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

Abstract

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.

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: 𝑢^′′(𝑥)+𝑎𝑥𝑢^′(𝑥)=𝑔(𝑥,𝑢(𝑥));0<𝑥<∞. (1)

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 [14–18]. 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.

2. Problem Formulation

As per Spivak et al. [24, 25], mass balance, linear momentum balance, and electric charge balance equations describe polymer fibers electrospinning process ∇⋅^→𝑢𝜌=0,_→𝑢⋅∇^→𝑢=∇𝐹^𝑚+∇𝐹^𝑒,∇⋅^→𝐽=0, (2) 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 𝑢𝜕𝑢(𝑥)=(𝑥)𝐸(𝑥)𝜕𝑥2𝜎,𝜌𝑟 (3) 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.

Figure 1 

Scheme of the studied setup.

By introducing the following charge balance equation: 2𝑟𝜎(𝑥)𝑢(𝑥)+𝑟²𝑘𝐸(𝑥)=𝐼, (4) 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 𝑢𝜕𝑢(𝑥)=𝐸(𝑥𝜕𝑥)𝐼−𝑟²𝑘𝐸(𝑥)𝜌𝑟²𝑢. (5) Then, by introducing the variable 𝑦=−6Ln(𝑢) (6) it gives 𝜕𝑦(𝑥𝜕𝑥=−6𝐸)𝐼−𝑟²𝑘𝐸(𝑥)𝜌𝑟²𝑒^𝑦/2. (7)

By differentiating the last equation, along with assuming weak r-dependence of the variable x, we have 𝜕²𝑦𝜕𝑥²−−6(𝑑𝐸(𝑥)/𝑑𝑥)𝜌𝑟²𝐼𝑒^𝑦/2+12𝑑𝐸(𝑥)/𝑑𝑥𝜌²𝐸(𝑥)𝑒^𝑦/2−3𝐸(𝑥)𝐼−𝑟²𝑘𝐸(𝑥)𝜌𝑟²𝑒^𝑦/2𝜕𝑦𝜕𝑥=0 (8) and by choosing the exogenous electric field profile so that 3𝐸(𝑥)𝐼−𝑟²𝑘𝐸(𝑥)𝜌𝑟²=2𝑥𝑒^{−3Ln(𝑢)/2}−6(𝑑𝐸(𝑥)/𝑑𝑥)𝜌𝑟²𝐼𝑒^𝑦/2+12𝑑𝐸(𝑥)/𝑑𝑥𝜌²𝐸(𝑥)𝑒^𝑦/2=−6Ln(𝑢)(−6−4Ln(−6Ln(𝑢)))𝑒^{−3Ln(𝑢)/2} (9) it gives, by annexing boundary conditions 𝜕²𝑦𝜕𝑥²+2𝑥𝜕𝑦(𝑥)(𝑥)𝑦(𝑥),𝑦(0)𝑦𝜕𝑥−6𝑦=4𝑦Ln=1,_′(0)=0. (10)

3. Modified Jacobi-Gauss Collocation (MJGC) Method

Let 𝛼>−1,𝛽>−1, and 𝑃_𝑘^(𝛼,𝛽) be the standard Jacobi polynomial of degree 𝑘. We have 𝑃_𝑘^(𝛼,𝛽)(−𝑥)=(−1)^𝑘𝑃_𝑘^(𝛼,𝛽)(𝑥),𝑃_𝑘^(𝛼,𝛽)()=()−1−1^𝑘Γ𝑘+𝛽+1,𝑃𝑘!Γ𝛽+1_{𝑘(𝛼,𝛽)}(1)=Γ𝑘+𝛽+1.𝑘!Γ𝛽+1 (11) Then, let 𝑇>0, then the shifted Jacobi polynomial of degree 𝑘 on the interval (0,𝑇) is defined by 𝑃^(𝛼,𝛽)_𝑇,𝑘(𝑥)=𝑃_𝑘^(𝛼,𝛽)2𝑥𝑇−1. (12)

Consequently, we have 𝑃^(𝛼,𝛽)_𝑇,𝑘(0)=(−1)^𝑘Γ𝑘+𝛽+1,𝐷𝑘!Γ𝛽+1_𝑞𝑃^(𝛼,𝛽)_𝑇,𝑘(0)=(−1)^𝑘−𝑞Γ𝑘+𝛽+1𝑘+𝛼+𝛽+1𝑞𝑇^𝑞ΓΓ.𝑘−𝑞+1𝑞+𝛽+1 (13) For 𝛼=𝛽, one recovers the shifted ultraspherical polynomials (symmetric shifted Jacobi polynomials) and for 𝛼=𝛽=∓1/2,𝛼=𝛽=0 the shifted Chebyshev of the first and second kinds and shifted Legendre polynomials, respectively; for the nonsymmetric shifted Jacobi polynomials, the two important special cases 𝛼=−𝛽=±1/2 (shifted Chebyshev polynomials of the third and fourth kinds) are also recovered.

Jacobi-Gauss interpolation starts by denoting 𝑥^(𝛼,𝛽)_𝑁,𝑗0≤𝑗≤𝑁, the nodes of the standard Jacobi-Gauss interpolation. The corresponding Christoffel numbers are 𝜛^(𝛼,𝛽)_𝑁,𝑗0≤𝑗≤𝑁. The nodes of the shifted Jacobi-Gauss interpolation on the interval (0,𝑇) are the zeros of 𝑃^(𝛼,𝛽)_𝑇,𝑁+1(𝑥), denoted by 𝑥^(𝛼,𝛽)_{𝑇,𝑁,𝑗}=(𝑇/2)^𝛼+𝛽+1𝜛^(𝛼,𝛽)_𝑁,𝑗. The corresponding Christoffel numbers are 𝜛^(𝛼,𝛽)_{𝑇,𝑁,𝑗}=(𝑇/2)^𝛼+𝛽+1𝜛^(𝛼,𝛽)_𝑁,𝑗. Let 𝑆_𝑁(0,𝑇) be the set of polynomials of degree at most 𝑁. Thanks to the property of the standard Jacobi-Gauss quadrature, it follows that for any Λ∈𝑆_2𝑁+1(0,𝑇)^𝑇₀(𝑇−𝑥)^𝛼𝑥^𝛽Λ(𝑥)=𝜏𝑑𝑥2^𝛼+𝛽+11₋₁(1−𝑥)^𝛼(1+𝑥)^𝛽𝑥^𝛽Λ𝑇2𝑥^(𝛼,𝛽)_𝑁,𝑗=𝜏+1𝑑𝑥2^{𝑁𝛼+𝛽+1}_𝑗=0𝜛^(𝛼,𝛽)_{𝑇,𝑁,𝑗}Λ𝑥^(𝛼,𝛽)_{𝑇,𝑁,𝑗}, (14) where 𝑥^(𝛼,𝛽)_{𝑇,𝑁,𝑗} and 𝜛^(𝛼,𝛽)_{𝑇,𝑁,𝑗} are the nodes and the corresponding weights of the shifted Jacobi-Gauss-quadrature formula on the interval (0,𝑇), respectively.

For solving (10), we first set 𝑆_𝑁(0,𝑇)=span𝑃^(𝛼,𝛽)_𝑇,0(𝑥),𝑃^(𝛼,𝛽)_𝑇,1(𝑥),…,𝑃^(𝛼,𝛽)_𝑇,𝑁(𝑥). (15) Thus, for any 𝑢∈𝑆_𝑁(0,𝑇), the norms ‖𝑢‖_𝑤𝑇^(𝛼,𝛽)_,𝑁 and ‖𝑢‖_𝑤𝑇^(𝛼,𝛽) coincide.

Associating with this quadrature rule, we denote by 𝐼^{𝑃𝑇(𝛼,𝛽)}_𝑁 the shifted Jacobi-Gauss interpolation as follows: 𝐼^{𝑃𝑇(𝛼,𝛽)}_𝑁𝑢𝑥^(𝛼,𝛽)_{𝑇,𝑁,𝑗}=𝑢𝑥^(𝛼,𝛽)_{𝑇,𝑁,𝑗},0≤𝑘≤𝑁. (16)

The shifted Jacobi-Gauss collocation method for solving (xxx) is to seek 𝑧_𝑁(𝑥)∈𝑆_𝑁(0,𝑇), such that 𝑧_𝑁^′′𝑥^(𝛼,𝛽)_{𝑇,𝑁,𝑘}+𝑧^′𝑥^(𝛼,𝛽)_{𝑇,𝑁,𝑘}²+2𝑥^(𝛼,𝛽)_{𝑇,𝑁,𝑘}𝑧^′𝑥^(𝛼,𝛽)_{𝑇,𝑁,𝑘}=4𝑧𝑥^(𝛼,𝛽)_{𝑇,𝑁,𝑘}𝑧+6;𝑘=0,1,…,𝑁−2,_𝑁(𝑖)(0)=0;𝑖=0,1. (17)

Let 𝑧_𝑁(𝑥)=^𝑁_𝑗=0𝑎_𝑗𝑃^(𝛼,𝛽)_𝑇,𝑗(𝑥),𝑎=𝑎₀,𝑎₁,…𝑎_𝑁. (18)

We first approximate 𝑧(𝑥),𝑧^′(𝑥), and 𝑧^′′(𝑥), as (17). By substituting these approximation in (17), we get 14^𝑁_𝑗=0𝑎_𝑗𝑗+𝛼+𝛽+1𝑃^{(𝛼+2,𝛽+2)}_{𝑇,𝑗−2}𝑥^(𝛼,𝛽)_{𝑇,𝑁,𝑘}+12^𝑁_𝑗=0𝑎_𝑗𝑗+𝛼+𝛽+1𝑃^{(𝛼+1,𝛽+1)}_{𝑇,𝑗−1}𝑥^(𝛼,𝛽)_{𝑇,𝑁,𝑘}²+2𝑥^(𝛼,𝛽)_{𝑇,𝑁,𝑘}12^𝑁_𝑗=0𝑎_𝑗𝑗+𝛼+𝛽+1𝑃^{(𝛼+1,𝛽+1)}_{𝑇,𝑗−1}𝑥^(𝛼,𝛽)_{𝑇,𝑁,𝑘}+4^𝑁_𝑗=0𝑎_𝑗𝑃^(𝛼,𝛽)_𝑇,𝑗𝑥^(𝛼,𝛽)_{𝑇,𝑁,𝑘}+6. (19)

Next, after using (11) and (13) at 𝑞=1, we have ^𝑁_𝑗=0(−1)^𝑗Γ𝑗+𝛽+1𝑎𝑗!Γ𝛽+1^𝑗=0,_𝑁𝑗=0(−1)^𝑗−1Γ𝑗+𝛽+1𝑗+𝛼+𝛽+1𝑇𝑎𝑗−1!Γ𝛽+2^𝑗=0. (20)

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.

4. The Boubaker Polynomials Expansion Scheme (BPES)

The Boubaker Polynomials Expansion Scheme (BPES) [26–35] is a resolution protocol which has been successfully applied to several applied physics and mathematics problems. The BPES protocol ensures the validity of the related boundary conditions regardless main equation features. The BPES is mainly based on Boubaker polynomials first derivatives properties ^𝑁_𝑞=1𝐵_4𝑞(𝑥)_𝑥=0=−2𝑁≠0,_𝑁𝑞=1𝐵_4𝑞(𝑥)_{𝑥=𝑟𝑞}=0,_𝑁𝑞=1𝑑𝐵_4𝑞(𝑥)𝑑𝑥_𝑥=0=0,_𝑁𝑞=1𝑑𝐵_4𝑞(𝑥)𝑑𝑥_{𝑥=𝑟𝑞}=^𝑁_𝑞=1𝐻_𝑞,with𝐻_𝑛=𝐵^′_4𝑛𝑟_𝑛=4𝑟_𝑛2−𝑟²_𝑛×∑^𝑛_𝑞=1𝐵²_4𝑞𝑟_𝑛𝐵_4()𝑛+1𝑟_𝑛+4𝑟³_𝑛. (21)

Several solution have been proposed through the BPES in many fields such as numerical analysis [26], material characterization [27], theoretical physics [28], mathematical algorithms [29], heat transfer [31, 32], and homodynamics [33–35].

The Boubaker Polynomials Expansion Scheme (BPES) is applied to the system (10) through setting the expression 𝑢(𝑥)=12𝑁₀^𝑁0_𝑘=1𝜆_𝑘×𝐵_4𝑘𝑥𝑟_𝑘, (22) where 𝐵_4𝑘 are the 4𝑘-order Boubaker polynomials, 𝑥∈[0,1] is the normalized variable, 𝑟_𝑘 are 𝐵_4𝑘 minimal positive roots, 𝑁₀ 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 12𝑁₀^𝑁0_𝑘=1𝜆_𝑘×𝑑²𝐵_4𝑘𝑥𝑟_𝑘𝑑𝑥²+1𝑥𝑁₀^𝑁0_𝑘=1𝜆_𝑘×𝑑𝐵_4𝑘𝑥𝑟_𝑘−3𝑑𝑥𝑁₀^𝑁0_𝑘=1𝜆_𝑘×𝐵_4𝑘𝑥𝑟_𝑘=2𝑁₀^𝑁0_𝑘=1𝜆_𝑘×𝐵_4𝑘𝑥𝑟_𝑘1Ln𝑁₀^𝑁0_𝑘=1𝜆_𝑘×𝐵_4𝑘𝑥𝑟_𝑘. (23)

The BPES solution is obtained by determining the non-null set of coefficients 𝜆_𝑘|_{𝑘=1…𝑁0} that minimizes the absolute difference Δ_𝑁0 as follows: Δ_𝑁0=12𝑁₀^𝑁0_𝑘=1𝜆_𝑘×Λ_𝑘−12𝑁₀^𝑁0_𝑘=1𝜆_𝑘×Λ^′_𝑘Λwith_𝑘=𝑟²_𝑘¹₀𝑑²𝐵_4𝑘𝑑𝑥²𝑥×𝑟_𝑘Λ𝑑𝑥,_′𝑘=−𝑟_𝑘¹₀2𝑥×𝑑𝐵_4𝑘𝑑𝑥𝑥×𝑟_𝑘+16+4Ln𝑁₀^𝑁0_𝑘=1𝐵_4𝑘𝑥×𝑟_𝑘×𝐵_4𝑘𝑥×𝑟_𝑘𝑑𝑥. (24)

The final solution is hence 𝑢(𝑥)=12𝑁₀^𝑁0_𝑘=1𝜆_𝑘×𝐵_4𝑘𝑥𝑟_𝑘. (25)

5. 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.

Figure 2 

Solution plots.

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).

Figure 3 

Mean error analyses.

6. 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.