Anisotropic wetting of droplets on stripe-patterned chemically

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Anisotropic wetting of droplets on stripe-patterned chemically heterogeneous surfaces: effect of length ratio and deposition position Yuxiang Wang, Meipeng Jian, Huiyuan Liu, and Xiwang Zhang Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b02491 • Publication Date (Web): 10 Oct 2018 Downloaded from http://pubs.acs.org on October 13, 2018

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Anisotropic wetting of droplets on stripe-patterned chemically heterogeneous surfaces: effect of length ratio and deposition position Yuxiang Wang, Meipeng Jian, Huiyuan Liu, Xiwang Zhang* Department of Chemical Engineering, Monash University, Clayton, VIC 3800, Australia

ABSTRACT: The equilibrium state of a droplet deposited on chemically heterogeneous surfaces is

studied by using many-body dissipative particle dynamics. The length ratio covers two orders from 0.01 to 1 and allows a systematical inspection of the changes of the droplet shape, contact angle and aspect ratio with this parameter. Moreover, a new parameter, global aspect ratio, is introduced to better characterize the distortion of the droplet. It is found that the droplet shape at the equilibrium stage strongly lies on the deposition position when the length ratio is beyond 0.1. Additionally, the lateral displacement is observed when depositing the droplet on the border of two stripes at large length ratios (over 0.1). On the other hand, Cassie area fraction also has significant effect on the wetting behaviors. When the droplet is driven by a body force with a 45° inclined angle to the stripes, the moving direction could be strictly in line with the force direction, deviating from the force direction, or totally in line with the stripes, depending on the length ratio.

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1. INTRODUCTION Wetting on heterogeneous surfaces is the most common phenomenon in nature and daily life as a result of the chemical or geometrical heterogeneity of most surfaces1. Many interests have been aroused on wetting on heterogeneous surfaces due to their fundamental and practical aspects2-17. To understand and control the wetting behavior on heterogeneous surfaces, a lot of efforts have been done by using patterned surfaces. For instances, by measuring contact angles, Xia et al.2 reported the extremely high droplet distortion and anisotropic wetting behavior of droplet on the well-defined micro- and nanopatterned grooved surfaces. A high contact angle gap of 79° between the directions parallel and orthogonal to the stripes was observed. In their experiments, the anisotropic wetting weakly depended on the intrinsic wetting properties of the substrates, indicating that the Cassie-Baxter model3 was still applicable. Carmeliet et al.4 conducted systematical simulations of wetting on checkboard-patterned surfaces with different pitch sizes by using lattice Boltzmann model (LBM). It was found that when the patch ratio (ratio of the patch size to the initial droplet radius) is lower than the critical ratio, the CassieBaxter model was applicable. However, when the patch ratio is higher than the critical value, the contact angle was not fixed anymore and a set of contact angle values were observed due to the strong heterogeneous effect of the surfaces. Thus, a beyond-Cassie-Baxter model was proposed to predict an equivalent contact angle. Kooij et al.5-9 did intensive studies on droplet anisotropic wetting on chemically stripe-patterned surfaces by employing both experiments and simulations. By fabricating well-controlled surfaces with alternating hydrophilic and hydrophobic stripes, they investigated the droplet wetting dynamics and shape evolution on such surfaces. The relative width of the two kinds of stripes was found to be the key factor to the anisotropic wetting of the droplet when the droplet size is one or two orders larger than the width of the stripes5. They also observed two distinct spreading regimes: the first regime is an inertial regime when the shape of a droplet can still keep circular from a


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top view regardless of what the surface underneath the droplet is, indicating that at this stage, the kinetics dominates the spreading. Another is the viscous regime, the properties of the underlying surface begin to dominate the spreading of the droplet, the droplet needs to adjust its shape according to the confinement of the hydrophobic stripes6. The dependence of droplet shape on relative width ratio of hydrophilic to hydrophobic stripes7, kinetic energy8 and deposition position9 were also investigated to deeply understand the wetting behavior on stripe-patterned surfaces. Yeomans et al. also performed quite a lot work in anisotropic wetting both numerically and experimentally10-13. Deposition positions10,11, impact velocity11, deposition methods12 and pattern shapes13 were found to influence the anisotropic wetting significantly, various droplet shapes were therefore identified. By decreasing stripe width to nanoscale, Damle et al. indicated that the anisotropic wetting nearly disappeared and the aspect ratio almost maintained unity even during an impact process14. Furthermore, by using dissipative particle dynamics (DPD), Suttipong et al. found the stripe-patterned heterogeneous surfaces also had a significant influence on the adsorption and aggregation of self-assembled surfactants15,16. More details about droplet wetting on heterogeneous surfaces, including the fabrication approaches, wetting behaviors and potential applications can be found in a recent review17. On the other hand, the droplet motion on patterned heterogeneous surfaces has also drawn an increasing of attentions18-30. A smart design of substrate which allows the spontaneous motion of droplet was demonstrated by Kooij et al.18: by narrowing the relative width of the hydrophobic stripes, a wettability gradient was generated to induce the directional motion of droplet to the low fraction of hydrophobic stipes area. In Sbragaglia et al.’s series of study19-21, a stick-slip droplet sliding manner was clearly demonstrated by setting a large wettability contrast (70°) on the patterned surfaces with alternating hydrophilic and hydrophobic stripes. The speed of the motion was found to be one order of magnitude smaller than that on the homogeneous surface which has the same apparent contact angle. 3 ACS Paragon Plus Environment

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The pinning-depinning effect was believed to cause the slowing down of the droplet motion19. Apart from stripe-patterned surfaces, the stick-slip motion also took place on heterogeneous surfaces with square or triangular hydrophobic spots, although they are not so pronounced21. Kusumaatmaja et al. employed Lattice Boltzmann Method to investigate the shape evaluation of droplets sliding on stripepatterned surfaces22,23. By studying the droplet sliding across the pillar-patterned rough surfaces, Wang et al.24 numerically found the central part of the surfaces underneath the droplet has no influence on the determination of the apparent contact angle. It seems the contact line is independent of this part of surfaces, and only surfaces properties around the three-phase contact line can affect the droplet moving. The simulation conclusions are well matched with observations in experiments and recently proposed wettability theories25-27. Lin et al.28 simulated the droplet sliding on chessboard-patterned smooth surfaces. Three sliding routes (straight, zigzag and oblique) were observed by simply altering the inclined angle of the substrates. Li et al.29 examined the flow structure inside a droplet when it was moving on a surface with linear wettability gradient and found that the rolling motion on hydrophobic areas was transited to the combination of rolling and sliding motion on hydrophilic areas. Semprebon et al.30 demonstrated four distinct motion manners: pinning, crossing, gliding and gliding & pinning by performing both experiments and simulations of a droplet sliding across a chemical step from hydrophilic area to hydrophobic area. Surprisingly, the contact angle hysteresis was regarded as a profitable factor to the control of droplet motion. To our best knowledge, although the droplet wetting on stripe-patterned surfaces with kinds of length ratios has been studied previously, they were too discrete to present the continuous varying of the anisotropic wetting behavior. For example, the limited length ratio range cannot show the details of a transition from Cassie-Baxter regime to Beyond-Cassie-Baxter regime. Therefore, the diversity of droplet shapes cannot be well captured. Besides, the aspect ratio itself is insufficient for good 4 ACS Paragon Plus Environment

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characterization of the droplet shape. In the present work, we aim to provide a systematic presentation of anisotropic wetting behavior and better characterization of the droplet shape. To this end, the stripepatterned chemically heterogeneous surfaces with a wide and continuous range of length ratio, covering from 0.01 to 1 with an exponential distribution, are thus modeled to investigate the scale effect on wetting behavior. Three typical deposition positions, the border of two alternating wettability stripes, middle of a hydrophobic stripe and middle of a hydrophilic stripe are also considered to inspect the dependence of droplet equilibrium state. A new parameter, global aspect ratio, is introduced to give a comprehensive characterization of the isotropic droplet shape. Effect of Cassie area fraction on the anisotropic wetting is also included. Different from the droplet sliding driven by forces orthogonal or parallel the stripes in the literature, forces with an inclined angle of 45° to the stripes are exerted on the droplet to investigate the droplet sliding behavior, novel droplet motion manners are thus observed and analyzed.

2. NUMERICAL METHOD 2.1. Many-body dissipative particle dynamics (MDPD) Dissipative particle dynamics (DPD) is a particle-based and mesh-free numerical technique which has been intensively applied to address problems at mesoscale due to its coarse-graining nature31,32. After the original version of DPD, many new features have been developed to enrich the DPD family33-35. In this work, a widely used DPD variant which considers the local density of particles, many-body DPD (MDPD)36, is employed to study the wetting behavior. By introducing a van der Waals loop into the equation of state, MDPD can mimic the vapor-liquid interface in the phase separating process. MDPD inherits the main features from the original DPD, only the conservative force is redefined to introduce


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the mentioned force loop, as shown in Eq. (1). The dissipative force and random force are left unchanged, as shown in Eq. (2) and Eq. (3). = + ( + ) = − ·

=

(1)

(2)

(3)

In Eq. (1), and are the attractive and repulsive coefficients in conservative force which are used to generate a force loop. = ∑%'& ω# (r%& ) is used to calculate the local density of fluid particles with its weight function ω# r%& =

()

*+,- .

(1 − r%& /r1 )*. r%& = 1 − r%& /r2 , ω1 r%& = 1 − r%& /r1 , =

31 − r%& /r2 and = 4 are another weigh functions in Eq. (1-3), respectively. is the real-time distance between two interacting particles, r2 is the cut-off radius for all particle pairs and r1 is the cut-off radius for fluid particle pairs only. and are two unit vectors related to the particle position and velocity. The dissipative coefficient is coupled with the random coefficient by function * = 267 8 to keep the kinetic energy in DPD system constant, 67 8 is the temperature term. The values of the mentioned parameters and other parameters in the calculation are listed in Table 1. It is worthy to mention that it is not prudent to specify a fluid generated by these parameters to a real fluid in experiments arbitrarily due to the high level of coarse-graining degree in DPD method. However, by focusing only on the concerned fluid property, such as the wettability, a pair of fluid/solid in DPD systems can represent a specific fluid/solid pair in experiments. For example, if we want to study deionized water wetting on surfaces coated with SU-8, by carefully adjusting the parameters in Table 1, a fluid which can wet a smooth solid surface with an intrinsic contact angle of 63°. Then we can safely 6 ACS Paragon Plus Environment

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say the fluid and solid in simulations represent deionized water and SU-8, respectively, but only available for the wetting-related studies. In this work, we would not like to specify the fluids and solids to specific materials since they may match many material pairs in experiments regarding to wetting issues. Table 1. Parameters adopted in the simulations (all in DPD unit) Name of parameter Symbol of parameter

Attractive coefficient (between fluid particles) Attractive coefficient (between solid and fluid particles)

;<

Repulsive coefficient

=

Cutoff radius in

Cutoff radius in ω1 r%&

67 8

Temperature of the system Random coefficient

Time step

Tunable 25.0 1.0 0.75 1.0 1.0

∆8

0.01

?

Empirical velocity-Verlet coefficient

-80.0

Dissipative coefficient

Value

* /(267 8) 0.65

2.2. Boundary condition Boundary condition is an indispensable requirement in DPD systems since the soft repulsive interaction between particles cannot prevent the penetration of fluid particles into solid walls. Many boundary conditions have been therefore proposed to address this issue37,38. In this work, a recently proposed boundary condition38 is employed. This boundary condition takes four liquid/solid interface factors (distance between the interacting solid and fluid particles, the overall force acting on the approaching fluid particle, the initial approaching velocity of the fluid particle and the angle between the mentioned force and velocity) into consideration. Since the four factors are easy to obtain in the calculation, the boundary condition is quite easy to implement. It has been proved that this boundary 7 ACS Paragon Plus Environment

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condition is effective in keeping the liquid/solid interface properties and preventing the penetration. Some applications of this boundary condition can be found in previous work39,40. 2.3. Substrates modeling In this work, substrates consist of parallel stripes with equal width but alternating wettability are modeled. The non-equal stripe widths are only studied in Section 3.4. The length ration @ is defined as the ratio of the stripe width A; to the initial droplet radius B . @ ranges from 0.01 to 1 with an exponential distribution, nine values are selected: 0.01, 0.0178, 0.0316, 0.0562, 0.1, 0.178, 0.316, 0.562 and 1. Three typical surfaces with different @ are shown in Figure 1.

Figure 1. Surfaces with @ = 0.01, 0.1 and 1, droplet radius B = 9. Brown stripes are hydrophilic and cyan stripes are hydrophobic. 3. RESULTS AND DISCUSSIONS 3.1. Droplet shape It has been widely reported that the equilibrium droplet shape on heterogeneous surfaces is heavily sensitive to the length ratio4,9,11. If wetting on homogeneous surfaces (the length ratio can be regarded as infinitesimal), the contact line of the droplet will spread to all radical directions with unique speed, or


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isotropic wetting, leading to a perfectly hemisphere-like equilibrium shape. Also, a recent study showed that when a droplet wetting on substrates with heterogeneous stripes at nanoscale which is extremely small compared with the droplet size, the droplet would still prefer the isotropic wetting like that on homogeneous surfaces14. However, when wetting on heterogeneous surfaces with considerable pattern size, the droplet shape will exhibit distortions due to the confinement of the relatively highly nonwetting areas. The length ratio is always used as a dimensionless parameter to characterize the level of the heterogeneity of surfaces patterned by regular geometries, like square4, stripe5-12,14 or triangle13,28. To focus on the surface property itself and diminish the influence of kinetic energy or inertia, the gravity effect is neglected and an extremely slow deposition velocity is given to the initial droplet to make the contact with surfaces. Three deposition positions are considered: the border between two stripes (P1), middle of the hydrophobic stripe (P2) and middle of the hydrophilic stripe (P3). Stripes with apparent contact angles of 60° and 120° are used to model surfaces. Figure 2 shows the results of droplet equilibrium shapes on surfaces with different length ratio @ and deposition positions. The overall tendency of the droplet distortion in pace with the length ratio is consistent with the results in other experiments11 and simulations4,28. The droplet shape can still keep a hemisphere-like shape when @ ≤ 0.1, the same critical value of @ can be also found in Ref. 4. When the length ratio increases, the droplet is elongated and the anisotropic wetting becomes apparent. The stripes underneath the droplet become less. It is also interesting to notice that, the number of wetted stripes is always odd with the two surrounding hydrophobic stripes, revealing the droplet prefers to wet the hydrophilic stripes. This is in good agreement with other experiments 9. After wetting a certain number of stripes for surfaces with a given @, the excessive interfacial free energy which keeps the droplet away from a global minimum energy state has been mostly released. The energy barrier formed by the hydrophobic stripes cannot be overcome, and further wetting on hydrophobic stripes is blocked, leaving 9 ACS Paragon Plus Environment

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the droplet a metastable state or local minimum energy state. This metastable state could be multiple according to different length ratio, leading to various droplet shapes. The equilibrium droplet shape is found that also highly depends on the deposition positions. For low @, no obvious dependence on deposition positions can be observed from the equilibrium droplet shapes. However, when @ is higher, the dependence on deposition position becomes increasingly evident. For P1 and P3 positions, the droplet is elongated much more than that for P2 position, and this is especially evident when @ = 1. The “lozenge” and “butterfly” droplet shapes reported in Ref. 11 are perfectly observed in our simulations. Here we give a detailed analysis of the droplet shape on @ = 1 surfaces. For P2 position (corresponding to cases 2 and 4 in Figure 2a in Ref. 11), since the first contact area is the middle of the hydrophobic stripe, the spreading speed is very slow in all directions at the very beginning. Once the spreading fronts contact the hydrophilic stripes at the two orthogonal sides, the orthogonal wetting speed will be immediately boosted, like the droplet is stretched. And once the hydrophilic stripes are wetted, further and faster wetting on these two hydrophilic stripes in the parallel direction will also be induced while the wetting on the hydrophobic stripe is still very slow. Finally, a “butterfly” equilibrium droplet shape is formed. For P3 position (corresponding to case 1 in Figure 2a in Ref. 11), at first, the spreading at all directions are all fast. Once the spreading fronts contact with the hydrophobic stripes at the two sides, the further spreading orthogonally is slowed down largely. Meanwhile, since the width of the stripes is equivalent to the initial radius of the droplet, which provides sufficient space to allow the droplet spreading along the stripe direction rapidly, the excessive interfacial free energy can be still released by wetting the hydrophilic stripe. This preferred energy release pathway will further make the spreading to the hydrophobic stripes impossible due to insufficient excessive interfacial free energy left for the droplet to overcome the energy barrier.


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Figure 2. The droplet equilibrium shapes on surfaces with different length ratios @ and deposition positions. The wetting process occurs for P1 (corresponding to cases 3, 5 and 6 in Figure 2a in Ref. 11) is nearly identical with that for P3, but regular and evident lateral displacements in direction orthogonal to stripes can be observed for high length ratio @, as shown in Figure 3. Such as for @ = 0.316, the lateral displacement is 1.4 and for @ = 1 is 4.5. The displacements are half of the corresponding stripe widths and the droplet always prefers to move to the hydrophilic stripe side. However, for all deposition positions, it is also found that for low @, random and slight displacement can be observed, and the direction and distance have no any regularity. Similar phenomenon was also observed for square and triangle patterned heterogeneous surfaces. The inner disturbance of the droplet due to the particle-based numerical approach is believed to cause this random displacement for low @ 28.


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Figure 3. Lateral displacements in direction orthogonal to stripes on surfaces with @ = 0.316 and 1. The red dash lines indicate the initial position of the droplet mass center before deposition and the red solid lines indicate the equilibrium position of the droplet mass center. At P2 and P3, the dash lines and solid lines coincide with each other. 3.2. Contact angle The wetting behavior in the directions orthogonal and parallel to stripes are different and more confinement effect occurs in the orthogonal direction due to the hydrophobic stripes. As a result of that, the contact angles in the two directions are quite different and lead to strong anisotropic wetting, especially when the stripe width is comparative to the initial droplet radius2,5-13. The anisotropic wetting makes Cassie-Baxter theory no longer applicable globally. However, for very low @ (≤ 0.1), since the droplet is not sensitive to the narrow stripes with alternating wettability, making the wetting processes in the two directions similar, such as the same spreading speeds, thus the droplet shape can still be hemisphere-like and contact angles measured in these two directions can still be identical, as shown in Figure 4. Experimental results also shows that if the stripe size is extremely small compared with droplet size, the wetting could be still isotropic14. In our simulations, all contact angles measured with @ from 0.01 to 0.1 in both directions are around 90°, agreeing with Cassie-Baxter equation predicted value very well. Note, the Cassie-Baxter equation is KLMN= = O( KLMN( + O* KLMN* , with O( = O* = 0.5 and


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N( = 60° N* = 120° in this work. Therefore, this regime with @ ≤ 0.1 can be referred to as the CassieBaxter regime.

Figure 4. The contact angle dependence on length ration and deposition position. The left inset shows the equilibrium state of a droplet deposited at P2 on the surface with @ = 1. NQ denotes the contact angles measured from the direction orthogonal to the stripes and N∥ denotes the contact angles measured from the direction parallel to the stripes. The right inset shows the wave-like contact line when @ S 0.1, indicating by red solid lines (only the bottom part of the droplet is visible as black area). When @ S 0.1, the stripe is adequately wide and the wettability difference can be felt by the droplet. In the parallel direction, though the droplet can spread on both hydrophilic and hydrophobic stripes, the contact line will move further on hydrophilic stripes than on hydrophobic stripes, leading to a wave-like contact line, as shown in the right inset in Figure 4. This interesting feature is in agreement with many observations in experiments14 and simulations by other approaches7,9,10. Multiple contact angles can be observed along this wave-like contact line, making Cassie-Baxter predicted value inaccurate. Thus, in Figure 4 it is unlikely to give the corresponding contact angles in the parallel direction, only contact angles measured in the direction orthogonal to the stripes are given. It should be noticed that a local contact angle set with continuously distributed values can be measured along the contact line in the 13 ACS Paragon Plus Environment

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parallel direction, as shown in the left inset in Figure 4 (between position B and C). Taking the surface with @ = 1 and deposited from P2 case as an example (the inset case), the maximum value of contact angle of 116° can be found at position B, located in the middle of the hydrophobic stripe, the minimum value 61° can be found at position C, located on the hydrophilic stripes. Both of the values for the two local contact angles are very close to the intrinsic contact angles on the corresponding stripes. The slight discrepancies from the intrinsic contact angles are caused by the competing effect from the surrounding stripes with different wettability. One can imagine that if the stripes are extremely wide compared with the initial droplet radius, or @ is infinity, the droplet can only sit on one stripe, contact angles measured on this stripe will exactly equal to the intrinsic value, here we say the competing effect from surrounding stripes is completely ignored. By contrast, if the stripes are extremely narrow, or @ is infinitesimal, the droplet can sit on an infinite number of stripes with alternating wettability, the contact angles measured on any of these stripes will exactly equal to the Cassie-Baxter predicted value, not the intrinsic contact angles of any of the stripes. Here we say the competing effect from surrounding stripes is fully considered. Just as @ = 0.01 in Figure 4, in which case all the measured contact angles are almost equal to 90°. It is worthwhile to note that the contact angles on surfaces with different @ don’t always follow a monotonous increase or decrease tendency with the increase of @ when @ S 0.1 and the actual tendency is largely dependent on the deposition positions. Here we should emphasize one of the advantages of our simulation work compared with other experimental or simulation work10,11,14: we inspect anisotropic wetting on substrates with a high density of exponentially distributed length ratio from @ = 0.01 to 1, which enables us to identify more accurate wetting features which may be out of expectations, such as the fluctuation of the contact angle distribution. Based on this ability, we give a detailed analysis of the difference between wetting on surfaces with @ = 0.316, 0.562 and 1 deposited from P3, the similar 14 ACS Paragon Plus Environment

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wetting mechanism can apply to other cases. Since the stripe widths are considerable to the initial droplet radius, the confinement effect becomes evident. The value of contact angle witnesses a sudden drop to 79° at @ = 0.562, thus wetting on this surface is deserved to be analyzed first. Since the width of the hydrophilic stripes of @ = 0.562 surface is not wide enough as that on @ = 1 surface, the contact line cannot spread much rapidly along the first contact hydrophilic stripe to allow a fast release of excessive interfacial free energy. Thus, the unreleased excessive interfacial free energy is still enough for the contact line to overcome the energy barrier and cross through the neighboring two hydrophobic stripes. Once the contact line contacts the new hydrophilic stripes after crossing the hydrophobic ones, as discussed in Section 3.1, the orthogonal wetting speed will be immediately boosted, leading to a full release of the excessive interfacial free energy. Finally, three hydrophilic stripes are wetted, as shown by the equilibrium droplet state in Figure 2 (red rectangle box in the third line). The same process can also happen on @ = 0.316 surface. Though also three hydrophilic stripes are covered, since the width is narrower than that on @ = 0.562 surface, the droplet is confined to a orthogonally narrower space and thus the contact angle is bigger, 111.2° to be exact. Wetting on @ = 1 surface has been discussed in Section 3.1. Since the width of stripes is wide enough, no more hydrophilic stripes are needed to facilitate the energy release. Here we should remark that the width of the total wetted hydrophilic stripes is critical for the release of excessive interfacial free energy. From all the cases we can conclude that the total width should be around or above the value of the initial droplet radius to allow the timely and sufficient energy release. Once the excessive interfacial free energy could be released, no additional wetting (on hydrophilic or hydrophobic stripes) is required. This interpretation can be well proven by the fact that for all cases (as shown in Figure 2), the product of hydrophilic stripe width with the number of wetted hydrophilic stripes is always larger than a critical value. This explains why the droplet shapes are highly dependent on the deposition position and length ratio. In the wetting process, the droplet is


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always trying to find adequate space to release the excessive interfacial free energy. The adequate space is highly associated with the deposition position and length ratio. 3.3. Aspect ratio Since on stripe-patterned heterogeneous surface, the droplet shape is always elongated from a hemisphere-like shape, the aspect ratio, defined as the ratio of the length U1 to the width V1 of the contact area at the bottom of the droplet (as shown in the left inset of Figure 5a), is thereby introduced to characterize the elongation degree from perfect hemisphere-like shape5,6. However, this definition can only describe the elongation at the bottom of the droplet. From a top view, the droplet may still keep a nearly hemisphere-like shape even when the contact area at the bottom is largely elongated, and this phenomenon could widely happen when the contact angles orthogonal or parallel to the stripes are over 90°, in which the maximum length and width are located above the bottom of the droplet. Such as the two insets in Figure 5a, they are from the same simulation case but with different viewpoints. To better characterize the comprehensive distortion of the droplet, a new aspect ratio, global aspect ratio, defined as the ratio of the maximum length of the whole droplet U2 to the maximum width of the whole droplet V2 (as shown in the right inset of Figure 5a), is therefore introduced. Note: the maximum length and width are not necessary to be in the same cross-section of the droplet parallel to the substrate. In some cases, they are measured in different cross-sections, as a typical example shown in Figure 6. In this work, the aspect ratio measured at the bottom is termed as local aspect ratio or BW , the aspect ratio measured from the whole droplet is termed as global aspect ratio or BX . It is significant for a more accurate characterization of the droplet distortion by employing the aspect ratios. Figure 5 plots BW and BX as a function of length ratio @, the three deposition positions are also considered. Two contact angle pairs, (60°, 120°) for Figure 5a and (90°, 150°) for Figure 5b are used in 16 ACS Paragon Plus Environment

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this section. It should be stated here that, in nature, perfectly smooth surfaces with intrinsic contact angle over 120° can hardly exist. In real experiments, the substrates should be coated with micro/nanostructures to allow air cushions formed, then the air cushions together with the solid materials can support the droplet to achieve a superhydrophobic state according to Cassie-Baxter theory. In this simulation work, the stripes with a contact angle of 150° can be seen as coated surfaces with nanostructures in experiments, but the nanostructures are extremely small compared to the smallest width of the stripes and the droplet size. Thus, the stripes with the contact angle of 150° are still referred to as smooth stripes. We first analyze the (60°, 120°) case. For very low @ (≤ 0.1), since the droplet cannot feel the narrow stripes with alternating wettability well, the droplet still stays in a hemispherelike shape, leading to both of the BW and BX with values around 1. When @ S 0.1, the local and global aspect ratios may be greater or smaller than 1 and they strictly follow the variation tendency of the contact angles shown in Figure 4. It should be emphasized here that the global aspect ratios BX are slightly smaller than the corresponding local aspect rations BW in some cases, such as @ = 0.316 deposited from P1 or P2. This indicates that the elongation degree of the whole droplet is slightly lower than elongation occurs at the bottom of the droplet, sharp shape transition can be expected from the middle part to the bottom of the droplet. We term this phenomenon as “dual-elongation effect”. This effect strongly depends on the length ratio and deposition position.


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Figure 5. The local and global aspect ratio dependence on length ratio and deposition position. The two used contact angle pairs are (a) (60°, 120°); (b) (90°, 150°).


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Figure 6. A typical example (droplet deposited at P3 on the surface with @ = 1) shows the maximum length and width are in different cross-sections of the droplet parallel to the substrate. The green dash line shows the plane where the maximum width is measured while the red dash line shows the plane where the maximum length is measured. We further inspect this effect on a (90°, 150°) contact angle pair, the results are plotted in Figure 5b. When @ ≤ 0.1, both of BW and BX still keep around 1 but with a less fluctuation compared with (60°, 120°) contact angle pair. The left inset of Figure 5b shows a typical droplet state (from P2 @ = 0.1 case as indicated by green dash circle and arrows): the lower one is a top view of the whole droplet and the upper one shows only the contact area at the bottom of the droplet by making other parts invisible. From this inset we can see both of the contours are almost perfect circles, indicating the elongations at every cross section parallel to the surface are very consistent. When @ S 0.1, the deviation of the two aspect ratios become more evident than that happens in the same regime of (60°, 120°) contact angle pair, like in P3 @ = 0.562 case, BW = 1.75 and BX = 1.15. The right inset in Figure 5b gives a typical droplet state to show the deviation (from P2 @ = 0.316 case). It is also interesting to find that BX only slightly increases when the droplet is deposited from P1 and P3 while in (60°, 120°) contact angle pair, BX fluctuates for the same deposition positions. This indicates that the average wettability of the two kinds of stripes can heavily influence the ability for the whole droplet shape to keep a hemisphere-like shape. In a contact angle pair with high hydrophobicity, though the contact area at the bottom of the droplet could be largely elongated, the upper part of the droplet can still keep at a nearly hemisphere-like shape. 19 ACS Paragon Plus Environment

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3.4. Cassie area fraction

Contact Angle

Table 2. Simulation results of contact angles, droplet shapes and aspect ratios at different Cassie area fraction φ.

Y = Z. [\]

0.25

0.375

0.5

0.625

0.75

0.875

^ = Z. Z[ 0.1

Droplet Shape

0.0316 0.316

Aspect Ratio

1

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The Cassie area fraction φ (defined as the ratio of the hydrophobic stripe width to the sum of a hydrophobic width and a hydrophilic width), also has significant influence on the droplet wetting 20 ACS Paragon Plus Environment

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behaviour5,7,11,20. In this section, 7 Cassie area fractions under different length ratios have been investigated. The droplet is deposited at the border between two stripes (P1) for all cases. Firstly, the simulated contact angles are compared with the Cassie-Baxter predicted contact angles N , as shown in the first row of Table 2. Still, when length ratio is low (β ≤ 0.1), the simulated contact angles at both parallel and orthogonal directions to the stripes can follow the corresponding Cassie-Baxter predicted values very well. For high length ratio (β = 0.316), significant deviation can be observed and it becomes more remarkable when the length ratio is increased to β = 1. This fact indicates the scope of availability of the Cassie-Baxter theory, which is important to the design of relevant applications. The second row shows the droplet shapes for all the cases. Even the Cassie area fraction φ is introduced, the overall trend of the shape variation is still similar to that in Figure 2. Here we emphasize some new features in which the parameter φ matters. For high β (0.316), large elongation of the droplet shape can be observed for relatively small φ (≤ 0.625), but the shape becomes hemisphere-like gradually when increasing φ (or the fraction of hydrophobic area), similar trend can also found for β = 1, but less evident. Triangle shapes are formed for β = 1 at low Cassie area fraction φ = 0.25 and 0.375 (as shown in red rectangle), and same shape was also reported in experiments11. The third row presents the aspect ratios for all cases. Length ratio still dominates the values of aspect ratio, as when β ≤ 0.1, both of the local and global aspect ratios almost keep at 1 while only when β S 0.1 the aspect ratios begin to deviate from unity and fluctuate along with Cassie area fraction. It is interesting to find that for β = 0.316 and 1, both the global and local aspect ratios witness an increase from φ = 0.375 to φ = 0.5 and then a decrease from φ = 0.5 to φ = 0.875 and become closer to unity. This fact indicates that the most elongated droplet shape is more likely to happen when the width of the hydrophilic and hydrophobic stripe widths are equal.


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3.5. Inclined sliding In this section, we study the droplet sliding behavior on stripe-patterned surfaces. The contact angle pair in this section is (90°, 150°). Different from most studied sliding behaviors in which the droplet is forced to slide in the directions orthogonal or parallel to the stripes, in the present work the driving force (a body force exerted on every droplet particle) is set with an inclined angle of 45° to the stripes to inspect how the sliding will follow the force direction and its dependence on the length ratio. Like other units in DPD systems, the driving force is also dimensionless and mimics the in-plane component of gravity in experiments30, the magnitude is selected carefully to ensure the droplet can be driven on homogeneous surfaces with contact angles of 90° and 150°, respectively. Similar work has been done by Semprebon et al.30, but the concerned aspects are different from this work. In their study, the body force (the in-plane component of gravity generated by tilting the substrates) and in-plane inclination angle of the body force to the chemical step boundary are varied and parametrically studied. While in our work, the body force and in-plane inclination angle are fixed, only the length ratio is varied to study the scale effect. Three typical sliding manners highly depending on the length ratio can be identified, as shown in Figure 7. When the length ratio is quite low, like the surface with @ = 0.0316 in Figure 7a, the droplet motion will strictly follow the direction of the driving force. In this sliding manner, the contact line of the droplet cannot feel the border between two stripes since they are too narrow, thus no obvious resistance change in the moving of advancing and receding contact lines to alter their directions, leading to the rectilinear sliding of the droplet which strictly follows the direction of the body force. If increasing the length ratio to some extent, like equal or just less than 0.562, the sliding direction will have a displacement angle with the force direction and show a zig-zag motion trajectory, as shown in Figure 7b. The similar phenomenon can also be observed in experiments41. This sliding manner is also quite similar to the crossing motion manner mentioned in 22 ACS Paragon Plus Environment

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Semprebon et al.’s work30. In their experiments, the substrate consists of only one border by jointing two parts with hydrophilic and hydrophobic areas. With high Bond number (≥ 1.3) and proper in-plane inclination angle, the motion direction of the droplet will deviate from the body force direction and prefer to glide along the border when it is trying to cross the border. Then, the droplet will continue to follow the force direction when it is totally on the lower part of the substrate. In our simulation, the stripe-patterned surface can be seen as the border or chemical step repeats itself several times. Therefore, the zig-zag motion trajectory can be well explained: it can be seen a repetition of the “gliding-following” motion. When continue to increase the length ratio to @ = 1, the stripe width equals to the initial droplet radius, as shown in Figure 7c, the droplet sliding will totally be blocked by the hydrophobic stripe and can only slide in the direction parallel to the stripes. This kind of motion manner corresponds to the gliding motion in Semprebon et al.’s work30. This result is also consistent with the fact in the wetting process of a droplet on a surface with @ = 1, where the barrier energy from the wide hydrophobic stripe is very high and even adding a driving force is still not enough to overcome it. Thus, the droplet can only slide on the hydrophilic stripe under the parallel component of the driving force. However, from the snapshots of Figure 7c, it can be still observed that the droplet is still keeping a potential to cross the border. Though the motion manners in our simulations share some similarities with work in Ref. 30 in terms of phenomena and behind mechanism, the significant difference should also be emphasized. In our simulation, the various motion manners only depend on the length ratio while in Ref. 30 the body force and in-plane inclination angle are key factors to vary the motion manners. Different control methods of droplet sliding could lead to different applications. For example, by selecting proper stripe width, droplets with different radii can have different length ratios: droplets with large radii have small length ratio and droplets with small radii have a large length ratio. For small length ratio, sliding manner showing in Figure 7a is preferred, for intermediate length ratio, the droplet motion could slide with a


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displacement angle while for large length ratio, the droplet can only slide along the stripes like that shown in Figure 7c. As a consequence, the droplets can be separated according to their radii.

Figure 7. Three typical sliding manners depending on the length ratio: (a) @ = 0.0316, strictly following the force direction; (b) @ = 0.562, sliding with a displacement angle; (c) @ = 1, only sliding along the stripes. In all panels, the black solid lines indicate the trajectory of the droplet mass center, the red dash lines indicate the force direction and the colorful droplet snapshots shows the typical droplet states at different moments. Figure 8 shows the time-dependent variations of the position of droplet mass center in the direction orthogonal to the stripes (X direction) and the contact area of the droplet bottom for the surface with @ = 0.562 from Figure 7b. Both of them show strong periodicity feature and the periodicity highly depends on the width of the stripes. In one cycle, the contact area (calculated by the number of particles) decreases to a bottom first while the droplet mass center undergoes a slow sliding in X direction (from time 1 to 3 as marked on the contact area curve), then the contact area increases to a peak while the droplet mass center experiences a rapid sliding (from time 3 to 5). In a whole circle, the distance of sliding is the length of a pattern periodicity (~10, including a hydrophilic stripe and a hydrophobic stripe). The insets show the snapshots of the contact area at the five typical moments. From the inset, it can be observed that at time 1 or 5, the droplet fully wets two hydrophilic stripes and a hydrophobic stripe, but at time 3, the droplet only fully wets one hydrophilic stripe and partially wets one hydrophilic stripe and one hydrophobic stripe. Based on the droplet states on the surface, the first slow and then fast 24 ACS Paragon Plus Environment

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displacement can be well explained. For the slow displacement, from moment 1 to 3, the receding contact line of the droplet is trying to leave the hydrophilic stripe while the advancing contact line is trying to wet the hydrophobic stripe. This process is obviously difficult since the hydrophilic stripes can pin the receding contact line while the hydrophobic stripes can resist the advancing contact line, a high contact angle hysteresis can be expected in this process. However, for the fast displacement, from moment 3 to 5, the receding contact line is leaving the hydrophobic stripe while the advancing contact line is wetting the hydrophilic stripe, this process is much easier than the first one due to less contact angle hysteresis occurring.

Figure 8. Time-dependent variations of the position of droplet mass center in the X direction and the contact area of the droplet bottom for a surface with @ = 0.562. 4. CONCLUSION In this work, the anisotropic wetting behavior of droplet on stripe-patterned heterogeneous surfaces is studied. The range of the length ratio from 0.01 to 1 is focused to demonstrate the transition from isotropic wetting to anisotropic wetting. When the length ratio is low, the droplet can maintain an almost hemisphere-like shape, with both local aspect ratio and global aspect ratio equal to one, as well as the 25 ACS Paragon Plus Environment

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contact angles measured from both directions, orthogonal and parallel to the stripes, are almost identical. At this range of length ratio, the Cassie-Baxter theory can still be well applied and the deposition position has no influence on the equilibrium droplet state. When increasing the length ratio, the droplet becomes elongated evidently. At this range of length ratio, the deposition position becomes to play an important role in the elongation. When deposited at the middle of hydrophilic stripe or border or two stripes, the droplet prefers to be elongated along the stripe while when deposited on the middle of a hydrophobic stripe, the droplet may be elongated more in the parallel or in the orthogonal directions and this depends on the exact length ratio. The droplet shapes in the simulations fall in good agreement with relevant experimental results. A key length ratio @ = 0.1 is therefore identified to separate the CassieBaxter regime and Beyond-Cassie-Baxter regime. This key parameter could be very significant to the design of functional interfaces where the wetting state of a droplet is critical. For why the droplet has to wet a certain number of stripes for a given length ratio to reach a stable state, to find the pathways where the droplet can fast release the excessive interfacial free energy is proposed to explain this. It is interesting to find that the change of aspect ratio along with the increasing of length ratio is not a monotonic function, some fluctuations can be identified when the length ratio is beyond 0.1. It is also found when the average hydrophobicity of the surface is high, the deviation of the global aspect ratio from local aspect ratio becomes larger for surfaces with large length ratio. Cassie area fraction φ is found also has significant influence on all droplet shape, contact angle and aspect ratio. The most elongation of the droplet is found at φ = 0.5 for high length ratios. By setting an inclined angle, three sliding manners are identified by forcing the droplet to slide on surfaces with different length ratios. The sliding trajectory of the droplet can strictly follow the force direction when the length ratio is small, or slide with a displacement angle to the force direction when the length ratio is intermediate, or totally slide on the hydrophilic stripe along the border of two stripes when the length ratio is sufficiently large.


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The numerical findings indicate that it is possible to utilize the stripe-patterned substrates to filter or separate droplets according to their radii in microfluidics systems. A possible application for the droplet separation according to their radii has been discussed in section 3.5. Other applications for droplet manipulation could also be expected to be inspired by our simulation findings.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Funding Sources This project is supported by the Australian Research Council (ARC) through the ARC Research Hub for Energy-efficient Separation (IH170100009). Notes The authors declare no competing financial interest. ACKNOWLEDGMENT


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The authors gratefully acknowledge the support of the Australian Research Council under IH170100009. Y. Wang would like to specially thank Monash University and the Monash Centre for Atomically Thin Materials (MCATM) for their scholarships. REFERENCES (1) Bonn, D.; Eggers, J.; Indekeu, J.; Meunier, J.; Rolley, E. Wetting and spreading. Reviews of modern physics 2009, 81, 739. (2) Xia, D.; Brueck, S. R. J. Strongly anisotropic wetting on one-dimensional nanopatterned surfaces. Nano letters 2008, 8, 2819−2824. (3) Cassie, A. B. D.; Baxter, S. Wettability of porous surfaces. Transactions of the Faraday society 1944, 40, 546−551. (4) Carmeliet, J.; Chen, L.; Kang, Q.; Derome, D. Beyond-Cassie mode of wetting and local contact angles of droplets on checkboard-patterned surfaces. Langmuir 2017, 33, 6192−6200. (5) Bliznyuk, O.; Vereshchagina, E.; Kooij, E. S.; Poelsema, B. Scaling of anisotropic droplet shapes on chemically stripe-patterned surfaces. Physical Review E 2009, 79, 041601. (6) Bliznyuk, O.; Jansen, H. P.; Kooij, E. S.; Poelsema, B. Initial spreading kinetics of high-viscosity droplets on anisotropic surfaces. Langmuir 2010, 26, 6328−6334. (7) Jansen, H. P.; Bliznyuk, O.; Kooij, E. S.; Poelsema, B; Zandvliet, H. J. Simulating anisotropic droplet shapes on chemically striped patterned surfaces. Langmuir 2011, 28, 499−505. (8) Jansen, H. P.; Sotthewes, K.; Ganser, C.; Teichert, C.; Zandvliet, H. J.; Kooij, E. S. Tuning kinetics to control droplet shapes on chemically striped patterned surfaces. Langmuir 2012, 28, 13137-13142. (9) Jansen, H. P.; Sotthewes, K.; Ganser, C.; Zandvliet, H. J.; Teichert, C; Kooij, E. S. Shape of picoliter droplets on chemically striped patterned substrates. Langmuir 2014, 30, 11574−11581. (10) Dupuis, A.; Yeomans, J. M. Lattice Boltzmann modelling of droplets on chemically heterogeneous surfaces. Future Generation Computer Systems 2004, 20, 993−1001. (11) Leopoldes, J.; Dupuis, A.; Bucknall, D. G.; Yeomans, J. M. Jetting micron-scale droplets onto chemically heterogeneous surfaces. Langmuir 2003, 19, 9818−9822. (12) Kusumaatmaja, H.; Vrancken, R. J.; Bastiaansen, C. W. M.; Yeomans, J. M. Anisotropic drop morphologies on corrugated surfaces. Langmuir 2008, 24, 7299−7308. (13) Vrancken, R. J.; Blow, M. L.; Kusumaatmaja, H.; Hermans, K.; Prenen, A. M.; Bastiaansen, C. W.; Broer, D. J.; Yeomans, J. M. Anisotropic wetting and de-wetting of drops on substrates patterned with polygonal posts. Soft Matter 2013, 9, 674−683. 28 ACS Paragon Plus Environment

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(31) Hoogerbrugge P. J.; Koelman J. M. Simulating Microscopic Hydrodynamic Phenomena with Dissipative Particle Dynamics. Europhysics Letters 1992, 19, 155. (32) Groot R.D.; Warren P. B. Dissipative Particle Dynamics: Bridging the Gap between Atomistic and Mesoscopic Simulation. The Journal of chemical physics 1997, 107, 4423−4435. (33) Li, Z.; Yazdani, A.; Tartakovsky, A.; Karniadakis, G. E. Transport dissipative particle dynamics model for mesoscopic advection-diffusion-reaction problems. The Journal of chemical physics 2015, 143, 014101. (34) Deng, M.; Li, Z.; Borodin, O.; Karniadakis, G. E. cDPD: A new dissipative particle dynamics method for modeling electrokinetic phenomena at the mesoscale. The Journal of Chemical Physics 2016, 145, 144109. (35) Liu, M.; Meakin, P.; Huang, H. Dissipative particle dynamics with attractive and repulsive particle-particle interactions. Physics of Fluids 2006, 18, 017101. (36) Warren, P. B. Vapor-liquid coexistence in many-body dissipative particle dynamics. Physical Review E 2003, 68, 066702. (37) Li, Z.; Bian, X.; Tang, Y. H.; Karniadakis, G. E. A dissipative particle dynamics method for arbitrarily complex geometries. Journal of Computational Physics 2018, 355, 534−547. (38) Zhang, D.; Shangguan, Q.; Wang, Y. An easy-to-use boundary condition in dissipative particle dynamics system. Computers & Fluids 2018, 166, 117−122. (39) Wang, Y.; Chen, S.; Liu, Y. Spontaneous Uptake of Droplets into Non-wetting Capillaries. Computers & Fluids 2016, 134, 190−195. (40) Wang, Y.; Chen, S.; Wu, B. Self-driven Penetration of Droplets into Non-wetting Capillaries. Computers & Fluids 2017, 154, 211−215. (41) Suzuki, S.; Nakajima, A.; Tanaka, K.; Sakai, M.; Hashimoto, A.; Yoshida, N.; Kameshima, Y.; Okada, K. Sliding behavior of water droplets on line-patterned hydrophobic surfaces. Applied Surface Science 2008, 254, 1797−1805.


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Anisotropic wetting of droplets on stripe-patterned chemically

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