Establishment of a one-dimensional physical model for electrospinning multiple jets of spinnerets

Abstract

Electrospinning is a technique used to prepare nanofiber material where an electrically charged jet of polymer solution is accelerated and stretched by electrostatic forces. There are two stages in electrospinning, the stable jet and bending instability. Almost all electrospinning research involves the theory and model of a single jet’s movement. Many devices for the mass production of nanofibers have been developed and studied to date. The multiple jets of spinnerets can form throughput nanofibers. However, a greater understanding is needed on the formation mechanism of these multiple jets, which would allow researchers to control nanofiber productivity and instruct spinneret design. In this work, we investigated the formation mechanism of multiple jets based on an annular spinneret. A one-dimensional physical model was established and multiple equations were combined; the wavelength $λ$ with respect to the applied voltage $U$ was obtained based on the derivation of the theoretical formulation. The theory order relationship was verified with experiments using multiple jets containing the annular spinneret.

Keywords

Multiple jets formation mechanism nanofiber theoretical formulation experimental verification

Introduction

Electrospinning is an effective technique for producing nanofiber material because of its simplicity, convenience, and applicability.¹ Compared to the traditional textile fibers, nanofibers have many advantages, such as a high ratio of surface area, the quantum size effect, and high porosity, which can be used to develop advanced materials and devices such as air filtration,^2,3 water treatment,⁴ thermal and moisture management textiles,⁵ flexible batteries,⁶ sensors,⁷ tissue engineering scaffold materials,⁸ and wound dressings.^9,10 In order to provide a theory on which to base a scientific guide for nanofiber preparation, it is necessary to study the formation of an electrospinning jet.

Theoretical research on electrical jet movement has been a topic of interest for a decade. Previous efforts to present the instability of an electrified liquid surface were made in the work of Zeleny.^11,12 In these experimental studies, the phenomenon of electric discharge from a liquid surface was observed and recorded. Zeleny’s work could be regarded as the precedent of electrospinning and electrospraying. For electrospinning, Taylor¹³ observed that there are two different stages in the process of electrospinning, which were an initial stable jet and a later bending instability. Feng.^14,15 presented a slender-body theory for an initial stable jet and explored the effects of extension thinning and thickening. The instability of a jet was analyzed and explained using a bead-spring model,¹⁶ which was employed to calculate the three-dimensional paths of continuous jets. A more detailed conservation of the charge, the surface tension forces in the jet axial direction, and the presence of a Newtonian viscous solvent were added to the bead-spring model,¹⁷ and the jet profile predictions given by the modified model greatly improved agreement with the experimental data. Spivak et al.¹⁸ proposed a continuum-type model for a stable jet during electrospinning. Hohman et al.^19,20 established an electrospinning instability theory.

The above studies demonstrate the large amount of research on the theory and model of a single jet movement during electrospinning. It is well known that the lower throughput of nanofibers has limited their development using a conventional needle electrospinning apparatus. Therefore, efforts have concentrated on improving the productivity of the electrospinning technique. Lu et al.²¹ introduced super-high throughput with a needleless electrospinning apparatus that used a rotary cone as a spinneret to produce nanofibers. A needleless electrospinning setup called needleless twisted wire electrospinning was developed to produce nanofibers.²² One stepped pyramid-shaped spinneret was used for high throughput production of nanofibers and core-shell nanofibers.^23,24 A novel method for a micro-bubble solution system was also investigated based on free surface electrospinning.²⁵ In addition, many different needleless spinnerets have been designed and studied.^26–31 These methods form multiple jets that significantly improve the throughput of nanofibers. However, few reports exist regarding the formation mechanism of multiple jets. Electrohydrodynamics instability plays an important role in the formation of a polymer solution jet. The research on electrohydrodynamics instability has been previously studied.^32–34 Forward et al.^35,36 analyzed a system in which a wire electrode was swept (in a rotary motion) through a bath containing a polymeric solution, as well as the formation process of liquid droplets on the wire and electrostatic jetting from each liquid droplet. A generalized approach for the self-organization of jets in electrospinning with a free liquid surface was studied.³⁷ Further research reports on the formation mechanism of multiple jets has not been reported. Hence, more in-depth extensive studies on the formation mechanism of multiple jets are needed to enable throughput control and structure adjustment of nanofibers.

In this study, we investigated the formation mechanism of multiple jets based on a novel spinneret. A one-dimensional physical model was established. The Navier-Stokes equation, electrohydrodynamic equation, free surface continuity equation, and wave motion equation were combined. The relationship equation of the wavelength $λ$ with respect to the applied voltage $U$ based on the derivation of the theoretical formulation was obtained. Finally, an experiment with multiple jets was conducted using the novel spinneret, and the theory relationship equation was verified by consistency between the experimental and theoretical values.

Experimental section

Materials

Polyacrylonitrile (PAN) from the Shanghai Chemical Fibers Institute at a molecular weight of 75,000 g·mol⁻¹ was used as the model polymer. N, N-Dimethylformamide (DMF) was used as a solvent and purchased from the Shanghai Lingfeng Chemical Reagent Co., Ltd. The PAN powder was dried at 80°C in a vacuum for 24 h, then dissolved in DMF at a solution concentration of 10 wt%. The obtained solutions were stirred magnetically over a period of time until they became a homogeneous solution.

Needleless electrospinning setup

The annular spinneret electrospinning setup has been previously reported by our group.³⁸ This setup includes four parts: The annular spinneret, automatic liquid supply system, high voltage supply power (GMMA, USA), and nanofiber collector. In needleless electrospinning experiments, the PAN polymer solution was transported from an insulated infusion tube to the narrow slit of the annular spinneret. The inner diameter and the outer diameter of the insulated infusion tube were 5 mm and 3 mm, respectively. The annular spinneret was connected to the positive electrode of the high-voltage power supply. The nanofiber collector could rotate at 70 r/min. Additionally, the nanofiber collector was connected to a ground line to ensure that a high-voltage electric field (between 50 and 70 kV) was formed between the annular spinneret and the nanofiber collector.

Measurement and characterization

The formation process of multiple jets was observed by a digital camera (Cannon 650D, Japan). The surface tension of the polymer solution was determined by a DCAT11 surface tensiometer (Data physics, Germany).

Results and discussion

The formation mechanism of multiple jets

In the process of electrospinning, we found that multiple jets with even stability could be formed on top of the annular spinneret. A high static electricity voltage was applied to the novel spinneret to form an electric field between the spinneret and the grounded drum collector. When the applied voltage gradually increased to a critical value, the electric force overcame the surface tension and gravity of the polymer solution, and the liquid became unstable and formed small wave peaks and valleys. A large number of small wave peaks could generate multiple jets on the surface of the polymer solution. The formation of multiple jets greatly enhanced nanofiber throughput.

To study the formation mechanism of multiple jets on the annular spinneret, a one-dimensional multiple jets model was established. The one-dimensional static and dynamic models of the free surface electrospinning can be seen in Figure 1(a) and (b), respectively. First, the polymer solution is categorized as a viscous fluid, so its characteristics obey the typical hydrodynamics equation, which is also named the Navier-Stokes equation, as follows:

\frac{\partial u}{\partial t} + (u \cdot \nabla) u = f - \frac{1}{ρ} \nabla p + μ Δ u

(1)

Where

u

is the solution velocity,

t

is the time,

f

is the external force of the polymer solution,

p

is the solution pressure,

μ

is the kinematic viscosity,

ρ

is the density of polymer solution,

\nabla

is the Hamiltonian,

∆

is the Laplace Operator, and

\nabla = i (\partial / \partial x) + j (\partial / \partial y) + k (\partial / \partial z), ∆ = \partial^{2} / {\partial x}^{2} + \partial^{2} / {\partial y}^{2} + \partial^{2} / {\partial z}^{2}

Figure 1.

One-dimensional (a) Static model and (b) Dynamic model of free surface electrospinning.

According to the theory of wave motion, the ratio of the migration acceleration and local acceleration can be expressed as follows:

\frac{(u \cdot \nabla) u}{\partial u / \partial t} \sim \frac{a}{λ}

(2)

Where

a

is the amplitude and a first-order quantity,

λ

is the wavelength and a first-order small quantity.

The wave motion of the polymer solution on the free surface belongs to the situation of small amplitude, where the amplitude $a$ is much smaller than the wavelength $λ$ . Therefore, $(u \cdot \nabla) u$ is much less than $\partial u / \partial t$ , thus allowing it to be ignored in equation (1). In addition, the polymer solution is assumed to be an incompressible homogeneous fluid. Therefore, $μ Δ u$ can be assumed to be zero. The external force of $f$ can be grouped into the solution pressure $p$ . Finally, equation (1) is simplified to the following equation.3³⁷

ρ \frac{\partial u}{\partial t} + \nabla p = 0

(3)

As mentioned above, the polymer solution is assumed to be an incompressible homogeneous fluid, and the free solution has its potential. Therefore, a small disturbance is categorized as irrotational motion with the following equation:

\nabla \times u = 0

(4)

According to Laplace’s irrotational motion, if $\nabla \times u = 0$ , then there must exist a velocity potential function $φ$ , so $u = \nabla φ$ . We could obtain $\nabla \times (\nabla φ) = 0$ , that is to say, a velocity potential that satisfies the Laplace equation. Therefore, a velocity potential function was introduced into equation (3), and the continuity equation of the wave motion could be obtained as follows:

\nabla (ρ \frac{\partial φ}{\partial t} + p) = 0

(5)

According to equation (5), the total pressure is related to the velocity potential function. The total pressure $p$ determines the movement of the polymer solution on the gas-liquid two-phase interface, which includes capillary pressure, the electric field force, and gravity. Therefore, the total pressure $p$ can be expressed as:

p = p_{l} + p_{e} + p_{g}

(6)

Where

p_{l}

is the capillary pressure,

p_{e}

is the electric field force, and

p_{g}

is gravity.

According to the Laplace equation, the capillary pressure $p_{l}$ can be expressed as:

p_{l} = - γ \frac{\partial^{2} ξ}{{\partial x}^{2}}

(7)

Where

ξ

is the height of the wave.

Based on the theory of electrohydrodynamics, the electric field force $p_{e}$ can be expressed as:

p_{e} = - \frac{1}{2} ε_{0} E_{0}^{2} - \frac{1}{2} ε_{0} ε_{p} (ε_{p} - 1) E_{p}^{2} k ξ

(8)

Where

ε_{0}

is the vacuum dielectric constant,

ε_{p}

is the dielectric constant of the polymer solution,

E_{0}

is the electric field strength of the spinneret,

E_{p}

is the electric field strength of the solution surface,

k

is the wave number, and

ξ

is the height of the wave.

According to the solution pressure, the gravity $p_{g}$ can be calculated as follows:

p_{g} = ρ g ξ

(9)

Where

ρ

is the density of the solution,

g

is the gravity acceleration, and

ξ

is the height of the wave.

According to Equations (5)–(9), we can obtain the final continuity equation of the wave motion:

ρ \frac{\partial φ}{\partial t} + ρ g ξ - γ \frac{\partial^{2} ξ}{{\partial x}^{2}} - \frac{1}{2} ε_{0} ε_{p} (ε_{p} - 1) E_{p}^{2} k ξ = 0

(10)

Based on this expression, we can introduce the wave equation to describe the movement of the polymer solution. The wave equation is expressed as:

ξ = A e^{[i (k x - ω t)]}

(11)

Where

A

is the amplitude,

i

is the pure imaginary number,

k

is the wave number, x is the horizontal displacement,

ω

is the frequency, and

t

is the time.

Based on a previous report,³⁵ the velocity potential function obeys the wave equation, which is set as follows:

φ (x, z, t) = B e^{k z} ξ

(12)

According to the definition of the velocity potential, we found that the velocity of the z axis was equal to the velocity potential of the critical surface z, and the equation below could be obtained:

\frac{\partial ξ (x, t)}{\partial t} = \frac{\partial φ (x, z, t)}{\partial z}

(13)

The coefficient B can be obtained according to equation (13),

B = - \frac{i ω A}{k}

(14)

According to Equations (10), (12), and (14), the square of the frequency $ω^{2}$ can be obtained as follows:

ω^{2} = [ρ g + γ k^{2} - \frac{1}{2} ε_{0} ε_{p} (ε_{p} - 1) E_{p}^{2} k] \frac{k}{ρ}

(15)

Equation (15) is the wave frequency. We found that the square of the frequency $ω^{2}$ was closely related to solution density, surface tension, wave number, and electric field of the solution. When $ω^{2} > 0$ , it indicated that there was no wave formation. In contrast, when $ω^{2}$ < 0, it indicated that there was wave formation. To analyze $ω^{2}$ , we set $ω^{2} = 0$ , $\frac{k}{ρ} \neq 0$ , and the following equation could be obtained:

ρ g + γ k^{2} - \frac{1}{2} ε_{0} ε_{p} (ε_{p} - 1) E_{p}^{2} k = 0

(16)

According to equation (16), the wave number $k$ was obtained as follows:

k = \frac{{ε_{0} ε_{p} (ε_{p} - 1) E}_{p}^{2}}{4 γ}

(17)

According to the relationship of the wave number $k$ and the theoretical wavelength $λ_{t},$ the following equation could be obtained:

λ_{t} = \frac{2 π}{k}

(18)

According to Equation (17) and Equation (18), the theoretical wavelength $λ_{t}$ is:

λ_{t} = \frac{8 π γ}{ε_{0} ε_{p} (ε_{p} - 1) E_{p}^{2}}

(19)

According to equation (19), the order relationship between $λ_{t}$ and $E_{p}$ becomes:

λ_{t} \sim E_{p}^{- 2}

(20)

In addition, the electric field strength of the solution

E_{p}

could be obtained through a previously reported formula,³⁰ and

E_{p}

is then expressed as follows:

E_{p} = \frac{U}{ε_{p} d - (ε_{p} - 1) h}

(21)

According to equation (21), the order relationship between $U$ and $E_{p}$ can be estimated as:

U \sim E_{p}

(22)

According to Equation (20) and Equation (22), the order relationship between $λ_{t}$ and $U$ becomes:

λ_{t} \sim U^{- 2}

(23)

Equation (23) represents the order relationship between the theoretical wavelength $λ_{t}$ and the applied voltage $U$ . The next section describes how the experimental details were used to verify the theoretical results of the wavelength and applied voltage.

Effect of applied voltage on multiple jets during electrospinning experiments

Figure 2(a)–(f) shows that multiple jets were recorded using a digital camera when the applied voltage increased from 50 kV to 75 kV. The SEM images the prepared PAN electrospun fiber could be seen in our previous research report.³⁸ From Figure 2(a)–(f), we found that multiple jets were successfully formed on the top of the annular spinneret, which demonstrated that annular spinneret needleless electrospinning had great potential in improving the multiple jets. The images of the digital camera showed that the interval distance between the two jets was not completely equal. This situation was caused by the instability and randomness of the polymer solution in a high-voltage electric field. It was difficult for us to distinguish and observe every jet when the annular spinneret produced multiple jets. Therefore, to clearly record the number of multiple jets, half of the annular slit was used to produce multiple jets, and the other half of the annular slit was blocked. The current recorded jet number was multiplied by two to obtain the total number of jets for a certain applied voltage.

Figure 2.

Digital camera images of needleless electrospinning with multiple jets with different applied voltages (a) 50 kV, (b) 55 kV, (c) 60 kV, (d) 65 kV, (e) 70 kV, (f) 75 kV.

Another notable issue was the interval distance between jets. We found that the applied voltage had significant importance for the number of jets. To further study the multiple jets, we first defined the interval distance as the experiment wavelength, $λ_{e}$ . This was approximated by the perimeter of the annular spinneret and the number of multiple jets. For the diameter of the annular spinneret $r$ , the perimeter of the annular spinneret was presented with the formula of $2 π r$ . Therefore, the wavelength of the experiment $λ_{e}$ could be expressed as follows:

λ_{e} = \frac{2 π r}{n}

(24)

Where

n

is the number of multiple jets,

r

is the diameter of the annular spinneret, and

λ_{e}

is the wavelength of the experiment.

Figure 3(a) shows the effect of the applied voltage on the multiple jets. The formation of multiple jets is a dynamic process, that often leads to a variety of values of $n,$ so a standard error was used to express the wavelength of the experiment. Figure 3(a) displays the relationship between the number of jets and the applied voltage. When the applied voltage was 50 kV, the number of jets was 9 ± 2. However, more jets could form from the solution surface with an increase in the applied voltage. The number of jets could reach 21 ± 1 when the applied voltage increased to 75 kV. The results thus indicate that the number of multiple jets increase with increasing applied voltage. The reason for this finding is that the instability of the solution surface becomes more apparent when the applied voltage is increased, more small peaks on the solution surface produce multiple jets. In addition, the polymer solution was categorized as a free surface liquid, and the status of the liquid continuously changed with the effect of different applied voltages. We found that the float range for the number of jets was reduced with an increase in the applied voltage. This result shows that the formation of multiple jets tends towards stability for the higher applied voltage during electrospinning. Figure 3(b) shows the effect of the applied voltage on the wavelength of the experiment. These data could be obtained according to equation (24). The wavelength of the experiment gradually decreased when the applied voltage increased. Furthermore, the wavelength of the experiment decreased from 11.58 mm to 4.96 mm when the applied voltage increased from 50 kV to 75 kV. The reason for this finding was that the interval between two jets decreased when the number of jets increased.

Figure 3.

Effect of applied voltages on the (a) Number of jets and (b) Wavelength of the solution during the electrospinning experiment.

Verification of the relationship between the applied voltage and wavelength with dimensionless processing

The formation of multiple jets was largely determined by the electric force on the top of the annular spinneret. The throughput of the nanofiber depended on the number of multiple jets in the process of needleless electrospinning. Therefore, it was very important to find and verify the specific relationship between the applied voltage and wavelength. Multiple jets were investigated, the experimental details were determined, and the wavelength $λ_{e}$ in the experiment was calculated with equation (24). In addition, the effect of the applied voltage on the number of jets and the wavelength in the experiment is shown in Figure 3(a) and (b). The theoretical wavelength $λ_{t}$ was obtained with a series of formula deductions. We adopted the experimental wavelength results and a dimensionless processing method to discuss and verify the theoretical wavelength. The relationship between the applied voltage and the wavelength with the dimensionless processing method is shown in Figure 4. The straight line fitting was drawn according to six different points. The equation of the straight line was $y = - 1.98 x - 0.85$ , and the square of the correlation coefficient was 0.99. For the dimensionless processing, the slope of 1.98 expressed that the relationship between the applied voltage and the wavelength in the experiment had a power of −1.98. Based on the above theoretical analysis, we determined that the relationship between the applied voltage and the theoretical wavelength had a power of −2. It was noteworthy that the experimental value was very close to the theoretical value. The results demonstrated the rationality of the theoretical analysis for the relationship between the applied voltage and wavelength. This finding could further expose the inner mechanism of multiple jets formation in needleless electrospinning. In addition, it was very essential for us to control the productivity of the nanofiber and design the structure of needleless electrospinning spinneret.

Figure 4.

The relationship between the applied voltage and the wavelength with dimensionless processing.

Conclusions

In this work, the formation mechanism of multiple jets was investigated based on the annular spinneret. The one-dimensional static state and dynamic state models were established. The wavelength theory, the Navier-Stokes equation, and the dispersion relationship equation were used to derive the relationship between the theoretical wavelength and the applied voltage. We determined that the relationship between the applied voltage and the theoretical wavelength had a power of −2. What surprised us was that the relationship between the applied voltage and the wavelength in the experiment had a power of −1.98. It was noteworthy that the experimental value was very close to the theoretical value. The findings from this work will enable further exploration and understanding of the formation mechanism of multiple jets for needleless electrospinning. Meanwhile, the results will further guide the spinneret design and industrial production of nanofibers.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by The Scientific Research Program Funded by the Shaanxi Provincial Education Department; 21JY015, The National Key Research and Development Program of China; 2018YFC2000903, The Qin Chuangyuan Scientists & Engineers Team Construction Project of Shaanxi Province; 2022KXJ-034, The Natural Science Basic Research Program of Shaanxi; 2022JQ-016.

ORCID iDs

Liang Wei

Sun Runjun

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