Unidirectional Self-Driving Liquid Droplet Transport on a Monolayer

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Surfaces, Interfaces, and Applications

Unidirectional self-driving liquid droplet transport on a monolayer graphene-covered textured substrate Zhong-Qiang Zhang, Xin-Feng Guo, Huayuan Tang, Jianning Ding, Yonggang Zheng, and Shaofan Li ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.9b09219 • Publication Date (Web): 15 Jul 2019 Downloaded from pubs.acs.org on July 20, 2019

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ACS Applied Materials & Interfaces

Unidirectional self-driving liquid droplet transport on a monolayer graphene-covered textured substrate Zhongqiang Zhang1 ,2, 3, 4, Xinfeng Guo1, Huayuan Tang3, Jianning Ding1, 4, Yong-Gang Zheng3 *, and Shaofan Li2 1Micro/Nano

*

Science and Technology Center, Jiangsu University, Zhenjiang, 212013, P.R.

China 2Department

of Civil and Environmental Engineering, University of California, Berkeley, CA

94720-1710, USA 3State

Key Laboratory of Structural Analysis for Industrial Equipment, Department of

Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116024, P. R. China 4Jiangsu

Collaborative Innovation Center of Photovoltaic Science and Engineering,

Changzhou University, Changzhou, 213164, P.R. China

ABSTRACT: Controllable directional transport of liquid droplets on a functionalized surface has been a challenge in the field of microfluidics since it does not require energy supply, and the physical mechanism of such self-driving transport exhibits extraordinary contribution to fundamental understanding of some biological processes and the design of microfluidic apparatus. In this paper, we report a novel design of surface microstructure that can realize unidirectional self-driving liquid mercury (Hg) droplet transport on a graphene-covered copper (Cu) substrate with three-dimensional surface microstructure. We have demonstrated that a liquid Hg droplet spontaneously propagates on a grooved Cu substrate covered by a monolayer graphene without any external force fields. Classical molecular dynamics (MD)

*

Corresponding author. Tel.: +86-411-84708751 E-mail address: [email protected] (Yong-Gang Zheng) * Corresponding author. Tel.: (510) 642-5362 E-mail address: [email protected] (Shaofan Li) 1

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results provide a profound insight on self-driving process of Hg droplets. It shows that the Hg droplet undergoes acceleration, deceleration and return stages successively from the narrow to wide ends of the gradient groove. Intriguingly, Hg droplets can move continuously and unidirectionally on the three-dimensional graphene-covered surface microstructure when they accumulate enough kinetic energy from the gradient groove to break the energy barrier at the step junctions between the two neighboring unit cells. The design of the zigzag textured surface covered by a monolayer graphene artfully uses the facts; (1) the monolayer graphene can effectively reduce the droplet pinning on the textured surface, (2) the hydrophobic graphene layer reduces the friction between Hg droplets and the substrate, and (3) the textured surface can permeably interact with the droplets through the monolayer graphene, to achieve a continuous self-driving process. The findings reported here open a door to explore the graphene-covered functional surface to directional transport of liquid droplets and provide an in-depth understanding of the self-driving mechanism for liquid droplets on graphene-covered textured substrates.

KEYWORKS: self-driving, monolayer graphene, liquid droplet, gradient groove, molecular dynamics

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1. INTRODUCTION Droplet transport is a very important physical chemistry process in many scientific and engineering fields, including high resolution inkjet printing, adhesion, self-lubrication, self-cleaning and spraying, and bio-fluidic devices. At microscale, with the growing interest in developing smaller and more accurate micro/nano-fluidic systems, the understanding of the driving mechanism and motion dynamics of droplets is of great importance in fabricating microfluidic chips, micro-mixers, micro-actuators, micro-sprinklers, and microfluidic transmissions. Since the uphill motion of the droplets on the silicon wafer surface with the gradient wettability was found by Chaudhury in 1992,1 driving of the droplets on the functionalized surfaces has attracted considerable attention. Recently, the trampolining of the droplets on the superhydrophobic surfaces,2 the fantastic chasing game of the droplets with different surface tension and evaporation rate,3 and the topological liquid diode based the pinning theory of contact lines4 were successively reported. These interesting findings triggered a boom of both theoretical and experimental researches on the driving and motion behaviors of the droplets on the textured surfaces.5,6 At present, the efficient approaches to drive droplets on the solid surface are mainly focused on the droplet propulsion caused by the gradient external fields including temperature,7,8 light,9 electric force10-13 and mechanical vibration,14 and the self-driving force caused by the gradient surface energy stemming from chemical gradient,1,

15-17

gradient

morphology18,19 and gradient stiffness.20,21 The underlying philosophy of the two approaches is to break the wetting symmetry of droplets on the surface, resulting in the gradient wettability and further driving the droplet. However, the contact line pinning due to the surface defect usually hinders the motion of the droplet on the surface. To overcome the contact line pinning, a situation above the Leidenfrost temperature is required to form a vapor layer which can hold up the droplet away from the surface.22 Unfortunately, this method 3



cannot accurately control the moving direction of the droplet due to the inhomogeneous temperature field and ultralow friction at the vapor-liquid interfaces. Nevertheless, combining the Leidenfrost method with the surface texture, a novel driving mechanism of a Janus droplet on a topographically patterned surface was discovered,5 which indicates that a gradient of Laplace pressure is generated in the Janus droplet attributed to the two concurrent wetting states, i.e., Leidenfrost and contact-boiling states. Interestingly, although the contact line pinning is usually annoying in droplet driving, a U-shaped topological structure was designed to achieve the reverse motion of the droplet via strong pinning ingeniously.4 It should be noted that the proposed topological structure is convenient to achieve the continuous liquid transport without getting rid of the contact line pinning at the posterior boundary of the droplet. Though extensive efforts have been directed toward driving mode and mechanism of droplets on functionalized surfaces, the accurate control of the droplet motion and the understanding of the competition between the driving force and energy dissipation in droplet-moving process are still in their infancy. The hydrophobic surface is propitious to drive the droplet because of its low friction at the solid-liquid interface. The natural hydrophobicity of graphene23 makes it an ideal candidate for constructing nanofluidic systems. However, the wettability of the monolayer graphene is quite dependent on its substrate,24 implying that the monolayer graphene is of wetting transparency, and the number of layers is an important parameter for the wettability of graphene.25 Regarding the fluid-graphene system, the previous efforts were mainly devoted to the static and dynamic wettability, the driving and fast diffusion of water droplets, the liquid-graphene interfacial friction, and the fluid boundary slip confined in graphene channel, etc.26-28 Recently, however, it was demonstrated experimentally29 that the monolayer graphene, as a coating of the nanochannel, can significantly increases the boundary slip of the confined fluid. It was found that the slip length of the fluid is explicitly sensitive to the 4


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substrate of the monolayer graphene. The finding suggests that the substrate can permeably interact with the fluid on the monolayer graphene and plays an important role in both the wettability and boundary slip of the fluid, which motivates us to put the monolayer graphene on the textured surface to drive droplets. In this work, we artfully integrate the low frictional property of the graphene surface into the droplet-driving function of the gradient textural surface. A novel self-driving model for liquid droplets on a graphene-covered Cu groove is proposed, in which liquid mercury (Hg) droplets can be driven by the graphene-Cu gradient groove directionally. Classical molecular dynamics (MD) method is used to investigate the self-driving behaviors of Hg droplets on the graphene-Cu composite substrate. The self-driving mechanism is clarified with considering the competition between the change in the solid-liquid interaction energy and the variation in the surface energy of Hg droplets during the moving process. The continuum model is utilized to reveal the influence rules of the structural parameters of the groove and the size of the Hg droplet on the driving properties. A functionalized textured surface is finally designed to drive liquid droplets unidirectionally based on the clarified self-driving mechanism. 2. MODEL AND METHODE The MD model is composed of a copper substrate with a gradient groove, a monolayer graphene, and a liquid mercury droplet, as schematically shown in Figure 1. The monolayer graphene is freely covered on the grooved copper substrate. A liquid mercury droplet is put upon the narrow end of the gradient groove of the graphene-Cu composite substrate. To ensure the initial model to be as an equilibrium state, first we equilibrate the grooved copper substrate and the liquid mercury droplet for 500 000-time steps (time step Δt = 1 fs) separately in MD simulations. Then, a monolayer graphene moves to the grooved copper substrate along - z direction gradually until the graphene is adsorbed freely on the copper substrate. The monolayer graphene can be covered on the internal surface of the groove, as 5



shown in Figure 1b. At last, an equilibrium Hg droplet moves to the copper-graphene composite substrate until it is adsorbed on the grooved surface at an initial location. In addition, a reflecting wall is fixed at the +x boundary (right edge) of the droplet to make it static prior to the simulation for the droplet self-driving process.

Figure 1. (a) Top view and (b) sectional view of the simulation model for a liquid mercury (Hg) droplet on a graphene-covered copper (Cu) substrate with a gradient structured groove. The green molecular cluster is the liquid Hg droplet, the gray sheet is the monolayer graphene, which is covered on the pink grooved Cu substrate. The size of the Cu block is 25 × 12 × 2.8 nm3. The depth of the groove is D while the diameter of the Hg droplet is d. The gradient width of the groove is characterized by an angle of θ. The initial location of the Hg droplet is 4.5 nm far from the left edge of the substrate, where the width of the groove is 3.4 nm which is kept constant as the gradient is varied by changing the angle θ.

In the simulation domain, the size of the Cu block is 25 × 12 × 2.8 nm3, the depth of the groove is D while the diameter of the Hg droplet is d, and the width gradient of the groove is characterized by a gradient angle of θ. Two layers of Cu atoms at the bottom of the copper substrate are fixed to make the whole substrate static. The fixed boundary conditions with reflecting walls to restrict the evaporated Hg atoms from running out of the simulation box. The initial location of the liquid mercury droplet is 4.5 nm far from the left edge of the substrate, where the groove width is 3.4 nm which is kept constant as the width gradient is varied by changing the angle θ. As for the choice of potentials, the interactions among carbon atoms of graphene are modeled by the second-generation reactive empirical bond order (REBO) potential.30 The embedded atom method (EAM) potential is used to describe the 6


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interaction among Cu atoms.31 The interactions between mercury atoms are approximated by a scaled ab initio mercury potential, which can reproduce some basic properties including melting point and liquid density of mercury quite well.32 A pairwise Lennard-Jones (LJ)



potential U LJ (r )  4  / r    / r  12

6



is used to simulate the Hg-C, Hg-Cu and C-Cu

interactions, where  and  are the potential parameters, and r is the distance between a pair of atoms. The LJ parameters 

and  for Hg-Cu and C-Cu listed in Table 1 are

calculated by applying Lorentz- Berthelot mixing rules in terms of the parameters of LJ potential for Cu-Cu,33 C-C,34 and Hg-Hg35 atomic pairs. The energy parameter ε for Hg-C has been experimentally demonstrated to reproduce the contact angle of 152.5°.36 Table 1 The parameters of LJ potential for Hg-C, Hg-Cu, and C-Cu atomic pairs

Atomic pairs

ɛ (meV)

σ (Å)

Hg-C

1.267

3.321

Hg-Cu

7.673

2.774

C-Cu

27.581

3.083

3. RESULTS Prior to the study on the self-driving behaviors of a droplet on the graphene-covered textured copper surface, we first put an Hg droplet on the gradient grooved copper substrate without a covering monolayer graphene. However, the Hg droplet moves with an extremely low speed and unpredictable oscillations, as shown in Figure 2a. (see more in Supporting Information SI, Section 1) This result arises from the fact that the driving force caused by the gradient copper groove cannot overcome the adhesion between the droplet and the substrate. The smooth low-frictional surface of the monolayer graphene raises a great hope to develop the graphene-covered textured copper surface to drive the liquid droplets. Figure 2b shows the self-driving process of a liquid Hg droplet with the diameter d = 5.8 nm on the 7



graphene-covered grooved copper substrate for the width gradient θ = 7° and the depth of groove D = 1.08nm. Interestingly, the Hg droplet undergoes the acceleration, deceleration and the return stages successively. Concretely, the droplet moves along the groove from the narrow end to the wide one in the acceleration stage, where the moving velocity increases up to a maximum of ~ 7 m/s. Then, the velocity decreases almost linearly to zero with the maximum displacement of 7.2 nm in the deceleration stage. After that, the Hg droplet begins to go back from the wide end of the graphene-covered groove in the return stage. The result indicates that the Hg droplet can be driven by the graphene-covered copper groove without any external fields, implying that the graphene plays a dominant role in the self-driving process of the Hg droplet.

Figure 2. (a) Self-driving velocity versus time for an Hg droplet on the graphene-covered Cu groove (Cu-Graphene) in comparison with that on the pure Cu groove. (b) The velocity (black solid curve) and the displacement (blue solid curve) of the Hg droplet on the graphene-covered Cu groove as a function of time. The diameter of the Hg droplet d = 5.8 nm, the width gradient θ = 7°, and the depth of Cu groove D = 1.08nm. The three zones with different background colors represent the acceleration, deceleration, and return stages, respectively. The horizontal black and blue dashed lines denote the zero and maximum velocity and displacement, respectively. The vertical red dashed lines denote the critical time nodes corresponding to three stages in the self-driving process of Hg droplet.

To interpret the self-diving behaviors of the Hg droplet on graphene-covered copper groove, the force on the Hg droplet in x direction Fx and the interaction energy between droplet and the graphene-Cu substrate versus time are plotted in Figure 3. The variation of Fx

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on the Hg droplet determines the variation of the driving velocity (Figure 3a). Concretely, Fx decreases to zero at ~1.04 ns and then increases in the opposite direction. As a result, the velocity increases up to a maximum and then decreases at an inflection point of t = 1.04 ns. To further explore the derivation of the force on the droplet, we calculate the interaction energy between Hg droplet and the graphene-Cu substrate as a function of time, as shown in Figure 3b. The monolayer graphene is of wetting transparency,24,25 i.e., the Hg droplet can interact with the copper groove through the monolayer graphene. But the magnitude of the interaction energy between Hg droplet and Cu groove EHg-Cu is much smaller than that between Hg and graphene EHg-C. It indicates that the effect of the substrate on the Hg droplet mainly derives from the monolayer graphene, whereas the gradient copper groove be a template to fabricate the gradient grooved monolayer graphene. The interaction energy between droplet and substrate decreases sharply initially and then exhibits a slow decrease with time. However, we have not found the turning point within the curves of the interaction energy versus time. It suggests that the self-driving behaviors of the Hg droplet are not only dependent on the interfacial energy, but also sensitive to the variation in the energy of droplet in the driving process induced by the gradient groove. To address this, the snapshots of the sectional view of the Hg droplet moving on the substrate at different times are illustrated in Figure 4. An obvious adsorption-induced deformation of the droplet can be observed especially for the droplet at the narrow end of the groove (t = 0.2, and 0.7 ns). With the increase in the width of the groove, the topography of the droplet first becomes spherical approximately (t = 1.04 ns), and then changes to ellipsoidal due to the adsorption from the side walls of the groove. At nanoscale, the variation in the surface area of the liquid droplet can significantly change its potential energy attributed to the role of the surface tension, which can be demonstrated by the potential energy of Hg droplet as a function of time plotted in Figure 3b. The potential energy of the droplet increases remarkably after t = 1.04 ns. 9



Hence, we can conclude that the variation from acceleration to deceleration of the Hg droplet is determined by the competition between the liquid-solid interfacial energy and the potential energy of Hg droplet since the droplet necessarily moves from the higher to the lower energy zones.

Figure 3. (a) The force (blue) in x direction Fx and velocity (black) of Hg droplet as functions of time. The blue dashed line denotes the zero-driving force, and black dashed line denotes the maximum of the velocity. (b) The interfacial interaction energy between Hg droplet and graphene-Cu substrate versus time. EHg-Cu is the interaction energy between Hg droplet and Cu substrate, EHg-C is the interaction energy between Hg droplet and the monolayer graphene, EHg-sub is the interaction energy between Hg droplet and the graphene-Cu composite substrate. The width gradient θ = 7°, the groove depth D = 1.08 nm, and the droplet diameter d = 5.8 nm.

Figure 4. Top sectional views across the horizontal plane A-A of the Hg droplet on the graphene-covered copper groove in the self-driving process at (a) t = 0.2 ns, (b) t = 0.7 ns, (c) t = 1.04 ns, (d) t = 1.7 ns and (e) t = 2.0 ns for the width gradient θ = 7°, the groove depth D = 1.08 nm, and the droplet diameter d = 5.8 nm. The corresponding displacements of the Hg droplet from figure (a) to (e) are 0 nm, 1.8 nm, 3.9 nm, 6.9 nm, and 7.2 nm.

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Figure 5. Schematic illustration for the adsorbed portion of a Hg droplet (green) within the gradient groove (black solid lines) on a graphene-Cu substrate (pink) in top view. Figures (a) and (b) present the contours of the adsorbed portion of a Hg droplet in acceleration (narrow end of groove) and deceleration (wide end of groove) stages, respectively. θ is the angle to characterize the groove gradient, and α represents the equilibrium contact angle of a Hg droplet on smooth planar graphene-Cu substrate, x1 and x2 are the distances from the apex to the meniscus-solid intersecting points. R1 and R2 are the radii of the curvature for both advancing and trailing menisci.

To further delineate the self-driving behaviors of Hg droplets on the gradient graphene-Cu groove, the force analysis based on the surface tension theory is carried out aiming at an analysis of

the pressure difference within the droplet in acceleration and

deceleration stages, respectively. For a droplet confined inside the concave structure, the curvatures of its trailing and advancing menisci can be obtained in terms of the geometric relationships (Figure 5a), i.e., R1-1 = -cos(α-θ)/(x1sinθ), and R2-1 = -cos(α+θ)/(x2sinθ), where α = 155.6° represents the equilibrium contact angle of the Hg droplet on smooth planar graphene-Cu substrate, which can be calculated from the average contour of Hg droplets (Supporting Information SI, Section 2). x1 and x2 represent the distances from the apex to the meniscus-solid intersecting points. Here, we assume that the surfaces of the adsorbed portion of the liquid droplet confined by two lateral groove walls have approximately cylindrical shapes (see Fig. 4a) with radii of curvature R1 and R2, and the surface curvature is positive when it bends to the outside. Then, in terms of the Young-Laplace equation, the pressures inside the droplet corresponding to the trailing and advancing menisci P1 and P2 can be

11



calculated by 𝑃1 = ―𝛾

cos (𝛼 ― 𝜃) 𝑥1sin 𝜃

and 𝑃2 = ―𝛾

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cos (𝛼 + 𝜃) 𝑥2sin 𝜃

, respectively. As last, the pressure

different ∆𝑃 within the Hg droplets in +x direction can be written as37,38 (see more in Supporting Information SI, Section 3) ∆𝑃 = 𝛾

[

cos (𝛼 + 𝜃) 𝑥2sin 𝜃

―

cos (𝛼 ― 𝜃) 𝑥1sin 𝜃

],

(1)

where 𝛾 is the surface tension of the liquid Hg. If we define a ratio λ = x1/x2, then the eq 1 can be expressed as ∆𝑃 = 𝛾·

𝜆cos (𝛼 + 𝜃) ― cos (𝛼 ― 𝜃) 𝑥1sin 𝜃

.

(2)

Obviously, the moving direction of a Hg droplet, i.e., ∆𝑃 is positive or negative, is dependent on the contact angle α, the groove gradient θ, and the ratio λ.38 In this work, 𝛼 = 155.6° > 90° + 𝜃 (here θ is in the range of 3~11 ° ), then cos (𝛼 + 𝜃)< 0 and cos (𝛼 ― 𝜃)< 0, both of which are constants in the whole driving process, implying that the sign of ∆𝑃 is determined by the ratio λ which varies continuously with the change in the contour of Hg droplets. In the acceleration stage, the contact length in x direction (x2-x1) is comparable to x2, as shown in Fig. 5a, resulting in 𝜆cos (𝛼 + 𝜃) > cos (𝛼 ― 𝜃), i.e., ∆𝑃> 0. However, in the deceleration stage, the droplet moves to the wider region of groove and it is adsorbed to be ellipsoid (see MD results in Fig. 4d, e) by the bilateral groove walls, schematically shown in Fig. 5b, where x1 is almost equal to x2. As a result, the ratio λ = ~1, and then cos (𝛼 + 𝜃) < cos (𝛼 ― 𝜃), i.e., ∆𝑃< 0. In a word, the moving direction of Hg droplets in both acceleration and deceleration stages is caused by the variation in the pressure difference inside the droplet, which is inherently determined by the competition between the liquid-solid interfacial energy and the potential energy of Hg droplet. Now that self-driving behaviors of the Hg droplet are sensitive to the liquid-solid interfacial energy and the potential energy itself, the structural parameters of the groove

12


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including the width gradient and the depth should be the key issues to affect the self-driving properties of Hg droplets. Figure 6 illustrates the self-driving behaviors of the Hg droplet for the width gradient θ of the grooves in the range of 3° - 11°. It shows obviously that all the Hg droplets undergo the acceleration and deceleration stages for the graphene-covered grooves with different width gradients. Nevertheless, the return stages are difficult to capture for the small width gradient of 3° and 5° due to the computational cost. Importantly, the driving velocity increases with the width gradient θ in the acceleration stage, whereas the maximums of the driving velocity are almost identical to each other. Consequently, the maximum displacement of the Hg droplets increases with the decrease in the width gradient θ, as shown in Fig. 6a. It can be concluded that the magnitude of the acceleration in the driving process increases with the width gradient according to the slope of the curves in both acceleration and deceleration stages in Fig. 6b. In other words, the driving force in the acceleration stage can be enhanced by increasing the width gradient.

Figure 6. Influence of the width gradient θ of the copper grove on the motion behaviors of the Hg droplet. (a) Displacement and (b) velocity of the Hg droplet versus time for the width gradient θ in the range of 3° 11°. The depth of the copper groove is D = 1.08 nm, and the diameter of Hg droplet is d = 5.8 nm. The section of curves confined in the dashed square in figure (b) denote the initial velocity oscillation of the Hg droplet at the original location.

Moreover, the influence of the groove depth on the self-driving behaviors of Hg droplet is also considered for the width gradient of θ = 7°, as shown in Fig. 7. Obviously, the driving 13



velocity in the acceleration stage, the maximum displacement, and the magnitude of the acceleration increase with increasing the groove depth D in the range of 0.72 nm - 1.44 nm. From our simulations, the monolayer graphene cannot be adsorbed in the bottom of the groove completely for the groove depth D larger than 1.44 nm, i.e., the monolayer graphene cannot completely cover the internal surface of the copper groove for D > 1.44 nm. Therefore, a relatively deeper graphene-covered gradient copper groove with a relatively larger width gradient θ is more propitious to drive the Hg droplet. Interestingly, the droplet can break away from one side wall of the groove in the deceleration stage since the deeper groove provides the droplet more driving momentum to overcome the absorption of one of the two side walls (the inset of Fig. 7a). Another important parameter is the diameter of the liquid Hg droplet d while the contact area at the liquid-solid interface and the surface morphology of the droplet vary dynamically in the self-driving process. From Fig. 8, the increase in the diameter of Hg droplets effectively improves the maximum displacement of the droplets but decreases the driving velocity and the driving acceleration in the acceleration stage. The maximum driving velocity seems to be independent of the diameter of Hg droplets.

Figure 7. Influence of groove depth D on the motion behaviors of the Hg droplet. (a) Displacement and (b) velocity of the Hg droplet versus time for the groove depth D in the range of 0.72 - 1.44 nm. The width gradient θ of the copper groove is 7°, and the diameter of Hg droplet is 5.8 nm. The embedded graph in figure (a) illustrates that the Hg droplet breaks away from one of the lateral groove walls at t = 3.0 ns in the case of D = 1.44 nm.

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Figure 8. Influence of diameter of Hg droplet d on the motion behaviors of the Hg droplet. (a) Displacement and (b) velocity of the Hg droplet versus time for the diameter of the droplet d in the range of 5.8 – 8.8 nm. The width gradient and the depth of the copper groove are 7° and 1.08 nm, respectively.

4. DISCUSSIONS Based on the above results, we can conclude that the self-driving behaviors of the Hg droplet are sensitive to the width gradient θ, the diameter of the droplet d, and the depth of the groove D. The change of these parameters can possibly lead to the variation in the potential energy of the droplet itself and the interaction energy between Hg droplet and graphene-covered gradient copper groove, thereby resulting in the change of the self-driving behaviors. Especially, the motion of the Hg droplet is determined by the competition between the liquid-solid interfacial energy and the potential energy of the droplet (see Fig. 3). These simulation results motivate us to control the driving trajectory of the droplet by adjusting the geometric parameters of the graphene-covered gradient groove and the volume of the droplet. Hence, we need find out the influence factors on the commensurability between the geometric parameters of gradient groove and the size of Hg droplet during the self-driving process. A continuum model is built to study the underlying mechanism of the self-driving behaviors of the droplet.39 In this model, an Hg droplet with a constant volume is adhered on the graphene-copper grooved substrate with a constant width of w. Moreover, a series of continuum simulations with different droplet diameters and groove widths are performed to 15



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further understand the size and gradient effects on the self-driving behaviors. Since the change in the potential energy of droplet is mainly caused by the changes in the surface tension energy and the solid-liquid interfacial energy. Thus, the total energy of the droplet– substrate system is given by 𝐸 = 𝐸𝑡 +𝐸Hg energy of Hg droplet and

𝐸Hg ― S

- S,

where 𝐸𝑡 is the equivalent surface tension

is the interfacial energy between droplet and

graphene-covered copper substrate. The equivalent surface tension energy can be approximately expressed as 𝐸𝑡 = 𝛾𝐴Hg, where 𝛾 and 𝐴Hg are respectively the surface tension coefficient and the surface area of the droplet, and the interfacial energy 𝐸Hg

-S

is

obtained by integration of the Lennard-Jones potential over the surfaces of the droplet and the substrate. To calculate the equilibrium configuration, the surfaces of the droplet and the substrate are divided into triangular elements and the total energy of the system is minimized with respect to the coordinates of the vertices via Surface Evolver.40 The details of the continuum model can be found in Section 4 of Supporting information SI.

Figure 9. (a) Number density of the Hg droplets as a function of the radius of Hg droplets for different side length l = 5 -10 nm of the initial Hg cubes. The dashed lines are used to denote the radii of the Hg droplets at the positions with the number density of 1/σHg3. (b) Internal pressure of the Hg droplets as a logarithmic function of the diameter of the droplets d.

In order to obtain the surface tension coefficient γ of the Hg droplets at nanoscale, the relationship between the internal pressure P and the diameter of the droplets d should be figured out in terms of the Laplace equation P = 4γ/d.41 If the Laplace equation is applicable 16


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at nanoscale, the surface tension coefficient can be obtained from the logarithmic equation of lg(P) = lg(4γ) – lg(d). In our simulations, the Hg droplets are generated from the initial Hg cubes with the side length in the range of 5 - 10 nm. The radii of the droplets are calculated from the mean radial number density within 1.5 ns due to the dynamic variation of the droplet profiles, as shown in Figure 9a. The shell bin with the number density of 1 per σHg3 is defined as the boundary of the droplets, where σHg is the distance parameter in LJ potential. Fortunately, lg(P) is linearly decreased with lg(d) (Figure 9b), and the surface tension coefficient γ = 3.495 GPa-nm. The parameters of the interfacial energy 𝐸Hg

-S

including the

interfacial strength and equilibrium distance between the droplet and the substrate, which are also derived from MD simulation results (Supporting information SI, Section 5). The total energy profiles of the droplet–substrate system as functions of the groove width w are plotted in Figure 10a,b. The energies are relative to the configuration with 𝑤 = 3.4 nm, which is consistent with the MD model. All the total energies firstly decrease and then increase as the groove width increases. This result implies that the droplet can move along the groove from the narrow end to the wider end initially until it reaches the location with minimum total energy, then the droplet begins to decelerate along the energy-rising direction. Thus, the motion behaviors of droplet predicted by the continuum model agree well with the MD results. As indicated by the evolutions of the equilibrium configurations with d = 5.8 nm and D = 1.08 nm in Figure 10c, the decrease of the total energy in the first stage corresponds to the increase of the adhesive area between the droplet and the bottom of the groove. The groove width is about 4.8 nm for the droplet at the equilibrium state of minimum energy while it is ~4.2 nm corresponding to the position with the maximum velocity in the MD result (Figure 11a). The discrepancy in the energy-minimum position between MD and continuum model is attributed to the constant groove width and static results in the continuum model. As the groove further broadens, the side walls of the groove pull the droplet out of equilibrium 17



and the surface area of the deformed droplet increases, resulting in a higher surface tension energy. Therefore, the motion of the droplet on the gradient substrate is due to the competition between the droplet–substrate interfacial energy and the potential energy of droplet, which aligns with the MD results. Furthermore, the minimum energy of the droplet– substrate system occurs at larger width of groove for the system with a deeper groove (Figure 10a) and a larger droplet (Figure 10b), which also agree well with the MD simulation results (Figure 11b,c).

Figure 10. Continuum model of the Hg droplet adhered on the substrate with a groove. Influences of (a) the depth of the groove and (b) the diameter of the droplet on the variations of the total energy of the system. (c) The evolution of the equilibrium configurations with 𝑑 = 5.8 nm and 𝐷 = 1.08 nm, which correspond to the circles in the energy profiles.

From Figure 11a, an interesting result is that the maximums of driving velocity vmax for different groove gradients are almost identical to each other, which can also be observed for different diameters of Hg droplets (Figure 11c). Here, we assume the acceleration of Hg 18


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droplet a to be a constant in the acceleration stage since the driving velocity increases almost linearly with time (Figures 6b and 8b). Then, in terms of Newton’s laws of motion, vmax = atvm, and svm = 1/2atvm2, where tvm and svm are the time and displacement, respectively, corresponding to the velocity maximum of vmax. Also, the groove width wvm corresponding to vmax can be obtained from the geometric relation of the gradient groove as 𝑤𝑣m = 𝑤0 + 2𝑠𝑣mtan𝜃,

(3)

where w0 = 3.4 nm is the initial width of the groove at the beginning of droplet (Figure 1a). Substituting the motion equations into eq 3, the groove width wvm can be written as 𝑤𝑣m = 𝑤0 + 𝑣2max·

tan 𝜃 𝑎

.

(4)

For the influence of the groove gradient tanθ on the droplet driving behaviors in the acceleration stage, we conclude that the ratio of

tan 𝜃 𝑎

is kept constant while wvm, w0, and vmax

are constants (Figure 11a). It suggests that the driving acceleration of an Hg droplet is linearly dependent on the groove gradient.

19



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Figure 11. The velocity of Hg droplets versus the groove width w for (a) the width gradient θ in the range of 3° - 11°, D = 1.08 nm and d = 5.8 nm, (b) for the groove depth D in the range of 0.72 - 1.44 nm, θ = 7° and d = 5.8 nm, and (c) for the diameter of the droplet d in the range of 5.8 – 8.8 nm, θ = 7° and D = 1.08 nm, respectively. The red dashed line in figure (a) denotes the groove width corresponding to the velocity maximum.

To explore the general rule of driving behaviors affected by the droplet diameter, the mass of Hg droplet m should be introduced in eq 4 in terms of Newton’s second law as follows 𝑤𝑣m = 𝑤0 + 𝑣2max·

𝑚·tan 𝜃 𝐹

,

(5) 1

where F is the driving force on an Hg droplet, and 𝑚 = 𝜌·6π𝑑3 (ρ and d are density and diameter of the Hg droplet, respectively). Then eq 5 can be written as 𝑑3

1

𝐹 = 6𝜋𝜌𝑣2max ∙ tan 𝜃·𝑤𝑣m ― 𝑤0.

(6)

Thus, for a fixed gradient of groove tan θ, the driving force on an Hg droplet is dependent on the commensurability ratio

𝑑3 𝑤𝑣m ― 𝑤0

between the volumetric size of Hg droplets and the

relative width of the groove. Continuous and directional water transportation on surfaces is of great scientific interest and immense technological importance in micro/nano fluidic systems. Based on the explicit explanation for the self-diving mechanism of Hg droplets on the graphene-covered copper groove, here we try to design a graphene-covered textured substrate composed of gradient grooves to drive the droplet unidirectionally without imposing any external fields. Now that 20


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the droplet can escape from the restriction of the groove lateral walls if the momentum is large enough (Figure 7), this motivates us to pick out the acceleration region of a groove to be a periodic unit cell to fabricate the textured substrate, schematically shown in Figure 12a, where a zigzag groove surface is specially designed to reduce the energy barrier at the junctions among unit cells. Surprisingly, the topographical zigzag surface can drive the Hg droplet unidirectionally (Supporting Information SII and SIII: videos for the self-driving process (lateral view and top view) of an Hg droplet on the proposed zigzag surface), as shown in Figure 13. The self-driving speed periodically increases within each unit cell, while a sharp decrease appears at the junctions among periodic units (see Figure 12b). Attributed to the attraction of each groove end, the droplet requires more kinetic energy to overcome the energy barrier at the junctions. Moreover, the stepped junctions can wrinkle the monolayer graphene locally and induce the droplet bouncing, which will make the motion behaviors of liquid droplets uncertain at the junctions of zigzag surface. Although the relationship between the energy barrier at junctions and the geometric parameters of graphene-covered zigzag surface, as well as the control of the motion trajectories of liquid droplets, should be further identified theoretically in future, at least we can conclude that the monolayer graphenecovered texture can significantly reduce the effect of the contact line pinning and drive the Hg droplet unidirectionally. This finding may open a door for a broad range of novel applications in utilizing self-driving of droplets on gradient topographical surfaces. The energy barrier at the junctions of the proposed zigzag model is the key issue to achieve unidirectional self-driving of Hg droplets since the solid edges at junctions will create a pinning effect on droplet movement.

42

Considering the dynamic and thermal oscillation of

Hg droplet in the self-driving process, a single junction is taken out from the above zigzag model to make an Hg droplet move uniformly across it, as shown in Figure 14a. We make the Hg droplet move across the junction with a slow constant velocity of 2.5 m/s to calculate the 21



interfacial energy between droplet and substrates. Figure 14b plots the interfacial energy versus the displacement of Hg droplet for both the graphene-Cu and pure Cu substrates. From the comparison, an obvious energy peak can be observed for Hg droplet moving across the pure Cu junction while an extremely small energy fluctuation appears at junction for graphene-Cu model. It suggests that the monolayer graphene can significantly reduce the pinning effect on droplet movement, which can also be demonstrated from the configuration of graphene-Cu junction shown in Figure 14b (partial enlarged graph). Concretely, the monolayer graphene covered on Cu junction makes the surface relatively smoother without sharp edge and corner, thereby decreases the energy barrier at junctions significantly (~ 42.25% in this case, from the data of Figure 14b). Consequently, the Hg droplets can successfully overcome the energy barrier at the junctions after an acceleration stage to reach the next groove unit.

Figure 12. Unidirectional motion of Hg droplet on the gradient textured graphene-Cu substrate with three periodic unit cells. (a) Schematic of the MD model, in which is a Hg droplet (green sphere) locates on the graphene (grey)-covered copper textured substrate (pink). The periodic unit cell is picked up from the acceleration stage of the droplet with the groove length of 25 nm, i.e., from the left end to the location of vmax (dashed blue line). The dot dashed blue line is the schematic of the moving trajectory of droplet. (b) The velocity of Hg droplet versus the position x in the case of D = 1.26 nm, θ = 2.68°, d = 7.2 nm. The original groove width at the starting point of droplet is 3.7 nm.

22


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Figure 13. Snapshots of the spontaneous propagation process of an Hg droplet on a grooved Cu substrate covered with a monolayer graphene.

Figure 14 (a) Graphene-Cu (G-Cu) junction model for an Hg droplet moving from x = 5 nm to 15 nm with a constant velocity of 2.5 m/s. The size of the kink model in lateral x and y directions is 20×14 nm2 (b) Interfacial energy between Hg droplet and substrate as a function of displacement of droplet for the G-Cu and pure Cu substrates, respectively. The diameter of Hg droplet and structural parameters of grooves are consistent with that in Figure 12. The embedded partial enlargement graph is the cross-sectional configuration of the G-Cu junction. All the MD data are average values in 100 000-time steps (0.1 ns).

5. CONCLUSIONS To summarize, we have found that a liquid Hg droplet can have spontaneous and unidirectional self-propagation on a grooved Cu substrate covered by a monolayer graphene

23



continuously without any energy input. The MD simulation results indicate that the Hg droplet undergoes acceleration, deceleration and return stages successively from the narrow to wide ends of the graphene-Cu groove. The transition from acceleration to deceleration of a Hg droplet is caused by the pressure difference inside the droplet in the moving direction, which is intrinsically determined by the competition between the liquid-solid interfacial energy and the potential energy of the Hg droplet. The self-driving velocity of the Hg droplet increases with the gradient and depth of grooves and decreases with the increase in its diameter. The magnitude of driving acceleration of an Hg droplet increases linearly with the gradient of Graphene-Cu grooves, whereas the magnitude of driving force for different size of Hg droplets is determined by the commensurability ratio between the volumetric size of Hg droplets and the relative width of the groove. A continuum model is adopted to interpret the influence mechanism of the groove depth, the gradient of groove and the diameter of droplets on self-driving behaviors, implying that the energy minimum point of droplet-substrate system occurs at larger width of groove for the system with a deeper groove and a larger droplet. Intriguingly, the Hg droplet can be driven unidirectionally by a zigzag gradient textured surface without imposing any external fields. We conclude that the monolayer graphene-covered texture can significantly reduce the effect of the contact line pinning on the droplet self-driving process. The results will provide a fundamental theoretical basis for state-of-the-art design of the graphene-covered textured surface to drive liquid droplets directionally and spontaneously. Supporting Information The Supporting Information is available free of charge on the ACS Publications website. It includes the comparison of self-driving behaviors of Hg droplets on both graphene-Cu and pure Cu grooves; computational method of contact angle; calculation of pressure difference within Hg droplets; the details of continuum model, and the calculations of equilibrium 24


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distance and interfacial strength (PDF) Supporting Information SII file: Video for the self-driving process (lateral view) of an Hg droplet on the proposed zigzag surface. (Video 1) Supporting Information SIII file: Video for the self-driving process (top view) of an Hg droplet on the proposed zigzag surface. (Video 2)

ACKNOWLEDGEMENTS This work was supported by the National Natural Science Foundation of China (NSFC) with Grants (11872192, 11772082, 51675236 and 11672063).

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Unidirectional Self-Driving Liquid Droplet Transport on a Monolayer

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