Rapid programmable nanodroplet motion on a strain-gradient surface

Jan 2, 2019 - When a nanodroplet is placed on a lattice surface, an inhomogeneous surface strain field perturbs the balance of van der Waals force bet...
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Rapid Programmable Nanodroplet Motion on a Strain-Gradient Surface Baidu Zhang,† Xiangbiao Liao,*,‡ Youlong Chen,§ Hang Xiao,‡ Yong Ni,*,† and Xi Chen‡,∥

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†

CAS Key Laboratory of Mechanical Behavior and Design of Materials, Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China ‡ Yonghong Zhang Family Center for Advanced Materials for Energy and Environment, Department of Earth and Environmental Engineering, Columbia University, New York, New York 10027, United States § International Center for Applied Mechanics, SV Lab, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, China ∥ School of Chemical Engineering, Northwest University, Xi’an 710069, P.R. China S Supporting Information *

ABSTRACT: When a nanodroplet is placed on a lattice surface, an inhomogeneous surface strain field perturbs the balance of van der Waals force between the nanodroplet and surface, thus providing a net driving force for nanodroplet motion. Using molecular dynamics and theoretical analysis, we study the effect of strain gradient on modulating the movement of a nanodroplet. Both modeling and simulation show that the driving force is opposite to the direction of strain gradient, with a magnitude that is proportional to the strain gradient as well as nanodroplet size. Two representative surfaces, graphene and copper (111) plane, are exemplified to demonstrate the controllable motion of the nanodroplet. When the substrate undergoes various types of reversible deformations, multiple motion modes of nanodroplets can be feasibly achieved, including acceleration, deceleration, and turning, becoming a facile strategy to manipulate nanodroplets along a designed two-dimensional pathway.

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INTRODUCTION Surfaces capable of directionally transporting water droplets are observed in ubiquitous biological systems, including rice leaves, striders’ legs liquid droplets, and cicada wings, resulting from directional and gradient nanostructures on these organism surfaces.1−6 Inspired by the nature, engineered strategies to manipulate liquid droplet motion have attracted extensive attentions because of potential applications in microfluidics,7,8 energy conversion,9 and smog removal.10 A variety of strategies based on wettability gradient were reported, including permanent chemical (e.g., doping of functional groups11,12) and physical (e.g., surface roughness,13−15 structural topology,16 combination of multiple gradients, etc.17−19) modifications to surfaces or droplet. These irrecoverable and passive treatments only for specific purposes are not suitable for reconfigurable platform of droplet manipulation, and slow droplet velocity of micrometers per second was induced. Additionally, droplet motions can be responsive to external stimuli by electromagnetic field, pH, temperature, and light.20−24 However, additions of chemicals into liquids were employed and thus involved either undesirable contaminations or slow response time. Recently, the gradient of substrate curvature presented a strategy for fast droplet transporting with the velocity up to 0.4 m/s,25 but the droplet may stagnate on the conical substrate instead of moving forward at a critical ratio of the local substrate radii to droplet radii.26 Up to now, facile fabrication of such curvature© XXXX American Chemical Society

gradient substrates and active manipulation on two-dimensional (2D) platform are challenging, which is the gap we aim to bridge in this study. Recently, strain engineering has been widely studied experimentally and theoretically for tuning material properties.27,28 Externally applied strain to the substrate was experimentally adopted to actuate the migration of fibroblasts in the direction of principal strain, and the rolling motion of biological organisms, such as zoosperms, was driven by the strain gradient induced by muscle contraction.29,30 Not limited to biological systems, as one of the effective strategies, the strain-gradient-based method has been realized in nanosystems, including transport of 2D nanoflakes on graphene.31,32 To the best of our knowledge, no explicit study has been reported on manipulating the motion of water nanodroplets in a more controllable way through deforming substrates. In this study, we propose a mechanism for rapid transporting nanodroplets through strain engineering and achieve controllable manipulation of both speed and direction of the nanodroplet along linear and nonlinear paths. Through molecular dynamics (MD) simulations and theoretical modeling, the strain gradient externally applied to substrates is able to shift the balance of van der Waals (vdW) force Received: November 8, 2018 Revised: December 27, 2018 Published: January 2, 2019 A

DOI: 10.1021/acs.langmuir.8b03774 Langmuir XXXX, XXX, XXX−XXX

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wettability, continuous variation of wettability may be achieved by applying strain with directional gradient, becoming a tunable driving force for water droplet motion. The motion of droplet on a monolayer graphene with constant strain gradient is investigated herein. Shown in the top of Figure 2a, the simulation system includes a nanodroplet of 500 water molecules and a monolayer graphene with the dimension of 197 Å × 84 Å. The system is then first relaxed in the NVT ensemble with 300 K for 500 ps with the mass center of nanodroplet fixed in both x and y direction with all graphene atoms fixed. Then, the atoms on the two boundaries of y direction are slowly stretched along y direction to a given position. Their displacements are proportional to their xcoordinates, from zero to dmax, thus creating a constant strain gradient Gx = dmax/wL along the x direction. The continuous variation of strain leads to an asymmetrical configuration with different contact angles along the direction of strain gradient in the bottom of Figure 2a, providing the driving force for droplet motion. The droplet is subsequently released, and its motion was simulated using the NVE ensemble. To save the computational cost and clearly demonstrate the droplet motion on a timescale of hundreds of picoseconds, the strain gradient Gx = 9.15 × 10−4/Å is taken as a representative example. The time histories of the displacement and velocity of the mass center of the droplet are shown in Figure 2b,c, respectively. The droplet speeds up along the x direction with the acceleration of ax = 3.31 × 10−3 Å/ps2, and the displacement along the y direction can be neglected, indicating that the net driving force is along the opposite direction of the strain gradient (see Movie S1). In the present study, the velocity of the nanodroplet can reach 50 m/s. It is evident from Figure 2d that the interactive vdW potential between water droplet and graphene decreases with the movement of droplet, and the analysis of the temperature curve of the droplet (Supporting Information) indicates that the potential energy is mainly converted to translational kinetic energy of the droplet. The driving force for nanodroplet motion can be qualitatively explained by the wettability gradient resulting from the strain gradient. For a spherical-cap drop on a solid surface induced by wettability gradient, the driving force can be written as37

between the substrate (graphene or copper) and the nanodroplet, thus providing the driving force. The programmable strategy suggests broad applications of directional droplet transport.

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MODELING AND METHOD All of the MD simulations were carried out on the LAMMPS package.33 The AIREBO potential field34 was used to describe the graphene, and the long-range TIP4P water model35 was used to model the droplet. The 12-6 Lennard-Jones pair 12 6 6 potential, V = 4eij(σ12 ij /rij − σij/rij), was used to describe the interactions between water molecules and graphene,36 with eCO = 4.063 meV, σCO = 3.19 Å, eCH = 0, and σCH = 0. The cutoff radius of Lennard-Jones pair potential and Coulombic potential is 12 Å, which well balances the simulation speed and accuracy comparing with other cutoff radii. All simulations were performed using a time step of 1 fs.

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RESULTS AND DISCUSSION Wetting Behavior of the Strained Substrate. We first study the wetting behavior of water droplet on a strained substrate, where graphene is employed as an illustrative example. A constant displacement is applied to the boundary atoms to achieve a uniaxial strain; then, the graphene is fixed. Figure 1a shows the schematics of a 500-water-molecule

Figure 1. (a)Top view and side view of a 500-water-molecule droplet on a monolayer graphene with a constant uniaxial strain ε. (b) Variation of the contact angle with the strain ε. The insets show the configurations of the droplet at two different strains representatively.

droplet on a monolayer graphene with constant uniaxial strain ε. After equilibrium in the NVT ensemble with 300 K for 500 ps, the configuration of water molecules is extracted and we calculate the contact angle by fitting the profile of the outermost molecules of water droplet. As shown in Figure 1b, when no strain is applied, the corresponding contact angle of water droplet is 86.95°, consistent with previous calculations using the same parameters.36 With the strain increasing from 0 to 0.2, the contact angle increases monotonously from 86.95° to 98.99°, indicating the wettability of water droplet on graphene changing from hydrophilicity to hydrophobicity. The reason is that the larger tensile strain applied to graphene induces smaller areal density of carbon atoms, which results in the decrease of liquid−solid surface energy and increase of contact angle. Note that no significant change in the contact angle is observed for larger droplets with more water molecules involved in simulations (see Figure S1). Thus, the contact angle of water droplet can be controlled by external force applied to the substrate. Nanodroplet Movement Driven by the Strain Gradient Substrate. On the basis of the strain-dependent

Fdriving = πR2γ

∂cos θc ∂cos θc = πR2γ Gx ∂x ∂ε

(1)

where R is the radius of the drop, γ is the liquid−gas interfacial tension, Gx is the strain gradient, and θc is the contact angle between drop and solid surface. On the basis of this relationship, the driving force is proportional to the strain gradient and the contact area of droplet, qualitatively agreeing with our simulation results (Figure 3). However, the term ∂cos θc/∂ε is hard to be accurately determined, and more insights need to be explored from molecular mechanism. Molecular Mechanism. An analytical model based on vdW interaction is developed (Supporting Information), from which one may derive the net driving force exerted on a spherical-cap droplet (with a molecular density of ρ and radius of R) sitting on a substrate with a strain gradient εy = ε0 + xGx B

DOI: 10.1021/acs.langmuir.8b03774 Langmuir XXXX, XXX, XXX−XXX

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Figure 2. (a) Top view and side view of a 500-water-molecule droplet on a monolayer graphene with a strain gradient Gx = dmax/wL. The side view shows the difference of advancing and receding contact angles. (b) Time history of the displacement of the mass center of the droplet along the x direction (black line) and y direction (red line). (c) Time history of the velocity of the mass center of the droplet along the x direction (black line) and y direction (red line). (d) Variation of interactive vdW potential energy (black line) and the system total potential energy (red line) with respect to the transport distance x.

Figure 3. (a) Time history of the displacement of the mass center of the 500-water-molecule droplet for various strain gradients Gx. (b) Acceleration of the mass center of the droplet with respect to strain gradient Gx for various sizes of the droplet.

Fdriving +

ÄÅ ÅÅ −(R2 − z 2)σ 12 0 = −4π σ0eρ(1 − ν)GxÅÅÅÅ 9 ÅÅ 45 z ÅÇ 2

(R2 − z 0 2)σ 6 6z 3

ÉÑ z 0σ 12 z 0σ 6 ÑÑÑ σ 12 σ6 ÑÑ + − − + 2z 35z7 20z 8 2z 2 ÑÑÑÑÖ

which shares the same scaling law with the previous phenomenological contact angle theory. m is the droplet mass. The acceleration of the droplet is proportional to the strain gradient and inversely proportional to the radius of droplet, meaning that the motion of the droplet can be tunable through changing the strain gradient of the substrate. Note that the present driving force may be similar to that induced by the stiffness gradient of the substrate,38 in the sense that the driving force is resulted from the redistribution of the vdW potential energy, nevertheless, the strain gradient is readily tunable in practice, but the surface stiffness gradient is not. To verify the theory, several MD simulations are conducted by varying the strain gradients and nanodroplet sizes. The relevant parameters σ0 (0.382/Å2), ν (0.186), R (18 Å for 500water-molecule droplet and 22.5 Å for 1000-water-molecule droplet), and ρ (0.0415/Å3) can be deduced from simulation. Figure 3a shows the displacement of a 500-water-molecule droplet as a function of time for different strain gradients, and the curves of acceleration versus strain gradient for the 500water-molecule and 1000-water-molecule droplets are plotted in Figure 3b. The simulation trends are consistent with our

R+z0

Δ

(2)

in which Δ is the distance between droplet’s bottom surface and substrate, ν is the Poisson’s ratio of the substrate, σ0 is the areal density of atoms on the unstrained substrate, ϵ and σ are LJ parameters, and z0 is the height of the center of the droplet with respect to the substrate surface Z = 0 (Figure S2). Through minimization of the interactive vdW potential energy between bulk water and graphene, one obtains Δ = 0.858σ. Therefore, the driving force and acceleration can be written as Fdriving ≈ −0.703π 2σ0eρ(1 − ν)GxR2σ 3 ∼ GxR2

(3)

a = Fdriving /m ∼ Gx /R

(4) C

DOI: 10.1021/acs.langmuir.8b03774 Langmuir XXXX, XXX, XXX−XXX

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Generalized Substrate. The above principles are not limited to the surface of graphene. To illustrate the broader applications, motion of water droplet on a (111) copper surface was simulated. The EAM potential39 is used to describe the copper substrate, and 500 water molecules are contained in the droplet. Because of small failure strain of copper (less than 8% in EAM potential), the length of the substrate is reduced to achieve the same strain gradient (strain gradient: 9.15 × 10−4/ Å) as that of graphene, as shown in Figure 5a. Compared with

theoretical model, and both the simulation and theory suggest that the acceleration is in directly and inversely proportion with the strain gradient and the droplet radius, respectively. Actuation by 2D Strain Gradient: Nonlinear Pathway. The driving force is relevant to the strain gradient and the size of the droplet, thus offering a way to precisely control the linear straight movement of the droplet. Certain applications, however, such as drug delivery, fog collection, and cell motility control, may require the movement of droplet in complex curved paths; this can be achieved adjusting strain gradients in two directions and loading sequences. The fundamental pathway of turning is illustrated here. Considering a 150 Å × 147 Å single-layer graphene and a droplet containing 1000 water molecules, it first experiences equilibrium at 300 K in the NVT ensemble for 500 ps, whereas the mass center is fixed, so that an initial stable configuration of the droplet (State A) is obtained. Next, displacement loading along the y direction is slowly applied to the atoms on the two boundaries along the x direction of graphene in 50 ps to obtain a 1.16 × 10−3/Å strain gradient along the x direction, after which the displacement boundary condition is fixed and the droplet is released (see the schematics in Figure 4). The strain gradient drove droplet to

Figure 5. (a) Top view and side view of 500-water-molecule droplet on the copper (111) surface. (b) Comparison of displacement−time curves of the 500-water-molecule droplet on graphene and copper (111) surface with same strain gradient Gx = 9.15 × 10−4/Å.

graphene, copper has smaller areal density of atoms σ0 (0.176/ Å2), larger Poisson’s ratio ν (0.3), and larger interactive vdW potential with water eCuO (7.394 meV).40 Also, the water droplet on copper has smaller radius R (16 Å), leading to a smaller driving force of the droplet on copper (111) than that on graphene. Shown in Figure 5b, the movement of droplet on the (111) copper surface is about 40% slower than that on graphene. This example indicates that the motion of water droplet can be controlled on any general surface with properly applied strain gradient.

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Figure 4. Trajectory of water droplet on a monolayer graphene with tunable strain gradient direction. The insets show the change of strain gradient direction from x direction to y direction and the variation of velocity direction of the droplet. The blue dashed line represents the expected pathway of the droplet, and the black line is the trajectory in the MD simulation.

CONCLUSIONS A novel method to control the movement of a nanodroplet on a substrate using strain gradient is proposed and verified using both MD simulations and theoretical analysis. The driving force, originating from the imbalance of vdW force between the substrate surface and droplet, is in the opposite direction of the applied strain gradient, and its magnitude is proportional to the magnitude of the strain gradient and the square of droplet radius. By properly adjusting the strain gradient and its direction, as well as loading sequence, one may precisely control the acceleration and deceleration the droplet, and even its trajectory. The study may inspire broad applications of directional droplet transport including fog collection41 and thermal management of devices.9

move along the x direction from the high strain end to the low strain regime for 200 ps (State B). Afterward, the strain gradient in the graphene is slowly released to the pristine state within 50 ps, whereas the droplet keeps moving in the x direction (see Figure S3). Thereafter, the graphene is slowly stretched again (within 50 ps), achieving a new strain gradient in the y direction of 1.16 × 10−3/Å and then the boundary is fixed, and this new strain gradient drove the droplet to accelerate along the y direction (see Figure S4). Figure 4 shows the trajectory of the droplet in a parabolic path (see Movie S2), which agrees quite well with parallel theoretical prediction (Supporting Information). Note that the decay of velocity in x direction is attributed to the nonuniaxial strain gradient and thus distorted areal density of carbon atoms near boundaries, subjected to future studies. This example suggests that by adjusting the loading amplitude (gradient), direction, and sequence, it is possible to manipulate a small droplet moving along complex pathways.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.8b03774. Variation of the contact angle with the strain and size of the droplet; sketch of a spherical-cap drop in the coordinate system; time history of velocity component in the x and y directions for water droplet on a D

DOI: 10.1021/acs.langmuir.8b03774 Langmuir XXXX, XXX, XXX−XXX

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monolayer graphene with tunable strain gradient direction; and time history of temperature of a 500water-molecule droplet in motion on a monolayer graphene (PDF) Droplet speeds up along the x direction and the displacement along the y direction can be neglected, indicating that the net driving force is along the opposite direction of the strain gradient (AVI) Trajectory of the droplet in a parabolic path which agrees quite well with parallel theoretical prediction (AVI)

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (X.L.). *E-mail: [email protected] (Y.N.). ORCID

Xiangbiao Liao: 0000-0001-8214-454X Yong Ni: 0000-0002-8944-5764 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The work of X.L. and X.C. was supported by the Yonghong Zhang Family Center for Advanced Materials for Energy and Environment. X.L. acknowledges support from the China Scholarship Council (CSC) graduate scholarship. This work was supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (grant no. XDB22040502), the Collaborative Innovation Center of Suzhou Nano Science and Technology, and the Fundamental Research Funds for the Central Universities (grant no. WK2090050043).

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DOI: 10.1021/acs.langmuir.8b03774 Langmuir XXXX, XXX, XXX−XXX