A model to core-shell structured polymer nanofibers deposited via coaxial electrospinning is presented. Investigations are based on a modified Jacobi-Gauss collocation spectral method, proposed along with the Boubaker Polynomials Expansion Scheme (BPES), for providing solution to a nonlinear Lane-Emden-type equation. The spatial approximation has been based on shifted Jacobi polynomials with was n the polynomial degree. The Boubaker Polynomials Expansion Scheme (BPES) main features, concerning the embedded boundary conditions, have been outlined. The modified Jacobi-Gauss points are used as collocation nodes. Numerical examples are included to demonstrate the validity and applicability of the technique, and a comparison is made with existing results. It has been revealed that both methods are easy to implement and yield very accurate results. 1. Introduction Polymer nanofibers have gained much attention due to their great potential applications, such as filtration, catalysis, scaffolds for tissue engineering, protective clothing, sensors, electrodes electronics applications, reinforcement, and biomedical use [1–6]. Particularly, polymeric nanofibers with core-shell structure have been attractive in the past decades [4, 5]. Coaxial electrospinning, which has emerged as a method of choice due to the simplicity of the technology and its cost effectiveness, provides an effective and versatile way to fabricate such nanofibers [6–8]. This technique uses a high electric field to extract a liquid jet of polymer solution from the bot core and shell reservoirs. The yielded jet experiences stretching and bending effects due to charge repulsion and, in the process, can reach very small radii. Coaxial electrospinning cannot only be used to spin the unspinnable polymers (polyaramid, nylon, and polyaniline) into ultrafine fibers, but also ensures keeping functionalizing agents like antibacterial and biomolecules agents inside nanofibers [9–11]. In this paper, a mathematical model to coaxial electrospinning dynamics, in a particular setup, is presented. The model is based on solutions to the related Lane-Emden equation on semi-infinite domains as follows: Lane-Emden-type equations model many phenomena in mathematical physics and nanoapplications. They were first published by Lane in 1870 [12], and further explored in detail by Emden [13]. In the last decades, Lane-Emden has been used to model several phenomena such as the theory of stellar structure, quantum mechanics, astrophysics, and the theory of thermionic currents in the neighbourhood of a hot body in thermal
References
[1]
P. Carlone, G. S. Palazzo, and R. Pasquino, “Modelling of film casting manufacturing process longitudinal and transverse stretching,” Mathematical and Computer Modelling, vol. 42, no. 11-12, pp. 1325–1338, 2005.
[2]
I. I. Kudish, “Modeling of lubricant performance in Kurt Orbahn tests for viscosity modifiers based on star polymers,” Mathematical and Computer Modelling, vol. 46, no. 5-6, pp. 632–656, 2007.
[3]
M. Fang, R. P. Gilbert, and X. G. Liu, “A squeeze flow problem with a Navier slip condition,” Mathematical and Computer Modelling, vol. 52, no. 1-2, pp. 268–277, 2010.
[4]
T. Danno, H. Matsumoto, M. Nasir et al., “Fine structure of PVDF nanofiber fabricated by electrospray deposition,” Journal of Polymer Science B, vol. 46, no. 6, pp. 558–563, 2008.
[5]
W. Li, C. Liu, Y. Zhou et al., “Enhanced photocatalytic activity in anatase/TiO2(B) core-shell nanofiber,” Journal of Physical Chemistry C, vol. 112, no. 51, pp. 20539–20545, 2008.
[6]
H. Yoshimoto, Y. M. Shin, H. Terai, and J. P. Vacanti, “A biodegradable nanofiber scaffold by electrospinning and its potential for bone tissue engineering,” Biomaterials, vol. 24, no. 12, pp. 2077–2082, 2003.
[7]
G. E. Wnek, M. E. Carr, D. G. Simpson, and G. L. Bowlin, “Electrospinning of nanofiber fibrinogen structures,” Nano Letters, vol. 3, no. 2, pp. 213–216, 2003.
[8]
J. A. Matthews, G. E. Wnek, D. G. Simpson, and G. L. Bowlin, “Electrospinning of collagen nanofibers,” Biomacromolecules, vol. 3, no. 2, pp. 232–238, 2002.
[9]
C. Y. Xu, R. Inai, M. Kotaki, and S. Ramakrishna, “Aligned biodegradable nanofibrous structure: a potential scaffold for blood vessel engineering,” Biomaterials, vol. 25, no. 5, pp. 877–886, 2004.
[10]
M. D. Pierschbacher and E. Ruoslahti, “Cell attachment activity of fibronectin can be duplicated by small synthetic fragments of the molecule,” Nature, vol. 309, no. 5963, pp. 30–33, 1984.
[11]
M. Hakkarainen, “Aliphatic polyesters: abiotic and biotic degradation and degradation products,” Advances in Polymer Science, vol. 157, pp. 113–138, 2002.
[12]
J. H. Lane, “On the theoretical temperature of the sun under the hypothesis of a gaseous mass maintaining its volume by its internal heat and depending on the laws of gases known to terrestrial experiment,” The American Journal of Science and Arts, vol. 50, pp. 57–74, 1870.
[13]
R. Emden, Gaskugeln, Teubner, Leipzig, Germany, 1907.
[14]
S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover Publications, New York, NY, USA, 1957.
[15]
H. T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover Publications, New York, NY, USA, 1962.
[16]
O. W. Richardson, The Emission of Electricity from Hot Bodies, Longmans, Green and Company, London, UK, 2nd edition, 1921.
[17]
D. C. Biles, M. P. Robinson, and J. S. Spraker, “A generalization of the Lane-Emden equation,” Journal of Mathematical Analysis and Applications, vol. 273, no. 2, pp. 654–666, 2002.
[18]
G. Bluman, A. F. Cheviakov, and M. Senthilvelan, “Solution and asymptotic/blow-up behaviour of a class of nonlinear dissipative systems,” Journal of Mathematical Analysis and Applications, vol. 339, no. 2, pp. 1199–1209, 2008.
[19]
J. P. Boyd, Chebyshev and Fourier Spectral Methods, Dover, New York, NY, USA, 2nd edition, 2000.
[20]
J. Shen, “Stable and efficient spectral methods in unbounded domains using Laguerre functions,” SIAM Journal on Numerical Analysis, vol. 38, no. 4, pp. 1113–1133, 2000.
[21]
Y. Maday, B. Pernaud-Thomas, and H. Vandeven, “Reappraisal of Laguerre type spectral methods,” Office National D'Etudes et de Recherches Aerospatiales, vol. 6, pp. 13–35, 1985.
[22]
H. I. Siyyam, “Laguerre Tau methods for solving higher-order ordinary differential equations,” Journal of Computational Analysis and Applications, vol. 3, no. 2, pp. 173–182, 2001.
[23]
B. Y. Guo, “Jacobi spectral approximations to differential equations on the half line,” Journal of Computational Mathematics, vol. 18, no. 1, pp. 95–112, 2000.
[24]
A. F. Spivak, Y. A. Dzenis, and D. H. Reneker, “Model of steady state jet in the electrospinning process,” Mechanics Research Communications, vol. 27, no. 1, pp. 37–42, 2000.
[25]
A. F. Spivak and Y. A. Dzenis, “Asymptotic decay of radius of a weakly conductive viscous jet in an external electric field,” Applied Physics Letters, vol. 73, no. 21, pp. 3067–3069, 1998.
[26]
P. Barry and A. Hennessy, “Meixner-type results for Riordan arrays and associated integer sequences,” Journal of Integer Sequences, vol. 13, no. 9, pp. 1–34, 2010.
[27]
M. Agida and A. S. Kumar, “A Boubaker polynomials expansion scheme solution to random Love's equation in the case of a rational Kernel,” Electronic Journal of Theoretical Physics, vol. 7, no. 24, pp. 319–326, 2010.
[28]
A. Yildirim, S. T. Mohyud-Din, and D. H. Zhang, “Analytical solutions to the pulsed Klein-Gordon equation using Modified Variational Iteration Method (MVIM) and Boubaker Polynomials Expansion Scheme (BPES),” Computers and Mathematics with Applications, vol. 59, no. 8, pp. 2473–2477, 2010.
[29]
S. Slama, J. Bessrour, M. Bouhafs, and K. B. B. Mahmoud, “Numerical distribution of temperature as a guide to investigation of melting point maximal front spatial evolution during resistance spot welding using boubaker polynomials,” Numerical Heat Transfer A, vol. 55, no. 4, pp. 401–408, 2009.
[30]
S. Slama, M. Bouhafs, and K. B. Ben Mahmoud, “A boubaker polynomials solution to heat equation for monitoring A3 point evolution during resistance spot welding,” International Journal of Heat and Technology, vol. 26, no. 2, pp. 141–146, 2008.
[31]
S. A. H. A. E. Tabatabaei, T. Zhao, O. B. Awojoyogbe, and F. O. Moses, “Cut-off cooling velocity profiling inside a keyhole model using the Boubaker polynomials expansion scheme,” Heat and Mass Transfer, vol. 45, no. 10, pp. 1247–1251, 2009.
[32]
S. Fridjine and M. Amlouk, “A new parameter: an abacus for optimizing PVT hybrid solar device functional materials using the boubaker polynomials expansion scheme,” Modern Physics Letters B, vol. 23, no. 17, pp. 2179–2191, 2009.
[33]
A. Belhadj, J. Bessrour, M. Bouhafs, and L. Barrallier, “Experimental and theoretical cooling velocity profile inside laser welded metals using keyhole approximation and Boubaker polynomials expansion,” Journal of Thermal Analysis and Calorimetry, vol. 97, no. 3, pp. 911–915, 2009.
[34]
A. S. Kumar, “An analytical solution to applied mathematics-related Love's equation using the Boubaker polynomials expansion scheme,” Journal of the Franklin Institute, vol. 347, no. 9, pp. 1755–1761, 2010.
[35]
A. Milgram, “The stability of the Boubaker polynomials expansion scheme (BPES)-based solution to Lotka-Volterra problem,” Journal of Theoretical Biology, vol. 271, no. 1, pp. 157–158, 2011.
[36]
S. A. Theron, E. Zussman, and A. L. Yarin, “Experimental investigation of the governing parameters in the electrospinning of polymer solutions,” Polymer, vol. 45, no. 6, pp. 2017–2030, 2004.
[37]
J.-H. He, Y. Wu, and W.-W. Zuo, “Critical length of straight jet in electrospinning,” Polymer, vol. 46, no. 26, pp. 12637–12640, 2005.
[38]
J.-H. He and H.-M. Liu, “Variational approach to nonlinear problems and a review on mathematical model of electrospinning,” Nonlinear Analysis: Theory, Methods and Applications, vol. 63, no. 5–7, pp. e919–e929, 2005.
[39]
L. Xu, Y. Wu, and Y. Nawaz, “Numerical study of magnetic electrospinning processes,” Computers and Mathematics with Applications, vol. 61, no. 8, pp. 2116–2119, 2011.
[40]
C. J. Thompson, G. G. Chase, A. L. Yarin, and D. H. Reneker, “Effects of parameters on nanofiber diameter determined from electrospinning model,” Polymer, vol. 48, no. 23, pp. 6913–6922, 2007.
[41]
E. Zhmayev, D. Cho, and Y. L. Joo, “Nanofibers from gas-assisted polymer melt electrospinning,” Polymer, vol. 51, no. 18, pp. 4140–4144, 2010.
[42]
Y. M. Shin, M. M. Hohman, M. P. Brenner, and G. C. Rutledge, “Experimental characterization of electrospinning: the electrically forced jet and instabilities,” Polymer, vol. 42, no. 25, pp. 9955–9967, 2001.
