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

Authors: K. Boubaker
Journal: ISRN Polymer Science
 (2012)

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## 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 polynomialsPT,N(α,β)(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.


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## Body

## 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:(1)u′′(x)+axu′(x)=g(x,u(x));0<x<∞.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 Section2 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
(2)∇·u→=0,ρ(u→·∇)u→=∇Fm+∇Fe,∇·J→=0,
where u→ is the axial velocity, J→ is the electrical current density, ρ is material density, and Fm and Fe 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(3)u∂u(x)∂x=2σ(x)E(x)ρr,
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:(4)2rσ(x)u(x)+r2kE(x)=I,
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
(5)u∂u(x)∂x=E(x)(I-r2kE(x))ρr2u.
Then, by introducing the variable
(6)y=-6Ln(u)
it gives
(7)∂y∂x=-6E(x)(I-r2kE(x))ρr2ey/2.By differentiating the last equation, along with assuming weakr-dependence of the variable x, we have
(8)∂2y∂x2-(-6(dE(x)/dx)ρr2Iey/2+12dE(x)/dxρ2E(x)ey/2-3E(x)(I-r2kE(x))ρr2ey/2∂y∂x)=0
and by choosing the exogenous electric field profile so that
(9)3E(x)(I-r2kE(x))ρr2=2xe-3Ln(u)/2-6(dE(x)/dx)ρr2Iey/2+12dE(x)/dxρ2E(x)ey/2=-6Ln(u)(-6-4Ln(-6Ln(u)))e-3Ln(u)/2
it gives, by annexing boundary conditions
(10)∂2y∂x2+2x∂y∂x-6y(x)=4y(x)Ln(y(x)),y(0)=1,y′(0)=0.


## 3. Modified Jacobi-Gauss Collocation (MJGC) Method

Letα>-1,β>-1, and Pk(α,β) be the standard Jacobi polynomial of degree k. We have
(11)Pk(α,β)(-x)=(-1)kPk(α,β)(x),Pk(α,β)(-1)=(-1)kΓ(k+β+1)k!Γ(β+1),Pk(α,β)(1)=Γ(k+β+1)k!Γ(β+1).
Then, let T>0, then the shifted Jacobi polynomial of degree k on the interval (0,T) is defined by
(12)PT,k(α,β)(x)=Pk(α,β)(2xT-1).Consequently, we have(13)PT,k(α,β)(0)=(-1)kΓ(k+β+1)k!Γ(β+1),DqPT,k(α,β)(0)=(-1)k-qΓ(k+β+1)(k+α+β+1)qTqΓ(k-q+1)Γ(q+β+1).
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 denotingxN,j(α,β)0≤j≤N, the nodes of the standard Jacobi-Gauss interpolation. The corresponding Christoffel numbers are ϖN,j(α,β)0≤j≤N. The nodes of the shifted Jacobi-Gauss interpolation on the interval (0,T) are the zeros of PT,N+1(α,β)(x), denoted by xT,N,j(α,β)=(T/2)α+β+1ϖN,j(α,β). The corresponding Christoffel numbers are ϖT,N,j(α,β)=(T/2)α+β+1ϖN,j(α,β). Let SN(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∈S2N+1(0,T)(14)∫0T(T-x)αxβΛ(x)dx=(τ2)α+β+1∫-11(1-x)α(1+x)βxβΛ((T2)xN,j(α,β)+1)dx=(τ2)α+β+1∑j=0NϖT,N,j(α,β)Λ(xT,N,j(α,β)),
where xT,N,j(α,β) and ϖT,N,j(α,β) 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
(15)SN(0,T)=span{PT,0(α,β)(x),PT,1(α,β)(x),…,PT,N(α,β)(x)}.
Thus, for any u∈SN(0,T), the norms ∥u∥wT(α,β),N and ∥u∥wT(α,β) coincide.Associating with this quadrature rule, we denote byINPT(α,β) the shifted Jacobi-Gauss interpolation as follows:
(16)INPT(α,β)u(xT,N,j(α,β))=u(xT,N,j(α,β)),0≤k≤N.The shifted Jacobi-Gauss collocation method for solving (xxx) is to seekzN(x)∈SN(0,T), such that
(17)zN′′(xT,N,k(α,β))+(z'(xT,N,k(α,β)))2+2xT,N,k(α,β)z'(xT,N,k(α,β))=4z(xT,N,k(α,β))+6;k=0,1,…,N-2,zN(i)(0)=0;i=0,1.Let(18)zN(x)=∑j=0NajPT,j(α,β)(x),a=(a0,a1,…aN).We first approximatez(x),z′(x), and z′′(x), as (17). By substituting these approximation in (17), we get
(19)14∑j=0Naj(j+α+β+1)PT,j-2(α+2,β+2)(xT,N,k(α,β))+12∑j=0Naj(j+α+β+1)PT,j-1(α+1,β+1)(xT,N,k(α,β))2+2xT,N,k(α,β)12∑j=0Naj(j+α+β+1)PT,j-1(α+1,β+1)(xT,N,k(α,β))+4∑j=0NajPT,j(α,β)(xT,N,k(α,β))+6.Next, after using (11) and (13) at q=1, we have
(20)∑j=0N(-1)jΓ(j+β+1)j!Γ(β+1)aj=0,∑j=0N(-1)j-1Γ(j+β+1)(j+α+β+1)T(j-1)!Γ(β+2)aj=0.Finally, from (17), (19), and (20), we get (N+1) nonlinear algebraic equations which can be solved for the unknown coefficients aj by using any standard iteration technique, like Newton’s iteration method. Consequently, zN(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
(21)∑q=1NB4q(x)|x=0=-2N≠0,∑q=1NB4q(x)|x=rq=0,∑q=1NdB4q(x)dx|x=0=0,∑q=1NdB4q(x)dx|x=rq=∑q=1NHq,withHn=B4n′(rn)=(4rn[2-rn2]×∑q=1nB4q2(rn)B4(n+1)(rn)+4rn3).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
(22)u(x)=12N0∑k=1N0λk×B4k(xrk),
where B4k are the 4k-order Boubaker polynomials, x∈[0,1] is the normalized variable, rk are B4k minimal positive roots, N0 is a prefixed integer, and λk|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
(23)12N0∑k=1N0λk×d2B4k(xrk)dx2+1xN0∑k=1N0λk×dB4k(xrk)dx-3N0∑k=1N0λk×B4k(xrk)=2N0∑k=1N0λk×B4k(xrk)Ln(1N0∑k=1N0λk×B4k(xrk)).The BPES solution is obtained by determining the non-null set of coefficientsλ~k|k=1…N0 that minimizes the absolute difference ΔN0 as follows:
(24)ΔN0=|(12N0∑k=1N0λ~k×Λk)-(12N0∑k=1N0λ~k×Λk′)|withΛk=rk2∫01d2B4kdx2(x×rk)dx,Λk′=-rk∫01((6+4Ln(1N0∑k=1N0B4k(x×rk)))2x×dB4kdx(x×rk)+(6+4Ln(1N0∑k=1N0B4k(x×rk)))×B4k(x×rk)(6+4Ln(1N0∑k=1N0B4k(x×rk))))dx.The final solution is hence(25)u(x)=12N0∑k=1N0λ~k×B4k(xrk).


## 5. Results Plots and Discussion

Plots of the solution obtained by the modified Jacobi-Gauss collocation spectral method are presented in Figure2, 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.Figure2 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% (Figure3).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.



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*Source: 603108-2012-10-14.xml*