Sphere-To-Tube Transition toward Nanotube Formation: A Universal Route by Inverse Plateau−Rayleigh Instability Long Ma,†,§ Jing Peng,‡,§ Changzheng Wu,*,‡ Linghui He,† and Yong Ni*,† †
CAS Key Laboratory of Mechanical Behavior and Design of Materials, Department of Modern Mechanics, and ‡Division of Nanomaterials & Chemistry, Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China S Supporting Information *
ABSTRACT: Nanotube formation in low-temperature solution has attracted intense interest since the 1990s. How to disclose the in-depth physicochemical nature of nanotubes and pursue new available chemical strategies is still highly desirable but remains a challenge. Here, we report that sphere-to-tube transition triggered by inverse Plateau−Rayleigh instability can be a chemical route for scalable production of nanotubes. As a proof of concept, formation of a phosphorus nitride (PN) nanotube and various hierarchical nanotube architectures by coalescence of the PN hollow spheres is achieved under systematic solvothermal reaction. The combination of theoretical analysis and dynamic simulation elucidates that the inverse Plateau−Rayleigh instability driven by the competition between curvature elasticity and surface energy is responsible for the PN nanotube formation observed in experiments. We anticipate that the sphere-to-tube transition provides a paradigm for nanotube synthesis for practical applications. KEYWORDS: sphere-to-tube transition, nanotube synthesis, solvothermal reaction, hollow spheres, Rayleigh instability
I
As shown in Figure 1, a sphere-to-tube transition can be seen as the signature route toward a tubular structure in the solution process. Natural phenomena of sphere coalescence into nanotubes have been widely observed in biological fields, such as lipid vesicle coalescence,13−15 and the study of the tubulation in lipid vesicles shows that the curvature elasticity modulated by anchoring amphiphilic polymers favors the tube formation.16 However, the tubular structure may undergo a pearling instability under the competition between curvature elastic energy and surface energy.17−19 The tube-to-sphere transition is known as Plateau−Rayleigh instability, which first considers liquids based on surface energy minimization and is later applied to solids in the sense that elasticity is sometimes not ignored,20−24 while the sphere coalescence into a tubular structure needs an inverse Plateau−Rayleigh instability. Actually, the signature of sphere coalescence into carbon nanotubes has been found in small proportion samples.25 However, so far, there is no detailed investigation on the nature of sphere-to-tube transition and no development of it into a large-scale nanotube preparation as functional nanomaterials.
norganic nanotubes, as prototype forms of one-dimensional nanostructures, have attracted significant attention in recent years. Cylindrical nanotubes usually have a long hollow structure with controlled wall thickness in the nanoscale range, which leads to extensive applications owing to their exceptional strength and stiffness as well as extraordinary thermal conductivity and electrical properties.1−5 Great progress has been made in the synthesis of nanotubes since the early 1990s, and a high-temperature reaction has promoted nanotube synthesis, in that nanotube formation conventionally requires overcoming the energy barrier of layer crimping, bond incorporation, and breaking at increased temperatures.1,6−8 Meanwhile, the low-temperature solution also triggers much interest for nanotube formation due to scalable production and low-energy consumption. Up to now, a few available strategies for nanotube formation without high-temperature reactions have been reported, including strain-induced rolling up of thin layers,9−11 a template sacrificing method,12 and so on. As is known, a solution process could facilitate diffusion and reaction of molecules or ions, providing an alternative way to lower the energy barrier of nanotube formation. Even though substantial progress has been made for nanotubes by a solution-based process, disclosing the in-depth physicochemical nature of nanotubes and pursuing a new available chemical strategy is still highly desirable but remains a challenge. © 2017 American Chemical Society
Received: December 8, 2016 Accepted: March 8, 2017 Published: March 8, 2017 2928
DOI: 10.1021/acsnano.6b08248 ACS Nano 2017, 11, 2928−2933
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reaction time. Scanning electron microscopy (SEM) and transmission electron microscopy (TEM) are utilized to investigate the growth process and structural transition of PN. As can be seen in Figure 2A,D, the product mainly consists of hollow spheres with an average size of 100 nm after reacting for 1 h. Subsequently, PN nanotubes arise and randomly intercross with hollow spheres by increasing reaction time (Figure 2B,E). Specifically, as the signature of the sphere-tonanotube coalescence, the worm-shaped nanotube, indicating that the nanotubes are formed via coalescence of primitive hollow spheres, is clearly observed in Figure 2E; as the coalescence continues, the spheres completely convert into nanotubes and spontaneously evolve into a nanotube network, which is demonstrated in Figure 2C,F. These PN nanotubes usually have a typical 200 nm diameter with the length ranging from several to tens of micrometers. Their closed ends also indicate that these tubes are formed through sphere-tonanotube coalescence. Intriguingly, the cross-linked nanotubes are found to be interconnected by a linked angle of 90°. From the above sequential morphologies, it is suggested that the whole reaction can be divided into two processes: The PN hollow spheres can be formed within a short reaction time and then gradually merge into sphere chains and finally coalesce into nanotubes over a relatively long time. During the evolution of nanotube formation, the nanotubes cross-link with each other and consequently self-assemble into hierarchical nanotube architectures. Linear Stability Analysis. To elucidate why the sphere-totube transition occurs and how it dominates such a selfassembly process, we perform a shape instability analysis as well as numerical simulations based on the phase-field method. Herein, we consider that a cylindrical membrane tube of radius R undergoes a sinusoidal perturbation in the form of r = R[1 + ε cos(kz)], where ε is the perturbation amplitude, k = 2π/λ is the wavenumber of such sinusoidal undulation, and z is the coordinate along the height axis in cylindrical coordinates. The total free energy of the tube based on the Canham−Helfrich model26 includes the curvature elastic energy and the surface tension term.
Figure 1. Sketch of typical routes of nanotube synthesis without high-temperature reaction: (A) nanotube formed by strain-induced rolling up of thin layers; (B) nanotube formed by eliminating the core of a core−shell nanowire; (C) sphere-to-tube transition toward nanotube formation.
Herein, we report a clear physicochemical panoramic scene for sphere-to-tube transition directing nanotube formation for the first time. As a proof of concept, phosphorus nitride (PN) hollow spheres coalescence into nanotubes and various hierarchical nanotube architectures is achieved under systematic solvothermal reaction. We propose that the inverse Plateau− Rayleigh instability driven by the competition between curvature elasticity and surface energy is responsible for the formation of interconnected tubular network observed in experiments by using theoretical analysis and phase-field simulation. We anticipate that the sphere-to-tube transition provides a paradigm for nanotube synthesis and its practical application.
RESULTS AND DISCUSSION Synthesis and Evolution of PN Nanotubes. In the facile solvothermal route, white PN of diverse characteristic shapes is obtained by reacting P3N3Cl6 with NaNH2 under progressive
Figure 2. Morphology of PN with different reaction times. SEM images of PN at reaction times of (A) 1, (B) 3, and (C) 9 h. TEM images of PN at reaction times of (D) 1, (E) 3, and (F) 9 h. 2929
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∫Γ H2dS + σ ∫Γ dS
Nonlinear Dynamic Simulation Based on the PhaseField Method. The energetic instability analysis explains why the hollow sphere-to-tube transition occurs and predicts the parameter range of the transformation from hollow spheres to cylindrical nanotubes. However, it does not shed light on how the sphere chain coalesces into the nanotube. To take a closer look at the dynamic coalescence process, we performed a Ginzburg−Landau based phase-field simulation27 to study the morphological evolution during the coalescence process of hollow spheres. In the phase-field model, a phase-field variable ϕ = ϕ(x) is introduced to label the surface profile of the hollow sphere assembly Γ: the set ϕ(x) = 0 locates the hollow sphere surface, whereas ϕ(x) = 1 is inside the hollow sphere and ϕ(x) = −1 is outside the hollow sphere. Tracking the coalescence process of a hollow sphere chain is obtained by solving the Cahn−Hilliard equation for the evolution of the phase-field variable driven by minimizing the total free energy. The total free energy contributed by curvature elastic energy and surface energy is expressed as a function of the phase-field variable (refer to the Methods). Unless otherwise stated, the simulations are conducted under σR2/κ = 0.125, at which the curvature elastic energy dominates and favors the nanotube formation. Figure 3B plots the free energy changes versus the dimensionless time t/τ with τ = R2/κD, a characteristic time scale, and D > 0 characterizes the mobility parameter during the morphological evolution from a hollow sphere chain to a cylindrical nanotube, indicated by the insertions. It shows that at t/τ = 0 hollow spheres are roughly arranged in a line, and then they connect together into a corrugated nanotube rapidly driven by significantly releasing the curvature elastic energy after t/τ ≈ 5. Over a relatively long time, the corrugation of the nanotube tends to be flattened. A steady state of a straight nanotube is obtained after t/τ ≈ 200, wherein the curve of the total free energy with respect to time exhibits a plateau. Figure 3C displays the energy changes versus the scaled time during the coalescence of two randomly interlaced hollow sphere chains. One can see that first the two chains both form corrugated nanotubes in a short time and join with a junction by t/τ ≈ 20. After a relatively long time, the two nanotubes become straight and the junction becomes orthogonal, which can more efficiently relax the curvature elastic energy after t/τ ≈ 200. Finally, a cross-shaped nanotube structure forms, and the total free energy almost reaches a minimum. Figure 4A−C further illustrates the simulated morphology evolution of a cluster of hollow spheres with random arrangement in a three-dimensional case. It shows that the adjacent spheres coalesce into rugged nanotubes with an irregular junction rapidly by t/τ = 5 (Figure 4B), and each rugged nanotube in the structure tends to be straight; meanwhile, the adjacent nanotubes near the junctions are arranged perpendicularly after t/τ = 400 (Figure 4C). The SEM image of an ordered nanotube structure in Figure 4D indicates that the simulation is in good agreement with the experimental result. Our simulation indicates that in the case of no spontaneous curvature if two nanotubes are non-orthogonally connected, the distribution of the thermodynamic driving force is asymmetric on each side of the junction because the acute angle side possesses larger curvature than that at the obtuse angle side, and the difference of curvature elastic energy on each side drives the junction to be orthogonal. However, if there is nonzero spontaneous curvature, the curvature elastic energy attains its minimum as the local curvature is equal to the spontaneous curvature. This may lead to a non-orthogonally
(1)
where κ is the bending rigidity and σ is the surface tension coefficient. H is the mean curvature of the surface. In this expression, the first term represents the curvature elastic energy and the second term that is proportional to the surface area represents the surface tension energy. The free energy is integrated over the whole surface Γ. For the given sinusoidal configuration, by introducing two dimensionless variables x = kR and s = σR2/κ, the variation of the total free energy associated with the perturbation can be expressed by ΔW = κπ2ε2f/2 with f = 3/x + x(−3 + 2x2) + 2s(x2 − 1)/x, a dimensionless expression of the energy variation per wavelength between the perturbed and initial cylindrical tube. A positive value of f indicates that the perturbed state is energetically unfavorable. Based on the shape instability analysis (refer to the Methods), a phase diagram for the morphology from a cylindrical nanotube to a pearling (close to sphere chain) state is shown in Figure 3A. It is found that below a
Figure 3. Stability analysis and dynamic simulation of the morphological transition from a hollow sphere chain to a cylindrical nanotube: (A) phase diagram for the sphere-tonanotube transition; (B) evolution of a sphere chain into cylindrical nanotube; (C) evolution of hollow sphere chains into a perpendicularly aligned nanotube junction.
critical value of σR2/κ, curvature elastic energy dominates and favors a cylindrical nanotube structure regardless of all wavelengths. However, the cylindrical nanotube becomes unstable when σR2/κ increases. Therefore, carefully adjusting the value of σR2/κ is important to guarantee that the cylindrical nanotube is thermodynamically stable. 2930
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addition, the simulated results in Figure S2d−f indicate that the transient corrugation state can be formed during the coalescence of a single hollow sphere to an existing cylindrical nanotube. The simulated results in Figures S3 and S4 also show that the rich transient corrugation states can be induced by variation of position and arrangement of the initial hollow spheres. Good agreement between the experimental observation and the simulation is achieved. The agreement further supports that the sphere-to-tube transition plays a key role in the PN nanotube formation. In previous reports, the carbon nanotubes coalesced by carbon hollow spheres usually have a narrow range of diameter and wall thickness. The diameter and wall thickness of the PN tubular structure reported here are in a relatively broad range. Figure S5 shows that the diameter can be from 140 to 190 nm and the wall thickness can be from 21 to 45 nm. In addition, the diameter of the PN nanotube structure monotonously increases with the increase of the wall thickness. This can be roughly understood in the following. The dimensionless parameter σR2/κ indicates that the diameter of the PN nanotube is related to a characteristic size R ∼ κ /σ . By the way, if we note that the bending rigidity κ of multilayered material is related to its thickness h according to mechanics of the plate, κ ∼ h3, there is a relation between the radius and wall thickness R ∼ h3/2, which can explain the positive correlation of R versus h shown in Figure S5.
Figure 4. Evolution sequence of hollow spheres with random arrangement coalescing into a three-dimensional ordered nanotube structure at t/τ = 0, 5, and 400.
connected nanotube structure. For example, in the carbon nanojunctions, the spontaneous curvature at the junction site can be induced by the aggregation of topological defects, such as pentagonal, heptagonal, and octagonal ring-like defects. Therefore, “X” and “Y” crossed-nanotube junctions can be observed in experiments. In addition, the fact that the orthogonal nanotube junction is less observed in the membrane tubular structures is possible due to the emergence of spontaneous curvature induced by adsorption of particles. The above results indicate that the nanotube formation through the sphere-to-tube transition is triggered by inverse Plateau−Rayleigh instability. The instability analysis based on the energy variation confirms that when the curvature elastic energy dominates, the straight nanotube is more energetically favorable than the hollow sphere chain because one of its two principle curvatures in the cylindrical nanotube structure is zero. This requires that the value of σR2/κ should be below a critical value (σR2/κ ∼ 1.5), as shown in Figure 2. When the value of σR2/κ is larger than the critical value, the surface tension energy dominates and the results in Figure S1 show that a pearling state would break into a hollow sphere chain, as a consequence of Plateau−Rayleigh instability, and a cluster of hollow spheres tends to evolve into a big hollow sphere instead of forming the nanotube structure. Herein, we investigate the parameter σR2/κ of PN tubes observed in the experiments. The surface energy σ is about 1 J/m2 for most inorganics, and the bending rigidity of nanotube can be approximately expressed by κ=
Eh3 , 12(1 − v 2)
CONCLUSIONS In summary, formation of a PN nanotube and various hierarchical nanotube architectures by coalescence of PN hollow spheres is achieved under systematic solvothermal reaction. The diameter and the wall thickness of the formed PN nanotube structure are tunable in a relatively broad range. An energetic calculation elucidates how sphere-to-tube transition as a fundamental mechanism of the PN nanotube formation observed in experiments occurs through the inverse Plateau− Rayleigh instability driven by the competition between curvature elasticity and surface energy. The dynamic simulation based on the phase-field method further recovers the microstructure development during sphere-to-tube transition that eventually leads to hierarchical nanotube structures, in good agreement with the experimental observations. The obtained results provide an efficient way for scalable fabrication of interconnected nanotube architectures with orthogonal orientation and controllable size. METHODS Synthesis of PN Nanotubes. In a typical solvothermal reaction, P3N3Cl6 and NaNH2 (molar ratio of 1:6) were added into Teflon-lined autoclave with 30 mL of cyclohexane. The sealed autoclave was kept heated at a temperature of 220 °C for 1, 3, and 9 h, washed by ethanol and distilled water three times, and finally dried in vacuum at 60 °C for 6 h. Characterization of PN Nanotubes. The field emission scanning electron microscopy images were obtained on a JEOL JSM-6700F SEM, and the TEM images were taken on a JEOL-2010 transmission electron microscope at an acceleration voltage of 200 kV. Linear Stability Analysis of the Tubular Structure. To check the shape instability, we assume that a tube with initial radius R0 is under a small shape perturbation of the sinusoidal form
where E is Young’s modulus, h is the wall
thickness of nanotube, and v is Poisson’s radio. Using R = 90 nm, h = 40 nm, and v = 0.3, we can obtain σR2/κ ∼ 109 Pa/E. Meanwhile the Young’s modulus of nitride-based materials is hundreds of GPa,28 so the value of σR2/κ in experiments is much less than the critical value which allows the sphere-totube transition. We note that besides orthogonal interconnected nanotube structures, many intermediate corrugated nanotube structures are also observed in the experiments. The results in Figure S2a−c demonstrate that coalescence of spheres with different radii could produce a significantly rugged surface, which lasts longer than that in the case of spheres with the same radii. In
r = R[1 + ε cos(kz)]
(2)
where ε is the perturbation amplitude, k = 2π/λ is the wavenumber, and z is the coordinate along the height axis in cylindrical coordinates. 2931
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employed. Model-B-like dynamics is applied for the dynamic evolution equation. Denote
2
R 0 = R 1 + ε /2 by comparing the initial and disturbed volume of the tube. With two principle curvatures as k1 and k2, the elastic energy and surface tension energy per wavelength before perturbation are, respectively
Wel =
κ 2
∫0
2π / k
(k1 + k 2)2 dS =
ε2 ⎞ 2κπ 2 2κπ 2 ⎛ = ⎜1 − ⎟ kR 0 kR ⎝ 4⎠
⎧ 1 2 ⎪ g = ϕ(ϕ − 1) ε ⎪ ⎪ 1 ⎨f 2 ⎪ = εΔϕ − ε ϕ(ϕ − 1) ⎪ ⎪ h = 3ϕ2 − 1 ⎩
(3)
and
(
and B(ϕ) = 2κ −Δ2g −
4π 2R 0 4π 2R ⎛ ε2 ⎞ Wsf = σ =σ ⎜1 + ⎟ k k ⎝ 4⎠
κ 2
∫0
2π / k
(4)
with
⎡1 ⎤2 2 ⎢⎣ [1 − ε cos(kz)] + εRk cos(kz)⎥⎦ dS R
the
area
∫0
2π / k
1dS ≈
(5)
element
σπ 2R(4 + ε 2k 2R2) k
ϕ(n + 1) − ϕ(n) = εκDΔ3ϕ(n + 1) + DB[ϕ(n)], n = 0, 1, 2, ... Δt (12) where Δt > 0 is the time step and the superscript represents the values at t(m) = mΔt. So this equation can be solved numerically as long as the initial configuration ϕ(0) ∈ Ω of the hollow spheres is given in a computational domain. In all of our simulations, the unit length is chosen as l = ε and a uniform grid of 128l × 128l × 128l is employed. The radii of spheres are R = 10l, and the elastic energy is expected to be the dominant factor (σR2/κ = 0.125) unless otherwise stated. The time step is set by DΔt = 10−5, and periodic boundary condition is always used. In addition, we use τ = R2/κD as a characteristic time during the morphological evolution.
(6)
(7)
Using two dimensionless parameters, s = σR /κ and x = kR = 2πR/λ, the variation of the total free energy associated with the perturbation is 2
ΔW tot =
κπ 2ε 2 [3/x + x(− 3 + 2x 2) + 2s(x 2 − 1)/x] 2
(8)
ASSOCIATED CONTENT
2 2
κπ ε
Because 2 > 0, we only need to estimate the value f = 3/x + x (−3 + 2x2) + 2s (x2 − 1)/x, a dimensionless expression of the energy variation per wavelength between the perturbed and initial tube. A positive value of f indicates that the perturbed state is energetically unfavorable. Phase-Field Method To Simulate Evolution of Hollow Spheres Coalescence. The phase-field method based on the minimization of the total free energy is widely used to study the morphological evolution of microstructure. In the phase-field method, the microstructure is described by phase-field variables which are continuous and smooth. In this work, we define a phase-field variable ϕ = ϕ(x) on the computational domain Ω in R3 to label the position of the hollow sphere: the set ϕ(x) = 0 locates the hollow sphere surface, whereas ϕ(x) = 1 gives the inside of the hollow sphere and ϕ(x) = −1 the outside bulk. The total free energy which includes curvature elastic energy and surface tension energy is expressed as a function of phasefield variable and its gradient:27 ⎛ ⎞2 1 W (ϕ) = κε⎜Δϕ − (ϕ2 − 1)⎟ dx ⎝ ⎠ Ω ε ⎛ε ⎞ 1 2 + σ ⎜ |∇ϕ|2 + (ϕ − 1)2 ⎟dx ⎠ Ω ⎝2 4ε
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.6b08248. Observation and simulation of intermediated tubular structures and the correlation between wall thickness and diameter in the observed PN tubular structures (PDF)
AUTHOR INFORMATION Corresponding Authors
*E-mail:
[email protected]. *E-mail:
[email protected]. ORCID
Changzheng Wu: 0000-0002-4416-6358 Author Contributions §
L.M. and J.P. contributed equally to this work.
∫
∫
(11)
where t is the time, D > 0 is the mobility parameter of the microstructure evolution, and δW/δϕ is the variational derivative of the total free energy. In this equation, the higher-order differential term is spaced out because the semi-implicit Fourier spectrum scheme is chosen as the discretization algorithm. Finally, we obtain the discrete evolution equation:
The surface tension energy can be expressed by Wsf ′ = σ
)
Δ(fh) − σ Δf , the driving equation is
⎛ δW ⎞ ∂ϕ 3 = DΔ⎜ ⎟ = εκDΔ ϕ + DB(ϕ) ∂t ⎝ δϕ ⎠
dS = 2πR 1 + (dr /dz)2 dz ≈ 2πR[1 + (εRk sin(kz))2 /2]dz . W e can obtain an explicit expression of this energetic integration by neglecting the small quantity o(ε4) κπ 2 Wel′ ≈ [4 + ε 2[2 + k 2R2(− 3 + 2k 2R2)]] 2kR
ε2
(10)
shown as
After the perturbation, the two principle curvatures of such surface are k1 = [1 − ε cos(kz)]/R and k2 = εRk2 cos(kz), respectively, so the curvature elastic energy associated with shape perturbation can be written as Wel′ =
1
Notes
The authors declare no competing financial interest. (9)
ACKNOWLEDGMENTS Y.N. was supported by the National Natural Science Foundation of China (Grant Nos. 11472262 and 11132009), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB22040502), the Collaborative Innovation Center of Suzhou Nano Science and Technology. C.Z.W. was supported by the National Young
where κ is the bending rigidity, σ is the surface tension coefficient, and ε is related to the width of hollow sphere interface. In the expression, Δ is the Laplacian operator, ∇ϕ is the gradient of the phase-field variable, and the energy is a bulk integral defined on the whole computational domain Ω. To obtain the equilibrium configuration of the microstructure, a partial differential equation which guarantees the total free energy to decrease monotonically until it reaches a plateau is 2932
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