Int. J. Experimental and Computational Biomechanics, Vol. 1, No. 1, 2009 45 Modelling of mechanical properties of electrospun nanofibre network Xiaofan Wei, Zhenhai Xia, Shing-Chung Wong* and Avinash Baji Department of Mechanical Engineering, The University of Akron, Akron, OH 44325 3903, USA Fax: (330) 972 6027 E-mail: xiaofan@uakron.edu E-mail: zxia@uakron.edu E-mail: swong@uakron.edu E-mail: avinashbaji@yahoo.com *Corresponding author Abstract: Electrospun nanofibres are widely investigated as extra-cellular matrix for tissue engineering and biomedical applications. Little is understood on the deformation mechanics of spun fibre mats. A model is developed to predict the deformation behaviour of randomly-oriented electrospun nanofibre network/mats with the fibre-fibre fusion and van der Waals interaction. The nanofibres in the mat are represented by chains of beads; the interactions between the beads are described by bonded (stretch, bending and torsion) and non-bonded (van der Waals) potentials. Stress-strain curves and dynamics fracture are predicted by this model. The results show that the fibre-fibre fusion has a significant effect on the tensile strength of the mats. Increasing the number of fusion points in the mat results in an increase in strength, but over-fusion may lead to lower fracture energy. The predicted stress-strain relationships are consistent with the experimental results. Keywords: electrospinning; nanofibres; molecular dynamics simulation; MD; fibre fusion; mechanical properties. Reference to this paper should be made as follows: Wei, X., Xia, Z., Wong, S-C. and Baji, A. (2009) Modelling of mechanical properties of electrospun nanofibre network, Int. J. Experimental and Computational Biomechanics, Vol. 1, No. 1, pp.45 57. Biographical notes: Xiaofan Wei received his MS and PhD in Mechanical Engineering from the University of Akron. He is working as a Postdoctoral Research Fellow in the Department of Mechanical Engineering at the University of Akron. His research interests include structural analysis, FEA, stochastic analysis and optimisation. His recent research focus is on modelling of mechanical properties of electrospun nanofibres. Zhenhai Xia, PhD, is an Assistant Professor of Mechanical Engineering at the University of Akron. His research interests are focused on nanomechanics and nanomaterials, including polymer, ceramic and metal composites, multifunctional materials, carbon nanotubes, thin films and fibrillar materials. Shing-Chung Wong received his PhD in Mechanical Engineering from the University of Sydney. He is an Associate Professor in the Department of Mechanical Engineering, University of Akron. His research interests include bio-inspired composite design, electrospinning of polymer nanofibres and tissue scaffolds. Copyright 2009 Inderscience Enterprises Ltd.
46 X. Wei et al. Avinash Baji received his MS in Biomedical Engineering from the University of Akron. He is currently a Doctoral student in Mechanical Engineering at the University of Akron. His research interest includes electrospinning of polymer nanofibres and their applications in the field of tissue engineering. 1 Introduction Electrospun nanofibres are widely studied as extracellular matrix for tissue engineering (Sell et al., 2007; Burger et al., 2006; Norman and Desai, 2006). Electrospinning (Wong et al., 2007b, 2007c) occurs when electrical forces at the surface of a solution or melt overcome the surface tension and cause an electrically charged jet to be ejected. The electric force accelerates and stretches the polymer jet resulting in a decrease of the diameter and a concomitant increase in the length. The cross-sectional area of the fibre decreases by a factor as large as a million and vice versa for the fibre length. In view of the versatile applications of electrospun fibres in drug delivery, wound dressing and tissue engineering (Wouterson et al., 2007a, 2007b), mechanical properties of the materials are critical to their success in future applications. Little work is done on the mechanical deformation of electrospun nanofibre mats and fabrics. There is little understanding on how spun fabrics deform and fail under mechanical load. How does the fused fibre network affect the deformation mechanisms? This lack of understanding may ultimately prevent a material from being applied regardless of its other merits. In our research group we have experimentally examined the tensile deformation properties of electrospun polymers as a function fibre diameter (Wong et al., 2007b). Modelling the deformation mechanics can provide fruitful insights in: 1 the tensile strength variation of woven fabrics without resorting to handling the micro and nanoscale fibres and measuring small loads in a laboratory 2 how fabrics deform and fracture in a mechanistic fashion. Our study will also create new frontiers for research to understand the non-woven fibre fabrics in load bearing fabrications such as in nanostructured composite materials (Wouterson et al., 2007a, 2007b; Wong et al., 2007a). Due to the difficulty in handling micro and nanoscale fibres and measuring small loads for deformation, mechanical properties of electrospun nanofibres have not been characterised with accuracy until recent years. Theoretical analyses based on continuum mechanics approach and experimental investigations are unable to address the effects of molecular and chemical interactions at materials interfaces, such as the interaction between fibres. Molecular dynamics (MD) simulation provides the needed molecular interactions at the interfaces and the effect of interfacial strength can be included in analyses. Fusion among the electrospun nanofibres is a sophisticated process. It is considered as a messy process because of the drastic disruption and combination of the molecules under electrical, extensional, gravitational and chemical forces. Kozlov and Markin (1983) proposed the classical model of membrane fusion to mimic a fusion pathway and calculate the free energy barrier. Noguchi and Takasu (2001) studied the fusion of two small bi-layer vesicles using a solvent-free model. Wang et al. (2003) developed a 2-D
Modelling of mechanical properties of electrospun nanofibre network 47 nano-scale finite element method to simulate polymer field deformation using the bead model. A coarse-grained model for a set of three polyimide isomers is developed by Clancy and Hinkley (2004). Curgul et al. (2007) presented the simulation results of amorphous polymer nanofibres to investigate their size-dependent and thermal properties by using MD simulations and the bead model. In the present work, a simple model considering the tensile deformation of the geometrically confined electrospun nanofibres is being presented. The uniqueness of our model is the network consisting of many randomly oriented filaments. Each filament may be linked to each other due to external forces. An MD simulation under the assumption of fibre fusion at intersection of fibres and interaction by van der Waals forces is presented. Using this model, the strength of fibre-fibre fusion and its effect on tensile deformation is investigated. The fracture energy and strength of the fibre fabrics are calculated. The process of dynamic deformation and fracture is observed in the MD simulation. This model provides insights in the failure mechanisms of the nanofibre network. 2 MD simulation of electrospun nanofibre network 2.1 Model description In electrospinning process (shown in Figure 1(a)), nanofibres are deposited onto a surface to form nanofibre network/mats, see Figure 1(b). Since the jet trajectory in the electrospinning process of nanofibres is in a form of complicated three-dimensional whipping way, most nanofibres obtained so far are in non-woven fabrics. The non-woven mat is usually hot-calendered or embossed at the softening point of fibres as shown in Figure 1(c). Figure 1(c) shows randomly oriented filaments in the non-woven fibre mat. Figure 1 A schematic diagram illustrating electrospinning of randomly oriented nanofibres, (a) the electrospinning process (b) the actual electrospinning setup in our laboratory and (c) the scanning electron micrograph (SEM) of the electrospun nanofibre network (see online version for colours) (a) (b) (c) A model is presented to study the randomly distributed nanofibre network. In this model, the nanofibres are represented by a chain of beads or coarse grain ; each bead is therefore a small segment of nanofibres. Figure 2 shows an example of the bead model.
48 X. Wei et al. The aspect ratio of the segment l/d is chosen to be a unit such that the beads represent the fibre segments more closely. In the bead model, the chains are held together by bond interactions. This model takes beads as joints and bonds as rods in the equivalent frame-like structure. Figure 2 A schematic illustrating the model of randomly-oriented nanofibres in the nanofibre network (see online version for colours) In electrospun nanofibre mats, the inter-fibre contact could lead to fusion due to solvent evaporation and chemical reaction. Such fibre-fibre fusion is assumed to be a chemical bond like the intra-molecular bonding in the model. The fusions are randomly distributed in the nanofibre network/mats. Non-fused type fibre-fibre interaction is subjected to van der Waals forces. Figure 3 shows the fibre-fibre fusion model in the simulation. Figure 3 Fibre fusion model in the simulation (see online version for colours) 2.2 Potential energy for bead model Molecular dynamics can predict accurately the interactions between constituent phases at the atomic scale. The time evolution of a set of interacting atoms and the motion equation integration are the important components in modelling. The motion equations follow Newton s second law as
Modelling of mechanical properties of electrospun nanofibre network 49 F = ma (1) i i i where F i is the force acted on the atom (or particle) i, and m i and a i are the mass and acceleration of atom i in a system of N atoms, respectively. The motion trajectories of atoms are calculated in a phase space by numerical methods of integration. In the bead model, the bead trajectories can be calculated using similar methods. Generally, the potential energy describing the atomic interactions in an organic material has many forms. For a carbon and hydrogen system, force field is a simple and effective approach for describing the atomic potential of interacting atoms consisting of many different atom types. In our simulations, the bead model for the generic (united-atom) hydrocarbon polymer molecules is utilised. Based on the nature of the restoration force, moment and torque, the covalent bond chain is considered as a chain of elastic rods connected by elastic joints. Rods support the restoration force; elastic joints support restoration bending force and torque. Thus, the force field can be determined by the sum of the individual energy contributions from each degree of freedom of the bead model system of N atoms as follows: Total potential energy: U = Us + Ub + Ut + Uvan (2) where the first three terms U s, U b and U t on the right-hand side are bonded energies. U s is the bond stretching energy, U b the bending energy, U t the torsion energy, U van the non-bonded energy. The bonded energy can be expressed as follows: Stretching energy: 1 Us = k r r 2 ( ) 2 s 0 (3) bonds Bending energy: 1 Ub = k θ θ 2 ( ) 2 b 0 (4) angles Torsion energy: 1 Ut = k ϕ ϕ 2 ( ) 2 t 0 (5) angles where k s is the bond stretching force constant, r the distance between atoms, r 0 the equilibrium distance between atoms, k b the bond angle bending force constant, θ the bond angle, θ 0 the equilibrium bond angle, k t the torsional resistance constant, φ the bond twisting angle and φ 0 the equilibrium bond twisting angle. As shown in Figure 4, the local geometry of chemical bonds of most materials can be characterised by a bond length r, bending angle θ and torsion angle φ. Chemical bond deformation includes bond stretches, bond angle bends and bond angle torsion.
50 X. Wei et al. Figure 4 Configurations of polymer chains (see online version for colours) From the viewpoint of structural mechanics and assuming bead bond as round section, the strain energy of the deformation of fibre network can be given by 1 EA 2 1 EI 2 1 GJ 2 U = US + UB + UT = ( ΔL) + ( Δα ) + ( Δβ ) (6) 2 L 2 L 2 L where U S, U B and U T are strain energies for axial tension, bending and torsion, respectively. L is the length of the equivalent beam, Δ L is the axial stretching deformation, Δα denotes the rotational angle at the end of the beam and Δβ is the relative rotation between the ends of the beam. EA, EI and GJ are the tension resistance, flexural rigidity, torsional stiffness, respectively. Considering the energy equivalence from equation (2) to equation (6), the constants k s, k b and k t are related to materials constants as follows: EA EI GJ = ks = kb = kt (7) L L L For randomly oriented electrospun nanofibre fabrics, nonlinear deformation behaviour is assumed, despite experimental evidence from uniaxially loaded electrospun fibres and aligned fibres show a high degree of linear elasticity prior to nonlinear deformation at failure. To account for the nonlinear deformation of the nanofibres, we modify the bond stretching constant k s in stretching energy term in equation (2) when the stretching deformation exceeds a yield strain. The modified stretching constant represents the deformation in plastic deformation regime. The non-bonded energy U vdw in equation (2) is induced by fibre-fibre interactions due to van der Waals effects. The fibre-fibre interaction energy is determined by integrating Lennard-Jones potential (Leckband and Israelachvili, 2001) and can be approximated as: 2 2 A B U( D) = π ρ R + 7 1260D 6D (8) where A and B are Lennard-Jones parameters, ρ and R are the density and radius of the fibre, respectively and D is the surface distance between beads. 2.3 Simulation procedures Generating the randomly distributed nanofibre network that can represent the structure of the electrospun fibre mats is the first step to the success of the model to predict the mechanical properties. To create a randomly-distributed fibre network, we first generate the fibres with different orientations, as shown in Figure 5(a) and then apply force on each bead of the fibres, forcing them to move toward a target surface and deposit on it.
Modelling of mechanical properties of electrospun nanofibre network 51 Such a procedure simulates electrospinning process and can generate the uniform structure with certain thickness (shown in Figure 5(b)) very close to electrospinning fibre fabric morphology (shown in Figure 1(c)). Figure 5 Simulation steps of the randomly-oriented nanofibre network, (a) force field applied on the randomly-oriented nanofibre network and (b) nanofibre network appearance in the simulation (a) (b) Figure 6 Loaded nanofibre network under tension Table 1 The parameters used in the calculation Radius, R (nm) 100 Young s modulus, E (GPa) 2 Poisson s ratio 0.35 Yield strain (%) 0.1 Failure strain (%) 200 Modulus after yielding 0.2
52 X. Wei et al. In our simulations, the nanofibre network is generated into a square sample consisting of several hundred randomly oriented filaments. The thickness of the network model for the calculation of stress is determined by averaging the distance of all the beads to the deposit surface. After the sample is generated, the fibre-fibre fusion points are introduced randomly. To introduce the fusion points, we search all the possible fibre crosses in the nanofibre network and then randomly choose some of them. The fusion degree is defined as the ratio of the number of fusion contact points and the total cross points in the nanofibre network. The ratio of selected to total crossing points is used as a measure of the fusion in the mat. The fibre beads at these selected fibre crosses are bonded with each other by chemical bonds. We assume that these interfibre bonds are the same as the bonds in the fibres. To test the nanofibre networks, the beads at one end of the sample are fixed, while uniform displacement is applied to the beads at the other end of the sample (shown in Figure 6). During the loading, force can be calculated and the effects of varying levels of fusion in the network/mats on the stress, strength and fracture energy are then predicted. The parameters used in the calculation are listed in Table 1. 3 Results and discussion Figure 7 shows the simulation results of fibre-fibre fusions with three different conditions: 1 complete fusion 2 half fusion 3 no fusion. Figure 7 Simulation results of three different levels of fibre-fibre fusion
Modelling of mechanical properties of electrospun nanofibre network 53 It is observed that Young s modulus and tensile strength of the nanofibre network with no fusion are only half of those with fused network either in full or half fusions. What is very interesting is both full and half fusions show little difference in both the Young s modulus and tensile strength. These results are consistent with the experimental work on the electrospun polymer nanofibre mats (You et al., 2006), where they observed that fibre fusion leads to a significant increase in tensile strength. Our work suggests that to enhance tensile properties of electrospun nanofibre mats/fabrics, it is critical to provide fusion contact points between fibres. It does not, however, require fully fused fibre network to obtain substantially higher tensile properties. We attempt to elucidate the deformation mechanics using this simple stick-and-bead model with the following assumptions: 1 The stick and bead can provide insights in how a nanofibre composed of complex polymer morphology deforms. 2 The stick and beads are orders of magnitude larger than the radius of gyration (~10 nm) of polymers, constituting a nanofibre consisting of tens of thousands of polymer chains in crystalline and amorphous distributions. 3 The usual nanofibre length is in the same order of magnitude as the simulated stick-and-bead length. 4 The fused bonding is representative of fibre-fibre interaction due to electrospun solvent evaporation in addition to van der Waals interaction. At the beginning of applied tension, the inter-atomic forces from stretching, bending and torsion have an approximate linear relationship with the deformation in the low strain region. Thus, the tensile properties of the nanofibre network exhibit the initial linear stress-strain relationship. With the increasing strain, the fibres deform plastically, leading to the nonlinear deformation. Figure 8 is a representative tensile stress-strain curve of a polycaprolactone fibre mat using a nanoforce tensile tester. As can be observed, Figures 7 and 8 share comparable similarity in the stress-strain deformation. Note that Figures 7 and 8 are derived from different polymer systems. The load reaches a maximum prior to a rapid reduction and failure ensues. In our simulation, we can observe individual fibres breaking as load increases. The formation of fibre fractures during loading contributes to the nonlinear deformation behaviour of the overall fibre mats/fabrics. The increased tensile strength is attributed to the enhanced load transfer between the fibres due to half or full fusions. In as-fabricated mat, the fibre-fibre load transfer occurs through weak van der Waals interactions, as well as mechanical inter-locking. The introduction of the fusion points significantly enhances the load transfer between the fibres, thus enhancing the tensile strength. Fracture energy is calculated by integrating the area under the stress-strain curves, as shown in Figure 7. The fracture energy and tensile strength relationships with the levels of fusion in the nanofibre network are shown in Figure 9. It should be noted that the fracture energy in the nanofibre initially rises and then decrease with an increase of the normalised fusion, whereas the failure strength increases continuously with an increase in fusion. When a fibre is broken in the mat, the load on the fibre must be transferred to its neighbouring fibres. This observation is indicative of the fracture mode operative with increased fusion. The highly fused fibres create stress concentration as individual fibres break and triaxial tension leads to a catastrophic failure of the mats and thereby reducing
54 X. Wei et al. the overall fracture energy. In the simulated fibre network with increased fibre-fibre fusion, the additional load due to fibre rupture with rising crack-tip triaxiality cannot be effectively transferred to neighbouring fibres, providing weak links and resulting in catastrophic failure and reduced fracture energy. This is different from the trend of failure strength whereby the maximum stress continues to rise as the fusion point s increase. Figure 8 A representative stress-strain curve of randomly oriented polycaprolactone fibres mats tested using an MTS NanoBionix Figure 9 The fracture energy and strength vs. the fusion levels (see online version for colours)
Modelling of mechanical properties of electrospun nanofibre network 55 To elucidate the failure mechanisms of the nanofibre mats with different fusion degrees, the nanofibre network deformation is examined through controlled loading steps. Figure 10 shows the deformation of the nanofibre network with full fusion in the simulation. Before the loading reaches its maximum, an initial crack-like damage consisting of a cluster of fibre rupture points becomes visible in the studied sample. At maximum load, the small cluster of fibre rupture points grows spontaneously without further increasing the applied load. This can be identified as the onset of crack growth with the total energy release rate is sufficient to overcome that required for crack propagation. At last, the nanofibre network ruptures along the crack as a rhombus with an approximate 45 angle against the tensile axis. Figure 10 Fracture process in the nanofibre network with full fusion 4 Conclusions A model for the randomly-distributed electrospun nanofibre network was created to investigate the complex fibre-fibre interaction, deformation and failure of nanofibre mats. The deformation mechanics and macroscopic properties are correlated. The model consists of chains of beads to represent the nanofibres, where fibre-fibre interaction and fusion can be readily controlled and varied for our study. The influence of the fibre-fibre fusion on tensile properties of the nanofibre network is predicted. Our results show that the tensile strength is significantly increased with an increase in the fibre-fibre fusion level. Network with zero fused strength except for the van der Waals forces exhibits less than half of the tensile load in deformation. Fracture energy density shows an initial rise followed by a sustained decrease as fusion increases. The latter is indicative of the fracture mode operative as individual nanofibres break contributing to the stress concentrations and crack-tip triaxiality, thus resulting in catastrophic failure. Our results suggest that the fibre-fibre fusion plays an important role in the overall deformation and fracture mechanisms of spun nanofibre network. The deformation process of the nanofibre network was examined in controlled loading steps. Direct visualisation of deformation process enables a good understanding of the failure mechanism of nanofibres. The model captures the important deformation mechanics of the nanofibre network and provides a reasonable description of tensile properties of nanofibres consistent with
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