Effect of electrostatic incitation on the wetting mode of a nano-drop over a pillar-arrayed surface

Abstract

The influence of the external electric field on the transition of the wetting mode over pillar-arrayed surfaces is investigated through a molecular dynamics study. The interplay between the energy barrier and the electrostatic force is analyzed by varying the pillar arrayed texture on a charged base substrate. At low density pillar texture, apart from a lesser energy barrier, the sagging of the liquid–gas interface also affects the switching of the wetting mode. A pattern map is developed to show the variation of the threshold voltage required for wetting transition at different surface topographies. The mechanism of the wetting transition on heterogeneous pillar-arrayed surfaces has also been analyzed from a molecular study. In light of the rapid development in the field of nanotechnology, the analysis of the wetting transition by an external electric field may provide significant insight towards the advancement of tunable engineered surfaces for nano-scale heat transfer and antifouling applications.

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Introduction

The recent trend of miniaturization in the field of biological and engineering applications has produced a considerable thrust towards the exploration of the anomalies in the dynamic wetting of droplets over solid surfaces. The wetting properties of a substrate not only depend on the chemical affinity of the material but also are an inviolable function of the intrinsic surface morphology.¹ A surface texture based non-wetting surface is ubiquitous in natural systems like lotus leaves² and water strider legs.³ It has its significance also in engineering applications like microfluidic⁴ devices to promote the mobility of the wetting media.

The rough surfaces with protruded nano-structures show a phenomenological difference in the physical modes of wetting from that of comparative smooth ones. The energy barrier⁵ between the substrate and the liquid medium is responsible for a droplet to be either in Cassie–Baxter state⁶ or in Wenzel state⁷ over a rough surface. In Cassie–Baxter state, the secondary medium got entrapped inside the valleys of the surface morphology. Thus, the droplet is situated over air cushioned composite (solid–gas) surface and exhibits high contact angle. On the other hand, in Wenzel state, the wetting media penetrates inside the grooves of the surface topography and attains higher surface wettability. Idealization of the random nano-structures over the substrate can be considered as pillar-arrayed texture on the surface for understanding the wetting mode and can be manufactured for industrial utilities. For different engineering applications, change of nominal wetting mode becomes necessary either permanently or instantaneously in tuneable fashion.⁸ These modes of wetting have their own applications which range from microfluidics,⁴ anti-fouling,^9,10 developing tuneable structured surfaces for desired heat and mass transfer rates through contact surface of the liquid¹¹ etc. Cassie state is useful for better mobility of a drop, whereas, Wenzel state is preferred in the applications where droplet–surface residue time needs to be increased. Imposition of external mechanical,^12–14 magnetic,¹⁵ thermal^16,17 or electrostatic^18–21 excitations are being proposed for wetting mode transition on the rough surface. The interplay between the energy barrier and the forces generated by external excitation decides the wetting mode in each case.

Over the years, the application of the electric field has been proven to be the most proficient way amongst all other means for manipulation of liquid volume in smaller scales.^22–28 The influence of external electric field on the transition of wetting state, over surfaces with rough topography is well addressed in the literature for sub-millimetre scales.^{18–21,29–31} But the scenario of wetting transition due electric potential is different at nano-scales; as the diameter of the droplet is smaller than the electric double layer. In case of pure water, the magnitude of electric double layer is in order of 10³ nm.³² Thus, for a size lesser that this, the electric field permeates through the whole droplet. Unlike macroscopic case, the polarization of the molecules in the entire droplet causes the electric energy to be accountable with the thermal energy of the nano-scale droplet–substrate system.^32,33 The polarization of molecules causes variation of hydrogen bonds in the interfacial regions. As a consequence, the interfacial tension is also altered with it.^32,34 A very few studies^34,35 have been targeted to the nano-scale wetting transition under the influence of external electric field. These molecular dynamic studies carry significant indication to the fact, that electric field has a definite influence on the wetting properties like state transition and shape deformation (of a nano-droplet) over pillar-arrayed surfaces at nano scales.

The study of Yen³⁴ reveals the influence of electric field on the wetting property of a nano droplet over textured surface. His analysis effectively illustrates the change in the shape and contact angle over the rough surface due to electric field. However, the transition of the wetting state is not addressed in this study. In the work of Yuan and Zhao,³⁵ they successfully analyzed the transition of wetting state on rough surface due to external electrostatic influence and produce significant contribution to understanding of the physics. However, in their study the whole surface (including the pillars of the surface texture) is charged. Textured surface are often prepared with grafted polymers,³⁶ and thus, the nano-pillars do not possess charge along with the base substrate upon electrification. At this scenario, variation in surface topography could lead the characteristics of the electric force driven wetting to a different direction. Thus, in the present study, we performed molecular dynamic simulations to analyze the wetting transition of nano-droplets over structured silicon surface under the influence of assigned charge on the base plate only. Surface morphology is varied to check its influence on the wetting characteristics. Both homogeneous and heterogeneous pillar-arrayed distributions are targeted and wetting dynamics is explained. Our study could aid the decisive factors for the development of micro and nano-fluidic devices.

Method

In the present study, extended simple point charge model (SPC/E)³⁷ is considered to simulate the nano-sized water droplet over textured surfaces. This model (SPC/E) considers three point masses (at the two oxygen and one hydrogen sites) with the OH bond distance of 1 Å. The charges at the hydrogen and oxygen sites are q_H = +0.4238e and q_O = −0.8476e respectively. The HOH angle is 109.47°. The inter-atomic forces are estimated based on Lennard-Jones (L-J)³⁷ and coulombic interactions. For the simulation of the liquid, the L-J energy and distance parameters are employed only for oxygen–oxygen interactions with ε_O–O = 0.1556 kcal mol⁻¹ and σ_O–O = 3.166 Å. The droplet consists of 3375 water molecules. The solid surface is considered to be composed of silicon atoms in diamond cubic structure with a lattice constant of 5.43. The L-J parameters for silicon atoms are assumed as σ_s–s = 3.408 Å and ε_s–s = 0.584 kcal mol⁻¹. The solid atoms are considered to be non-polar. These do not get charge due to polarization at the presence of the polar molecules like water. To analyze the effect of electric field, charge is assigned (ranging from +0.0005e to +0.018e) to the individual solid atoms of the base substrate during the simulation after equilibration. The solid molecules are kept frozen to minimize the computational expenses. Lorentz–Berthelot³⁸ mixing rule is employed to estimate the LJ interaction parameters between liquid and solid atoms. In the present system, the droplet is allowed to coexist with its saturated vapour. At nano-scale the entrapped gas (between pillar spacing) acts as a rarefied medium. Based on Avogadro's law, at standard temperature and pressure, the estimated value of the number of any gas molecule present inside the pillar spacing will be in fraction. Thus, following the earlier works,^39–45 the presence of the secondary gas medium is neglected in this nano-scale study. The open source molecular dynamic code, LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator)⁴⁶ is used to carry out the simulation at a constant temperature of 300 K. The time integration is performed by Verlet algorithm with a step size of 1 fs. We used Nose–Hoover⁴⁷ thermostat to keep the system temperature constant. SHAKE⁴⁸ algorithm is employed to constrain the geometry of the water molecules. The LJ interactions are truncated smoothly at a distance of 14 Å. Since, charge is assigned to the solid atoms during simulation to mimic the electric field; consideration of truncated electrostatic interaction only (neglecting the effect at the far field) can produce spurious results. Thus, in the present study, short range electrostatics is handled by coulombic interactions with a real space cut off of 11 Å and a particle–particle particle mesh solver (pppm)⁴⁹ is used for the long range interactions (to reduce computational cost). Periodic boundary condition is assigned to all the faces of the simulation domain. The vertical dimension of the simulation domain is kept high to minimize the interactions between the periodic images in the same direction.

Result and discussion

In the present study, the substrate is considered to be composed of diamond cubic silicon crystal. The simulation of the droplet over the smooth silicon substrate shows a contact angle of 86.96°, which is in a close agreement with the value (86.47°)⁵⁰ reported in the literature. The physical configuration of the system for the analysis of wetting transition is shown in Fig. 1. To study the interplay between the surface morphology and electrostatic influence, the height of the pillar-array and the spacing between two consecutive pillars are varied from 5.43 Å to 21.724 Å and 7 Å to 21 Å respectively. To produce electric fields, the atoms of the bottom plate are charged with +0.0005e to +0.018e. The equivalent electric field can approximately be estimated as E = Q/(2Aεε₀),³⁵ where Q and A are the total charge and area of the surface. ε and ε₀ are the dielectric constant and permittivity of water and free space respectively. Thus, the electric field is ranging from 0.0041 V nm⁻¹ to 0.1489 V nm⁻¹. The simulation is performed in two stages. Initially the molecules are stacked in a cubic structure and equilibrated to a hemispherical shape in a separate simulation. Then the droplet is placed on the main simulation domain (on pillared surfaces) and equilibrated. After the equilibration, charge is assigned to the bottom plate as shown in Fig. 1. As the surface of the pillars do not carry charges, the electro-mechanical force acts on the interfacial water molecules (near the solid surface) mainly governed by their remoteness from the charged plate. Thus the spacing and height of the pillars has a significant consequence on the wetting transition by electrostatic influence.

Fig. 1 Schematic illustration of the physical configuration considered for molecular simulation.

The state of wetting at different spacing (S) and heights (h) of the micro pillars is shown in Fig. 2 for a constant assigned charge (+0.003e). For low pillar height (h = 5.43 Å) the water molecules at the interface are closer to the charged electrode. Thus, less electro-mechanical force (hence a low charge concentration) is sufficient to achieve a Wenzel state (Fig. 2(a), (d) and (g)) irrespective of spacing. However, for the pillars with larger elevations, the transition becomes dependent on the pillar spacing.

Fig. 2 Modes of wetting at different spacing and heights of the pillar for a charge concentration of +0.003e.

The scenario can be comprehended by analyzing the time averaged density contour plot (shown in Fig. 3) of the droplet for different column spacing. When the pillars are arranged at higher density, the three phase contact line got pinned at the top surface of the pillar posts and liquid–gas interface sagged down at the gap between two consecutive columns. Due the sagging of the interface, it gets closer to the bottom plate. To get into a more clear picture the path lines of the representative molecules from different zones is depicted in Fig. 4. The initial positions of the molecules are shown by spherical markers with corresponding colours. The pathlines near the solid surface are conglomerated together, proving a low diffusion due to the constraint imposed by the solid surface. However, the unrestrained water molecules far from the pillar surface quickly dispersed towards the bottom plate creating larger streaks (upon application of electric field). Thus, relatively lesser electric potential is required to bring down the unimpeded water molecules from the sagged region in cases with larger pillar spacing (as shown in Fig. 2(h) and (i)). For denser spacing of the pillars, the mechanism of the wetting transition alters from the earlier situation. In this case, the three phase contact line got unpinned from the top surface of the nano pillars and slides down along its surface under the external electric field. This requires higher electrostatic force for the collapse of the air pocket formed by the droplet and surface morphology. Thus, for a low charge concentration (+0.003e) the nano droplet remains at Cassie–Baxter state on denser pillar forest as shown in Fig. 2(b) and (c).

Fig. 3 Time averaged (for 30 000 time steps) density contour of the droplet over pillar structure; (a) h = 16.29 Å and S = 5 Å, (b) h = 16.29 Å and S = 18 Å.

Fig. 4 Trajectory of the representative molecules from different zones of the droplets during the transition of the wetting state.

The contour map of the electrostatic potential around the droplet–surface system at the different stages of wetting transition is depicted in Fig. 5. The potential distribution elucidated in the figure is averaged over 20 000 time steps and in a scale of kT/e. Where, k and e are Boltzmann constant and charge of electron, respectively. T is the temperature of the system (300 K). In the presence of electric field, the bipolar water molecules try to re-orient themselves. As a consequence the electric field gets screened at the periphery of the droplet and the bulk became less affected from it. Once the peripheral water molecules penetrated inside the grooves, the high charge at the oxygen and hydrogen sites (compared to the charged silicon sites) dominates on the local electric field. Toward the end of the transition, the potential decreases at the vicinity of the charged substrate. Thus, the resultant electrostatic force on the water molecules is diminished by a substantial amount at this region. It leads towards a state of saturation. This is in accordance with the earlier study by Yuan and Zhao.³⁵ They have observed and analyzed the saturation of the electrowetting effect over fully charged pillared surfaces.

Fig. 5 Time averaged (for 20 000 time step) contour map of the electrostatic potential. The scale is in the units of kT/e. The time averaging is done around t₁, t₂ and t₃, where, t₁ < t₂ < t₃. Charge concentration, spacing and pillar height are +0.007e 10 Å and 16.29 Å.

Fig. 6 depicts the variation threshold charge (q_t) required for wetting transition as a function of pillar spacing for different column heights. Development of functional relationship to predict transition in terms of height, spacing and width requires further analysis of the system. To analyze the variation of threshold charge, a simplistic expression of energy barrier per unit area (underneath the wetted area) can be derived. However, it is worthwhile to mention that droplet (small liquid volume) undergoes significant deformation in the presence of external electric field. Thus, a full scale model of total interfacial energy considering the droplet deformation can provide useful insight to the prediction of the threshold electric parameter for wetting transition over rough surfaces. But at the same time this model will be quite complicated involving near wall and far wall dynamics together along with coupling. As way out, in this work we only consider the localized analysis and does not consider change of drop shape. In Fig. 6, we have shown cartoons very near to the pillared surfaces and omitted the free surface of the droplet which undergoes shape change. In this microscopic domain, we analyzed the energy barrier per unit area.

Fig. 6 Variation of minimum charge density required for wetting transition as a function of pillar spacing for different pillar heights.

The total interfacial energy underneath the liquid surface can be expressed as:

E = A_slγ_sl + (A − A_sl)γ_lg + (A − A_sl + A·d·4·H·a)γ_sg (1)

where, A is the total apparent contact area underneath the liquid surface. γ_sl, γ_lg and γ_sg are the solid–liquid, liquid–gas and solid–gas interfacial energy per unit area. H and a are the height and width of the nano-pillars. d is the density of the nano-pillars (d can be calculated as considering the inset of Fig. 6).

For smaller spacing of the pillars, the sagging of the liquid–gas interface can be neglected. Thus, considering the pillars to be rectangular parallelepipeds, the interfacial energy per unit area below the liquid surface can be expressed as

where, ϕ is the ratio of the solid–liquid interfacial area to the total apparent contact area underneath the droplet. After a penetration of the wetting front by a distance h, the estimation of the interfacial energy per unit area will be

Thus, at Wenzel state (h = H) the interfacial energy can be expressed as

As the interfacial energy between solid–liquid interface is higher than that of solid–vapour boundary, the energy will rise with the penetration of the wetting front. The maximum increase in the energy can be considered as the energy barrier.⁵ From the above expression the energy barrier can be estimated as

It can be noted from eqn (5) that, for a constant height the energy barrier is proportional to 1/(a + S)². Thus, the variation of threshold charge for wetting transition is steeper at low spacing. At low pillar height the liquid–gas interface remain close to the bottom plate. Thus, the requirement of charge for wetting transition is less at beginning. Moreover, a little increase in the column spacing, the hanged down portion of the interface touches the valleys of the surface topography without the application of charge. This makes the curve to remain steeper for low pillar height. For higher pillar altitudes, after a certain spacing the hang down of the liquid–gas interface (between pillar spacings) has an definite influence on the transition. As a result, the curve for h = 16.29 Å, in Fig. 6, has become flatter at larger S. Due to the dipolar property of the water molecules, electric field has a strong influence on its orientation. Thus, the internal structure of the droplet changes with the electric field. Energy barrier is larger for the pillar forest with higher density. Thus, higher electric field is required to suffice the transition. This large electric field also affects the bulk of the droplet; it got elongated in the direction of the electric field. As a consequence the average density of the droplet reduces. This, in turn, increases the potential energy of the droplet. Fig. 7 shows the variation of the potential energy at the threshold charges (q_t) with the corresponding pillar spacing. The potential energies at the Cassie states (in absence of charge, q = 0) for different pillar spacing are also plotted. For q = 0, potential energy does not vary much with the gap between the columns. However, it increases with the reducing spacing owing to the lesser density at high electric field. For a pillar height of 10.86 Å, the droplet is at Wenzel state without the assignment of charge near 10 Å spacing. To analyze the effect of electric field further on the droplet shape, we increased the charge after the droplet reached at its Wenzel state. Fig. 8 depicts the shape of the droplet on a surface where the texture is created by pillars with 8 Å spacing and 10.86 Å height. As the magnitude of the charge increases, the droplet changes its mode of wetting from Cassie to Wenzel state. Eventually, its length got increased with higher charge. To dig down further, the electric potential distribution (shown at the inset of Fig. 8(d)) is analyzed inside the computational domain. The potential distribution reveals that, even at high surface charge, the bulk of the droplet is less affected due to the local electric field produced at the presence of high charge at the oxygen and hydrogen sites. The water molecules at the interface are less constrained compared to the bulk. Thus, these molecules can reorient themselves easily according to the external electric field. The polarization of the interfacial molecules creates dissimilarity in the electric potential distribution with the bulk. This produces interfacial electric stress. The competition between the electric stress and the surface tension determines the shape of the droplet. Due to the higher electrostatic force near the solid surface (due to the gradient in the electric field in the lateral direction), the water molecule spreads inside the pillar forest (Fig. 8(d)). This causes a reduction of the contact angle at higher electric field. At high electric field dissimilarity in the stress condition is created at the top of the droplet. Thus, the interface got stretched out in the direction of electric field (shown at the inset by white lines), resulting a spike like structure (Fig. 8(d)).

Fig. 7 Variation of the potential energy at threshold charge (required for wetting transition) as a function of pillar spacing.

Fig. 8 The shape and state of wetting of the droplet at different charge density (H = 10.86 Å and S = 8 Å); (a) q = +0.002e, (b) q = +0.0045e, (c) q = +0.007e, (d) q = +0.014e.

From the forgoing discussion it is evident that, in nano-scale the influence is much profound compared to the macroscopic system. In case of sub-millimeter droplet, the electric field gets screened by the electric double layer³² but the local influence of electric field is still present. The body forces due gravity and convective motion of the bulk also have significant contribution to the system at this scale. Thus, the energy due to polarization is less significant compared to the thermal energy. The ratio of affected interfacial molecules to the bulk molecules is much higher for nanoscopic system. Hence, the changes in the structure of the interfacial water molecules⁵¹ have significant consequences.

The practical implementations of engineered surfaces utilized in several microfluidic applications may encounter non-homogeneity in the surface morphology. Thus, the analysis of the wetting transition on the surfaces with diversified topography is certainly relevant in the present study. Here, the non-homogeneity in the surface texture is constructed either by varying pillar height (keeping the pillar spacing constant) or by altering the pillar spacing (for same pillar height). Fig. 9(a) and (b) represent the non-homogeneous surfaces created by varying the pillar heights from 5.431 Å to 21.724 Å. Once the droplet is placed over these surfaces, the water molecules penetrate into the grooves at some portion of the droplet due to the low local roughness or because of sagging of the interface (Fig. 9(a)). Thus, it exhibits a mixed mode of wetting at the absence of external influence. The application of the electric field alters the wetting mode. The different stages of wetting during the transition are shown in Fig. 10 and 11 for a concave and convex heterogeneity respectively. In this case, after the initial penetration locally at the places with low roughness, the water molecules moves in the lateral direction inside the forest (shown in the inset of Fig. 10(b)) of pillars due to the van der Wall and electrostatic attraction forces between the water molecules. This phenomenon favours the electromechanical forces to fill the air pockets in the grooves.

Fig. 9 Wetting over non-homogenous surface morphology constructed with constant pillar spacing (8 Å); (a) over a heterogeneity of concave shape (number of water molecules = 7650), (b) over a convex morphology (number of water molecules = 17 640).

Fig. 10 Evolution of the wetting front with time over a concave shaped non-homogeneous surface texture at q = +0.009e; (a) t = 0.086 ns, (b) t = 0.191 ns, (c) t = 0.6 ns.

Fig. 11 Evolution of the wetting front with time over a convex shaped non-homogeneous surface morphology at q = +0.010e; (a) t = 0.02 ns, (b) t = 0.084 ns, (c) t = 0.528 ns.

Fig. 12 depicts traversal of the wetting front on a textured surface with an uneven distribution of pillars. Here, the non-homogeneity is created by increasing the pillar spacing gradually from 8 Å to 14 Å in a direction left to right. The mix mode wetting can be observed in Fig. 12(a). The wetting transition, similar to the earlier case is observed upon application of charge at the bottom plate.

Fig. 12 Different stages of wetting over a non-homogeneous surface texture created by varying column gapping at q = +0.0055e (number of water molecules = 20 000); (a) t = 0.0 ns, (b) t = 0.31 ns, (c) t = 0.50 ns.

Conclusions

The influence of external electric field on wetting transition of nano-sized droplets over variable nano-post configurations is analyzed by molecular dynamic study. The study unravels that, instead of the remoteness of the molecules from the base plate, the sagging of the liquid–gas interface between two consecutive pillars is also a determining factor of the required charge for wetting transition. The study also shows a large variation of threshold charge with nano-column spacing for denser pillar forest. At high charge (required for wetting transition at low pillar spacing), formation of a liquid bridge is observed due to the alignment of the bipolar water molecules with the electric field. Wetting transition on non-homogeneous surface texture reveals a traversal of a lateral wetting front inside the nano-grooves (at the application of external electric field) due to intermolecular forces, which promotes the transition further. The study can be used as a decisive factor for designing textured surfaces at nano-scale applications.

Acknowledgements

Financial support of this work was received from Department of Science and Technology, India (Grant No. SB/FTP/ETA-84/2013).

Notes and references

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