Clumping Stability of Vertical Nanofibers on Surfaces - Langmuir (ACS

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Clumping Stability of Vertical Nanofibers on Surfaces Ming Zhou, Kaikai Chen, Xiao Li, Lincong Liu, Youtang Mo, Long Jin, Liangchuan Li, and Yu Tian Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b02009 • Publication Date (Web): 20 Aug 2018 Downloaded from http://pubs.acs.org on August 21, 2018

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Clumping Stability of Vertical Nanofibers on Surfaces Ming Zhou, 1,* Kaikai Chen, 2, *, † Xiao Li,3 Lincong Liu,1 Qunfeng Zeng,4Youtang Mo,1 Long Jin,1 Liangchuan Li,1 Guoshi Su,1 Jiangliu Che,1 Yu Tian, 2,* 1

School of Mechanical Engineering, Guangxi University of Science and Technology,

Liuzhou 545006, Guangxi, China 2

State Key Laboratory of Tribology, Tsinghua University, Beijing 100084, China

3

Chengdu Carbon Co., Ltd, No.88 South2 Road, Economic and Technological

Development Zone, Chengdu 610100, Sichuan, China 4

Key Laboratory of Education Ministry for Modern Design and Rotor-Bearing System,

Xi’an Jiaotong University, Xi’an, 710049, China

ACS Paragon Plus Environment

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ABSTRACT: The clumping behavior of nanofibers, including nanowires and nanotubes, is a challenge to their fabrication, which may diminish their optical, electrical, and mechanical performance. However, the stability of the clumping status, especially the unstable clumping state, was rarely discussed to give a deep understanding on clumping criteria. In this study, an energy-based analysis of the nanofiber system was introduced to analyze the deformation of the fibers, providing a novel method to define the thermodynamic stability and the kinetic stability of clumping. The clumping stability design map was proposed, further the stability of the clumping status and the criteria of the five states (the stable, the thermodynamic stable, the kinetic stable, unstable and the non-clumping state) were discussed. The theoretical criteria provide new insights in to designing nanofiber arrays on surfaces to achieve desired clumping or non-clumping states.

KEYWORDS: nanofiber arrays, nanowires, nanotubes, clumping, stability


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INTRODUCTION Since its birth in 1990s, micro- and nanotechnology has aroused great repercussions in the field of science and technology and has received extensive attention. The design and manufacturing technology of micro- and nanostructures is one of the bottlenecks for the application of micro- and nanoscience and technology. The most common micro nanostructures are nanofibers (nanowires, nano-hair, nano-pillars, nano-filaments, etc.), which exhibit a wide range of potential applications such as ultraviolet nano-lasers, solar cells, Micro-Electro-Mechanical Systems, lab-on-a-chip devices, biotechnology, and micro/nano-sensors1-15. The increase of the aspect ratio could improve the performance, while this also increases the compliance of fibers, leading to the clumping behavior and unable to separate by itself. The undesired clumping of fibers with high aspect ratio caused by their mechanical flexibility and relatively strong attractive adhesion forces, external vibration disturbances and electrostatic forces16, 17, can seriously diminish their optical and other properties and may even lead to the failure of nanofiber array systems18-20. It is important for nanofiberbased applications to avoid the undesired clumping of nanofibers. The properties of materials are always determined by their chemical properties and their configurations. The clumping behaviors of nanofiber arrays depend on the competition between the adhesion force and the bending, determined by their geometry parameters (such as length, radius and spacing distance), surface energy, and bending rigidity, which accordingly determine the clumping conditions. In terms of experiments, Geim et al.21 fabricated the polyimide fibril array inspired by gecko and reported the clumping


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geometry. In view of the critical clumping behavior of nanowires, Khorasaninejad et al.20 prepared a group of silicon nanowires with the same length, the same spacing but different radii, and observed that with decrease of the silicon nanowires, the nonclumping status switched to the side-side clumping, then the top-side and the top-top clumping, and finally the clumping of multiple fibers. Aizenberg et al.22 observed the clumping configuration of multiple fibers, which exhibited twining, annular, helix and hierarchical structures. For experiments, based on the model of the fibril cantilever beams, the relationship between critical clumping spacing and the other parameters such as the length, radius and the Young’s modulus was proposed by Sitti23. Hui et al.24 proposed the critical clumping length of the fibers by modeling the fibers as rectangle cantilevers and considering the differential of the strain energy to the surface energy of laterally adhesion. Then Gao et al.25 understood the non-clumping property of the steal arrays of geckos using this model. Glassmaker et al.26 used the Johnson–Kendall–Roberts (JKR) theory to obtain the elastic energy of the two parallel cylindrical fibers adhered along the axis and proposed the critical clumping length. Tang et al.27 proposed the adhesion model of the two singlewalled carbon nanotubes. By introducing the elliptical contact model, Zhou et al.28, 29 proposed a universal model for the critical clumping criteria for three contact geometries of nanofiber arrays, including the side-side, side-tip, and tip-tip clumping status. As reviewed above, recent advances focused on the contact configurations of the fibers, and the predictions of the critical clumping geometric parameters. To further understand the process of the clumping behavior, the stability of the clumping status should be discussed, as there is an intermediate state of unstable clumping between the non-


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clumping and the clumping state. Basically, the stability of the clumping status could be described as the trend of the deformation of the fibers, which depends on the variation of the system energy. In this article, the system energy formulas of the simplified models of sphere-plane and then the spring-sphere-plane in the whole contact process are theoretically analyzed. Then by discussing the stability of the vertical nanofibers, the criteria of the three states including the stable clumping state, the stable non-clumping state and the unstable clumping state is proposed.

MECHANISM AND DEFINITION In chemistry, chemical stability is described in two aspects: thermodynamic stability and kinetic stability. The Thermodynamic stability is determined exclusively by the difference of free energy between the initial state and the final state, while the kinetic stability described as the activation energy decides the reaction rate in practice under the given environmental conditions. When the initial barrier of activation energy that must be overcome to set the reaction in motion is relatively high, the reaction tends to proceed slowly. Basically, energy is the fundamental measure of the transformation of physical motion. Inspired by the theory of the chemical stability, the clumping stability could be described by introducing the concepts of thermodynamic stability and kinetic stability. For the system of the two neighboring fibers, the forces doing work in the processes of clumping and anti-clumping (i.e. restoring the non-clumping state from clumping state) include the bending elastic force of the fibers, the surface forces between the two neighboring fibers and the contact elastic force in the contact region of the fibers. Particularly, the surface


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forces and the contact elastic force in the contact process could be described by contact mechanisms, which are analyzed in the contact system in the following section. Thus, the energy of a contact system could be defined as the sum of the energies of the two objects that may change due to contact. Obviously, surface force play importan role in the contact process, which dominates the change of the surface energy during contact. Besides, the contact objects would deform, which leads to the change of the elastic deformation due to contact. Then, the the energy of contact system could be described by contact mechanisms. Therefore, the bending elastic energy and the energy of contact system contribute to the clumping system of the two neighboring fibers. Based on the analysis above, the clumping criteria could be defined in the following. The typical processes of clumping (the solid line) and anti-clumping (the dotted line) were illustrated in Figure 1. U is the energy of contact system and the bending elastic energy of the clumping system of the two neighboring fibers, while δ is the contact displacement of the two neighboring fibers (i.e. positive when compressed and negative when stretched or separated). The clumping energy state could characterize the thermodynamic stability of clumping, i.e. the clumping behavior was defined as thermodynamically stable when the clumping energy is less than zero. On the other side, the difficulty in the process of clumping or anti-clumping should be considered as the kinetic stability. There are two energy barriers of clumping and anticlumping in Figure 1. The distance δ approaching to negative infinity, represents the objects infinitely far away from each other, while δ>0 represents the objects contact and compressed. When they clumped, the energy of the objects should overcome the energy barrier of clumping and then get to the clumping energy state as shown in Figure 1. When


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they seperated to the non-clumping state, the energy of the objects should overcome the energy barrier of anti-clumping and then return to the natural state. If the energy barrier of clumping is less than the energy barrier of anti-clumping, the clumping behavior is defined as kinetically stable.

Figure 1. The typical processes of clumping and anti-clumping.

MODEL AND METHODS In order to discuss the stability of nanofibers, the energy of adjacent fibers is to be analyzed. The contact area of the tip-tip clumping of fibers with sphere ends is a circle, thus the contact configuration is simplified as that of a sphere on a plane. In this case, the ends of adjacent fibers jump into contact and adhere to each other, which is equivalent to the transformation from a free state to a sphere-plane (ball-on-flat) adhesion in the spring-sphere-plane system, as shown in Figure 2, where the sphere-sphere contact is


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further simplified as sphere-plane contact. We analyze the energy curves in this simplified system from a free state to the adhesion state.

Figure 2. Schematic of the spring-sphere-plane model for the tip-tip clumping of nanofibers. (a) The free states without clumping. (b) The contact state.

RESULT AND DISCUSSION Energy curves in the sphere-plane model In the analysis of the energy change from a free state to the adhesion state in the sphereplane model, the sphere and the plane are regarded as ideal elastic bodies so the adhesion hysteresis induced by the viscoelastic effect is neglected. The sphere is driven by a force P to move close to the plane from an infinite distance and is then compressed, and it detaches as the force is uploaded. If the force P is changing to keep the ball in an equilibrium state so it is written as


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P = FE + FS

（1）

where FE is the elastic force of the ball and Fs is the surface force between the sphere and the plane. P is positive when the sphere is compressed along the approach direction while FE and Fs are negative along the approach direction. If the force is integrated as

∫

δ

δ

Pdδ = ∫ ( FE + FS )dδ

−∞

−∞

（2）

the equation can be also represented as

WP = U E + U S

（3）

where WP = U E + U S equals the energy of the contact energy UB-P of the sphere-plane system. There are several contact theories to describe the relationship between these energies (UE , US and UB-P ) and the displacement δ . Thus, we could discuss the transformation of the contact energy during the process of contact and moving back to an infinite distance. Generally, the phenomena of “Jump into contact” during contact and “Jump out of contact” during moving back to an infinite distance were discussed both in theories and experiments previously. Take the most commonly used Johnson-Kendal-Roberts model(i.e. JKR theory) for example to understand the role which the UE and US played30. Based on the JKR model,

U E and U S are given as30


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UE =

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 1 53 1 2 13   P1 + P P1  2 1   15 3  K 3 R3  1

∗

 R P1   U S = −π∆γ BS   K 

（4）

2 3

（5）

where P1 is

P1 = P + 3πR∆γ BS + 6πR ∗ ∆γP + (3πR∆γ )

In the above equations, K is represented as K =

modulus:

2

（6）

4 ∗ E . E ∗ is the equivalent contact 3

1 1 − v12 1 − v22 = − , where E1 , E 2 , v1 and v 2 are the elastic modulus and the E1 E2 E∗

Poisson's ratios of the two contact bodies. R ∗ is the equivalent contact radius and ∆γBS is the adhesion energy in unit surface area. In the ideal JKR model, the dimensionless force P and the dimensionless energy U s , U E and U s + U E as functions of the dimensionless displacement δ are shown in Figure 3. P and δ are calculated as

P=

δ P and δ = ∗ 2 ∗ 2 πR W12 9π R ∆γ BS 16E ∗2

(

)

13

. After dimensionless treatment, δ is then

1  1 23 2   δ =  P1 + P P1 3  3 3 

（7）

Where P1 is calculated as


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（8）

P1 = P + 3 + 6P + 9

The sphere jumps into contact form point O to point Q (shown in Figure 3a). A part of the adhesion energy is converted to elastic energy, and another part is dissipated. The energy of the contact system also suffers a loss from O' to Q' in Figure 3b. In Figure 3b, it is shown that both the surface energy and the elastic deformation due to contact play critical roles in the energy of a contact system.

Figure 3. Force and energy curves from a free state to the adhesion state in the sphere-plane model based on the ideal JKR theory. (a) The dimensionless force

P

as a function of the dimensionless displacement

δ . (b) The dimensionless energy U s , U E and U E + U s as functions of δ . Besides, there are also other adhesive contacat model proposed by other scholars, which 1

were suitable under different Tabor number µT =

2

R ∗ 3 ∆γ 3 2 ∗3

. To be more precise, a

E ε

numerical adhesive model of elastic spheres was proposed by Greenwood31, 32. Actually, the surface force is not 0 as the sphere is approaching from an infinite distance, and


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Greenwood showed the process is actually described as a “S” curve, as shown by the F1B1-E1-C1 curve in Figure 4a31, 32. The sphere approachs form an infinite distance (point A1) to B1, jumps to that of C1, and then reaches the minimum energy at D1, so the system is in an equilibrium at this point after unloading the forces. On the other hand, when the sphere is moved back to an infinite distance, the state jumps to F1 from E1. The energy curve can be integrated from the force curve in Figure 4a using Equation 2, and the result is shown in Figure 4b. The jumps of force in these two occasions leads to the energy dissipation. The contour of the energy curve integrated from the S curve is similar to that predicted by the ideal JKR theory. The differences between the numerical results and the classical models (including JKR theory and Derjaguin-Mulla-Toporov theory) were discussed in ref 27. It is worth noting that the numerical results showed that with different Tabor number, the shape of the “S” curve varied. When applied to actural cases, the Tabor number should be calculated with the parameters of R*, ∆γ , E* and ε . However, the “S” shape of the curves was unifying, so that we could discuss the common jump of the contact force during contact and moving back to an infinite distance. Thus, take the typical Tabor number of 3 for example, the further force and energy curves were discussed in this manuscript.


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Figure 4. The force and energy curves in the sphere-plane model and the spring-sphere-plane model from a free state to the adhesion state. (a) and (b) show the force and energy in the sphere-plane model with a

typical Tabor number

µT =

2 3

1 ∗3

R ∆γ 2 ∗3

= 3 . ε is the distance between atoms. (c) and (d) show the force

E ε and energy curves in the spring-sphere-plane model.

Energy curves in the spring-sphere-plane model For the clumping behavior of nanofibers, the laterl bending force is approximately proportional to the disturbance. Thus, the bending deformation of the nanofibers could be modeled as the elongation deformation of the spring, which the contact system is fixed on.


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With the elongation deformation of the spring considered, the sphere-plane model is replaced by the spring-sphere-plane model by assuming that the sphere is fixed by a spring, as shown in Figure 1. The energy of the system is thus described as

UT = U K + U E + U S , including the elastic energy of the spring U K . For the deformation of the energy, the constitute relation is determined using a dimensionless method similar to that of the JKR model,

∆δ

Fk =

2 3

1 3

∗

2 3

K π R ∆γ ks

1 3 BS

（9）

where ∆δ is the dimensionless displacement of the sphere with regard to the position in a 2 3

1 3

∗

2 3

K π R ∆γ free state. We define κ = ks

1 3 BS

2 3

∗

1 3

∗

2 3

1 3 BS

, and κ = K π R ∆γ , which has a unit of

N/m of a spring constant and is named as the equivalent spring constant. So κ is the ratio of the equivalent spring constant to the spring constant for quantifying the rigidities of the contact surface and the spring. The dimensionless elastic energy of the spring is then

1 Ek = 2

∆δ 2 3

1 3

2

∗

2 3

K π R ∆γ ks

1 3 BS

（10）

The dimensionless force P , energy of the system U T as functions of δ are shown in Figures 4c and 4d. Here an extra force is needed to overcome the energy barrier at B2' to achieve the force equilibrium state at D2, and the system has the minimum energy at D2'. So the energy barriers at A2' and B2' are to overcome to reach the minimum energy state


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at D2' from the state with the free spring and the detached sphere from the plane. On the other way around, from D2' to the free state, barriers at D2' and E2' are to overcome. Therefore, the criteria of the stable adhesive contact in the spring-sphere-plane is described as: the energy barrier from a free state to the contact state is less than that in the other direction. This is explained by the equation

U B 2′ − U A2′ ≤ U E 2′ − U D 2′ ,

（11）

which is defined as the kinetically stable condition. Figures 5a and 5b show the force and energy curves with the same A2 but different κ values at the free state of the spring. With the decrease of κ , the position with equilibrium force (and the minimum energy) shifts to the left, and the minimum energy increases even above 0. Meanwhile U B 2′ − U A2′ is increasing while U E 2′ − U D 2′ is decreasing. Thus, with the same δ of A2, the smaller the κ , the easier for the system to maintain a free state.

Figures 5c and 5d show energy curves with the same κ but different A2 values. With the increase of the distance between the spring-sphere and the plane at free states, the force equilibrium state also shifts to the left, and the minimum energy at stable contact increases. U B 2′ − U A2′ is increasing while U E 2′ − U D 2′ is decreasing. Thus, with the same κ , the smaller δ of A2, the easier for the system to maintain an free state.


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Figure 5. Force and energy curves in the spring-sphere-plane model with different parameters. (a) and (b) show the plots with the same value but different values of

δ of A2 but different κ values. (c) and (d) show the plots with the same κ δ of A2.

The critical value of κ is calculated to satisfy Equation 11 with different δ values of A2, as shown in Figure 6. The distance between the sphere and the plane is defined as δ 0 at the free state of the spring. With the increase of the absolute value of δ 0 , κ should be increased to equal the energy barrier in the approaching and detaching processes. κ values are fitted with a power function as


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( )

κ = 0.467 − δ0

1.815

（12）

With κ values below the calculated critical values, as shown in the shadowed area in Figure 6a, it is easier for the system to switch from a free state to a contact state than in the reverse process, i.e., the system tend to stay in a free state rather than a contact state with load disturbances. So the adhesion force is fitted as

P = −1.392κ −0.374

（13）

To avoid the adhesion of the sphere to the plane, the force should be

Fk 0 =

− δ0 3 > κ 2

（14）

( )

（15）

So κ is

κ
−δ0 × 2 + 2R = 2.4766

∆γ f

0.8503 1.6526

L

π 0.2995 (1 − v 2 )

E 0.8503 R1.5029

0.2995

+

（19）

Based on Equation 15, the criteria is then described as


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a>2

L3∆γ + 2R R3E

（20）

We also verified the results by calculating the values using Equation 1 in ref 24 and the parameters of the tip-side contact shown in table S128. The calculated values agree with those obtained from Equation 19, showing that both the methods work for the calculation.

Applicability and limitation of proposed model Previously, other criteria of clumping of nanofibers based on the maximum adhesion force, predicted whether the clumping behavior could happen. This theory based on the curve of the energy of nanofibers under different contact distances, which is the integral of the contact force and the bending forece to the displacement, discussed difficulty and the stability of the clumping behavior. Basically, the shape of the adhesive force curve played important roles. The capillary forces, the electrostatic forces and other surfaces forces would change the specific values of the critical stable clumping conditions 35

.Therefore, the direct numerical results in the manuscript is not valid when the forces

other than van der waals forces play a leading role. However, the jumps of force in the process of the clumping and anti-clumping still occur, and the theory of analysising the kinetical stability and the thermodynamical stability based on the energy barrier is suitable by taking these surface forces into account33, 34. Basically, this theory can be used to the analysis and then design of the clumping of carbon nanotubes, nanofibers of othe pillar array. According to the desired clumping stability state, the Tabor number should be calculated and then the numerial force curve and then the energy curve could be obtained. Then the clumping stability state figure


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similiar to Figure 6 can be otained to guide the design of the spacing distance or the bending rigidity. Further, based on the fundamental analysis of the kinetical stability and the thermodynamical stability, this priciple can be extended to the application on the many areas about pillar array clumping, such as the control of the capillary-induced selforganization, elasto-capillary fabrication3, 5, 8, 9.

CONCLUSIONS In this study, a universal model of the stability criteria of the clumping behavior of vertical nanofibers was presented based on the analysis of the system's energy change of the two neighboring fibers in the processes of clumping and anti-clumping state. A novel method to define the thermodynamic stability and the kinetic stability of clumping was proposed. Further, quantified distance criteria were presented for the thermodynamic stable and the kinetic stable state of nanofibers with tip-tip contact. According to the criteria, the clumping stability design map, which could be divided into five regions including stable, the thermodynamic stable, the kinetic stable, unstable and non-clumping states, were discussed. The theoretical model and the clumping stability design map proposed in this study have important implications in the design and analysis of clumping behavior of nanofiber arrays on surfaces.


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ASSOCIATED CONTENT Supporting Information. Table S1 showing the parameters of the tip-side contacts.

AUTHOR INFORMATION Corresponding

Authors.

*E-mail:

[email protected];

[email protected]; [email protected] Present Addresses. † Present address: Cavendish Laboratory, University of Cambridge, JJ Thomson Ave, Cambridge, CB3 0HE, United Kingdom. Notes. The authors declare no competing financial interest.

ACKNOWLEDGEMENTS This work is supported by the Natural Science Foundation of China (Grant No. 51505442)

and

Guangxi

Natural

Science

Foundation

under

Grant

No.

2018GXNSFAA138174.


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TABLE OF CONTENTS GRAPHIC


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Clumping Stability of Vertical Nanofibers on Surfaces - Langmuir (ACS

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