Geometrical potential and nanofiber membrane’s highly selective adsorption property

Abstract

The definition of a nanofiber is discussed, and the history of the development of bubble electrospinning is briefly elucidated, its main properties are emphasized, and the morphology of its products is summarized. A geometrical interpretation of the nanofiber membrane’s adsorption is given as a natural intrinsic property of the geometrical potential of nanoscale structure. Nano-effect, Casimir effect, capillary effect, Earth’s gravity difference, Newton’s gravity, and lotus leaf’s highly selective repulsion property can all be interpreted using the geometrical potential or boundary-induced force.

Introduction

Nanofiber membranes and nanoporous fibers are widely used for adsorption of hazardous substance due to their extremely high surface energy (Cheah et al., 2017; Kolbasov et al., 2017). Many special and unprecedented properties, such as super-hydrophilic property (Obaid et al., 2015), super-hydrophobic property (Kakunuri, 2017), high temperature resistance (Li et al., 2017), good electronic conduction (An et al., 2017), high mechanical strength (Lu et al., 2017; Ngadiman et al., 2017), and stab-proof property (Dou et al., 2016), are observed due to the high surface area, where many hidden laws are waiting for discovery. It is astonishing that some nanofiber membranes behave super-hydrophobically, while others behave in an opposite way. Such conflictive phenomena are caused by the complex surface of nanofiber membranes. The famous physicist Wolfgang E. Pauli once said that God made solids, but surfaces were the work of the devil. This quote implies that the surface science is enormously complex but tremendously fascinating and extremely challenging as well. It is now well-known that 95.5% of our universe’s total energy is the dark energy stored on boundary of our universe, resulting in the well-known cosmic expansion (Arun et al., 2017; El Naschie, 2016). The surfaces have been becoming the playground of many disciplines, such as physics, chemistry, material science, nanotechnology and modern textile science as well. Fractional calculus become reviving after about 300 years’ silence to deal with unsmooth boundary conditions (Hu and He, 2016; Sing, 2017; Wang et al., 2017a, 2017b), and nano reactors have been proposed to produce energy from “nothing” of the nanomaterials (El Naschie, 2015a, 2015b).

Most energy is stored on the surface, especially a sharp change of surface will remarkably increase interface action, for example, the capillary effect and the stress concentration, which can be reduced or relieved by a smoother geometry. Various properties will be changed rapidly on nanoscale surfaces, independent of their bulk materials (He et al., 2007c). Nanofibers become fascinating mainly due to their nanoscale surfaces, and are worth their weight in gold; therefore, there is an extremely longing trend of industrialization of nanofibers in China using a modification of bubble electrospinning called bubbfil spinning (Li et al., 2016a,2016b; Liu et al., 2015; Zhang et al., 2016b) and in other parts of the world as well.

It was reported that nanoscale pores have selective adsorption property (Alduhaish et al., 2017; Kahveci et al., 2017; Qiu et al., 2017; Sun et al., 2017; Yoon et al., 2017; Zhou et al., 2017a, 2017b); this induces a technology for fabrication of nanofiber membranes with controllable pore sizes, but its mechanism is still ambiguous, and a new theory is much needed to uncover the adsorption properties. This paper gives a brief introduction to the bubble electrospinning (He et al., 2007a, 2008, 2017) for fabrication of nanofiber membranes and elucidates their adsorption/desorption properties.

Bubble electrospinning

The classical approach to producing nanofiber members is electrospinning (He et al., 2008), but the definition of the nanofiber is unclear and something ambiguous. For example, the URL understandingnano.com defines a nanofiber as a fiber with a diameter of 100 nanometers or less; the URL Wikipedia.org defines a nanofiber as a fiber with diameter in the nanometer range, while many publications on nanofibers range from 100 nm to less than 1000 nm. For an example, a recently published article by Qureshi et al. (2017) named nanofibers ranging from 100 to 300 nm.

We give the definition as follows (Chen et al., 2014): Nanofibers are fibers with nano-effect properties or extreme/extraordinary surface properties independent of their bulk materials, which always result in, for example, excellent electronic, thermal or mechanical properties. Nano-effect generally occurs when the fiber diameter reduces to less than 200 nm. Sometimes by adding some nanoscale particles in the spun solution, e.g., carbon nano tube, graphene nanoribbons, or zirconia nanoparticle, the obtained fibers with larger diameters also have remarkable mechanical property, or high surface energy and surface reactivity, or excellent thermal property or good electric conductivity independent of their bulk materials. We also call such fibers as nanofibers though their diameters might be larger than 1000 nm. Nano-effect can also remarkably enhance some functional properties which are much superior to bulk materials.

In the bubble electrospinning, a bubble is ruptured by an electrostatic field, see Figure 1. The ruptured bubble can produce multiple jets as shown in Figure 1(d). As various shapes of fragments can be formed from a ruptured bubble, various morphologies of the products are predicted.

Figure 1.

Rupture process of a bubble under an electrostatic field. (a) A bubble is formed and enlarged; (b) Critical bubble electrospinning; (c) The maximal bubble with thinnest wall; and (d) Multiple jets are ejected from a broken bubble.

The initial thickness of a bubble wall ranges from 2000 nm to 3000 nm, its wall becomes thinner and thinner when the bubble is enlarged, the thinnest wall can reach as small as 100 nm (Chen et al., 2015), and it is affected greatly by environmental temperature (Cui and Liu, 2014), humidity (Liu and Dou, 2013), and bubble size (Ren et al., 2016). At such small scale, nano-effect occurs on the bubble surface, where the entangled macromolecules become disentangled, chemical bonds among macromolecules become weaker and a higher surface energy is predicted; this process can greatly enhance the chemical and physical properties of the bubble wall, which will greatly affect the product’s properties.

The bubble electrospinning is always used for the fabrication of smooth cylindrical nanofibers and nanoparticles (Dou and He, 2012); the latter can be easily realized when the molecular weight is not large enough. The smallest fiber so far obtained is as small as 5 nm (Kong and He, 2014), considering the average water diameter is 2.82 Å or 0.282 nm (Franks, 2000). When a polymer bubble is broken, each debris is stretched by external forces, such as the air flows, centrifugal force, and electrostatic force and others, to form a smooth fiber; other morphologies are summarized below.

Various nanofibers fabricated by bubble electrospinning

Nanoscale hollow fibers

When a bubble is broken, the fragments with various forms are formed. The minimization of surface energy during stretching process might result in nanoscale hollow fibers.

Consider a strip of a ruptured film as illustrated in Figure 2, and we assume the width of the strip is $\alpha$ and thickness is $h$ . According to the calculus of variations (He, 2017), the thin film has a strong trend to become a cylindrical one with the outer radius of r and inner radius of R

$$r=\frac{a}{2\pi }$$

(1)

$$R=r-h$$

(2)

Figure 2.

Minimization of surface energy of a fragment results in a nanoscale hollow fiber.

When $r-h>0$ , a hollow fiber can be formed. The inner surface is non-smooth in nanoscales because before solidification, the flow will interact with each other on the inner surface. Therefore, the nanoscale hollow fibers have fascinating optical, photocatalytic and photoelectricity properties and can be used for invisibility device (e.g. stealth plane) and radiation protection and clothes for extreme environments (e.g. space suit, Apollo suit, Moon suit).

Two-dimensional superthin nanobelts

When a bubble is broken and the fragments are fast solidified during stretching process in a high temperature environment, superthin strips are formed (He et al., 2012; Kong and He, 2012). Nanoscale graphite powders as additives in the spun solution can generally guarantee superthin nanobelts. During the spinning process, the graphite powders, which have something of graphene, a two-dimensional nanomaterial, have higher surface energy than that of the fragments of the broken bubble; this prevents the fragment from becoming a cylindrical one.

The superthin nanobelts can be used as a two-dimensional nanomaterial to economically replace the valuable graphene in various applications. For example, the conductive two-dimensional superthin nanobelts can be used as a bottom electrode in a molecular junction, which is a promising technology for molecular-scale device. (Zhang et al., 2016).

Beaded fibers

Beads in a nanofiber are formed due to the combination effect of the instability of the ejecting jet’s radius and minimization of surface energy during the spinning process. A rigorous mathematical analysis was given by He et al. (2012). The distance between adjacent beads scales as

$$T\propto k^{1/2}{\mu }^{1/2}{\rho }^{-1}$$

(3)

where $k$ is the conductivity of the liquid, $\mu$ is the fluid viscosity, ρ is the specific mass of the liquid. When $T\to 0$ nano/micro particles can be obtained; and when $T\to \infty$ smooth fibers are predicted. Some additives, i.e., NaCl, metal nanoparticles, carbon nanotube, graphene, and others, can remarkably improve the value of k, as the frequency of bead distribution will decrease. A lower concentration of the spun solution yields a lower viscosity; as a result, more beads on a fiber can be observed. The nanoscale beads can greatly affect fiber’s wettability (Liu et al., 2015).

Nanoporous spheres and fibers

Nanoporous fibers become a hot topic in both material science and chemistry due to extremely higher specific surface than smooth nanofibers, and have many applications in adsorption, catalytic and hydrogen-storage systems, sensors, catalysis, and electrode materials. Far-reaching applications include invisibility device, radiation protection, medical implants, cell supports, drug releasing and others (Liu et al., 2017; Wu et al., 2008; Xu et al., 2007, 2010). Fan et al. (2017) gave a fluid mechanic model for fabrication of nanoporous fibers.

Some nano scale porous materials have excellent photocatalytic and photoelectrochemical properties. Figure 3 is a nanoscale porous sphere, which is formed from a large solution droplet from a broken bubble. He et al. (2007a) elucidated how to fabricate micro sphere with nanoporosity. The sudden evaporation of the solvent from the droplet always forms a porous surface (Liu and He, 2017; Zhao et al., 2017).

Figure 3.

Nanoscale porous structure in a micro sphere. PLA/DMF (9 wt%) solution was fabricated by bubble-electrospinning with the applied voltage of 20 kV and the collector distance of 15 cm. The temperature is 25.4°C and the relative humidity is 68%.

When a bubble is broken, its solvent is evaporated immediately due to sharp change of pressure and temperature, and nanoscale porosity is formed. During the stretching process, sudden solvent evaporation will always produce nanoscale porous fibers (Liu and He, 2017; Peng et al., 2018; Zhao et al., 2017). A multiple solvent system always results in a low evaporation temperature, which can guarantee the occurrence of the sudden solvent evaporation to obtain nanoscale porous materials. He et al. (2007b) gave a mathematical model for the fabrication of the nanoporous surface.

Other approaches to effective fabrication of nanoporous materials include improvement of the ejecting velocity and negative pressure suction.

The ejecting velocity of a fragment can be expressed as (Sun et al., 2018)

$$u=\sqrt{\frac{2(P_i-P_0)}{\rho }}$$

(4)

If the pressure difference between inside and outside is 2 atmos with air density $\rho =2.3381\text{kg}/{\text{m}}^3$ , and the air inside is air, we have

$$u=\sqrt{\frac{2\times 2\times 101325}{2.3381}}=416\hspace{0.17em}\text{m}/\text{s}$$

(5)

That means a fragment is accelerated from zero velocity to about 400 m/s, according to the Bernoulli equation

$$\frac{1}{2}{u}^2+\frac{P}{\rho }=B$$

(6)

where $u$ is the velocity of the moving jet, $P$ is the fluid pressure, $\rho$ is the density, $B$ is the Bernoulli constant. A sudden pressure drop occurs when the fragment is ejected; this results in a sudden solvent evaporation in the spinning process.

Negative pressure suction is to decrease the air pressure near nozzles, lower than an atmosphere. At a lower pressure, the solvent can be attracted from the moving jets, and a porous fiber is formed, like Figure 4.

Figure 4.

Nanoporous fibers. PLA (10 wt%)/CHCl₃/DMF (8:2) solution was fabricated into nanoporous fibers with the average diameter of pores of about 50 nm by electrospinning on the applied voltage of 15 kV and collector distance of 20 cm with the temperature of 24.3°C and relative humidity of 52%.

Crimped nanofibers

Crimped nanofiber (Chen et al., 2013; Huang et al., 2014, 2015, 2017; Wang et al., 2017) can be fabricated by the bubble electrospinning and its modifications where the bubbles are used as spun medium; it can also be fabricated by electrospinning (Tang et al., 2011). The pore structure of nanoporous fibers was analyzed in our previous papers (He et al., 2007a, 2007b; Liu and He, 2017; Peng et al., 2018; Xu et al., 2007). During the bubble electrospinning process, the bubble’s wall becomes thinner and thinner, and a thin film from a broken bubble is extremely unstable; vibration will inherently occur.

Any thin structure will become unstable under a small perturbation, when a transverse vibration of an axially moving a slender fragment happens, crimped fiber might be predicted. There is a critical velocity of moving fragment, below which transverse vibration is exponentially damped

$$u_{cr}=\sqrt{2\left(\frac{T}{\rho }-B-\frac{EI}{\rho A}\frac{{\pi }^2}{L^2}\right)}$$

(7)

where T is the strain of the moving fragment, ρ is the density, B is Bernoulli constant, E is the modulus of viscoelastic fragment, I is the moment of inertia, A is the section area, and L is the length of the fragment.

A high initial strain of the fragment is needed; this requires enlargement of the bubble, as illustrated in Figure 1. When a bubble is enlarged, its strain becomes high. The frequency of crimp distribution is

$$\omega =\sqrt{\left(\frac{1}{2}{u}^2+B-\frac{T}{\rho }\right)\frac{{\pi }^2}{L^2}+\frac{EI}{\rho A}\frac{{\pi }^4}{L^4}}$$

(8)

Shorter fragment results in larger distribution frequency or smaller distribution period. The governing equations for the crimpled fiber fabrication are complex, and the frequency can only be approximated solved by some analytical methods (Andrés, 2017; El-Dib, 2017; He, 2016; Nimafar et al., 2017).

Geometric potential: An explanation of nanofiber membrane’s absorption

The original idea of the geometrical potential goes back to the Vujičić–He force action (Vujičić and He, 2004), which was further developed into a new concept of the morph-force or boundary-induced force (He, 2009, 2010). Nosonovsky and Bhushan (2013) proposed a similar concept of roughness-induced superhydrophobicity. The geometric potential implies that any shape will produce a force; it can be gravity, Casimir force, capillary forces and others.

Nano-effect or size effect in nanotechnology

With decreasing diameter of a fiber, the surface-to-volume ratio grows in a rocketing way, so that surface energy increases greatly, and the surface effects gradually dominate over the volume effects. The surface effect is called the nano-effect when the fiber size tends to nanoscales. Nanofibers behave remarkably different from their bulk due to the nano-effect or the size effect. Scale effects in dry friction at macro-to nanoscale are considered (Bhushan and Nosonovsky, 2004).

Before explaining the nano-effect in nanotechnology, we introduce first the grain-size hardening (or Hall–Petch strengthening) in materials science (Hall, 1951; Petch, 1953). A material can be strengthened by decreasing their average crystallite (grain) size. This size effect was found by Hall (1951) and Petch (1953), and the following Hall–Petch relationship is now widely accepted for various applications (Tian et al., 2018a, 2018b; Yu et al., 2017; Zhang et al., 2017)

$$\sigma ={\sigma }_0+\frac{k_{\sigma }}{d^{\alpha }}$$

(9)

where $d$ is the average grain diameter, $\sigma$ is the stress at a higher plastic strain, ${\sigma }_0$ is the corresponding stress for larger single crystals, $\alpha$ is a scaling exponent, $k_{\sigma }$ is the material constant. Equation (9) implies a smaller gain size results in a harder material.

Similar to the Hall–Petch relationship, the nano-effect implies that some mechanical or thermal properties of a nanofiber are size-dependent

$$\tau ={\tau }_{\text{0}}+\frac{k_{\tau }}{d^a}$$

(10)

where $\tau$ can be surface energy, strength, elastic modulus, or thermal resistance, ${\tau }_0$ is the bulk’s property, $k_{\tau }$ is a material constant, $d$ is the fiber diameter, $0<d<100\hspace{0.17em}\text{nm}$ , $\alpha >0$ is a scaling exponent. Nano-effect arises generally when $d$ tends to hundreds of nanometers. When the diameter of fiber gradually decreased, stress concentration of fiber will increase; when the diameter of the fiber decreased to nanometer, the stress concentration of the fiber will rapidly increase. Gu et al. (2005) measured the Young’s modulus of a single fiber with a diameter ranging from several hundred nanometers to tens of nanometers, and found that the nano-effect happens near 150 nm for the studied fiber. The nano-effect can explain well why the spider silk is almost five times stronger than steel of the same diameter. This is because the silk is a nanofiber assembly consisting of nanofibers with a diameter of about 20 nm (Kong and He, 2012).

The surface energy of a nanofiber can be also expressed as

$$E_{fiber}=\frac{k_{fiber}}{r^{\alpha }}$$

(11)

where $k_{fiber}$ is a constant, r is the fiber’s radius, $\alpha$ is a scaling parameter, which will be discussed below.

The potential of a nanoscale porosity is

$$E_{pore}=\frac{k_{pore}}{r^{\beta }}$$

(12)

where $r$ is the radius of the nano-porosity, $\beta$ is a scaling parameter.

Dimension gradient and geometrical potential

A force is produced when two bodies contact with each other. When a bullet acts a surface of a subject, a larger force can be produced than a blunt one. The force magnitude depends upon the acting and acted surfaces. The dimension gradient is the difference of the dimensions of acting and acted surfaces or boundaries. The geometrical potential can be expressed as (He, 2008, 2009, 2010a, 2010b; He and Mo, 2009; Liu and He, 2018)

$$E\propto \frac{1}{R^{D_1-D_2}}=\frac{1}{R^{\Delta D}} \text{for\hspace{0.17em}\hspace{0.17em}non}-\text{particle\hspace{0.17em}\hspace{0.17em}interaction}$$

(13)

or

$$E\propto R^{\Delta D} \text{for\hspace{0.17em}\hspace{0.17em}particle\hspace{0.17em}\hspace{0.17em}interaction}$$

(14)

where E is the potential of the acting boundary, R is the acting distance, $D_2$ is the boundary dimensions of the acting boundary, D₁ is the boundary dimensions of the acted boundary, $\Delta D=D_1-D_2$ is the dimension gradient.

Equation (14) is valid for the case when the acted subject is a point with zero dimension, and its boundary is negative one dimension.

The acting force is

$$F=\frac{\partial E}{\partial R}\propto -\frac{1}{R^{D_1-D_2+1}}=-\frac{1}{R^{\Delta D+1}}\text{ for }\text{non}-\text{particle}\text{interaction}$$

(15)

or

$$F=\frac{\partial E}{\partial R}\propto R^{\Delta D-1} \text{for\hspace{0.17em}\hspace{0.17em}particle\hspace{0.17em}\hspace{0.17em}interaction}$$

(16)

As an example, we consider the well-known capillary phenomenon. The capillary action results in the interaction of the tube boundary with $D_2=2$ and the solution surface $D_{\text{1}}=2$ ; as a result, we have

$$F\propto -\frac{1}{R^{D_1-D_2+1}}=-\frac{1}{R}$$

(17)

Balance of the boundary-induced force (capillary force) with gravity requires that

$$k\frac{1}{R}=\rho gh$$

(18)

where k is a constant, $\rho$ is density, $g$ is gravitational acceleration, h is the height of a liquid column, while the capillary theory predicts

$$h=\frac{2\sigma \cos \theta }{\rho gR}$$

(19)

where $\sigma$ is the liquid-air surface tension (force/unit length), θ is the contact angle. We can see that both equations (18) and (19) predict $h\propto 1/R$ .

As another example, we consider the motion of tectonic plate as illustrated in Figure 5. The boundary-induced force can be easily determined, and its direction should point to the earthquake centers, earthquake magnitude is inverse proportion to its curvature radius.

Figure 5.

Twin earthquakes due to deformation of tectonic plate.

The boundary-induced force is the main factor for earthquake; in most cases there might be more than one center of earthquake, and we call twin earthquakes or multi-earthquakes.

The motion of the tectonic plate will change the local gravity, which will cause animals’ unusual behaviors and some abnormal phenomena, for example, anti-gravity illusion that water moves suddenly upwards before the earthquake. Earthquake is a process to make the planet spherical just like a fragment of a bubble to form a cylindrical fiber in the bubble electrospinning.

The local gravity depends upon also its local curvature radius.

$$g\propto \frac{1}{R^2}$$

(20)

where R is the local curvature radius. The Earth’s equatorial radius is a = 6378 km, and the Earth’s polar radius is b = 6356 km. The curvature radius at polars and at the equator are, respectively, as

$$R_{polars}=\frac{b^2}{a}$$

(21)

$$R_{equator}=\frac{a^2}{b}$$

(22)

So gravitational acceleration ratio between the polars and the equator is

$$\frac{g_{polars}}{g_{equator}}=\frac{a^6}{b^6}={\left(\frac{6378}{6356}\right)}^6=\text{1}.0\text{2}0\text{9}$$

(23)

The observed values are, respectively, $g_{polars}=9.832\hspace{0.17em}\text{m}/{\text{s}}^2$ and $g_{equator}=9.780\hspace{0.17em}\text{m}/{\text{s}}^2$ , $g_{polars}/g_{equator}=1.0053$ , the relative error is 1.55%.

For the particle–particle interaction, for example the Sun–Earth system and two point charges, the acted earth is a zero-dimensional point, and its boundary is negative one dimensions, the Sun acts upon the Earth, and the force is along the straight line joining them, while the acting force is one dimension, and its boundary is zero dimensions. That is $\Delta D=(-1)-0=-1$ ; therefore, the force scales as

$$F\propto R^{(-1)-1}=R^{-2}$$

(24)

which agrees with that required in Newton’s gravity. Similarly, for the two point charges, our result corresponds to the Coulomb's law.

Casimir effect

The Casimir effect (Soroush and Yekrangi, 2017; Wongjun, 2015) arises when two plates without any charge are placed a few nanometers apart; it is found that the plates do affect photons, it is a quantum-like force (Otto, 2017a, 2017b). The Casimir force is due to the zero-point energy, and can be written as

$$F=-\frac{{\pi }^2\mathit{\hslash }c}{240}\frac{A}{H^4}$$

(25)

where $\mathit{\hslash }$ is the reduced Planck constant, $c$ is the speed of light, $H$ is the distance between the two plates, A is the plate area.

The plate has the dimensions of 2, while the photon is zero dimensions; its boundary is negative one dimension (He, 2010a)

$$D_{plate}=2$$

(26)

$$D_{particle}=-1$$

(27)

The dimension gradient is

$$\Delta D=(-1)-2=-3$$

(28)

The Casimir force scales as

$$F\propto H^{\Delta D-1}=\frac{1}{H^4}$$

(29)

which agrees with equation (22).

El Naschie (2015a, 2015b) gave a physical-mathematical connection between dark energy and the Casimir effect, and proposed a nano reactor to extract dark energy from its nano boundary of its holographic boundary. El Naschie’s nano reactor can be considered as a nano universe from which its 95.5% energy concentration could be extracted without actually reaching to the boundary of our universe (El Naschie, 2015a, 2015b).

Highly selective absorption

A nanofiber membrane has two geometrical structures, one is fiber-web, and the other is the porosity. We assume that the surface of the nanofiber membrane has two dimensions

$$D_{fiber-web}+D_{porosity}=2$$

(30)

where $D_{fiber-web}$ and $D_{porosity}$ are, respectively, the fractal dimensions of the fiber-web and the porosity. When $D_{fiber-web}$ =2, the Casimir effect is expected.

Nanofiber membrane can adsorb particles with diameter ranging from 100 nm to 5000 nm. The adsorbed particle has zero dimensions, and its boundary has negative one dimensions: $D_{particle}=-1$ (He, 2010b). We assume that the dimensions of the web-fiber are $D_{fiber-web}=\alpha$ . This yields the following potential for the fiber web

$$E_{fiber-web}\propto -x^{-(1+\alpha )}$$

(31)

where x is the height above the membrane’s surface.

The potential for the porosity

$$E_{porosity}\propto x^{-(3-\alpha )}$$

(32)

The absorbed particle is a combined result of the fiber-web and the porosity potentials

$$E(x)=E_{fiber-web}+E_{porosity}=-\frac{a}{x^{1+\alpha }}+\frac{b}{x^{3-\alpha }}$$

(33)

This is similar to the well-known Lennard-Jones potential for describing the interaction between a pair of neutral atoms or molecules, which was proposed by John Lennard-Jones (1924). The force acting on the absorbed particle is

$$F=\frac{dE}{dx}=\frac{a(1+\alpha )}{x^{2+\alpha }}-\frac{b(3-\alpha )}{x^{4-\alpha }}$$

(34)

When

$$\frac{dF}{dx}=-\frac{a(1+\alpha )(2+\alpha )}{x^{3+\alpha }}+\frac{b(3-\alpha )(4-\alpha )}{x^{5-\alpha }}=0$$

(35)

or

$$x_{\max }={\left[\frac{b(3-\alpha )(4-\alpha )}{a(1+\alpha )(2+\alpha )}\right]}^{1/(2-2\alpha )}$$

(36)

the force reaches its maximum

$$F_{\max }=k\frac{a^{(4-\alpha )/(2-2\alpha )}}{b^{(2+\alpha )/(2-2\alpha )}}$$

(37)

where

$$k=\frac{(1+\alpha ){[(1+\alpha )(2+\alpha )]}^{(2+\alpha )/(2-2\alpha )}}{{[(3-\alpha )(4-\alpha )]}^{(2+\alpha )/(2-2\alpha )}}-\frac{(3-\alpha ){[(1+\alpha )(2+\alpha )]}^{(4-\alpha )/(2-2\alpha )}}{{[(3-\alpha )(4-\alpha )]}^{(4-\alpha )/(2-2\alpha )}}$$

(38)

The maximal force attracts a particle onto its surface

$$F_{\max }=k\frac{a^{(4-\alpha )/(2-2\alpha )}}{b^{(2+\alpha )/(2-2\alpha )}}\propto w\propto D^{1/3}$$

(39)

where $w$ and $D$ are, respectively, the molecular weight and the diameter of the absorbed particle. Equation (39) implies that

$$w=w_{0}k\frac{a^{(4-\alpha )/(2-2\alpha )}}{b^{(2+\alpha )/(2-2\alpha )}}$$

(40)

$$D=D_0{k}^{\text{3}}\frac{a^{3(4-\alpha )/(2-2\alpha )}}{b^{3(2+\alpha )/(2-2\alpha )}}$$

(41)

where $w_0$ and $D_0$ are constants.

The maximal radius of the absorbed particles is

$$R_{\max }={\left[\frac{b(3-\alpha )(4-\alpha )}{a(1+\alpha )(2+\alpha )}\right]}^{1/(2-2\alpha )}$$

(42)

beyond which, the particle will contact with the membrane’s surface directly, and a reacting force is formed to push the particle away from its surface.

When

$$x={\left[\frac{b(3-\alpha )}{a(1+\alpha )}\right]}^{1/(2-2\alpha )}$$

(43)

the force is vanished completely F = 0. That means no any particles can be absorbed onto the surface when $x<{\left[\frac{b(3-\alpha )}{a(1+\alpha )}\right]}^{1/(2-2\alpha )}$ . Particles located at $x=R_{\max }$ are dominantly attracted or absorbed. When $x>R_{\max }$ the force decreases exponentially.

For a particle with radius of $R_{\min }$ given in equation (43), it cannot be absorbed onto the surface, owing to no absorption force.

$$R_{\min }={\left[\frac{b(3-\alpha )}{a(1+\alpha )}\right]}^{1/(2-2\alpha )}$$

(44)

The size of the absorbed particles lies between

$${\left[\frac{b(3-\alpha )}{a(1+\alpha )}\right]}^{1/(2-2\alpha )}<R<{\left[\frac{b(3-\alpha )(4-\alpha )}{a(1+\alpha )(2+\alpha )}\right]}^{1/(2-2\alpha )}$$

(45)

and the particles with radius of $R=R_{\max }-R_{\min }$ are dominated.

$$R=R_{\max }-R_{\min }={\left[\frac{b(3-\alpha )(4-\alpha )}{a(1+\alpha )(2+\alpha )}\right]}^{1/(2-2\alpha )}-{\left[\frac{b(3-\alpha )}{a(1+\alpha )}\right]}^{1/(2-2\alpha )}$$

(46)

That means a particle with radius of $R=R_{\max }-R_{\min }$ will be absorbed at $x=R_{\max }$ , where the attracting force reaches maximum. This is the mechanism of highly selective absorption, which has been observed everywhere.

To verify highly selective absorption of nanoporous structure, we picked randomly a leaf at the campus of Soochow University, Figure 6 is the SEM illustrations showing absorded particles with almost same size, the phenomenon was also observed before (Kong et al., 2013, 2014)

Figure 6.

Leaf’s highly selective absorption property. The absorbed particles are almost in the same size.

Highly selective repulsion

In the above section, we assume that $\alpha <1$ ; that means the porosity area ratio is larger than 50%. Figure 7 shows the geometrical potentials for different values of $\alpha .$ It can be found that when $\alpha >1$ the acting force becomes the opposite; that means when the porosity area ratio is less than 50%, the repulsion is dominant instead of the absorption as discussed in the above section.

Figure 7.

Geometrical potential vs. the fractal dimensions. (a) α = 0.2; (b) α = 0.5; (c) α = 1.5; and (d) α = 2.

We consider the geometrical structure of a lotus leaf as illustrated in Figure 8. Such bio-inspired surface can be fabricated by the bubble electrospinning (Liu et al., 2016a, 2016b; Zhao, 2016). The lotus leaf is famous for its super-hydrophobic or its self-cleaning property (Zhou et al., 2018). There are many protrusions on the rough surface due to the so-called papillose epidermal cells, which form asperities or papillae with diameter of about 50 nm. The maximal protrusion area ratio on the leaf surface is

$$\frac{4\times \pi r^2}{{(4r)}^2}=\frac{\pi }{4}>50\%$$

(47)

where $r$ is the radius of the protrusions. The minimal porosity area ratio is 21.46%, so the nanoscale protrusions surface satisfy $\alpha >1$ ; the maximal fractal dimensions for protrusions surface is

$${\alpha }_{\max }=\frac{\ln \pi }{\ln 2}=1.6515$$

(48)

Figure 8.

SEM illustrations of a lotus leaf (a) Fresh leaf: contact angle $\theta =151.6°$ , fractal dimensions $\alpha =1.6287$ ; and (b) dried leaf: contact angle $\theta =159.4°$ , fractal dimensions $\alpha =1.4283$ .

Instead of absorption, nanoscale protrusions surface has a highly selective repulsion. The fractal dimensions of the lotus leaf are calculated by the following mathematic formulation (Wang et al., 2015)

$$\alpha =\frac{\ln M}{\ln N}$$

(49)

where M is the number of new units with a new dimension (r) for a chosen unit with dimension (R), and N = R/r. Using ImageJ software, the total area of the protrusions surface in Figure 8(a) is 3326.309 µm², the minimal protrusion area is 15.853 µm², the total sample area given in Figure 8(a) is $123.007\times 86.539$ µm² = 1.0651 × 10⁴ µm².

We choose

$$R=\sqrt{\text{123}\text{.077}\times \text{86}\text{.539}}$$

(50)

and the new dimension

$$r=\sqrt{\text{15}\text{.853}\times \text{4/}\pi }$$

(51)

the number of new units

$$M=\frac{\text{3326}\text{.309}}{\text{15}\text{.853}\times \text{4/}\pi }$$

(52)

and the scaling ratio

$$N=\frac{R}{r}=\frac{\sqrt{\text{123}\text{.077}\times \text{86}\text{.539}}}{\sqrt{\text{15}\text{.853}\times \text{4/}\pi }}$$

(53)

The fractal dimensions are

$$\alpha =\frac{\ln \frac{\text{3326}\text{.309}}{\text{15}\text{.853}\times \text{4/}\pi }}{\ln \frac{\sqrt{\text{123}\text{.077}\times \text{86}\text{.539}}}{\sqrt{\text{15}\text{.853}\times \text{4/}\pi }}}=\text{1}.\text{6287}$$

(54)

which is very close to the golden mean.

When $D_{fiber-web}=\alpha =2$ , $F$ in equation (34) becomes Casimir force; this requires $b=0$ ; when $D_{fiber-web}=\alpha \to 0$ , geometrical potential vanishes completely; this requires $a=b=0$ . Therefore, the constants $a$ and $b$ in equation (34) can be expressed in a general form:

$$a=a_0{\alpha }^m,b=b_0{\alpha }^n{(2-\alpha )}^q$$

(55)

where $a_0$ , $b_0$ , $m$ , $n$ , and $q$ are constants.

$$F_{\max }=k\frac{a^{(4-\alpha )/(2-2\alpha )}}{b^{(2+\alpha )/(2-2\alpha )}}=2\pi r\sigma \sin \theta +mg$$

(56)

where $r$ is the radius of contacted surface on the leaf, $\sigma$ is the surface tension, $\theta$ is the contact angle, $mg$ is the gravity of water sphere. Considering Fmax ∝w, where w is the molecular weight, we have

$$w=\mu (2\pi r\sigma \sin \theta +mg)$$

(57)

 

where µ is a constant. The repulsion or the contact angle depends upon molecular weight as illustrated in Figure 9.

Figure 9.

Contact angle vs. molecular weight. (a) Water; (b) ethanol (c) starch solution; and (d) potassium chloride solution.

Conclusions

This paper introduces the bubble electrospinning and the fiber morphology and outlines the theoretical principle of the geometrical potential and basic design concepts of super-hydrophilic property or super-hydrophobic property of nanofiber membranes. In a nutshell, the theory and the actual design depend crucially upon the morphology or porous structure of the nanofiber membranes.

In our theory, only the nanoscale porous structure has either absorption or repulsion. Only a smooth fiber can attract a particle into its surface, but it cannot be absorbed on the surface, because when the particle is touched on the fiber’s surface, a reaction is produced which is colossally larger than the attraction force, and it will be rebounded back. Only a particle (e.g. a water molecule) is at the point where either the absorption or the repulsion force reaches its maximum, hydrophilic or hydrophobic property will be appeared. High selective absorption or repulsion properties of nanofiber membranes make it possible to fabricate oxygen-enrichment membranes (Shen et al., 2016) or membranes with high adsorption capacity and selectivity of CO₂ over N₂ (Zhou et al., 2017).