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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 Liouvilles 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 ma