A ConfirmedModel to Polymer Core-Shell Structured Nanofibers Deposited via Coaxial Electrospinning

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 P (α,β) T ,N (x) with α,β(−1,∞),T > 0 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.


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][2][3][4][5][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][7][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][10][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 equilibrium and the thermal behaviour of an isothermal gaseous sphere [14][15][16][17][18].Even if Lane-Emden problem was numerically challenging because of the singularity behavior at the origin, several methods have been used in order to solve it in semi-infinite domains.Boyd [19] used domain truncation method replacing the semi-infinite domain with [0, K] interval by choosing K, sufficiently large, while Shen [20] used spectral-Galerkin approximations based on Laguerre functions to perform analytical solutions and demonstrated that they were stable and convergent with spectral accuracy in the Sobolev spaces.Maday et al. [21] proposed a Laguerre type spectral method, Siyyam [22] applied two numerical methods using the Laguerre Tau method, and Guo [23] reformulated the original Lane-Emden problem to a singular problem in a bounded domain by variable transformation using the Jacobi polynomials.
This paper is organized as follows.In Section 2 we present an illustrated formulation of the problem, then, in Section 3, we give an overview of the modified Jacobi-Gauss collocation spectral method along with its applications, and in Section 4, we present the fundaments and the application of the Boubaker Polynomials Expansion Scheme (BPES).In Section 5, results are plotted and discussed along wirh comparison with some existing solutions.A conclusion is given in Section 6.

Problem Formulation
As per Spivak et al. [24,25], mass balance, linear momentum balance, and electric charge balance equations describe polymer fibers electrospinning process where u is the axial velocity, J is the electrical current density, ρ is material density, and F m and F e are terms which represent viscous and electric forces, respectively.
Under the assumptions of a steady state jet and a neglectable thermal effort, the electrically generated force is dominant, the monodimensional momentum equation is hence where u is the modulus of the axial velocity, r is the radius of the jet at axial coordinate x (Figure 1), σ(x) is the surface charge density, and E is the exogenous electric field in the axial direction.By introducing the following charge balance equation: where I is the electrical current intensity and k is a constant which depends only on temperature in the case of an incompressible polymer, it gives Then, by introducing the variable it gives By differentiating the last equation, along with assuming weak r-dependence of the variable x, we have ρr 2 e y/2 ∂y ∂x = 0 (8) and by choosing the exogenous electric field profile so that 3E(x) it gives, by annexing boundary conditions ISRN Polymer Science 3
Associating with this quadrature rule, we denote by the shifted Jacobi-Gauss interpolation as follows: The shifted Jacobi-Gauss collocation method for solving (xxx) is to seek z N (x) ∈ S N (0, T), such that Let We first approximate z(x), z (x), and z (x), as (17).By substituting these approximation in (17), we get Next, after using ( 11) and ( 13) at q = 1, we have Finally, from ( 17), (19), and (20), we get (N+1) nonlinear algebraic equations which can be solved for the unknown coefficients a j by using any standard iteration technique, like Newton's iteration method.Consequently, z N (x) given in ( 18) can be evaluated, and then the approximate solution of (10) can be obtained.
The Boubaker Polynomials Expansion Scheme (BPES) is applied to the system (10) through setting the expression where B 4k are the 4k-order Boubaker polynomials, x ∈ [0, 1] is the normalized variable, r k are B 4k minimal positive roots, N 0 is a prefixed integer, and λ k | k=1...N0 are unknown pondering real coefficients.Thanks to the properties expressed by (21), boundary condition are trivially verified in advance to resolution process.The system (1) is hence reduced to  The BPES solution is obtained by determining the nonnull set of coefficients λ k | k=1...N0 that minimizes the absolute difference Δ N0 as follows: The final solution is hence

Results Plots and Discussion
Plots of the solution obtained by the modified Jacobi-Gauss collocation spectral method are presented in Figure 2, along with BPES solution.
According to the results recorded by Theron et al. [36], He et al. [37,38], Xu et al. [39], and Thompson et al. [40], velocity at the beginning of the process is jugulated along a short path then is exponentially accelerated.These features are fully verified in the present results (Figure 2).It seems that the two methods ensure the preset boundary condition expressed in (10).From a methodological point of view, the BPES resolution process forces the validity of the boundary conditions prime to establishment of the resolution algorithm while the modified Jacobi-Gauss collocation spectral method takes boundary conditions into account at the same level with the main equation.
Figure 2 displays a velocity profile which matches perfectly the conditions evoked by Zhmayev et al. [41] and Shin et al. [42].In fact, along an unitary path, jet velocity increases of about 300 percent, which is a favorable condition for fiber formation [41,42].
Additional error analyses yielded a mean relative error below 0.4% (Figure 3).

Conclusion
The Lane-Emden-type equations describe a variety of phenomena in theoretical physics and astrophysics, including polymer fibres electrospinning process dynamics.In this paper, we tried to give a founded supply to recently proposed related models.Efficient and accurate resolution schemes based on a comparative study of a modified Jacobi-Gauss collocation spectral method and the Boubaker Polynomials Expansion Scheme (BPES) have been used.
Plots and error analysis illustrations were given to demonstrate the validity and applicability of the method.The results show that the model is not very far from real assumptions and considerations.

Figure 1 :
Figure 1: Scheme of the studied setup.