect solutions. Thisis illustrated, for example, by the severe limitation of organs for transplantation. Further-more, also surgical reconstruction can lead to problems in the long term, while mechanicaldevices cannot perform all functions of a normal organ. A relatively new treatment can befound in tissue engineering. This technique uses the combinations of cells, engineeringand materials to maintain or improve tissue functions [23]. Tissue engineering can be usedas treatment for many disorders, like liver and pancreas failure, but also to create artificialskin and bone.
Apart from the above mentioned disorders, tissue engineering can also be applied toknee ruptures. These ruptures are very frequent injuries. Inside the knee connective tis-sues between bones are present in the form of ligaments. They provide stability for the jointmotion. One of the most ruptured ligaments in a human knee is the anterior cruciate lig-ament (ACL) [9]. Since current treatments show numerous drawbacks, tissue engineeringhas gained a lot of attention as a new treatment method for this injury.
There are several ways to create new tissue. One of the strategies is the placement ofcells on or within matrices. These matrices are typically called scaffolds. These systems canbe implanted and become incorporated into the body.
Electrospinning is a technique that can be used to produce a scaffold. This processwas introduced in the field of tissue engineering over the past decade. Scaffolds createdby electrospinning can possibly meet the design criteria of an ideal engineered scaffold.Electrospinning is a technique to produce nanofibres in a cost effective way and has there-fore earned a lot of attention recently [26]. During the process, a liquid polymer solutionis charged by an electric field. Then the fluid is attracted by a collector of opposite polarity,which is placed at a certain distance from the device containing the solution. A pendentdroplet is formed and when the electric field strength exceeds surface tension of the fluid,a fibre is formed traveling towards the collector plate. In this way, polymer nanofibres areformed that can be used to construct the scaffold.
However, there is one big drawback. This is the unstable dynamical behaviour of the
3
liquid jet which is formed during the electrospinning process. This instability inhibits thefibres to be aligned in a regular way, which is crucial to satisfy the scaffolds specifications.In the past decade, many research has been done to study this jet behaviour. When theinstability of the jet can be controlled, electrospinning can not only be adapted to producehigh quality scaffolds for tissue engineering, but also for many other applications [25].
1.2 Aims
As mentioned in the previous section, much research has already been performed on theelectrospinning process. Especially the research group of Reneker from the University ofAkron, did a lot of research on electrospinning in the last 20 years. In one of these studiesthey made a mathematical model that describes the behaviour of the liquid jet [29]. Themain focus in this research assignment will be to use this mathematical model in a Matlabsimulation. Then this simulation can be used to study the dynamical behaviour and, inparticular, the instability of the jet and how this is affected by changes in the system’sparameters. This makes it easier to check assumptions about the process behaviour andstudy how the parameters affect the dynamics, without doing expensive experiments timeafter time. After this, the next step is to study the possibilities of a dynamical analysis ofthe system.
This study has as main aim to use the above mentioned mathematical model, for creat-ing the following products.
1. A Matlab program that simulates the dynamical behaviour of the jet.
2. Study the possibilities for a stability analysis of the nonlinear dynamical behaviourof the system. The actual stability analysis is meant to give information about thisdynamical behaviour of the jet. However, since the jet appears to be a complex system,not every route that is taken during the stability analysis will give satisfying results.
1.2.1 Matlab program
To begin the study, the first thing to do is to design aMatlab program that is able to simulatethe dynamical behaviour of the system. The mathematical model as mentioned before,is used as a starting point. It describes the liquid jet as a mechanical system consistingof little mass particles, called beads. These are interconnected by viscoelastic elements,representing the structure of the liquid jet. The Matlab script uses this model to simulatethe process while it will also be designed in such a way that it is able to cope with thechanging number of coordinates. This is done since new beads are being introduced in thesystem, while the ones that reach the collector plate need to be eliminated.
1.2.2 Dynamical analysis
The second aim is to study the possibilities for a dynamical analysis. The instability of thejet is a problem that has been studied by many others in the past years. With the availabilityof a Matlab script, it is now faster and easier to simulate the behaviour of the jet and howthis is affected by the system parameters. Studying the dynamical behaviour of the jet is
4
of interest for the development of a possible control system. Such a study should focus onthree parts, all of them producing different information about the jet instability. The totalproduct of the dynamical analysis then relies on the results of these sub studies. Thereforethis report will study the possibilities for a dynamical analysis consisting of:
• a study of the instability of the trajectory of the jet as time evolves,
• a study of the possible chaotic behaviour, and
• a study of the bifurcations of the dynamical system that represents the jet.
In this report, the terms stable or unstable behaviour will be used several times. Theseterms are based on the observations that can be made while looking at the fluid jet. The lat-eral displacements in the lower part of the jet, are thought to be the effects of the instabilityof the equilibrium trajectory where the jet moves along. Therefore these lateral displace-ments will be referred to as ’unstable behaviour’. Stable behaviour, on the other hand, isthe case when there are no observations of instability along the jet’s main axis, which issomething that occurs in the first part of the process, i.e. the upper part of the jet.
1.3 Overview of the report
In Chapter 2, a literature review will be given on the background of this project. Also theelectrospinning process will be explained in more detail.
In Chapter 3 the development of the Matlab programme to simulate the dynamicalbehaviour of the jet is given. It starts by discussing the mathematical model that forms thefoundation of the script. Then the transformation of this model to a useful script will bedescribed.
Chapter 4 gives the results of the different simulations and tries to give some explana-tions of the results. It shows how the different results represent the differences in dynami-cal behaviour as some parameters change.
In Chapter 5 more information will be given about an analytical study of the electro-spinning process. First a mathematical proof for the instability will be studied, looking atthe trajectories that the particles of the jet follow. Then the possibilities for studying thepossible existence of chaotic behaviour will be examined. The last section of the chapterwill give some brief information about the bifurcation behaviour of the jet.
Chapter 6 gives the conclusions that were drawn from the results shown, and the rec-ommendations that were given in relation to the outcome of Chapter 4 and 5.
5
Chapter 2
Literature review
2.1 Tissue engineering and the anterior cruciate ligament
In the field of medical engineering, tissue engineering is gaining more importance and alot of research in this area has been done recently. Since 1987, the number of researcherswho consider themselves to be working in tissue engineering has grown substantially [33].Tissue Engineering can be seen as ‘an interdisciplinary field that applies the principles ofengineering and life sciences toward the development of biological substitutes that restore,maintain, or improve tissue function or a whole organ’ [23]. Some examples where tissueengineering can be used, are bioartificial liver devices, artificial pancreas, artificial skin andartificial bone.
Tissue engineering can also be applied as treatment for a ruptured anterior cruciate lig-ament. The ACL is one of the most frequently ruptured structures within the knee. In theUnited States, the estimated prevalence of this injury is about 1 per 3000 Americans [9].With an increasing emphasis on sports activities, this number is increasing. Where cur-rent treatments show numerous drawbacks, like reduced mechanical integrity, morbidityof the donor site and risk of infection, there has been great interest in new reconstructivemethods. Tissue engineering appears to be the most promising solution [20].
Developments in tissue engineering have also yielded opportunities to fabricate tissuesin the laboratory from combinations of scaffolds. Scaffolds are porous implants made fromresorbable material. They can be used as temporary structures to support cells/tissues dur-ing regeneration and repair. Therefore the primary function of scaffolds is to provide astructure for organizing dissociated cells into appropriate tissues by creating an environ-ment that allows three-dimensional (3D) cell growth and tissue formation [20]. Therefore,scaffolds that are used for tissue engineering should fulfill certain requirements regardingthe scaffolds design and material. In Appendix A some additional information is givenabout scaffold designs and materials.
2.2 Electrospinning
In the previous section the need for tissue engineering and the required technogly for thiswere discussed. Scaffolds, which are necessary for creating a tissue engineered ACL can beproduced through fibre bonding or electrospinning. This latter technique was introduced
6
in the field of tissue engineering over the past decade. Producing scaffolds by electrospin-ning has some significant advantages. They have a wide range of pore size distribution,high porosity and high surface area-to-volume ratio. A method to produce a scaffold byelectrospinning, is to collect the electrospun fibres in such a way that they are arranged ac-cording to the geometry of a non-woven scaffold. Constructing the structure in such a way,makes that these electrospun structures can potentially meet the essential design criteriaof an ideal engineered scaffold [26]. In Appendix B some additional information is givenabout the history of the electrospinning process.
Pipette
Pendent drop
Instability region
High voltage
power supply
Collector plate
h
Straight
jet
y
z
x
Figure 2.1: Schematic diagram of the electrospinning process.
The process of electrospinning produces polymer fibres on a nano scale. A simplediagram of the process is provided in Figure 2.1. A polymer solution consisting of a polymerlike poly(D,L-lactide-co-glycolide) (PLGA) and a solvent, for example chloroform is charged.This usually happens by induction charging, caused by a contact between an electrode andthe fluid. Charge carriers are created, which can be in the form of ions, ion pairs or freeradicals. The production of charge carriers can be very sensitive to solution impurities [22].
When the charge is applied to the polymeric solution, an equilibriumwill be establishedbetween the electrical force caused by the charge pointing in the direction of the collectorplate and the surface tension force which works in opposite direction [12]. The cone that isformed as a result of this equilibrium is called a Taylor cone. It represents the initial stageof fibre formation [32].
After the Taylor cone has been established, the fluid will emit from the cone, forming aslender, continuous liquid filament. Because of the potential difference applied to the fluid,it has a high charge and therefore accelerates to the collector plate. This process can bemaintained for a longer period when the applied voltage is kept within certain limits [7].
When the filament accelerates down to the collector plate the fluid solidifies and the jetbecomes thinner. During this process the jet can show some unstable behaviour, perform-ing one or more instabilities. Reneker et al. [29] described that the bending segments of aloop in the instability region of the jet, can suddenly develop a new bending instability, sim-ilar to, but at a smaller scale than the first. However, the most common type of instabilityis the forming of a spiral with an increasing diameter. This is called the ‘whipping mode’
7
[29], [31].Finally the fibre will reach the collector plate. As a result of the described instability, the
fibres will be randomly orientated on the collector plate.Since the electrospinning process depends on a lot of different parameters, such as
material properties, applied voltage, material flow rate and capillary height, it is extremelycomplex. The most critical part in the electrospinning process is the instability. This projectfocuses on that jet instability. In the past years, much research has been performed on ex-perimental investigation, mathematical modeling and instability analysis. A few examplesof the studies performed are discussed in Appendix C.
8
Chapter 3
Development of the Matlabsimulation
3.1 Mathematical model
When developing a Matlab simulation, the first thing to do is to understand the mathemat-ical model that forms the foundation of the script. In the paper of Reneker et al. [29], amathematical model was given by some differential equations (DEs) , that together repre-sent the equations of motion. Furthermore, they modelled the fluid jet as a system of beads,which is shown in Figure 3.1. They found that the total force acting on each bead consists
Figure 3.1: Fluid jet modeled by a system of beads, connected by viscoelastic elements[29].
of a viscoelastic force, an electric force, a Coulomb force and a surface tension force. In thenext subsections the equations representing these forces are given.
To understand the mathematical model, first some details are given about the modelingof the system. The model of the fluid jet is described as a system of beads that are intercon-
9
nected by viscoelastic elements. These elements can also be called filaments. The forcesthat will be discussed in the next subsections are acting on each bead in the system. Beforethese forces can be discussed, first some notations have to be introduced. Consider againFigure 3.1, highlight one bead of the system depicted in this figure, and call this bead, beadi. The forces acting on bead i are dependent on the position of the other beads that arepresent in the system, since bead i is interconnected with those other beads.
Next the state of bead i can be described. It has 3 position coordinates: xi, yi, and zi,and 3 corresponding velocity coordinates: xi, yi and zi. If the forces acting on bead i areknown, then the acceleration of this bead can be calculated by the equations of motion,using Newton’s second law. However, not the acceleration, but the position is of interest,since the latter can be used to visualize the condition of the fluid jet. To obtain the positioncoordinates the acceleration of the bead needs to be integrated twice. This can be performedby using Matlab’s ode45 solver, which numerically integrates the differential equations.However to do this, the equations describing the state of the bead need to be written in astate space form.
The parameter values that are used in the simulations, which are shown in Chapter4, are obtained from experiments and simulations that were performed in other studies.These values are all given in Gaussian units, since this is also used in those studies, ratherthan the international system of units [Systéme International (SI)]. This Gaussian systemof units is based on centimeter-gram-second (cgs) units. For a list of the parameters, theirvalues and units used, the reader is referred to Appendix D.
3.1.1 Viscoelastic force
The viscoelastic force acting on bead i is due to a stress σ. This stress, which is acting on anelement between 2 beads, models a viscoelastic Maxwellian liquid jet. The Maxwell modeldescribes a spring-damper system and the equation of motion of this system is given by:
dǫ
dt=
1
G
dσ
dt+
σ
µ, (3.1)
with ǫ being the strain of the element,G and µ the elastic modulus respectively the viscosityof the material and σ the stress on the element [1]. The rate of strain dǫ
dt is defined by
Reneker et al. [29] to be: dldt
1l , with l the length of the element.
To adapt (3.1) to the mathematical model, again consider bead i. This bead is connectedby viscoelastic elements to 2 other beads, bead i + 1 that is above bead i, and bead i − 1,which is below it. This can also be seen when looking at Figure 3.1. The stresses betweenbead i and the two beads that are above, respectively below it, can then be defined by thetwo differential equations shown below:
dσuidt
= G1
lui
dluidt
−G
µσui,
dσbidt
= G1
lbi
dlbidt
−G
µσbi,
(3.2)
10
where t is the time, and G and µ represent the elastic modulus respectively the viscosityof the material. The distances between the beads and thus the length of the filaments, aregiven by lui and lbi, where the subscript ui denotes the filament between bead i and i + 1,while the subscript bi denotes the filament between bead i and i − 1. These distances aregiven by the Pythagorean Theorem, shown in the two equations below:
lui =√
(xi+1 − xi)2 + (yi+1 − yi)2 + (zi+1 − zi)2,
lbi =√
(xi − xi−1)2 + (yi − yi−1)2 + (zi − zi−1)2.(3.3)
The mathematical model that was developped by Reneker et al. [29], used some dimen-sionless descriptions, something that is also customary in fluid mechanics. Then a lengthscale, L can be defined, to render the filament length l dimensionless. This is length scaleis given by:
L =
√
e2
πa20G, (3.4)
with e being the charge of the bead and a0 the initial cross-sectional radius of the jet att = 0. If the assumption is made that the mass is conserved and evaporation is neglected,then these equalities apply:
πa2uilui = πa20L,
πa2bilbi = πa20L,(3.5)
yielding for the filament radii aui and abi:
aui = a0√
L/lui,
abi = a0√
L/lbi.(3.6)
Multiplying the stress that was defined in (3.2) by the cross-sectional surface of the jet, givesthe viscoelastic force. The net viscoelastic force acting on bead i results from the sum of theforce generated by bead i+1 that is defined to be positive and bead i−1, acting in oppositedirection, thus negative. It can then be described by the following equation:
fve =πa2uiσui
[
ixi+1 − xi
lui+ j
yi+1 − yilui
+ kzi+1 − zi
lui
]
− πa2biσbi
[
ixi − xi−1
lbi+ j
yi − yi−1
lbi+ k
zi − zi−1
lbi
]
,
(3.7)
11
where i, j, and k are the unit vectors along the x, y, and z axes respectively.It is obvious that using the above derived equation requires the solution of the differen-
tial equation shown in (3.2). However this latter mentioned equation also shows that thereare two time derivatives in these DEs, one in the left hand side of the function and one inthe right hand side of the function, giving the time derivative of the distance l. This meansthat before the integration of these DEs can take place, first the time derivative of the dis-tance l needs to be calculated. This is relatively easy to do, however a more general solutionto this problem can be found by rewriting the differential equation, which is shown below.This will also be useful when performing a dynamical analysis, as will be described in Sec-tion 5.1. Note that the subscripts ui and bi are removed here since rewriting the DEs shownin (3.2) is a similar process for both. As starting point the original differential equation isconsidered:
dσ
dt= G
1
l
dl
dt−
G
µσ. (3.8)
Now this equation can be rewritten by moving all time derivatives to one side of the equa-tion. This yields the following equation:
d
dt(σ −G ln(l))︸ ︷︷ ︸
σ
= −G
µσ, (3.9)
where a new variable σ is introduced. This directly gives for σ:
σ = σ +G ln(l). (3.10)
Then the differential equation can be further rewritten as:
dσ
dt= −
G
µσ −
G2
µln(l). (3.11)
This means that the viscoelastic force, as given in (3.7) now has to be rewritten, since theDEs shown in (3.2) are now of the form shown in (3.11). In this new form, not σ is thesolution of the DEs, but σ is. This yields for the viscoelastic force:
fve =πa2ui(σui +G ln(lui))
[
ixi+1 − xi
lui+ j
yi+1 − yilui
+ kzi+1 − zi
lui
]
− πa2bi(σbi +G ln(lbi))
[
ixi − xi−1
lbi+ j
yi − yi−1
lbi+ k
zi − zi−1
lbi
]
.
(3.12)
3.1.2 Coulomb forces
When again looking at bead i, another point of attention is the Coulomb force. This forceis influenced by the position of each bead in the system. Other forces on bead i are onlycaused by the location of the two beads that are above, respectively below bead i. However,the total Coulomb force on bead i is a summation of the Coulomb forces acting on i causedby every bead present in the system, except for bead i itself.
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Figure 3.2: Earnshaw instability, caused by the Coulomb forces, leading to the bendingof an electrified jet.
The magnitude of this force can be calculated according to Coulomb’s law. This lawstates the following [13].
Theorem 3.1. The magnitude of the electrostatic force between two point electric charges isdirectly proportional to the product of the magnitudes of each of the charges and inverselyproportional to the square of the total distance between the two charges.
Themagnitude of the Coulomb force acting on charge q1 due to the presence of a chargeq2 is then given by the following equation:
F =q1q2R2
, (3.13)
with R the distance between the 2 charges.The direction of the Coulomb force depends on the positions of the beads at which it
acts, since the force acts between two beads. This Coulomb force is thought to be the maincause for the instability of the jet [29]. This statement is based on the Earnshaw’s theorem[19], named after Samuel Earnshaw. This theorem states the following for a static electricfield.
Theorem 3.2. A charged body placed in an electric field of force cannot rest in stable equilib-rium under the influence of the electric forces alone.
This theorem can be adapted to the electrospinning process as shown below [29].Consider three beads, modeled as point masses, each of them having a charge e. Originallythey are vertically aligned underneath each other, having positions A, B and C as shown inFigure 3.2.
BeadB is placed between the two beads, A and C. The charges of beadsA and C create
a Coulomb force havingmagnitude F = e2
r2. This force then pushes from both sides against
bead B.Next, consider a perturbation that moves point B off the line formed by the jet axis. Thena distance δ is created between the original position B and the new position B′. Now a netforce F1 acts on B perpendicular to the jet axis, which has the magnitude:
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F1 = 2F cos(θ),
=2e2
r2cos(θ) =
2e2
r3δ.
(3.14)
This forces cause B to move further from the jet axis in the direction of the perturbation.The displacement of B generated by the force F1 can be derived from the second orderdifferential equation given by Newton’s second law:
md2δ
dt2=
2e2
l31δ, (3.15)
with the assumption being made that the perturbation δ << 1, such that length l1 ≈ r.The general solution of this equation is given by:
δ(t) = A exp[λ1t] +B exp[λ2t]. (3.16)
The 2 parameters λ1,2 can be found by calculating the roots of (3.15). The solution canthen be obtained very easily by substituting these roots in the general solution of (3.16).The roots of (3.15) and the solution of (3.16) are shown below:
λ1,2 = ±
√
2e2cml31
,
δ(t) = A exp
(√
2e2cml31
t
)
+B exp
(
−
√
2e2cml31
t
)
.
(3.17)
The solution calculated above shows that small perturbations increase exponentially [29].The total Coulomb force acting on bead i can now be derived by calculating the sum of
the Coulomb forces generated by all the beads present in the system, as was described inthe beginning of this section. First, consider (3.13). Assume that the charge of all beadsin the system are equal. Next the distance R in (3.13) can be defined as the distance Rij
between bead i and a bead j. This distance can be derived from the Pythagorean Theorem:
Ri,j =√
(xi − xj)2 + (yi − yj)2 + (zi − zj)2. (3.18)
With (i, j = 1, ..., N, j 6= i).Now the total Coulomb force acting on bead i, generated by all other beads j is given by:
fC =N∑
j=1j 6=i
e2
R2ij
[
ixi − xjRij
+ jyi − yjRij
+ kzi − zjRij
]
, (3.19)
14
3.1.3 Surface tension force
The surface tension force tends to restore the jet back into rectilinear shape. The surfacetension force is derived from the surface tension coefficient. This coefficient is typicallydefined as a force along a line of unit length. If this is multiplied with the surface of thejet and its curvature the surface tension force for a segment can be calculated. The surfacetension force acting on the ith bead is then given by:
fst = −απa2avki√
x2i + y2i
[ixi + jyi] . (3.20)
In this equation α is the surface tension coefficient and aav is the average radius of a seg-ment of the jet between two beads, given by:
a2av =(aui + abi)
2
4, (3.21)
with aui and abi are defined as in (3.6). The variable ki represents the jet curvature, calcu-lated using the coordinates of bead i−1, i and i+1. The curvature of the jet can be definedas:
ki =1
R, (3.22)
with R being the radius of the curvature at an arbitrary point. To approximate this radius,consider Figure 3.3. In this figure 3 beads, bead i− 1, bead i, and bead i+ 1 are shown ona curve representing the curved jet in a x− y plane. The whipping of the jet mainly occursin the x− y plane, thus in a two-dimensional (2D) space. It is then allowed to perform thecalculation of the curvature in the x−y plane, thus in 2D. This is also advantageous for thesimplicity of the calculation. Since the positions of these 3 beads are known, it is possible
y
x
r
r
r
i +1
i
i −1
a
b
Figure 3.3: Estimation of the curvature
to construct a circle through these positions. This circle is indicated by the dashed curve.It has its centre at the position (x, y) = (a, b) and has a radius r. This radius will be used
15
to estimate the curvature of the jet at the position of bead i. To determine this radius theinformation of all 3 beads is necessary. The equation of a circle is given by:
(x− a)2 + (y − b)2 = r2, (3.23)
with a and b being the position of the centre of the circle in x, respectively y direction. Thevariables x and y indicate the coordinates of a certain point on the circle. With this in mind,and the x and y positions of the 3 beads known, it is possible to calculate the radius of thecircle since the next system of equations is obtained:
(xi−1 − a)2 + (yi−1 − b)2 = r2,
(xi − a)2 + (yi − b)2 = r2,
(xi+1 − a)2 + (yi+1 − b)2 = r2.
(3.24)
The subscripts i − 1, i and i + 1 denote the x and y positions of the corresponding beads.For the complete derivation of the radius, the reader is referred to Appendix G. Now theradius of the circle constructed in Figure 3.3 is known, the curvature can be calculated veryeasily by (3.22). The curvature k is then given by:
k =1/R, where
R =
[(
xi −(yi−1 − yi)(x
2i+1 − xi2 + y2i+1 − y2i ) + (yi − yi+1)(x
2i−1 − xi2 + y2i−1 − y2i )
2((xi+1 − xi)(yi−1 − yi)− (xi−1 − xi)(yi+1 − yi))
)2
+
(
yi −(xi − xi+1)(x
2i+1 − xi2 + y2i+1 − y2i ) + (xi+1 − xi)(x
2i−1 − xi2 + y2i−1 − y2i )
2((xi+1 − xi)(yi−1 − yi)− (xi−1 − xi)(yi+1 − yi))
)2 ]1/2
.
(3.25)
3.1.4 Electric force
The electric force acting on bead i is derived from the potential difference between thependent drop and the collector. This difference yields an electric field. Since the potentialdifference in this system is only applied in the z direction, this electric field is a 1D field.The electric field can also be defined as the force it experiences per unit charge. With thisin mind, the force can then be given by:
fe = −eV0
hk. (3.26)
With h being the distance between the pendent drop and the collector.The gravitational force will be neglected, since the effect of this is very small. This can beverified by taking the mass of a particle, which is 2.83 × 10−9 kg. Then the gravitationalforce becomes Fg = m·g, yielding fg = 2.78×10−8 N. To compare, if the bead has a chargeof 2.83 × 10−9 C, a voltage of 10 kV is applied and the distance between the pendent dropand the collector is 0.2 m, then the electric field force has a magnitude of, 1.415 × 10−4N.
16
This means that the electric field force is more than 5000 times as large as the gravitationalforce.
3.1.5 Equations of motion
Since al the different forces acting on each bead are now defined, Newton’s second law canbe used to derive the equations of motion. The position vector ri = ixi + jyi + kzi of bead ican then be calculated by the following equation.
md2ridt2
=
N∑
j=1j 6=i
e2
R2ij
(ri − rj)− eV0
hk+
πa2ui (σui +G ln (lui))
lui(ri+1 − ri)
−πa2bi (σbi +G ln (lbi))
lbi(ri − ri−1)−
απa2avki√
x2i + y2i
[ixi + jyi] ,
(3.27)
with i = 1, ..., N . This equation represents the mathematical model in 3D space that willbe used to write the Matlab script. This will be discussed in the next section.
3.2 Transformation to Matlab script
To start the transformation of the mathematical model to the Matlab script, first (3.27)needs to be separated into 3 equations of motion, one for each of the 3 dimensions. Theprogram can be built by using the ode45 solver. In a separate function the differential equa-tions are defined in a state vector. That state vector now consists of 6 equations describingthe 3D positions and velocities of one bead. Furthermore, the 2 differential equations de-scribing the viscoelastic stresses σui and σbi, between bead i and i + 1 resp. bead i andi − 1 are also part of the state vector. When these differential equations are substituted inthe state vector, it will be in the form as given in (3.11), which was the form where all timederivatives were moved to one side of the equation.
3.2.1 Different ways of handling the state vector size
Since all the differential equations are defined, the Matlab script can be designed. There areseveral ways to do this. In each case, the main feature in the script is its ability to cope withthe introduction of new beads into the system and the removal of beads that have reachedthe collector plate, as was described in Section 3.1. There are a couple of ways to incorporatethis feature, as discussed below.
• New beads are introduced in the system and removed if they reach the collector.This is the most complex situation, since the state vector has a varying size. Moreover,another difficulty is that beads which have reached the collector must be removedfrom the state vector.
17
• New beads are introduced in the system and are reintroduced into the system after reachingthe collector.In this way the amount of beads and thus the amount of differential equations staysbounded within a specified amount of beads. The integrations starts with one beadand the system grows after the introduction of new beads each time. When the firstbead reaches the collector, it will be reintroduced into the system, which leaves thesize of the state vector unchanged.
• Integration starts with specified amount of beads which are reintroduced into the systemafter reaching the collector.This is the simplest way to describe the system, as the size of the state vector isspecified before the integration starts and does not change during the simulation.
For the practical use of the program, the best variant to use seems to be the last variant.This is because it consumes the least computational capacity. Furthermore, studying thedynamical behaviour of the jet will only take place at a steady state situation. This meansthat the transient behaviour of the jet, which is simulated in both the first and secondvariant is not of interest.
3.2.2 Transformation of differential equations to a state vector
The next thing to do in the construction of the Matlab script, is to transform the differentialequations that were given in this chapter into a state space notation consisting of first orderdifferential equations. This will be used as input for Matlab’s ode45 solver. To do so, all theposition, velocity and tension coordinates of the system are transformed to a new coordinatevector, called u. Note that the name of u is arbitrary. The size of u depends on the amount
18
of beads present in the system. The total state vector for one bead becomes as shown below:
u =
u1u2u3u4u5u6u7u8
ddt−→
u1 = u2
u2 =∑
j=1,Nj 6=i
e2
R2ij
(ui1 − xj1) +πa2ui (u19 +G ln (lui))
lui(u7 − u1)
−πa2bi (u20 +G ln (lui))
lbi(u1 − u13)−
απa2avki√
u21 + u23u1
u3 = u4
u4 =∑
j=1,Nj 6=i
e2
R2ij
(ui3 − xj3) +πa2ui (u19 +G ln (lui))
lui(u9 − u3)
−πa2bi (u20 +G ln (lui))
lbi(u3 − u15)−
απa2avki√
u21 + u23u3
u5 = u6
u6 =∑
j=1,Nj 6=i
e2
R2ij
(ui5 − xj5) +πa2ui (u19 +G ln (lui))
lui(u11 − u5)
−πa2bi (u20 +G ln (lui))
lbi(u1 − u13)− e
V0
h
u7 = G−G
µu19 −
G2
µln (l)
u8 = G−G
µu20 −
G2
µln (l).
(3.28)
Since this set of differential equations only applies to one bead, it must be multiplied by thenumber of beads present in the system to obtain the total state vector for the entire system.This means that the length of the total state vector is a multiple of eight. For a more detailedexplanation of the coordinate transformation and a list of the variables that are used in theMatlab simulation, the reader is referred to Appendix H.
3.2.3 Handling disappearing and new beads
The simulation of the process is able to cope with the introduction of new, and removalof old beads. To simulate this behaviour, the script must have the ability to detect beadsreaching the collector plate, and save the information of the bead at this moment. Matlab’sode45 solver has a special event detection function for this. It detects certain events, speci-fied by the user and saves the values of the state vector at that moment. By making use ofthis function, the ability arises to stop the integration immediately after a bead has reached
19
the collector. At this moment the information is saved and the bead will be reintroduced inthe system, while getting a ’clean’ initial position.
3.2.4 Dimensionless parameters
In Section 3.1.1 it was already mentioned that adopting dimensionless parameters is cus-tomary in fluid dynamics. The typical length scale L was already defined in that sectionas:
L =
√
e2
πa20G. (3.29)
Now also the time t can be made dimensionless by dividing it by the relaxation time u/G.The stress σ is made dimensionless by dividing it by the elastic modulus G, the velocityv by LG/µ and the radius a, by a0. Furthermore, W is denoted as −v. Then the nextdimensionless groups can be defined:
Q =e2µ2
L3mG2, (3.30a)
V =eV0mu2
hLmG2, (3.30b)
Fve =πa20µ
2
mLG, (3.30c)
A =απa20µ
2
mL2G2, (3.30d)
Ks =ωµ
G, (3.30e)
Kt =2πL
λ, (3.30f)
H =h
L. (3.30g)
In Appendix F, a list is given of all dimensionless parameters and the dimensionless groupsand how they are used in the equations of motion.
The dimensionless groups defined in (3.30), can now be applied to (3.27). Then thisequation is obtained in the following dimensionless form:
20
d2ridt2
=−∑
j=1,Nj 6=i
Q
l2(Lri − Lrj) + V k− Fve
σuilui · L
(Lri+1 − Lri)
+ Fveσbi
lbi · L(Lri − Lri−1) +A
1√
x2i + y2i
[ixi + jyi] .
(3.31)
The last 3 dimensionless groups that are shown in (3.30e), (3.30f) and (3.30g) areneeded for the perturbing displacements and initial conditions. Simulations can be per-formed by using the dimensionless equations. This gives a better view on the relationsbetween the lateral displacements and the vertical jet.
All the major features of the Matlab program have now been explained. In the nextchapter the results of this program will be shown and discussed.
21
Chapter 4
Results of electrospinningsimulations
With the Matlab program constructed, the jet can be simulated. In the next sections someresults of the different simulations are shown. In Chapter 3 the coordinates of the mathe-matical model were introduced. These coordinates will also be used in this chapter.
Simulations of the jet that are being made in this chapter, all have their specific parame-ter values and initial conditions. The parameter values that are used for the first simulationserve as reference material are given in Appendix D. If different values for parameters areused for certain simulations, these are given with the concerning simulation. Initial con-ditions are required for using the ode45 integration. The basic initial condition vector forone bead, has the following form:
u0 = [x0, x0, y0, y0, z0, z0, σui0, σbi0 ]T .
In the next sections, this vector will contain the values belonging to the initial conditions ofthe specific simulation discussed.
4.1 Simulating a straight jet
The first simulation that is made, is to check how the system reacts to its initial conditions.The Coulomb forces in the mathematical model are thought to cause the instability of thejet. In Subsection 3.1.2 a description was given on how these Coulomb forces push thebeads away from their equilibria after being perturbed. This means that in the ideal casewhere the beads are not perturbed, the jet will not show unstable behaviour. To check this,the initial positions x, y and corresponding velocities x, y of the beads is set to 0. The initialcondition vector for the jet at the moment the integration starts, then becomes:
u0 = [0, 0, 0, 0, h/N · i, 0, 0, 0]T ,
with h, the distance between the pendent drop and the collector. In all simulations h isset to 20 cm, N is the total amount of beads that together form the fluid jet, and i is thenumber of a specific bead in the system. After the jet is simulated, the graphs shown inFigure 4.1 are obtained.
22
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
10
5
10
15
20
xy
z
−1 −0.5 0 0.5 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x
y
Figure 4.1: Simulation with no x, y perturbation, N = 20, V0 = 10 kV, axis units are incm.
In Figure 4.1 a view on the collector plate is shown, which is the x − y plane in thefigure, next to a 3D view of the jet. In this figure it is clearly visible that all beads hit thecollector plate at the same position, i.e. they all hit the collector in the origin. This meansthat the jet is straight, hence there is no bending instability. These figures confirm thethoughts that the jet will remain straight if there are no perturbations on the jet.
4.2 Simulations with the first version of the script
When it is desired to simulate the unstable behaviour of the jet, it is necessary to introducea perturbation to the jet, as could be seen in the previous section. In Figure 4.2, the state ofthe jet is visualised before integration starts, i.e. it is a visualisation of the initial conditionsof the jet before the simulation starts. The initial conditions that belong to this figure are:
u0 = [10−3L sin(2πi/20), 0, 10−3L cos(2πi/20), 0, h/N · i, 0, 0, 0]T ,
withL the dimensionless length scale, defined Subsection 3.1.1. This initial condition vectorcauses each bead i in the system to be perturbed in such a way that the position of eachbead i in the x−y plane has different coordinates. This should give the jet a more ’realistic’behaviour.
The first simulation with this new initial condition vector, is based on the parametervalues that were given in the paper of Reneker et al. [29]. This is done since the simulationsdescribed in this chapter are dependent on these values because no parameter values areobtained with own work. The initial conditions for the following simulation are takensimilar to the initial condition vector shown above. Then the results shown in Figure 4.3are obtained. This figure shows a view on the collector plate, showing all incoming beads,next to a 3D visualisation of the trajectory of an arbitrary bead in the system. Furthermore
23
0 1 2 3 4
x 10−4
−4−202
x 10−4
0
2
4
6
8
10
12
14
16
18
20
Figure 4.2: Initial conditions of the jet, axis units are in cm.
an image can be made of the established jet at a certain moment during the simulation.This image is shown in Figure 4.4. When considering this figure some remarks can bemade. First of all one can see that the jet shown in this figure, has a more or less ’rough’curve. In other words: one of the properties of the visualised jet, is that the beads do notseem to follow each other in a smooth curve downwards, i.e., beads are passing each other.This is visualised in Figure 4.5.
The mathematical model that was used for the simulations shown in this chapter, doesnot restrict this behaviour. This is in accordance with real electrospinning experiments,something that is visualised in Figure 4.6.
Another property of the jet is the fact that it seems to restore itself to its equilibriumposition, i.e., it bends back to the origin. This property is not visible in the figures shown inthis section, however during simulations that were performed in this study, this behaviourwas discovered. The reason for this unexpected behaviour can be found in the modelingof the system. In this version of the script, the assumption was made that the simplestsimulation model of the system was one in which beads that reached the collector weredirectly removed from the system. This implies that a bead i, which is the bead closest tothe collector plate, does not have any connection anymore with its predecessor, bead i− 1,as illustrated in Figure 4.7. This is because bead i − 1 already has reached the collectorplate and thus is removed from the system. In first instance, modeling the simulationin this way, was considered to be a good assumption, since beads that have reached thecollector plate, have lost their electrical charge. Therefore they should not contribute to thebending instability of the other beads that are still present in the system. However, thisalso means that since there is no connection anymore between these 2 beads, there are noviscoelastic or surface tension forces present between them. In practice this implies that theviscoelastic and surface tension forces between between bead i and its follower, bead i+ 1,tend to restore the jet back into its equilibrium, since the Coulomb forces acting on bead ipartly vanish, as illustrated in Figure 4.7. In a real life experiment however, this behaviourwill not take place, since the resistance in the form of the viscoelastic and surface tensionforces between bead i − 1 and bead i are still present. In the next section the simulationswill be shown that are obtained by using a new program that was adapted to cope with the
24
−3
−2
−1
0
1
x 10−5
−0.02
0
0.02
0.04
0.060
5
10
15
20
z
xy−0.01 −0.005 0 0.005 0.01 0.015−0.01
−0.005
0
0.005
0.01
0.015
0.02
x
y
Figure 4.3: Simulation of the first version of the script, N = 10, V0 = 10 kV, axis unitsare in cm.
phenomenon, described above.
Coulomb force
viscoelastic and
surface tension forces
Connection broken
Bead i−1
Bead i
Bead i+1
FC
FC
Figure 4.7: Broken connection between bead i− 1 and bead i, axis units are in cm.
4.3 Simulations with the second version of the script
The second version of the script was adapted to cope with the shortcomings of the firstversion which were mentioned in the previous section. Performing the same simulationwith the new script yields the results shown in Figure 4.8. In this figure again a view isshown on the collector plate, showing the incoming beads. The area covered by the beadshitting the collector plate is approximately the same as was in the simulations performedin the previous section. However, adapting the script to remember the bead position as wasdescribed in the previous section, is thought to be a useful addition to the script. In Table
25
−6
−4
−2
0
2
x 10−3
−0.02
0
0.02
0.04−5
0
5
10
15
20
25
z
yx
Figure 4.4: Snapshot of the established jet at a certain moment during the simulation,axis units are in cm.
Influence of changing parameters
Parameter and original
value
Type of change New parameter value Effect on lateral
displacement
Amount of beadsN = 10 beads
Increase N = 100 beads Larger
Viscosityµ = 1× 105 g/(cm s)
Increase µ = 1× 107 g/(cm s) Approx. equal
Viscosityµ = 1× 105 g/(cm s)
Decrease µ = 1× 103 g/(cm s) Approx. equal
Surface tensionα = 700 g/s2
Increase α = 900 g/s2 Slightly smaller
Surface tensionα = 700 g/s2
Decrease α = 300 g/s2 Slightly larger
Elastic modulusG = 1× 106 g/(cm s2)
Increase G = 1× 107 g/(cm s2) Smaller
Elastic modulusG = 1× 106 g/(cm s2)
Decrease G = 1× 105 g/(cm s2) Larger
Initial jet radiusa0 = 150× 10−4 cm
Increase a0 = 200× 10−4 cm Larger
Initial jet radiusa0 = 150× 10−4 cm
Decrease a0 = 100× 10−4 cm Smaller
Applied voltageV0 = 10000 g1/2cm1/2/2
Increase V0 = 50000 g1/2cm1/2/2 Larger
Applied voltageV0 = 10000 g1/2cm1/2/2
Decrease V0 = 5000 g1/2cm1/2/2 Approx. equal
Table 4.1: Effects of changing parameters on bead positioning on the collector plate.
26
i
i
i+1
+1
Figure 4.5: Bead i+ 1 passes bead ion its way downward.
Figure 4.6: Photograph taken of the elec-trospinning proces, A with a capture timeof 4× 103 µs and B 18× 10−3 µs [25].
4.3 the effects are shown which were observed during simulations that were performedwhile changing some parameters. In Appendix E a more detailed discussion is providedon the effect of changing parameters on the simulation results. When considering theeffects shown in Table 4.3 and discussed in Appendix E, the first conclusion that can bedrawn is that not all parameters seem to have the same scale of influence on the simulationresults. The amount of beads has the largest effect on the dynamical behaviour of thejet, while the influence of the viscosity and surface tension coefficient on the dynamicalbehaviour of the jet seems to very limited. However, the effects of the elastic modulus andthe initial jet radius on the simulations seem to bemore noticeable. The effect of the appliedvoltage was already studied by Karunaratne et al. [20]. They discovered that increasing theapplied voltage seems to increase the lateral displacement of the jet, while decreasing thevoltage has a very limited effect. This is also confirmed by the simulation results shown inAppendix E. Therefore the next conclusions can be drawn from the simulation results:
• increasing the elastic modulus has a positive effect on the unstable behaviour of thejet, while
• increasing the initial jet radius yields the opposite effect.
The effect of the elastic modulus could be expected since a ’stiffer’ jet should be less sen-sitive for perturbations. The influence of the initial jet radius is the opposite of what onewould expect by common sense. Apart from the conclusions that were discussed above, thegeneral observation that can be made when considering the results, are the unexpected tra-jectories of the beads. Unexpected since one would expect that these trajectories would startin a straight line downward and then get the shape of a coil with increasing diameter. Thisis shown in experiments that were performed on the electrospinning process in other stud-ies. These trajectories however, show a bending curve that seems to move mainly in lateraldirection, direct from the start This behaviour can not be explained yet, however since there
27
are no experimental results obtained by own work, the assumptions of beads following astraigth line followed by a coil shaped path, as described above, cannot be verified either.
The results that were shown in this chapter gave some insight in the behaviour of theelectrospinning process and the effect that the process’ parameters have on the dynamicalbehaviour of the jet. However, one should notice that all simulations were performed with avery limited amount of beads. Since the mathematical model is based on a jet with 50,000beads it is quite obvious that the simulation results that were presented in this chapter donot give a realistic image of the influence of the process parameters on the jet behaviour. Itonly provides a rough sketch of how the process is influenced by the parameters. Further-more, it shows that jet behaviour is very dependent on the amount of beads in the system,confirming the statement that the validity of the simulation results is limited.
−2
0
2
4
6
x 10−6
−0.02
0
0.02
0.04
0.06−5
0
5
10
15
20z
xy−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01−0.035
−0.03
−0.025
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
x
y
Figure 4.8: Simulation of the second version of the script, N = 10, V0 = 10 kV, axisunits are in cm.
28
Chapter 5
Dynamical analysis
Since the system shows unstable behaviour, it is interesting to find mathematical proof forthis possible instability. Furthermore, examining the type of instability that occurs and howit is influenced by the different system parameters is also of interest. Since this system is,according to the differential equations, clearly a nonlinear system, some nonlinear tools arerequired to analyse it. Another question is whether the system possesses chaotic behaviourand if something can be found about the type of bifurcation that takes place.
5.1 Stability analysis
When looking at the dynamics of the system, an important aspect is the evolution of itstrajectories. This can be clarified when looking at the following equation:
x = f(x). (5.1)
This is a time invariant differential equation. Where x is a vector of the linearly independentsolutions x1(t), ..., xn(t) and x its time derivative.The stability of this system is given by the following definition [21].
Definition 1. The equilibrium point x = 0 of Equation (5.1) is
• stable if, for each ǫ > 0, there is δ = δ(ǫ) > 0 such that ‖x(0)‖ < δ ⇒ ‖x(t)‖ < ǫ, ∀t ≥ 0,
• unstable if it is not stable.
This means that a trajectory of a solution of (5.1) starts in a region δ it will stay theregion ǫ, for all t ≥ 0. So, by studying the trajectories, one can explain whether the systemis stable or not.
However the electrospinning system, which is discussed in this report, does not haveequilibria in the form of equilibrium points. It has an ’equilibrium trajectory’. This meansthat a bead imoves along a trajectory and shows stable or unstable behaviour in the neigh-bourhood of this trajectory. In the case of the electrospinning process there is one tra-jectory that can imaginary be drawn from the pendent drop, which has the coordinates(x, y) = (0, 0), straight down, in the negative z direction, which means that this trajectory
29
coincides with the z-axis. In Figure 5.1 this trajectory is shown together with the trajectorythat is described by a bead i. The theory that is discussed in Definition 5.1 is now not longerused for an equilibrium point, but for an equilibrium trajectory. This means that if (5.1)
Equilibrium trajectory
Trajectory
z
x y
Figure 5.1: Evolution of a trajectory around an ’equilibrium trajectory’.
is adapted to the electrospinning process, the coordinates belonging to z trajectory, i.e., theposition and velocity of the bead in the z direction need to be separated from the rest. Forall clarity the vectors x and x, which were given in (5.1), are transformed to the vectors uand u, since these vectors are also used in Chapter 3. Now, as discussed above, the ’z-part’ isseparated from the vector u. The original vector u is now separated in a vector v and z. Thevector z contains zi and zi which are the position and velocity information in z directionof bead i, while v contains the information of bead i in x and y direction and the stressesbetween bead i and its neighbours. Then the system of (5.1) becomes:
v = f(v, z). (5.2)
If the explanation of the equilibrium trajectory, which was given above, is kept in mind,then the following remark can be made. Again consider bead i. The state of this bead isdescribed by a trajectory in 3D space. During its movement on this trajectory it moves inx,y and z direction. However, since the equilibrium trajectory is drawn on the z-axis, the z-coordinate is not of interest, since the unstable behaviour of the system does not take placein this direction. Therefore studying the trajectories of the bead will only be performed inx and y direction. To study these trajectories, consider a begin volume, V0, containing thestarting points of the vector x and being surrounded by the surface Ω0 as can be seen inFigure 5.2. The volume V (t) is spanned by the trajectories of the solution vector x startingat x0 in volume V0. Now the evolution of the volume V (t) will be observed as time evolves.If V (t → ∞) = ∞, the system is unstable since the volume continues to grow as timeevolves, something that is caused by the trajectories that also grow to infinity. However, ifV (t → ∞) goes to zero, this does not necessarily mean that the system is stable. This canbe explained by looking at the figure. If, for example, x1 goes to infinity and x2 goes to zero
30
Figure 5.2: Begin volume V0, surrounded by the surface Ω0
as t → ∞, the volume V (t → ∞) does not grow. This might suggest that the system isstable, however it is clearly visible that this is not the case, since x1 goes to infinity. Withthis in mind, the conclusion that can be drawn, is that the only thing which can be saidabout stability, is that the system is certainly not stable if V (t → ∞) goes to infinity.
To study the behaviour of V when time evolves, the theorem of Liouville is used andextended for non-linear differential equations. For this theorem and the derivation for non-linear DEs, the reader is referred to Appendix I. By applying this derivation of Liouville’stheorem the next equation is obtained:
dV
dt=
∫
Ω(t)
divf(x)dx, (5.3)
with div(x) = ∂f1∂x1
+ ......+ ∂fn∂xn
= tr(∂f∂x ).In this equationΩ(t) is the surface, at a certain time t, which surrounds the volume V . Withthis equation it is possible to follow the evolution of the volume V (t), by simply looking atthe divergence of the system of differential equations. In this case, the z coordinate isnot of interest when studying the instability of the bead around a certain position on theequilibrium trajectory.
When considering 5.2 and applying (5.3) to it, the solution volume V , seems to shrink.This is due to the fact none of the differential equations shown in (3.28) give a value otherthan zero, since the time derivative of the state that each equation describes is not depen-dent on the state itself. In other words, the divergence of the system is zero, hence thevolume is shrinking according to (5.3). This means that nothing can be said in this caseabout the stability or instability of the system.
5.2 Chaos
The dynamical behaviour shown by the process might be chaotic. This statement is basedon the observations of the experiments and simulations that have been performed on theprocess in previous studies on this subject. Chaos is a term that is used in analysis ofdynamical systems. It is used to describe the complex behaviour of what is considered tobe a simple, well-behaved system [16]. The requirements for trajectories belonging to asystem with chaotic behaviour are:
31
1. bounded trajectories
2. and exponential divergence of nearby trajectories.
The above requirements cannot be satisfied simultaneously in one- or two-dimensionalstate spaces. Therefore chaotic behaviour can only occur in a 3D, or a higher-dimensionalstate space. These requirements imply that trajectories which are fulfilling these require-ments initially diverge and than curve back in the state space. It means that these trajecto-ries have the ability to remain within some bounded region while intertwining and wrap-ping around each other [16]. Experiments and simulations of the electrospinning processshow some similar behaviour.
To check whether a system is chaotic or not, the Lyapunov exponents (LEs) of the systemcan be calculated. A positive Lyapunov exponent can be an indication of a chaotic system.Lyapunov exponents provide ameasure of the degree of the instability of a trajectory [4]. Thetheory behind this is that the stability of a trajectory x(t) can be studied by looking at theevolution of its nearby trajectories x′(t), starting at an initial position x′(0). This positionis obtained by an infinitesimal vector x′(0) = x(0) + δx(0), and s(0) = |δx(0)| << 1. Ifthe distance s(t) between the reference trajectory and perturbed ones grows, it indicates oninstability of the trajectory x. However, when it grows asymptotically with time, the systemis chaotic.
So take a time dependent equation:
x(t) = f(x), (5.4)
then the rate of change between x(t) and x′(t) is given by:
s = x′ − x, (5.5)
= f(x′)− f(x), (5.6)
=df
dx
∣∣∣∣x
(x′ − x). (5.7)
Where (5.7) is the first derivative term of a Taylor series expansion.The distance∆(t) between the reference trajectory and perturbed ones grows exponentiallywith time when a system is chaotic. This relation is then given by:
∆(t) ∼ ∆(0)eγt, (5.8)
where γ is the local exponential rate of expansion.
5.3 Bifurcation
The change of the qualitative behaviour of the system is called bifurcation. This term isused to describe any sudden change in the dynamics of the system. Since the jet in theelectrospinning process starts with showing stable behaviour, but at some point changes
32
into a system that shows unstable behaviour, bifurcation theory could apply to these dy-namics. The point where the dynamics of the system change is called the bifurcation point.In the case of bifurcation, the system has a parameter that causes the stability properties ofthe system to change. In previous studies [14] [15], the length of the straight jet was alreadydiscussed. A relation was found between the jet radius and the position on the jet in thez-direction. This relation is based on scaling laws and states that the radius of the jet r isrelated to the axial distance z from the nozzle by: r ∼ zb. The exponent b is dependent onthe stage in which the jet is. In the initial (steady) stage the exponent b = −1/2. In theinstability stage b = −1/4 and in the terminal stage, which is the part of the process wherethere is no acceleration in z direction anymore, has as exponent b = 0. This accelerationis thought to be a critical factor in the length of the straight jet. According to He et al. [14]the acceleration of the jet decreases during the process. This is caused by the viscous resis-tance of the fluid during the acceleration of the jet. This resistance increases and when italmost reaches or surpasses the electric field force the acceleration becomes constant andthe jet might start to oscillate under the influence of a small perturbation. Therefore theacceleration of the jet could be identified as a possible bifurcation parameter.
33
Chapter 6
Conclusions and recommendations
6.1 Conclusions
The previous chapters described how to create a model that can simulate the electrospin-ning process and how to study the instability of the fluid jet. The different versions of theMatlab script that were used for these simulations seemed to have some shortcomings. Itwas adapted several times to cope with these shortcomings, yielding slightly better results.The electrospinning process is dependent on a lot of different parameters. Changing theseparameters will lead to changes in the process. The influence of the viscosity, the sur-face tension, the elastic modulus, the initial jet radius and the applied voltage was studied.Simulations that were performed with a different capillary height, do not show noticeablebehaviour, as was the case when changing the viscosity. Changing the surface tension co-efficient did not show noticeable differences in the dynamical behaviour. Increasing theelastic modulus did have some influence on this behaviour, something which can be ex-plained since increasing this modulus causes the jet to be ’stiffer’. The initial jet radiusseems to have a large effect on the dynamics of the jet. Increasing the radius leads to moreunstable behaviour, decreasing the radius yields the opposite. The relation between theinitial jet radius and the dynamical behaviour of the jet might be an interesting subjectfor further study, since this should be a parameter that can be changed relatively easily.Increasing the applied voltage also leads to more unstable behaviour, decreasing it doesnot have much influence on the dynamics. This all means that by changing some systemparameters, the dynamical behaviour can be influenced, at least slightly. If attempts are be-ing made to control the electrospinning process, it is important to know which parametershave a large influence on the systems dynamical behaviour and, in which order they can bechanged.
Furthermore, a start for a dynamical analysis was given. Attempts are made to find amathematical proof for the unstable behaviour of the jet. This behaviour was performed byadapting Liouville’s theorem for nonlinear differential equations. A volume containing thesolution trajectories of the system was studied. If this volume would increase with time, theinstability of the jet could be proved. However, the volume decreased, hence no statementscould be made about the stability of the jet, since a shrinking volume will not necessarilymean that the jet is stable.
34
6.2 Recommendations
Considering the conclusions that are given in this chapter, some recommendations forfuture research can be made. The first thing to do in a new study that considers the simula-tion of electrospinning processes should focus on optimizing the script, such that it is ableto perform simulations with a higher amount of beads. The simulations that were shown,were all performed with 10 or 20 beads, with one exception that was performed with 100beads. A script which is able to simulate at least 50 to 100 beads would not only increasethe quality of the graphs, but would also contribute to a more realistic simulation, since thewhipping of the jet is thought to be dependent on the amount of beads and their mutualdistances. Note that the ’reality’ of the simulation is based on comparisons with results ofexperiments that were shown in other studies. The simulation performed with 100 beadstook much computation time, something that is unwanted while performing simulations.Furthermore the script seems to be sensitive for changes in system parameters, resultingsometimes in unexpected and sometimes, very large computation times.
Further research of the instability of the jet needs to focus on finding mathematicalproof for the instability. A lack of time prohibited that this study could focus more on thistopic, as well as a mathematical proof for the existence of chaotic behaviour.Experiments on electrospinning have shown that the process starts with a straight jet. Thelength of this straight jet might be influenced by the jet radius. Future research should alsofocus on linking this length with a possible bifurcation parameter.
35
Appendix A
Scaffolds for ACL tissueengineering
Cells for tissues are often implanted into an artificial structure, called scaffold. Scaffoldsare matrices that are fashioned from natural materials such as collagen or from syntheticpolymers. They can promote cell and tissue development by providing a 3D environmentin which cells can proliferate, attach and deposit extracellular matrix [24]. Scaffolds shouldbe biodegradable, which means that they are removed by bioabsorption. This is necessaryfor the regeneration of the native tissue.
In Section 2.1 the need for a new approach in regenerating the ACL was already men-tioned. An ideal artificial ACL mimics perfectly the characteristics of a normal ACL, butunfortunately there are no synthetic grafts that have the qualifications for a lasting ACLsubstitute [10]. Scaffolds that are used for the regeneration of the ACL should ideally be re-sorbable and can temporarily withstandmechanical loads while new tissue is being formed.The load carrying capacity of the scaffold is gradually transferred to the newly formed tis-sue as the scaffold degrades. In the ideal situation, there is a balance between load transferand tissue regeneration [3]. This means that the ideal ACL replacement scaffold shouldbe biodegradable, porous, biocompatible, exhibit sufficient mechanical strength and ableto promote the formation of ligamentous tissue [5]. Here biocompatibility is defined as’acceptance of an artificial implant by the surrounding biological tissue and by the body asa whole’ [28]. The scaffold’s design is of great importance since it has to mimic the ACLfunctions according to these requirements. Furthermore, the architecture should providestructural support, whilst being porous in order to allow bioactivity throughout the struc-ture. Even though these properties by itself are already hard to achieve, one also has toconsider more factors like fibre dimensions, number of layers, pore size, biocompatibility,degradation rate and cell attachment. This makes the design of the scaffold a very compli-cated task [17]. When looking at current scaffold designs, they can be identified in fourmaincategories, which are: braided, knitted, woven and non-woven. These design are discussedbriefly in the next subsection.
36
a b c d
Figure A.1: Different scaffold designs: a) the braided scaffold, b) the knitted scaffold,c) the woven scaffold, d) the non-woven scaffold.
A.0.1 Scaffold designs
The first scaffold design to discuss, is the braided scaffold which is shown in Figure A.1a.Braided scaffolds consist of a formation of intertwined fibres. The Gore-Tex prosthesis isone of the most renowned applications of the braiding technique. It has very good mechan-ical properties and the ultimate strength of the graft is roughly 2.5 times that of the nativeACL [2]. However, the Gore-Tex prosthesis is not a scaffold. It is intended to be a permanentreplacement of the ACL. It therefore is of great importance that this prosthesis maintainsits tensile strength. Unfortunately in the years after the introduction of the Gore-Tex pros-thesis an increasing number of problems was reported. These problems included failuresof the prosthesis, ruptures and infections. Therefore the ligament was removed from themarket in 1993 [11]. However, since this prosthesis provides a good initial tensile strength,scientists optimised its architecture to use it as a scaffold. Unfortunately the new scaffolddesign shows a reduced tensile strength, making it not sufficient as scaffold for ACL tissueengineering [5], [20].
The knitted scaffold is formed by interlooped fibres in a mesh. It is shown in FigureA.1b. These loops form highly porous, square shaped scaffolds. It has excellent mechanicalproperties, however several studies reported high failure rates [34].
The geometry of woven scaffolds looks very similar to that of knitted scaffolds. It isshown in Figure A.1c. It has a geometry which is such that the fibres are interlaced. In thisway a mesh of square cells is produced. The advantage is that this structure provides initialmechanical properties that resemble the native ACL. The disadvantage is that structuralfailure is likely in the long term [27].
Non-woven scaffolds have a randomly orientated fibre structure, as is shown in FigureA.1d. In this case the fibres lay over each other. Due to the recent development of thefabrication techniques for this type of scaffold, very little is known about its mechanicalproperties. However the Young’s modulus of this structure has been measured at 230-299MPa [3]. This should be sufficient to use as an ACL scaffold since the ACL has a lowerYoung’s modulus. However since the fibre deposition is random, the architecture of thescaffold will vary from batch to batch. This will affect the mechanical properties of scaffoldsfrom different batches [20].
37
A.0.2 Scaffold materials
As was already mentioned earlier in this section, an ACL scaffold should be biodegradable,provide structural support and be porous in order to allow bioactivity throughout the struc-ture. These properties make some requirements for the material of the scaffold. In eachcase, all the materials that are used in tissue engineering should be biocompatible. Cur-rently, all materials applied in ligament tissue engineering are polymers. These polymerscan be separated in two groups: natural polymers and synthetic polymers. The most oftenused natural polymer in tissue engineering is collagen, while polylactic acid (PLA) is themost frequently used material in synthetic polymers. Collagen is reported to be so far thematerial that caused themost mature ACL regeneration. However there are some problemswith the crosslinking properties of this material. Crosslinks are essential in normal liga-ments for tensile strength characteristics, as well for resistance to chemical or enzymaticbreakdown. Crosslinking the collagen with the native ligament is difficult which causes afaster degradation of the collagen as well as a drop of mechanical properties of the scaffold.The synthetic polymer PLA has the property that it degrades within a period of between 10months and 4 years depending on its molecular weight, shape and implantation site. It hasbeen used in sheep ACL reconstruction. However, only 12.3% of the ultimate tensile loadhas been reported after 48 weeks implantation [10]. This makes PLA less useful as scaffold
Anothermaterial can be found in poly (D,L-lactide-co-glycolide) (PLGA). There has beendone some research to this material recently, to use it for electrospinning scaffold fabrica-tion [30]. Sahoo et al. [30] studied the use of this material for electrospinning nano fibresbetween loops of knitted scaffolds. It increased the surface area and reduced the pore sizeof the knitted scaffold. In this way the novel scaffold that they produced is able to facilitatecell seeding and promote cell proliferation [30]. Another study used PLGA in the electro-spinning process for non-woven scaffolds. They compared different collagen and PLGAsolutions to create long and straight nano fibre yarns. The best results were obtained witha 60% PLGA solution in a choloroform solvent [20].In the study of Li et al. [26] PLGA was used to study 3D homogeneous structures consist-ing of ultra fine fibres produced by electrospinning. They concluded that the texture of theproduced structure is similar to that of a natural extracellular matrix. It indicates that thePLGA electrospun scaffold is suitable as a tissue substitute [26].
38
Appendix B
Electrospinning history
The idea behind electrospinning is that an electrically charged fluid accelerates to a collec-tor of opposite polarity, forming a liquid jet. The fluid consists of a polymer and a solvent.While the jet travels through air, it causes the solvent to evaporate and the jet elongates.This means that with this technique, one-dimensional (1D) nanostructures can be created.Such structures have been subject of intensive research since they have unique properties.A large number of fabrication methods have been demonstrated to generate them [25].However, electrospinning seems to provide the simplest approach to create solid, as wellas hollow nanofibres. This technique is, unlike other 1D nanostructure generation tech-niques, based on the uniaxial stretching of a viscoelastic jet. If this process is comparedwith mechanical methods, the conclusion is that electrospinning is better suited to gen-erate fibres with a thin diameter. This is because the elongation of the fibre is caused byan external electric field. This technique is also suitable for producing high-volume fibres,since it is a continuous process [25].
Studying the behaviour of electrically charged thin liquid jets dates back to the work ofLord Rayleigh [29]. In 1902 the electrospinning process was patented twice, by J.F Cooleyand W.J. Morton respectively. In 1934 Anton Formhals patented several developments tofurther commercialize the process [8]. Since then over 30 U.S. patents have been issued[29]. In the 1960s several important contributions were made by Taylor. He calculated theconical shape of the pendent drop at the nozzle, now known as the ’Taylor cone’. In the1990s several publications were made, especially by the group of Reneker at the Universityof Akron. Until 1993 the process was known as electrostatic spinning. Reneker popular-ized the process by introducing the name ’electrospinning’. Since then, the amount ofpublications on the process have increased exponentially each year [24], [25].
Following the interest in this topic and the large amount of publications, electrospin-ning seems to be an interesting process, with some very interesting features, like nanofibre-reinforced composites, membranes, supports for enzymes and catalysts, sensors, materials,etc [25]. However it has a major drawback, which is the random orientation of the fibres onthe collector plate. This means that if electrospinning is used for producing ACL scaffolds,this random fibre orientation causes the architecture of the scaffold to be arbitrary. Hence,the scaffold design will vary from batch to batch. It will be clear that this causes a differencein mechanical properties of scaffolds from different batches. To fully realise the potentialof electropsinning, it is important to control the fibre arrangement on the structure. Fur-
39
thermore, it is desired to produce not only two-dimensional, but also three-dimensionalstructures. Only then electrospinning can be fully adapted to the vast range of applications.
40
Appendix C
Examples of previouselectrospinning studies
In the past years many studies have been performed on different aspects of the electrospin-ning process. Examples of some studies which focussed on the unstable behaviour of thejet and the resulting random fibre orientation are shown in below.
• Deitzel et al. [6] studied the influence of the applied voltage and the solution concen-tration on the process. They discovered that the morphology of the fibres is stronglydependent on parameters such as the feed rate of the solution, the applied voltageand material parameters, such as concentration, viscosity and surface tension.
• Reneker et al. [29] studied the reasons for the instability. Themain conclusion in theirpaper is that the instability is caused by the Coulomb forces acting on the elements ofthe jet. This is known as the Earnshaw theorem [19], [29]. It leads to the conclusionthat it is impossible to create a stable structure when the elements only interact bythe forces resulting from Coulomb’s law, as described above.
• He et al. [15] formulated a series of equations to calculate the length of the straightjet in the process, which can be seen in Figure 2.1. That paper describes that thejet, starting at the Taylor cone, accelerates due to the electrostatic force caused by theapplied voltage. However, during this acceleration, the viscous resistance decreasesthe acceleration. If the moment is reached where the acceleration becomes constant,then any small perturbation stops the straight movement and instability will be theresult.
The above mentioned research constitutes only a few examples of the large amount ofpublications in recent years. Many of the studies performed on the electrospinning processfocussed on the dynamical behaviour of the jet, and how to straighten it. This can be doneby studying the dynamics and instability, but also by finding solutions that align the fibresfrom the unstable jet. This means that instead of collecting the fibres randomly on thecollector plate, there are other methods that can be used to collect them in a more structuralway. This is desirable, since collecting the fibres in a structural way increases the numberof applications of the process. However, none of the studies so far has found a solution
41
to obtain a long continuous nanofibre yarn. Some of the possible methods that have beenattempted to align the fibres are discussed below [18].
• A cylinder with high rotational speed.
Figure C.1: A schematic rotating collector for electrospun ultrafine fibres [18].
Researchers from Virginia Commonwealth University collected the fibres using acylinder with a very high rotational speed, in the order of some thousands of rpm.The idea is that with the right speed, the fibres can be collected on the cylinder’ssurface and will be aligned in a regular pattern. This means that the cylinder has toturn fast enough. When the rotational speed of the cylinder is slower than the align-ment speed of the randomly deposited fibres, the fibres will be collected randomlyon the cylinder’s surface. However, if the rotational speed is too fast, then the speeddifference between cylinder and fibre will cause the fibrejet to break. The group hasmade some successful preliminary trials, however the chaotic motions of polymerjets make this process less controllable and a perfect alignment is therefore difficultto achieve.
• An auxiliary electrode/electrical field.
Figure C.2: Aligning electrospun fibres with an auxiliary electrical field [18].
Another method is described in US Patent 4689186. Here the fibres are wound ona tube, while employing an auxiliary electrical field. The set-up consists of a Teflontube placed in front of the jet. A charged grid, made of aluminium foil and having anegative charge was placed some 8 cm behind the collection mandrel. The mandrelrotates with a speed of 1165 rpm, above the charged grid. The effect on the alignmentwith the auxiliary electrical field was improved substantially .
• A thin wheel with sharp edge.
Theron et al. made some significant advancement in collecting aligned electrospunnanofibres. They described a new approach to collect and align nanofibres on a ta-pered and grounded wheellike bobbin. The sharpened edge of the bobbin attractedthe fibres by substantially concentrating the electric field. When the fibres are at-tached to the wheel tip and have not yet reached the electrically grounded target, they
42
Figure C.3: Aligning fibres with a thin wheel with sharp edge [18].
still have residual charges. This causes the fibre to exert a repulsive force on the nextfibre which results in a separation between the deposited fibres. The distance be-tween the fibres is dependent of the nanofibre’s diameter and residual charge. How-ever, this method is a signifcant advancement in collecting electrospun nanofibres.
Apart from the abovementioned collection techniques there are other alternative approaches.The studies that were discussed in this appendix all tried to find a solution to randomlyplaced fibres that result from the unstable behaviour of the jet. However, for the productionof tissue engineering scaffolds, these methods are not suitable to adapt to the electrospin-ning process. This is because these methods all collect a long fibre, the fibre cannot be usedany more to draw structures like a non-woven scaffold. To produce this type of scaffold thefibre needs to be aligned directly according to the scaffold design. This also means that it isrequired to control the number of fibres, each of them having a specified length.
43
Appendix D
Simulation parameters
In this appendix some parameters and their values are shown that were used during thesimulations.
System parameters
Symbol Definition Value Unit
a0 Initial cross-section radius 150× 10−4 cm
e Charge 8.48 (g1/2cm3/2)/sG Elastic modulus 1× 106 g/(cm s2)h Distance from pendent drop to grounded collector 20 cmm Mass 0.283× 10−5 g
V0 Voltage 10000 g1/2cm1/2/2α Surface tension 700 g/s2
µ Viscosity 1× 105 g/(cm s)ρ Density 1.21× 10−3 g/cm3
44
Appendix E
Effects of changing parameters onsimulation results
In this appendix more information is given about the effect of the parameter values on thesimulation results that were shown en briefly discussed in Section 4.3.
E.1 Increasing the amount of beads
The behaviour of the simulated jet is influenced by the value of the simulation parametersthat are used in the simulation. In Section 3.1 a description was given on how the Coulombforces that are acting on beads push other beads present in the system away from theirequilibria. The magnitude of this Coulomb force is dependent on the distance between 2beads and the amount of beads. This means that, when beads are closer to each other andmore beads are present in the system, the Coulomb forces have a higher magnitude. It alsomeans that the lateral movement of the jet is expected to become larger. This is verifiedwhen looking at Figure E.1. This figure shows a simulation of the jet performed with thesame parameters as the simulation belonging to Figure 4.3, except for the amount of beads.The number of beads is increased from N = 10 to N = 100. It is clearly visible that thearea covered by incoming beads, which is shown in the graph of the collector plate, is largerthan the area shown in Figure 4.8. The second graph shows the trajectory of an arbitrarybead, just as in the previous figures. However, in this figure one large trajectory is shownby the long continues curve next to some short curves. These short curves are a result ofthe way the simulation is programmed. They represent short sections of trajectories drawnby other beads, but are not part of the simulation results.
E.2 Influence of the viscosity
In Chapter 3, the cause of the instability was discussed. According to Earnshaw’s Theorem[19], a charged body in an electric field can not rest in stable equilibrium under the influenceof the electric forces alone. However, the beads in the system are not only influenced byelectric forces, but also by surface tension forces and viscoelastic forces. The viscosity of thefluid creates a certain resistance between the beads and is therefore thought to be a helpful
45
−2
−1
0
1
2
−1.5−1
−0.50
0.51
−30
−20
−10
0
10
20
z
y x−1 −0.5 0 0.5 1 1.5 2 2.5 3−2
0
2
4
6
8
10
x
y
Figure E.1: Simulation of the second version of the script, N = 100, V0 = 10 kV, axisunits are in cm.
factor in the possibilities to control the unstable behaviour of the jet. Simulations of the jetwith a higher and lower viscosity µ are shown in Figures E.2 and E.3.
In these figures one can see that a higher or lower viscosity µ does not have muchinfluence on the behaviour of the jet. In the study of Reneker et al. [29], the value of µ thatwas taken for the simulations was reported to be much larger than the zero-shear viscosityµ0 that was used for experiments. This was done since the strong longitudinal flows inthe jet lead to an increase by several order of magnitude compared to µ0. Furthermorethe viscoelastic forces are acting on both sides of a bead in opposite directions, partiallycompensating each other. Therefore, the influence of the viscosity seems to be small.
E.3 Influence of the surface tension
Another factor in the dynamics of the jet is the surface tension. The surface tension forcetends to restore the jet back to its equilibrium position. Therefore a higher surface tensioncoefficient should result in a smaller lateral displacement of the unstable jet. In Figures E.4and E.5 simulations with high, respectively low surface tension coefficients are shown.
In these figures one can see that the surface tension coefficient α seems to have someinfluence on the behaviour of the jet. The simulation with a high α show a smaller area cov-ered by incoming beads on the collector plate than the simulation with a low α. However,the influence of the surface tension coefficient in this simulation seems to be very limited.
E.4 Influence of the elastic modulus
The elastic modulus is present in both the surface tension terms as well as the viscoelasticterms of the differential equations. Unlike the viscosity, as discussed above, the elastic
46
−4−3
−2−1
01
x 10−5
−0.02
0
0.02
0.04
0.060
5
10
15
20
z
xy−0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03−6
−4
−2
0
2
4
6
8
10
12
14x 10
−3
x
y
Figure E.2: Simulation of the jet with high viscosity, N = 10, V0 = 10 kV, µ = 1 × 107
g/(cm s), axis units are in cm.
−5
0
5
10
x 10−6
−0.02
0
0.02
0.04
0.060
5
10
15
20
z
xy−0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035−4
−2
0
2
4
6
8
10
12
14x 10
−3
x
y
Figure E.3: Simulation of the jet with low viscosity, N = 10, V0 = 10 kV, µ = 1 × 102
g/(cm s), axis units are in cm.
47
−4−3
−2−1
01
x 10−5
−0.02
0
0.02
0.04
0.060
5
10
15
20
z
xy−0.01 −0.005 0 0.005 0.01 0.015 0.02−2
0
2
4
6
8
10
12
14
16x 10
−3
x
y
Figure E.4: Simulation of the jet with high surface tension coefficient, N = 10, V0 = 10kV, α = 900 g/s2, axis units are in cm.
−3
−2
−1
0
1
x 10−5
−0.01
0
0.01
0.02
0.03−5
0
5
10
15
20
z
xy−0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.06−0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
x
y
Figure E.5: Simulation of the jet with low surface tension coefficient, N = 10, V0 = 10kV, α = 300 g/s2, axis units are in cm.
48
−0.3
−0.2
−0.1
0
0.1
−0.5
0
0.5
1−5
0
5
10
15
20
z
xy−0.01 −0.005 0 0.005 0.01 0.015 0.02−0.015
−0.01
−0.005
0
0.005
0.01
0.015
x
y
Figure E.6: Simulation of the jet with high elastic modulus, N = 20, V0 = 10 kV,G = 1× 107 g/(cm s2), axis units are in cm.
modulus does have a noticable influence on the results of the simulations. Increasing thismodulus gives a smaller area of incoming beads on the collector plate, where the resultsof the simulation performed with a lower elastic modulus show a larger area. This can beseen in Figures E.6 and E.7. The first figure gives the results of experiments performedwith a high elastic modulus where the second gives the results with a low elastic modulus.
E.5 Influence of the initial jet radius
The study that was performed by He et al. [14], discussed the relation between the lengthof the straight jet and jet radius. By making use of scaling laws they found out that therelation between the radius r of the jet and the axial distance z, from the nozzle is r ∼zb, with b for each position of the jet being a negative constant or 0. This will briefly bediscussed in Section 5.3. However, for now it is important that there seems to be a relationbetween the jet radius and the length of the straight jet. To verify this 2 simulations havebeen performed, one with a larger radius, one with a lower radius. The results of thesesimulations are shown in Figures E.8 and E.9.
These figures show that increasing the jet radius leads to a larger area of incomingbeads on the collector plate, while decreasing the radius yields a smaller area.
E.6 Influence of the applied voltage
The applied voltage also has its effect on the dynamical behaviour of the jet. Althoughthe voltage is only explicitly visible in the electric force, which only acts in the negative z-direction, different experiments with different applied voltages, show the influence of the
49
−15
−10
−5
0
5
x 10−4
−0.15
−0.1
−0.05
0
0.050
5
10
15
20
z
xy−0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
x
y
Figure E.7: Simulation of the jet with low elastic modulus, N = 20, V0 = 10 kV, G =1× 105 g/(cm s2), axis units are in cm.
−4−3
−2−1
01
x 10−5
−0.02
0
0.02
0.04
0.06−5
0
5
10
15
20
z
xy−0.08 −0.06 −0.04 −0.02 0 0.02−0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
x
y
Figure E.8: Simulation of the jet with large initial jet radius, N = 10, V0 = 10 kV,a0 = 200× 10−4 cm, axis units are in cm.
50
−3
−2
−1
0
1
x 10−5
−0.01
0
0.01
0.02
0.03−5
0
5
10
15
20
z
xy−0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05−0.015
−0.01
−0.005
0
0.005
0.01
0.015
x
y
Figure E.9: Simulation of the jet with small initial jet radius, N = 10, V0 = 10 kV,a0 = 100× 10−4 cm, axis units are in cm.
applied voltage on the dynamics of the jet [20]. This is also verified by looking at FiguresE.10 and E.11.
Figure E.10 has an applied voltage which is 5 times the original voltage. In this figure,the larger area covered by incoming beads is clearly larger then in the original situation.Figure E.11 has an applied voltage which is half the original voltage. There is not a noticeabledifference between the simulations with 10 kV and 5 kV. According to the study performedby Karunaratne et al. [20], the length of the straight jet reaches its maximumwhen a voltageof 10 kV is applied. Below 10 kV the length of the straight jet decreases slightly, however itdecreases very fast when applied voltage is increased. If the length of straight jet is relatedto the magnitude of the lateral displacements in the lower part of the jet, the simulationsthat are shown in Figures E.10 and E.11 show the same behaviour as was shown in theabove mentioned study. Studying the relation between the length of the straight jet andthe lateral displacements in the lower part of the jet is interesting since a larger straightjet decreases the part where the jet shows unstable behaviour. This might give smallerlateral displacements, since the magnitude of these displacements grows until it reachesthe collector plate.
E.7 Dimensionless parameters
In Section 3.2 some dimensionless parameters were introduced. The above simulationscan also be performed with these dimensionless parameters. As mentioned in that section,doing this gives better insight in the relations between the length of the straight jet andthe lateral displacements in the unstable part of the jet. During this study some trials withdimensionless simulations were performed and they showed some nice results. However,these results are not reliable. In the script that simulated these results, some wrong as-
51
−2
0
2
4
6
x 10−6
−0.01
0
0.01
0.02
0.03
0.040
5
10
15
20
z
xy−0.02 0 0.02 0.04 0.06 0.08 0.1−0.02
−0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
x
y
Figure E.10: Simulation of the jet with high applied voltage, N = 20, V0 = 50 kV, axisunits are in cm.
−8−6
−4−2
02
x 10−3
−0.01
0
0.01
0.02−15
−10
−5
0
5
10
15
20
z
xy−0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
x
y
Figure E.11: Simulation of the jet with low applied voltage, N = 20, V0 = 5 kV, axisunits are in cm.
52
sumptions were made. This was discovered at a later moment in this study. Because ofa lack of time, the script could not be changed anymore, since it needed to be completelyrevised. In the next appendix, the definition of the dimensionless groups is provided.
53
Appendix F
Definition of dimensionless groups
The use of dimensionless descriptions is customary in fluid mechanics. To render thedifferent parameters dimensionless, the next steps are taken.To start the dimensionless description, first the length scale L is defined, which is given in
(3.4) and is defined as:
L =
√
e2
πa20G, (F.1)
where e is the charge, a0 is the initial cross-sectional radius at t = 0 and G is the elasticmodulus. With this length scale the length l can be rendered dimensionless, which gives:
l = l/L, (F.2)
with l is the dimensionless length scale. For the sequel, all dimensionless parameters aredenoted by bars. The dimensionless stress σ is given by:
σ = σ/G. (F.3)
The dimensionless time t is:
t = t/(µ/G), (F.4)
with µ the viscosity of the fluid.The dimensionless velocity v becomes:
v = LG/µ. (F.5)
The radius a is rendered dimensionless by a0, yielding:
a = a/a0. (F.6)
Now (3.27) shown in Section 3.1.5 and which is given by:
54
md2ridt2
=∑
j=1,Nj 6=i
e2
R2ij
(ri − rj)− eV0
hk+
πa2ui (σui +G ln (lui))
lui(ri+1 − ri)
−πa2bi (σbi +G ln (lbi))
lbi(ri − ri−1)−
απa2avki√
x2i + y2i
[ixi + jyi] ,
(F.7)
can be made dimensionless, performing the following steps.The acceleration term that is given in the left hand side of (F.7) can be made dimensionless
by first writing the second derivative of the position vector d2ridt2
, as dvdt . Then the left hand
side of the equation can be written as:
md(vLG
µ )
d(t µG)
Next, the non-dimensionless variables L,G and µ are eliminated from the term shownabove. Since the magnitude of the Coulomb force in (F.7) is given by e2
R2
ij
, this can be
rendered dimensionless by adapting the following dimensionless group:
Q =e2µ2
L3mG. (F.8)
This group is yielded by dividing the left hand side of the dimensionless form of (F.7) byLG/µ and multiplying it with u/G. The same procedure can be applied to the other forcespresent in the right hand side of (F.7), yielding for the other dimensionless groups:
V =eV0mu2
hLmG2, (F.9a)
Fve =πa20µ
2
mLG, (F.9b)
A =απa20µ
2
mL2G2, (F.9c)
(F.9d)
Next the parameter W is denoted as −V , then the dimensionless description of (F.7) be-comes as shown below:
55
dW
dt=−
∑
j=1,Nj 6=i
Q
l2(ri · L− rj · L) + V k− Fve
σuilui · L
(ri+1 · L− ri · L)
+ Fveσbi
lbi · L(ri · L− ri−1 · L) +A
1√
x2i + y2i
[ixi + jyi] .
(F.10)
This equation represents the dimensionless equation of motion.
56
Appendix G
Radius derivation for curvature
The radius of a circle is given by the next equation:
(x− a)2 + (y − b)2 = r2, (G.1)
with x and y the coordinates of an arbitrary point on the circle, a and b the x, respectively ycoordinate of the centre of the circle and r the radius.Since the positions of the beads are known, a circle can be drawn through 3 beads as isshown in Figure 3.3. The next equations are then obtained:
(xi−1 − a)2 + (yi−1 − b)2 = r2,
(xi − a)2 + (yi − b)2 = r2,
(xi+1 − a)2 + (yi+1 − b)2 = r2.
(G.2)
With the subscripts i − 1, i and i + 1 denoting the number of the bead. These equationscan be rewritten in the next form:
(xi − a)2 + (yi − b)2 = (xi−1 − a)2 + (yi−1 − b)2,
(xi − a)2 + (yi − b)2 = (xi+1 − a)2 + (yi+1 − b)2.(G.3)
Rewriting the equations, such that the brackets vanish, obtains the following form:
−2axi + xi2 − 2byi + y2i = −2axi+1 + xi+ 12 − 2byi+1 + y2i+1,
−2axi + xi2 − 2byi + y2i = −2axi−1 + xi− 12 − 2byi−1 + y2i−1.(G.4)
Moving all second order terms to one side of the equations gives:
57
a(xi+1 − xi) + b(yi+1 + yi) =1
2(x2i+1 − xi2 + y2i+1 − y2i ),
a(xi−1 − xi) + b(yi−1 + yi) =1
2(x2i−1 − xi2 + y2i−1 − y2i ).
(G.5)
Then the coordinates of the centre of the circle a and b can be determined by solving thefollowing system:
[ab
]
=1
2 ((xi+1 − xi) (yi−1 − yi)− (xi−1 − xi) (yi+1 − yi))·
[yi−1 − yi yi − yi+1
xi − xi+1 xi+1 − xi
] [x2i+1 − xi2 + y2i+1 − y2ix2i−1 − xi2 + y2i−1 − y2i
]
.
(G.6)
With the coordinates of the centre a and b known, the radius r can be calculated very easilyby substituting the values for a and b in one of the equations of (G.2). This gives:
r =
[(
xi −(yi−1 − yi)(x
2i+1 − xi2 + y2i+1 − y2i ) + (yi − yi+1)(x
2i−1 − xi2 + y2i−1 − y2i )
2((xi+1 − xi)(yi−1 − yi)− (xi−1 − xi)(yi+1 − yi))
)2
+
(
yi −(xi − xi+1)(x
2i+1 − xi2 + y2i+1 − y2i ) + (xi+1 − xi)(x
2i−1 − xi2 + y2i−1 − y2i )
2((xi+1 − xi)(yi−1 − yi)− (xi−1 − xi)(yi+1 − yi))
)2 ]1/2
.
(G.6)
58
Appendix H
Coordinate transformation
To use the equations of motion shown in (3.28), a coordinate transformation takes place,since the coordinates u = [u1....u8] need to be used for all beads present in the system.The size of u depends on the amount of beads present in the system. In each case, fora minimal simulation the number of beads should be at least three. This statement issupported by the equations of motion in (3.27), where the terms ri−1, ri and ri+1 representthe dependence of one bead on its two neighboring beads. This means that the minimalamount of coordinates present in the system, and thus the size of vector u, is 24 since thestate of each bead is determined by 8 differential equations.
TheMatlab script is constructed in such a way that the state vector grows, as more beadsare being introduced in the basic system. It has the minimum size of 8 DEs, while it has20 coordinates. This might be surprising since a state vector of 24 was described above.This will be explained in more detail below.
Consider again the differential equation that ware defined earlier this chapter. This setof differential equations only applies for one bead, say bead i. But also the coordinates of itsneighbours, bead i− 1 and i+ 1, are required to solve these equations. This is no problemas long as bead i has 2 neighbour beads. However, when in the simulation the first beadstarts to travel downward, it obviously does not have a predecessor, i.e. there is no beadi − 1. Therefore all the coordinates in the system that belong to bead i − 1 are given analternative value, i.e., in this case they get the same value as the coordinates of bead i itself.This is because there is no element between bead i and i− 1, since bead i is the first beadin the system. Note that for this situation, it is necessary to set some of the terms in thedifferential equations for bead i equal to zero, to prevent singularities in the equations.
Something similar happens when a new bead has just been introduced in the system.For this bead, the successor, bead i + 1, does not yet exist. This means that coordinatesin this case will get the values that belong to starting position of a new bead. To use allthese coordinates in theMatlab simulation, a coordinate transformations takes place, whereall coordinates together form a vector u, with u1 the x coordinate of bead i, u2 the timederivative of this x coordinate, u3 the y coordinate, and so on.
The next table shows how the coordinates are transformed.
59
System parameters
Coordinatein new eq.of motion
Definition Definition in orig-inal eq. of motion
Matlab coordinate
u1 x-position of bead i xi u(8 ∗ i− 7)u2 x-velocity of bead i xi u(8 ∗ i− 6)u3 y-position of bead i xi u(8 ∗ i− 5)u4 y-velocity of bead i yi u(8 ∗ i− 4)u5 z-position of bead i xi u(8 ∗ i− 3)u6 z-velocity of bead i zi u(8 ∗ i− 2)u7 x-position of bead i+ 1 xi+1 u(8 ∗ (i+ 1)− 7)u8 x-velocity of bead i+ 1 xi+1 u(8 ∗ (i+ 1)− 6)u9 y-position of bead i+ 1 xi+1 u(8 ∗ (i+ 1)− 5)u10 y-velocity of bead i+ 1 yi+1 u(8 ∗ (i+ 1)− 4)u11 z-position of bead i+ 1 xi+1 u(8 ∗ (i+ 1)− 3)u12 z-velocity of bead i+ 1 zi+1 u(8 ∗ (i+ 1)− 2)u13 x-position of bead i− 1 xi−1 u(8 ∗ (i− 1)− 7)u14 x-velocity of bead i− 1 xi−1 u(8 ∗ (i− 1)− 6)u15 y-position of bead i− 1 xi−1 u(8 ∗ (i− 1)− 5)u16 y-velocity of bead i− 1 yi−1 u(8 ∗ (i− 1)− 4)u17 z-position of bead i− 1 xi−1 u(8 ∗ (i− 1)− 3)u18 z-velocity of bead i− 1 zi−1 u(8 ∗ (i− 1)− 2)u19 stress between bead i and i+ 1 σui u(8 ∗ (i− 1)u20 stress between bead i and i− 1 σbi u(8 ∗ i)
In this table, the parameter i is used to denote the number of an arbitrary bead in thesystem.
60
Appendix I
Derivation of Liouville’s theoremfor nonlinear differential equations
Consider the homogeneous equation:
x = A(t)x, (I.1)
and the autonomous differential equation:
x = f(x). (I.2)
Next, the theorem of Liouville is used:
Theorem 1. Liouville
W (t) = exp(
t∫
t0
tr(A(τ))dτ). (I.3)
In this equation, W (t) is the Wronski-determinant of (I.1). Consider the system of(I.1). The n linearly independent solutions of this system are defined as x1(t), ..., xn(t).The solutions at time t = 0 are then given by x1(t0), ..., xn(t0) Now, define a matrixXt = (x1(t), ..., xn(t)). The determinant of this matrix can be seen as the volume of theparallelepiped spanned by the vectors x1(t), ..., xn(t). This volume is given by the Wronski-determinant. To calculate the Wronski-determinant, a n×nmatrix is constructed. With onthe first row the functions f1(x), ..., fn(x). On the second row the first derivatives of thesefunctions f ′
1(x), ..., f′n(x), the third row containing the second derivatives and so on until
the n − 1th derivative of each function. Furthermore, tr(A) is the the trace of A (tr(A)) isgiven by: tr(A) =
∑ni=1 aii.
Consider (I.2). The divergence of f(x), can be defined as
divf(x) =
n∑
i=1
∂fi∂xi
(x) (I.4)
If ξ(t, x) is the solution of the of the initial value problem of (I.2) and Ω(0) ⊂ Rn. Then
the next definition can be made.
61
Definition 1. The surface Ω(t) is equal to the solution ξ(t, x) with x being an element in Ω(0).
Ω(t) = ξ(t, x)|x ∈ Ω(0)
So the surfaceΩ(t) depends on the solution of (I.2). Now the volume V (t) in the surfaceΩ(t) can be defined as
V (t) =
∫
Ω(t)
dx (I.5)
With x is the statevector [x1....xn].Now since the integration of a surface Ω(t) yields the volume V (t) and Ω(t) is defined asstated in Definition I
V (t) =
∫
Ω(t)
det(∂ξ(t, x)
∂x)dx (I.6)
By making use of a Taylor series expansion, it can be proved that
∂ξ(t, x)
∂x= I+
∂f
∂x(x)t+O(t)(t → 0) (I.7)
Which directly leads to
det
(∂ξ(t, x)
∂x
)
= 1 + divf(x)t+O(t)(t → 0) (I.8)
Now, finally an expression for V (t) can be found. If (I.8) is substituted in (I.6) anddivided by t, the next equation is obtained:
dV
dt=
∫
Ω(t)
divf(x)dx (I.9)
With div(x) = ∂f1∂x1
+ ......+ ∂fn∂xn
= tr(∂f∂x )
62
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