Nanoscale adhesion and attachment oscillation under the geometric potential. Part 1: The formation mechanism of nanofiber membrane in the electrospinning

Accepted Manuscript Nanoscale adhesion and attachment oscillation under the geometr ic potential Par t 1: the for mation mechanism of nanofiber membr ane in the electr o spinning Xiao-Xia Li, Ji-Huan He PII: DOI: Reference:

S2211-3797(18)33397-7 https://doi.org/10.1016/j.rinp.2019.01.043 RINP 2009

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Results in Physics

Received Date: Revised Date: Accepted Date:

17 December 2018 13 January 2019 15 January 2019

Please cite this article as: Li, X-X., He, J-H., Nanoscale adhesion and attachment oscillation under the geometr ic potential Par t 1: the for mation mechanism of nanofiber membr ane in the electr ospinning, Results in Physics (2019), doi: https://doi.org/10.1016/j.rinp.2019.01.043

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Nanoscale adhesion and attachment oscillation under the geometric potential Part 1: the formation mechanism of nanofiber membrane in the electrospinning

Xiao-Xia Li and Ji-Huan He*

National Engineering Laboratory for Modern Silk, College of Textile and Clothing Engineering, Soochow University, 199 Ren-Ai Road, Suzhou 215123, China Abstract

An electrospinning process is to produce nanofibers from a polymer solution by a high electrostatic force, which is used to eject jets from a Taylor cone formed on a nozzle. When a moving jet approaches to a receptor within few nanometers during the electrospinning process, a Casimir-like attraction is formed, which attracts the jet’s end to the surface of the receptor. Thousands of thousands nanofibers will be attached on the receptor’s surface or attached on each other, as a result a nanofiber membrane is formed. The geometric potential was implemented to explain the nanoscale adhesion, which is greatly affected by the attachment oscillation. Mathematical models are established to describe the attachment oscillation under the Lennard-Jones potential and the geometric potential, respectively, and their amplitude-frequency relationship is elucidated analytically. The present theory gives an alternative explanation of the gecko effect and molecule scale forces.

Keywords: Lennard-Jones potential, van der Waals force, Casimir force, geometrical potential, molecule oscillation, smart adhesion, parthenocissus tricuspidata, gecko effect, macromolecular electrospinning

Introduction

Nanoscale or molecule scale attachment has been caught much attention recently due to the fast development of nanotechnology, especially MEMS[1,2] and molecule devices[3,4]. Gecko[5] and parthenocissus tricuspidata[6,7] have a unique adhesion property which enables the creature and the climbing plant to cling to any surfaces, smooth or unsmooth. The former can detach at will from the attached surface, this phenomenon is often called as the smart adhesion or the gecko effect. Their Corresponding author.

*

E-mail address: [email protected] Tel: 86-512-6588-4633

attachment systems are benefit from their nanoscale surface morphology, which has been mimicked by scientists to achieve nature’s optimal solutions achieved by millions of millions years. In this paper, we will elucidate the mechanism of nature’s attachment systems by the geometric potential[7], and then reveal the mechanism for forming a nanofiber membrane in an electrospinning process[8]. Considering the attachment oscillation affects greatly the adhesion property, a mathematical model will be established for the theoretical analysis.

Theory of geometric potential

The original idea of the geometric potential goes back to the Vujicic-He force action [9], which was further developed into a new concept of the morph-force or the boundary-induced force[10,11]. Nosonovsky and Bhushan also proposed a similar concept of the roughness-induced superhydrophobicity[12]. The geometric potential implies that any shape will produce a force, it can be gravity, Casimir force, capillary force or others[13]. A spherical Earth produces the gravity, and a concave lithosphere plate can produce anti-gravity phenomenon, which can induce volcanic explosion[10,11]. A sharp geometry produces a stress concentration, which can be weakened by smooth surface. When a block is placed on a ramp, the surface of the ramp produces a force perpendicular to the surface, which is generally called as the normal force in many text books. When a light meets a surface, a force is produced, which is perpendicular to the surface acting on the light, and refraction occurs, see Fig.1. Ferromagnetic ordering is extremely important for ferromagnetic materials and it can be also explained by the geometric potential as illustrated in Fig.2. Due to each ferromagnet is small, and it is a short-range force. The ferromagnet ordering can produce an extremely high interaction when the distance tends to be zero.

(a)

(b)

Fig.1 Refraction through an interface. The black arrows are vectors indicating directions of the boundary-induced forces.

(a)

(b)

Fig.2 Ferromagnet ordering in a ferromagnetic material According to the theory of geometric potential, every spherical nanoparticle attracts adjacent subjects, every concave surface produces a repulsion acting on adjacent objects[7,13]. In order to elucidate this, we did the following experiment[14]. Titanium dioxide(TiO2) nanoparticles were used as additives in an PVA solution, before electrospinning, the solution was stirred for half hours under a high speed to make the solution homogeneous. The obtained nanofibers were observed using a transmission electron microscope, see Fig.3, where the coagulation of nanoparticles was observed. The coagulation is due to the attraction among nanoparticles.

(a)

(b)

Fig. 3 Coagulation of nanoparticles. (a) TEM image of PVA/TiO2 nanofibers by electrospinning; (b) Interaction among nanoparticles.

The geometric potential can model the Hall–Petch effect[15,16], the inverse Hall– Petch effect[16], the nano-effect[16], and the lotus effect[17], it can also explain

effectively the permeability and wetting properties of a nanofiber membrane[18], and the selective adsorption property[13].

Micro/nano scale attachment in nature

The attachment systems of some creatures and climbing plants are so well-developed that many engineering surfaces have been designed to mimic nature’s property. Fig. 4 shows the attachment system of a climbing plant called parthenocissus tricuspidata, when the surface of the attachment pads approach to a solid surface a few nanometers apart, the geometric potential[7,13] can produce a high attraction force enabling the attachment pads to cling to a solid surface. Fig.2 illustrates the surface morphologies of gecko and parthenocissus tricuspidata, respectively, the unsmooth surface makes the creature and the plant easily to cling to any solid surfaces due to the geometric potential.

Fig. 4 A photo of a climbing plant (Parthenocissus tricuspidata)

(a)

(b)

Fig.5 Scanning electron microscopy (SEM) figures for the attached surface of gecko (a) and Parthenocissus tricuspidata (b)

According to the geometric potential, a convex surface can produce an attraction, its geometrical potential can be expressed as[7,13] E

k1 x

(1)

where x is the acting distance,  is a surface morphology factor, for a spherical surface,  =2, k1 is a constant. A concave surface can produce a repulsion, its geometric potential can be expressed as[7,13] E

k2 x

(2)

where  is surface morphology factor, for a concave spherical surface,  =2, k2 is a constant. For an uneven surface or a concave-convex surface, the geometric potential can be expressed as [7,13]: E

When   12 and   6 , potential[22,23].

Eq.(3)

k1 k2  x x 

becomes

(3) the

well-known

Lennard-Jones

Casimir-like effort and formation mechanism of nanofiber membrane

The electrospinning[19-22] and the bubble spinning[23-26] are widely used to fabricate of various functional nanofiber membranes. The electrospinning process is to produce nanofibers from a polymer solution by a high electrostatic force, which is used to overcome the surface tension of a Taylor cone formed on a nozzle. In the bubble spinning, it uses an external force to overcome the surface tension of a polymer bubble. During the spinning process, a Casimir-like force arises when a moving jet approaches to a surface within a few nanometers. The Casimir force arises when two plates are approaching to each other within a few nanometers. The Casimir potential can be written as[27-29] E

1 x3

(4)

and the Casimir force is[27-29] F

dE  2 c A  dx 240 x 4

(5)

where x is the distance between two plates,  is the reduced Planck constant, c is the speed of light, A is the plate area. It can be seen clearly that the Casimir potential is the geometric potential when  =3. Casimir force is the main factor causing the pullin instability in an MEMS system[1]. When a nanofiber flies to a receptor during an electrospinning process or a bubble spinning process [19-26], its end will be attracted when it is extremely closed to the receptor, and the whole nanofiber will randomly fall down from a high electric potential to the receptor’s surface where the receptor’s electric potential is the lowest. Fig.6 illustrates the attachment process of the nanofibers on the receptor. Thousands of thousands nanofibers can cling to the receptor or attached each other, as a result, a nanofiber membrane can be formed, see Fig.7.

Fig.6 Formation mechanism of nanofiber membrane during the electrospinning

Fig.7 Scanning electron microscopy (SEM) figure for a nanofiber membrane

Molecule oscillation under Lennard-Jones potential

The Lennard-Jones potential can be written in the form[30,31]

   E  4 ( )6  ( )12  x   x

(6)

where ε is the depth of the potential well, σ is the finite distance at which the interparticle potential is zero, x is the distance between the particles. The force produced by the Lennard-Jones potential is

F

dE  6 2 12  24 ( 7  13 ) dx x x

(7)

A free molecule oscillation under Lennard-Jones potential can be written as

6

2 12 mx  24 ( 7  13 )  0 x x

(8)

or  x

a b  13  0 7 x x

(9)

where m is the mass, a  24 6 / m , b  48 12 / m . If we consider the oscillated molecule is an elasticity, a forced molecule oscillator can be written in the form

 x  02 x 

a b  13  F (t ) 7 x x

(10)

where F(t) is the external force.

Attachment oscillation under the geometric potential

An attachment system is illustrated in Fig.8, the force due to the geometric potential[7,13] can be written as F

dE  k1  dx x 1

(11)

Fig.8 The attachment system due to the geometric potential, which can produce a force helping geckos walk effortlessly along walls and ceilings

The free oscillation due to the geometric potential can be written as mx 

 k1

x 1

0

(12)

If the attachment system is considered as a linear spring, a forced attachment oscillator can be written as mx  kx 

 k1

x 1

 F (t )

(13)

Eqs.(12) and (13) can be written in the forms, respectively,  x  02 x   x  02 x 

 x 1

 x 1

0

(14)

 f (t )

(15)

Frequency- amplitude relationship

In this section, we will apply the frequency formulation[32] to search for the amplitude-frequency relation. The frequency formulation is the simplest analytical approach to the nonlinear oscillator and has been widely applied[33-44]. We consider the attachment oscillator in the form u   u  1u 3 

2

0,

u3

(16)

with initial conditions u (0)  A u (0)  0

(17)

According to the frequency formulation[32], we define a residual as follows R(u )  u u 3  u 4  1u 6   2

(18)

We assume two trial solutions u1 (t )  A0  A1 cos 1t

(19)

u2 (t )  A0  A1 cos 2t

(20)

where 1 and 2 are two assumed frequencies, which can be chosen freely. and A1 are constants,

A0

We define the following integrals[44] T1 /4

R1 



R(u1 (t ))dt

(21)

0

T1 /4

R 2 



R(u2 (t ))dt

(22)

0

By the frequency formula[32], we have 2 

22 R1  12 R 2 R1  R 2

(23)

We choose 1  1 and 2  2 for simplicity, the frequency formula reads[32] 2 

22 R1  12 R 2 R1  R 2

and the approximate solution is



A0  A1 4 2 3 (1  1 ( A0  A1 ) 2  ) A1 4 3( A0  A1 ) 2

(24)

u (t )  A0  A1 cos t

(25)

We set R (u (0))  0

(26)

This leads to an additional equation  2 A1 ( A0  A1 )3  ( A0  A1 ) 4  1 ( A0  A1 )6   2  0

(27)

The initial conditions lead to the following requirement: A0  A1  A

(28)

Solving Eqs.(24), (27) and (28) simultaneously, we can obtain easily A0 , A1 and

 . When  2 =0, Eq.(16) becomes the Duffing oscillator and the obtained result is same with that by the homotopy perturbation method[45-53].

Conclusion

The geometric potential can explain many phenomena, from the normal force to refraction, from the ferromagnet ordering to the volcanic explosion, from molecules’ or nanoparticles’ agglomeration to Newton’ gravity, and from the smart adhesion(the gecko effect) to the nanofiber’s attachment. For the first time ever, the formation mechanism of a nanofiber membrane in electrospinning and bubble electrospinning is theoretically elucidated. We establish the molecule oscillator under Lennard-Jones potential and the attachment oscillator under the geometric potential of nanofibers. The attachment oscillation has never been appeared in the open literature, and this paper reveals the main factors affecting the oscillation property. We use the simplest analytical method called He’s frequency formula to insight into the periodic property of the attachment oscillator. To have a higher accuracy of the periodic solution, we have to use the homotopy perturbation method[45-53]and other analytical methods.

Acknowledgments: The work is supported by National Natural Science Foundation of China under grant No.51463021 and Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

Nomenclature

a, constant

A, the plate area or amplitude b, constant c, the speed of light E, geometrical potential F, force k1, constant k2, constant m, mass x, the acting distance

α, dimensionless surface morphology factor β ,dimensionless

surface morphology factor

ε (kg/s2﹒m4) the depth of the potential well σ (m) the finite distance at which the inter-particle potential is zero

 , reduced Planck constant

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