Experimental observations of drops of water with aniline dye softly located or impacting onto balsa wood substrates were used to elucidate the effect of an in-plane electric field (at a high voltage of 10 kV applied) on drop behavior. The top and side views were recorded simultaneously. The short-term recordings (on the scale of a few ms) demonstrated a slight effect of the applied in-plane electric field. In some trials, a greater number of finger-like structures were observed along the drop rim compared to the trials without voltage applied. These fingers developed during the advancing motion of the drop rim. The long-term recording (on the scale of ∼10 s) was used to evaluate the wettability-driven increase in the area-equivalent radius of the wetted area. These substrates had grooves in the inter-electrode or the cross-field directions. The groove directions affected the wettability-driven spreading and imbibition. The wettability-driven spreading in the long term was a much more significant effect than the effect of the electric field, because the imbibition significantly diminished the drop part above the porous surface, which diminished, in turn, the electric Maxwell stresses, which could stretch the drop. A simplified analytical model was developed to measure the moisture transport coefficient responsible for liquid imbibition in these experiments. Furthermore, the phase-field modeling of drops on balsa was used to illustrate how a change in the contact angle from hydrophobic to hydrophilic triggers drop imbibition into balsa wood.

The behavior of liquid drops on solid surfaces has been extensively studied in recent years. In particular, surfaces with different mechanical, chemical, and electrical properties as well as various levels of roughness, topographies, porosity, and wettability were employed. In practical applications, surface porosity and wettability are of great interest in spray painting, adhesion, spray coating, and micro-fluidics and comprise one of the most critical aspects of drop dynamics on solid surfaces during drop impact and spreading.1–4 Furthermore, an attractive approach to the control of surface wettability is by means of electrowetting (EW) or electrowetting-on-dielectric (EWOD), which do not involve the use of additional chemical agents or surface treatments.5–8 The physical reason for the electrowetting phenomenon is in the redistribution of ions in liquid, which results on the macroscopic level in a decrease in the contact angle at the triple line (i.e., an increase in wettability), formally described by the Young–Lippmann equation.3,5,7–10 Many studies have focused on drops on dry metallic surfaces, albeit drop behavior facilitated by electrowetting on dielectric surfaces also was tackled in recent years.

The experimental observations of the impact and evolution of aqueous liquid drops on different dielectric surfaces such as Teflon (stretched and unstretched), polypropylene, and parafilm, subjected to the in-plane electric field, revealed partial suppression of splashing and drop rebound (in some cases at higher electric field strengths, complete rebound suppression was observed).11,12 Moreover, drops of colloidal suspensions have also been studied on dielectric substrates without and with in-plane DC electric fields acting on the solid surface. Paints are common and widely used suspensions of different pigments (colloidal particles) in a wide range of solvents. They reveal different types of behavior when an in-plane electric field is applied.13–15 For instance, turpentine oil with suspended aniline dyes and water with water-soluble aniline dyes revealed fingering accompanied by their spreading on dielectric substrates facilitated by an in-plane electric field at different applied voltages. Additionally, the application of an in-plane electric field on dielectric surfaces with a different wettability increases the painted area compared to the case without voltage applied.14 Such an important result has direct implications for a painting of large surface areas. Furthermore, at least one type of solvents, silicone oils, does not react to an in-plane electric field because of an insufficient number of ions inside. Yet, the addition of such a pigment as titanium dioxide particles facilitates the stretching of silicone oil drops along the in-plane electric field lines, and the painted area can be gradually increased in time.16,17 Therefore, the application of an in-plane electric field facilitates drop spreading on different dielectric surfaces.

Despite extensive research of drop spreading on metallic and dielectric surfaces, including those without and with nanotexture,2,18,19 there is a limited number of works on drop impact and spreading on wood surfaces. Several recent works deal with impacts of drops of pure solvents onto wood surfaces and explore the effect of surface roughness, fluid properties, and impact velocity.20,21 Accordingly, an increased roughness resulted in violent splashing of impacting water drops with a greater likelihood of exhibiting recoil on rougher wood substrates implying a reduced contact between the liquid film and the wood surface. Therefore, a stronger influence of the capillary forces that drives the recoil stage is expected.21 Moreover, experimental results demonstrate that enhanced roughness of wood surfaces leads to a greater degree of rim deformation on spreading liquid lamella and a decrease in the number of fingers resulting from it.22 The data also revealed that an increase in the impact height affected the drop dynamics resulting in an increase in the number of secondary drops, and in their corresponding splash velocities at a fixed time moment after impact.23 Similarly, water drops impinging on different wood materials such as Jatoba and Shorea hardwood reveal a clear difference between the two wetting diameter/spread factors, and, hence, might hint that hydrophobicity of both surfaces could be different, despite similar roughness values. Hence, hydrophobicity of wooden materials could also play a role in the liquid drop spreading after impact onto such surfaces.24,25

Experimental, theoretical, and numerical studies conducted in the present work aim the effect of an in-plane electric field on the spreading of drops of aqueous suspensions of pigment particles. Multiple experimental observations reveal the formation of a number of pronounced finger-like structures at the applied voltage of 10 kV. Additionally, long-term recordings reveal the presence of the pore impregnation resulting in a diffusion-like spreading causing a significant increase in the painted area of wood surface under the action of the in-plane electric field. Sections II, III, and IV describe the experimental setup, theoretical estimates, and numerical methodology, respectively, and Secs. V and VI present the experimental and numerical results and discussion of the spreading of liquid drops on wooden substrates subjected to strong in-plane electric fields. Conclusions are drawn in Sec. VII.

Drop impact and spreading experiments were conducted using a balsa wood substrate (Bright Creations Balsa Wood Sheets) acquired from Walmart. Drops of water mixed with water-soluble gold oak aniline dye (2 wt. % concentration) were released by gravity from a needle tip fixed at a height of h = 12.5 cm and impacted at the middle of the dielectric wooden substrate, which was subjected to a DC electric field sustained by two in-plane powered electrodes. In all the cases studied experimentally, the electric field applied to the balsa wood substrate had been turned on before drop impact. Then, the drop impact dynamics and spreading were recorded at the rate of 1000 and 100 fps (to capture the short-term and long-term evolution, respectively) by two high-speed cameras, Phantom V210 and Phantom Miro 4, used for the side and top viewings, respectively. The setup used in the present experiments is sketched in Fig. 1. Balsa wood substrates and the in-plane electrodes made of a copper tape were overlaid over a glass slide. The distance between the copper electrodes (cf. Fig. 1) was kept constant at 1.5 cm in all experiments conducted. The balsa wood substrate pieces of 2 cm (length) × 2 cm (width) with a thickness of 1.5 mm were overlaid on top of the two in-plane copper electrodes, and the central 1.5 cm of it was used as the working space, with drops aiming right at the middle. As a precaution, Teflon tape was used to insulate the copper electrodes from liquid drops and prevent a short circuit when performing the experiments, especially at higher applied voltages. Hence, the drops were only in contact with the wooden substrate after impact and spreading.

FIG. 1.

Schematic of the experimental setup used in the experiments on drop impact and spreading with and without in-plane electric fields. The balsa wood substrates had surface grooves (probably resulting from the manufacturing process). Here and hereinafter, the substrates with grooves oriented along the electric field lines are denoted as wood substrates I, whereas those with grooves oriented across the electric field lines are denoted as wood substrates II.

FIG. 1.

Schematic of the experimental setup used in the experiments on drop impact and spreading with and without in-plane electric fields. The balsa wood substrates had surface grooves (probably resulting from the manufacturing process). Here and hereinafter, the substrates with grooves oriented along the electric field lines are denoted as wood substrates I, whereas those with grooves oriented across the electric field lines are denoted as wood substrates II.

Close modal

The glass slide with the balsa wood substrate (cf. Table I) on top of it equipped with the copper electrodes was mounted on an adjustable platform, which could be controlled in three directions aiming the drop impact at the inter-electrode center by fine-tuning in the horizontal direction (Fig. 1). One of the electrodes was connected to a positive high-voltage DC source (Glassman High Voltage Inc., EL40P1, custom built with a 0–20 kV range). The other copper electrode was grounded.

TABLE I.

Properties of balsa wood used in the experiments. Various pore shapes in balsa wood with equivalent sizes ranging between 7.41 and 52.86 μm were observed.

MaterialDensity ρbalsa (kg/m3)Relative dielectric permittivity εElectrical conductivity σ (S/cm)
Balsa wood 120–150 1.22–1.37 10−18–10−16 
MaterialDensity ρbalsa (kg/m3)Relative dielectric permittivity εElectrical conductivity σ (S/cm)
Balsa wood 120–150 1.22–1.37 10−18–10−16 

Water mixed with water-soluble gold oak aniline dye at a 2 wt. % concentration was used as a working fluid in the present experiments. A laboratory syringe pump (NE-1000 Series) supplied the liquid to a 90°-bent 27-gauge needle at a flow rate of 4 ml/h for the short-term and long-term recordings. The average drop diameters were ∼2.50 mm when a 27-gauge needle was used. The drops detached from the tip of the needle and fell onto the balsa wood substrate due to gravity, and thus, the impact velocity was controlled by the height of the needle tip h. The average impact velocities recorded by the side-view high-speed camera were ∼1.50 m/s, values slightly lower (due to the effect of the air drag) than the estimate of 2gh=1.57m/s, where g is the gravity acceleration. Only drop impact normal to the substrate was explored in all cases studied.

For the side and top views, a back lighting (LED lamp) was employed to illuminate the drop. For the side view, the high-speed camera was aligned at the same level as the balsa wood substrate. The images obtained from the high-speed recordings were analyzed in Adobe Photoshop, MATLAB, and ImageJ. All measurements and experiments were performed at room temperature.

For the short-term and long-term drop impact recordings, six and eight consecutive drop impact trials were conducted in each of the experiments without and with the applied voltage, respectively. It should be emphasized that the datasets (snapshots) reported here are fully representative of all cases, because similar drop evolution and spreading patterns were observed in the other corresponding trials in each case. Multiple repetitive results reported here were acquired for statistical purposes. The material properties of the balsa wood substrates used in this work are listed in the following Table I.

Additionally, the optical and SEM observations of balsa wood reveal various pore sizes and shapes (cf. Fig. 2). It should be emphasized that optical micrographs of balsa reveal a diffuse-porous microstructure, with certain uniformity in the distribution of cells throughout the grain cross section, whereas scanning electron micrographs in Ref. 26 and here in Fig. 2(d) show that the grains in balsa wood are highly oriented in the longitudinal direction.

FIG. 2.

Optical micrographs of the balsa wood substrates at: (a) 4 × and (b) 20 × magnification, and SEM micrographs of the balsa wood substrates at: (c) 100 × and (d) 500 × magnification.

FIG. 2.

Optical micrographs of the balsa wood substrates at: (a) 4 × and (b) 20 × magnification, and SEM micrographs of the balsa wood substrates at: (c) 100 × and (d) 500 × magnification.

Close modal

If an axisymmetric drop is spreading by wettability on a perfectly wettable substrate without any porosity and liquid imbibition, in the remote asymptotics, when drop is already significantly flattened, the dependence of its footprint radius a on time t is found using the Hoffman–Voinov–Tanner law, from the following differential equation:27 

dadt=0.0107σμ(4Vπ)31a9,
(1)

where σ denotes the driving force, the surface tension of the solid–gas interface, which is equal to the surface tension of liquid for perfectly wettable surfaces; μ is the liquid viscosity; and V is the drop volume.

If one assumes V = const (i.e., no imbibition into the substrate), Eq. (1) is integrated as

a=[0.107σμ(4Vπ)3t]1/10t1/10.
(2)

It should be emphasized that Eq. (2) describes a tremendously slow spreading compared, for example, with the one which stems from liquid imbibition in the substrate pores. Indeed, a self-similar solution for moisture content ζ as a function of depth into the substrate z and time t is given as28 

ζ=1erf(z2αmt),
(3)

where z = 0 corresponds to the substrate surface (and 0z in the substrate), and αm is the moisture transport coefficient.29,30 It should be emphasized that the moisture transport coefficient is a lumped characteristic of liquid (typically, water) imbibition due to the pore wall wettability, as well as the diffusion, adsorption, desorption, and precipitation of vapor (water vapor) inside the pores.29,30

Using Eq. (3), one can estimate the rate of liquid suction from the footprint of the spreading drop into a porous wettable substrate as

Q=πa2αmζz|z=0=παmta2.
(4)

Accordingly, the drop volume balance reads

dVdt=παmta2.
(5)

When both wettability-driven spreading and the imbibition into a wettable porous substrate take place, drop evolution is described by the system of two inter-related ordinary differential equations (1) and (5) for two unknowns a and V. One can expect that the wettability-driven change of a(t) according to Eq. (1) with volume V decreasing in time is much slower even than that of Eq. (2). Therefore, Eq. (5) could be integrated implying a slow quasi-static variation of a(t), which yields

V=V02a2παmt,
(6)

where V0 is an initial drop volume.

On the other hand, the Hoffman–Voinov–Tanner law implies that the current footprint radius and contact angle θ are related as

a3θ=4Vπ,
(7)

which yields together with Eq. (6) the following expression for the moisture transport coefficient

αm={V0π[a3(t)θ(t)]/4[a(t)]2}214πt.
(8)

Note that water wets wood, which is accounted by Eq. (7).

Using the drop side-view snapshots, Eq. (8) will be used below to determine the moisture transport coefficient from the experimental data, and to establish to what extent an approximate Eq. (8) guarantees its constancy in time.

Note also that purely wettability-driven spreading inside a porous wettable substrate is expected to follow the diffusion-like law2,18,19

a=const(αmt)1/2,
(9)

which describes a much faster spreading than the one of Eq. (2).

An axisymmetric drop model is considered here, and thus, a cylindrical coordinate system is used for the phase-field modeling (PFM), where a drop impacts normally onto a porous balsa substrate (the bottom layer in Fig. 3). The main governing equations, the computational domain, and the boundary conditions required for the simulations are outlined in Fig. 3 and listed in this section below. Based on the experiment (cf. Fig. 2), the equivalent pore sizes in the balsa wood ranging between 7.41 and 52.86 μm were adapted in the model to reproduce the experimentally observed geometry of the substrate. In the PFM, various pore shapes and sizes closely matching this experimental observation were assumed (in the 8–52 μm range). Also, in the numerical modeling, the thickness of the porous layer (d) was assumed to be ∼400 μm.

FIG. 3.

Median cross section of an axisymmetric PFM computational domain, the main governing equations, and the necessary boundary conditions. The red colored circle denotes liquid drop approaching the substrate, whereas air is shown in blue color. The porous balsa substrate is on the bottom, where a white space indicates porous structure. The pore space is implied to be occupied by air (blue). The insert in the right upper corner shows a zoom-in of the balsa substrate, illustrating the characteristic sizes of the porous area. The pore space varies between 8 and 52 μm, which is in the experimental range (cf. Fig. 2), albeit in the model the pores are axisymmetric, whereas in real wood, they are obviously not.

FIG. 3.

Median cross section of an axisymmetric PFM computational domain, the main governing equations, and the necessary boundary conditions. The red colored circle denotes liquid drop approaching the substrate, whereas air is shown in blue color. The porous balsa substrate is on the bottom, where a white space indicates porous structure. The pore space is implied to be occupied by air (blue). The insert in the right upper corner shows a zoom-in of the balsa substrate, illustrating the characteristic sizes of the porous area. The pore space varies between 8 and 52 μm, which is in the experimental range (cf. Fig. 2), albeit in the model the pores are axisymmetric, whereas in real wood, they are obviously not.

Close modal

The fundamental concepts of the PFM used to study drop impact on a porous balsa substrate were previously reported;12,31,32 thus, only a brief overview is given in this paper. The modeling domain consists of three phases, that is, the surrounding air, the liquid drop, and the porous balsa wood substrate. The PFM is a physically consistent method based on the thermodynamic treatment of the phase interfaces, where the free energy functional [F(c¯)] summarizes different types of energy contributions existing in the system12,31,32

F(c¯)=Vfch(c¯)+fex(c¯)dV.
(10)

In Eq. (10), the term fch(c¯) describes the chemical free energy density, with the phase-field variable c¯. This variable is used to model the two phases, with c¯=1 denoting the surrounding air phase and c¯=1 for the liquid drop phase. The function fex(c¯) denotes the excess free energy due to the inhomogeneous distribution of the volume fraction (c¯) in the interface region. The expressions for fch(c¯) and fex(c¯) are taken from Ref. 12. The Cahn–Hilliard (CH) equation, combined with the advective term, is used to express the evolution of the phase-field variable c¯.31 The phase-field equation is coupled to a fluid flow, that is, the Navier–Stokes (NS) equation, at the drop-air interface by means of the two-phase flow and phase-field coupling feature.33 The balsa-liquid and balsa-air interface is modeled by imposing the wetting boundary condition as described by Eq. (14) below. An improved conservation of the integral of the phase field variable (c¯) is obtained using the nonconservative form, providing that the discretization order of the phase field variable is equal to or lower than the order of the pressure. Thus, the best way is to use the non-conservative form initially and reduce the percentage of mass loss and then implement the conservative form. The phase-field mobility or the phenomenological mobility,34 which determines the interface relaxation time and the timescale of diffusion in the CH equation, is typically considered to be proportional to the square of the interface thickness.35,36

For the simulation of the two-phase fluid flow, the NS equations are used and are coupled with the CH equation. The mass balance (continuity) equation and the momentum balance equation are given, respectively, as follows:

u=0,
(11)
ρut+uu=p+μu+uT+Fst+ρg,
(12)

where p is the pressure, ρ is the density of the fluid, μ is the dynamic viscosity, and g is the gravity acceleration. The body-force term, Fst, is associated with the surface tension force and is used to couple the NS and the CH equations31 

Fst=φc¯,
(13)

where φ is the chemical potential (F(c¯)c¯). This surface tension force term arises from the compositional gradient at the interface and modifies the general NS equations to account for the two-phase flow effects in the PFM method.

The contact line dynamics in the phase-field formulation is imperative and several approaches in the prior literature have been proposed to study it.35,37–39 For the advective CH equation, the wetting (contact angle) boundary condition is expressed using the geometrical formulation, where for a given prescribed contact angle, θs, the wetting condition is given by38 

nεintc¯=εint cos(θs)c¯,
(14)

where n is the normal to the surface and εint is the interface thickness.

In order to solve for the chemical potential, the wetting boundary conditions in Eq. (14) are used. The wetting boundary conditions dictate the water-balsa interaction. The geometrical formulation is used here, as proposed in Ref. 38. The paint/dye particles/molecules do not participate in such interaction directly. Due to the fact that the PFM is the mean-field approach, individual particles inside a drop are not resolved. However, the change in the intensive properties of water when particles are added is taken into account. If additions lead to a change in viscosity, density, and surface tension, then the corresponding intensive properties are changed in the PFM. In this way, the effect of paint/dye particles/molecules participation in water–balsa interaction is taken into account.

Additionally, it is required to use the non-penetration boundary condition at the solid surface, while solving the CH equation

nφ=0.
(15)

Equation (15) implies that the chemical potential flux does not penetrate through the solid porous surface. At the solid porous surface, the no-slip boundary condition is applied

u=0.
(16)

The open boundary conditions are used at all other boundaries. This is implemented in the model by imposing a pressure at the free surface, that is, at the left and top boundary, while keeping the normal outflow velocity zero.

The model is simulated using the COMSOL Multiphysics software version 5.5, employing the computational fluid dynamics (CFD) laminar flow and two-phase phase-field modules. The adaptive mesh algorithm with the gradient of the PFM variable as the error indicator and the adaptive time stepping (backward differentiation formula) are used. The phase-field variable c¯ varies only across the interface between the liquid drop and air; thus, it is suitable to be used as an error indicator during the computation. While using the adaptive meshing, both longest edge refinement and general modification methods are considered. The MUMPS40 solver, which is used to solve a system of linear equations for a finite element problem, is used to solve the initialization stage and the transient stage.

High-speed video recordings of water with aniline dye on balsa wood revealed wettability-driven drop spreading accompanied by the imbibition into the porous substrate. The recorded evolution of the contact angle as a function of time on the balsa surface is depicted in the side-view images in Fig. 4. It should be emphasized that additional trials revealed a similar behavior of wettability-driven spreading and imbibition. Accordingly, the images presented in Fig. 4 are representative of the entire set of the experimental data.

FIG. 4.

Wettability-driven spreading and imbibition a drop of water with aniline dye on balsa wood substrate I. The drop was softly placed on the surface, rather than dripped. The top row of the images depicts the side view, whereas the bottom row of the images depicts the top view. Note that the side-view camera was positioned at the same level as the wood surface. The top-view images might give impression of a tilted camera (which was not) because of the light reflection from the back of the adjustable platform equipped with the glass slide (and the wood substrate and copper electrodes on top of it).

FIG. 4.

Wettability-driven spreading and imbibition a drop of water with aniline dye on balsa wood substrate I. The drop was softly placed on the surface, rather than dripped. The top row of the images depicts the side view, whereas the bottom row of the images depicts the top view. Note that the side-view camera was positioned at the same level as the wood surface. The top-view images might give impression of a tilted camera (which was not) because of the light reflection from the back of the adjustable platform equipped with the glass slide (and the wood substrate and copper electrodes on top of it).

Close modal

The data acquired from the high-speed recordings reveal the evolution of the contact angle of a liquid drop spreading (and being imbibed) on the balsa surface as a diminishing function of time (cf. Fig. 5). The drop images also reveal that the height of the spreading drop is decreasing in time (cf. the images in Fig. 5).

FIG. 5.

Contact angle θ of a drop of water with aniline dye on balsa wood substrate spreading (and being imbibed) measured as a function of time. The drop was softly placed on the surface.

FIG. 5.

Contact angle θ of a drop of water with aniline dye on balsa wood substrate spreading (and being imbibed) measured as a function of time. The drop was softly placed on the surface.

Close modal

Wettability-driven spreading of a drop of water with aniline dye accompanied by its imbibition into the porous balsa substrate also reveals the evolution of the effective footprint radius a as a function of time depicted in Fig. 6 (without an electric field applied). The drop spreads over the surface and the painted area increases as attested by the top-view high-speed camera imaging (cf. Fig. 5). The dependence of the effective (area-equivalent) radius of a drop of water with aniline dye a can be fitted by the equation a=C1t+C2, where C1 and C2 are constants (cf. Fig. 6). It should be emphasized that a is the area-equivalent radius, because the painted areas are not circularly symmetrical.

FIG. 6.

Area-equivalent radius a of a drop of water with aniline dye spreading (and simultaneously being imbibed) on balsa wood substrate. The drop was softly placed on the surface. The experimental data are shown by the symbols, whereas the line corresponds to the equation a=0.4863t+0.0002.

FIG. 6.

Area-equivalent radius a of a drop of water with aniline dye spreading (and simultaneously being imbibed) on balsa wood substrate. The drop was softly placed on the surface. The experimental data are shown by the symbols, whereas the line corresponds to the equation a=0.4863t+0.0002.

Close modal

Using the data from Figs. 5 and 6, the values for the moisture transfer coefficient are found using Eq. (8). The corresponding results are plotted in Fig. 7.

FIG. 7.

Moisture transfer coefficient measured as a function of time using the data for a drop of water with aniline dye on balsa wood substrate. The drop was softly placed on the surface.

FIG. 7.

Moisture transfer coefficient measured as a function of time using the data for a drop of water with aniline dye on balsa wood substrate. The drop was softly placed on the surface.

Close modal

The data in Fig. 7 reveal that during the initial time interval up to ∼8 s, the moisture transfer coefficient values vary, because the drop shape still had not achieved the remote-asymptotic regime, implied in Eq. (8). However, at t > 8 s, an approximately constant value of the moisture transport coefficient αm of the order of 10−6 cm2/s is measured, which is in the value range reported in the literature for several different types of wood including balsa. Specifically, Refs. 41–45 report αm in the 10−7 to 10−4 cm2/s range, and in particular, Refs. 44 and 45 report αm in the 10−7 to 10−6 cm2/s range.

Figure 8 illustrates the recorded short-term evolution and spreading of drops of water mixed with aniline dye dripped from 12.5 cm after they impacted onto the dielectric wooden surface at the applied voltages of 0, 7, and 10 kV (rows 1–3, respectively). The charge relaxation time of water expressed in CGS units is τC=ε/4πσ, where ε is the relative dielectric permittivity and σ is the electrical conductivity of the liquid. For water, τC=0.71×104ms. Note that the conversion of the SI unit for the electrical conductivity (S/cm) into the corresponding Gaussian (CGS) unit is done with the following factor: 1 S/cm = 9×1011 s−1.

FIG. 8.

Top views of the short-term evolution and spreading of drops of water with gold oak aniline dye. Drops impacted onto the dielectric balsa wood substrate I without an applied voltage at 0 kV (top row), and with the applied voltages of 7 and 10 kV corresponding to the electric field strengths of 4.6 × 105 and 6.6 × 105 V/m, respectively (middle and bottom rows, respectively). The Weber number We = 77.3, the Ohnesorge number Oh = 0.002, and the composite parameter K = 928.5. The electric capillary numbers are CaE = 0.415 and CaE = 0.847 for 7 and 10 kV, respectively. The left-hand side electrode is the anode, and the right-hand side one is grounded (the cathode). The time moments are listed on top of the columns. The average diameter of each drop before the impact was D 2.50 mm, and the impact velocity was V0 1.50 m/s. Scale bar is 3 mm.

FIG. 8.

Top views of the short-term evolution and spreading of drops of water with gold oak aniline dye. Drops impacted onto the dielectric balsa wood substrate I without an applied voltage at 0 kV (top row), and with the applied voltages of 7 and 10 kV corresponding to the electric field strengths of 4.6 × 105 and 6.6 × 105 V/m, respectively (middle and bottom rows, respectively). The Weber number We = 77.3, the Ohnesorge number Oh = 0.002, and the composite parameter K = 928.5. The electric capillary numbers are CaE = 0.415 and CaE = 0.847 for 7 and 10 kV, respectively. The left-hand side electrode is the anode, and the right-hand side one is grounded (the cathode). The time moments are listed on top of the columns. The average diameter of each drop before the impact was D 2.50 mm, and the impact velocity was V0 1.50 m/s. Scale bar is 3 mm.

Close modal

In the experimental data sets discussed here, the dimensionless parameters used to describe drop impact are the Weber number We=ρDV02/γ, the Ohnesorge number Oh=μ/ρDγ, and the dimensionless K number K=WeOh2/5,2,4 where ρ is the liquid density, D = 2r0 is the initial drop diameter, V0 is the impact velocity, and μ and γ are the liquid viscosity and surface tension, respectively. The effect of the electric field is accounted for by the dimensionless electric capillary number CaE=εairE2r0/γ (proportional to the electric Bond number BoE=E2r0/γ), where εair1 is the dielectric permittivity of air, E is the applied electric field strength between the electrodes, and r0 is the initial drop radius.

The impact onto the dielectric balsa wood surface I with grooves in the same direction as the electric field lines reveals the formation of finger-like structures and secondary drops in the short-term impact dynamics when no voltage is applied (row 1 in Fig. 8). It should be emphasized that this pattern is formed during the advancing, rather than receding motion, of drop contact line. Consecutive trials at 0 kV reveal a similar behavior. Furthermore, the application of a 7 kV voltage had a minimum effect on the observed pattern, as shown in row 2 in Fig. 8. Additional experiments with an applied voltage of 10 kV revealed a more pronounced fingering of drops of water with aniline dye during the advancing motion of the contact line. The snapshots at different time moments with the applied electric field of 10 kV (row 3 in Fig. 8) reveal the formation of longer fingers along the advancing drop rim compared to the case without voltage (row 1), especially at time moments t = 2 and t = 4 ms. Moreover, the subsequent snapshots at t = 6 and t = 16 ms also reveal that long fingers are still present and begin to be predominantly directed toward the neighboring electrodes, presumably due to the electric force. Note that here and hereinafter each consecutive trial was carried out using a new balsa wood sample.

The long-term observations of the same experiments at a lower acquisition frame rate of 100 fps revealed the progressive spreading of a drop of water mixed with aniline dye after impact onto the balsa wood substrate I without and with the electric field applied. This is presumably, a wettability-imbibition-driven phenomenon. It should be emphasized that here t = 0 s corresponds to the time moment when the drop had already impacted onto the dielectric surface and developed a short-term fingering of the type of Fig. 8, as seen in the first snapshots in Fig. 9. Therefore, the fast initial advancing and receding stages after impact had already been completed before the evolution depicted in Fig. 9.

FIG. 9.

Long-term evolution of drops of water with gold oak aniline dye on balsa wood substrate I. Drops impacted onto the dielectric balsa wood substrate I without an applied voltage at 0 kV (rows 1 and 3), and with the applied voltage of 10 kV corresponding to the electric field strength of 6.6 × 105 V/m (rows 2 and 4). The Weber number We = 77.3, the Ohnesorge number Oh = 0.002, and the composite parameter K = 928.5. The electric capillary number is CaE = 0.847 for the applied voltage of 10 kV. The left-hand side electrode is the anode, and the right-hand side one is grounded (the cathode). The time moments are listed on top of the columns. Top views are presented in rows 1 and 2, whereas the side views are shown in rows 3 and 4; the snapshots corresponding to the side views (rows 3 and 4) were taken in different trials from the top-view snapshots shown in rows 1 and 2. The average diameter of each drop before the impact was D 2.50 mm, and the impact velocity was V0 1.50 m/s. Scale bar is 3 mm.

FIG. 9.

Long-term evolution of drops of water with gold oak aniline dye on balsa wood substrate I. Drops impacted onto the dielectric balsa wood substrate I without an applied voltage at 0 kV (rows 1 and 3), and with the applied voltage of 10 kV corresponding to the electric field strength of 6.6 × 105 V/m (rows 2 and 4). The Weber number We = 77.3, the Ohnesorge number Oh = 0.002, and the composite parameter K = 928.5. The electric capillary number is CaE = 0.847 for the applied voltage of 10 kV. The left-hand side electrode is the anode, and the right-hand side one is grounded (the cathode). The time moments are listed on top of the columns. Top views are presented in rows 1 and 2, whereas the side views are shown in rows 3 and 4; the snapshots corresponding to the side views (rows 3 and 4) were taken in different trials from the top-view snapshots shown in rows 1 and 2. The average diameter of each drop before the impact was D 2.50 mm, and the impact velocity was V0 1.50 m/s. Scale bar is 3 mm.

Close modal

The contours tracing the drop wetted area corresponding to the top-view observations in Fig. 9 are depicted in Fig. 10. The superposition of the drop contours without the electric field (at 0 kV) and with the application of a 10 kV DC voltage (with the electric field strength of 6.6 × 105 V/m) reveals the effect of the electric force on the footprint area on the balsa wood substrate I. It is clearly visible that the wetted area increases in the case of 10 kV. The applied electric field stretches the drop along the substrate (in the direction of the field lines and grooves), and the footprint area achieved at 16 s (when the drop begins to extend beyond the frame limit of the top camera) slightly increases as the electric field strength increases, as shown in Table II. Note that recording with the top-view camera could continue up to 18 s. The centers of the drops analyzed in Fig. 9 are very close in all cases presented, which simplifies the analysis of the wetted areas (six trials and eight trials for 0 and 10 kV, respectively) obtained from the top-view camera.

FIG. 10.

Wetted area contours of a single drop of water with aniline dye after its impact onto the balsa wood substrate I: (a) 0 and (b) 10 kV. The left-hand-side electrode is the anode, and the right-hand-side electrode is grounded (the cathode). The Cartesian coordinates on the substrate are denoted as x (in the field direction) and y.

FIG. 10.

Wetted area contours of a single drop of water with aniline dye after its impact onto the balsa wood substrate I: (a) 0 and (b) 10 kV. The left-hand-side electrode is the anode, and the right-hand-side electrode is grounded (the cathode). The Cartesian coordinates on the substrate are denoted as x (in the field direction) and y.

Close modal
TABLE II.

Average area-equivalent radius a of water–aniline dye drop wetted area recorded on balsa wood substrate I without and with the applied voltage.

Time (s)a (mm), for 0 kVa (mm), for 10 kVPercentage increase (%) due to the applied voltage
1.72 ± 0.25 2.32 ± 0.21 34.9 ± 6.5 
2.29 ± 0.22 3.15 ± 0.18 37.6 ± 4.9 
2.66 ± 0.29 3.46 ± 0.15 30.1 ± 7.7 
3.17 ± 0.21 3.71 ± 0.19 17.1 ± 1.7 
10 3.41 ± 0.26 4.04 ± 0.14 18.5 ± 4.6 
12 3.68 ± 0.30 4.26 ± 0.24 15.8 ± 2.7 
14 3.76 ± 0.23 4.54 ± 0.19 20.7 ± 2.2 
16 4.01 ± 0.11 4.71 ± 0.13 17.5 ± 0.02 
18 4.08 ± 0.10 4.84 ± 0.10 18.6 ± 0.4 
Time (s)a (mm), for 0 kVa (mm), for 10 kVPercentage increase (%) due to the applied voltage
1.72 ± 0.25 2.32 ± 0.21 34.9 ± 6.5 
2.29 ± 0.22 3.15 ± 0.18 37.6 ± 4.9 
2.66 ± 0.29 3.46 ± 0.15 30.1 ± 7.7 
3.17 ± 0.21 3.71 ± 0.19 17.1 ± 1.7 
10 3.41 ± 0.26 4.04 ± 0.14 18.5 ± 4.6 
12 3.68 ± 0.30 4.26 ± 0.24 15.8 ± 2.7 
14 3.76 ± 0.23 4.54 ± 0.19 20.7 ± 2.2 
16 4.01 ± 0.11 4.71 ± 0.13 17.5 ± 0.02 
18 4.08 ± 0.10 4.84 ± 0.10 18.6 ± 0.4 

The area-equivalent radii from all the trials conducted at 0 and 10 kV (six and eight, respectively) are plotted at different time moments in Fig. 11. It should be emphasized that in Fig. 11 each data point represents an individual trial, and the corresponding mean values and standard deviations are listed in Table II.

FIG. 11.

Area-equivalent radius of drop wetted area (water with aniline dye) after impact onto the balsa wood substrate I as a function of time without and with the applied voltage of 10 kV. The experimental data are shown by symbols, the fitting by curves corresponding to equations a=0.8348t+0.5477 and a=0.8897t+1.0656 for 0 and 10 kV, respectively.

FIG. 11.

Area-equivalent radius of drop wetted area (water with aniline dye) after impact onto the balsa wood substrate I as a function of time without and with the applied voltage of 10 kV. The experimental data are shown by symbols, the fitting by curves corresponding to equations a=0.8348t+0.5477 and a=0.8897t+1.0656 for 0 and 10 kV, respectively.

Close modal

An additional analysis of the wetted area contours also revealed the corresponding effective dimensions in the horizontal direction (the electric field direction) Δx and the orthogonal direction Δy depicted in Fig. 12. These dimensions reveal a repeatable pattern of the spreading of drop in the field and cross-field directions (Δx and Δy, respectively) in all trials conducted at 0 and 10 kV [Figs. 12(a) and 12(b), respectively].

FIG. 12.

Characteristic dimensions of wetted area of drops (water with aniline dye) after impact onto the balsa wood substrate I: (a) 0 and (b) 10 kV.

FIG. 12.

Characteristic dimensions of wetted area of drops (water with aniline dye) after impact onto the balsa wood substrate I: (a) 0 and (b) 10 kV.

Close modal

Additional experiments on balsa wood substrate II with grooves directed approximately at 90° with respect to the direction of the electric field lines were conducted using the same working fluid: water mixed with aniline dye. Single drops were dripped onto such a substrate from the same fixed height of h = 12.5 cm. As before, the top views recorded by a high-speed camera (Phantom Miro 4) captured the drop evolution at the rates of 1000 and 100 fps to resolve the short-term and long-term dynamics, respectively. Figure 13 illustrates the short-term dynamics recorded at the applied voltages of 0, 7, and 10 kV (rows 1–3, respectively).

FIG. 13.

Top views of the short-term evolution of drops of water–gold oak aniline dye on balsa wood substrate II. Drops impacted onto the substrate without an applied voltage at 0 kV (row 1), and with the applied voltages of 7 and 10 kV corresponding to the electric field strengths of 4.6 × 105 and 6.6 × 105 V/m, respectively (rows 2 and 3). The Weber number We = 77.3, the Ohnesorge number Oh = 0.002, and the composite parameter K = 928.5. The electric capillary numbers are CaE = 0.415 and CaE = 0.847 for 7 and 10 kV, respectively. The left-hand-side electrode is the anode, and the right-hand-side one is grounded (the cathode). The time moments are listed on top of the columns. The average diameter of each drop before the impact was D 2.50 mm, and the impact velocity was V0 1.50 m/s. Scale bar is 3 mm.

FIG. 13.

Top views of the short-term evolution of drops of water–gold oak aniline dye on balsa wood substrate II. Drops impacted onto the substrate without an applied voltage at 0 kV (row 1), and with the applied voltages of 7 and 10 kV corresponding to the electric field strengths of 4.6 × 105 and 6.6 × 105 V/m, respectively (rows 2 and 3). The Weber number We = 77.3, the Ohnesorge number Oh = 0.002, and the composite parameter K = 928.5. The electric capillary numbers are CaE = 0.415 and CaE = 0.847 for 7 and 10 kV, respectively. The left-hand-side electrode is the anode, and the right-hand-side one is grounded (the cathode). The time moments are listed on top of the columns. The average diameter of each drop before the impact was D 2.50 mm, and the impact velocity was V0 1.50 m/s. Scale bar is 3 mm.

Close modal

Figure 13 in comparison with Fig. 8, essentially, demonstrates that the groove directions relative to the electric field lines have no discernable effects on the short-term drop evolution after impact without or with the electric field applied. Note that as before, each consecutive trial was carried out using a new balsa wood sample, so different groove patterns are clearly visible in the snapshots in Fig. 13. There is only a slight dependence of the drop evolution on the groove orientation relative to the electric field. The slight differences between the experiments with balsa wood substrates I with grooves oriented in the same direction as the electric field lines (Fig. 8) and substrates II with groves across the field lines (Fig. 13) are observed at times moments from t = 6 ms through t = 16 ms. Specifically, at 10 kV the snapshots for these moments depicted in Fig. 8 reveal the formation of more fingers along the rim of the drop, which stay visible during the experiment, than in Fig. 13. Note that this distinction was confirmed in the repetitive experiments.

The snapshots shown at t = 2 and t = 4 ms in Figs. 8 and 13 revealed finger formation due to prompt splashing in the cases without and with the electric field. The roughness of the balsa wood substrates increased prompt splashing at the advancing contact line. When voltage was applied to the dielectric wooden surface, trials with the applied voltages of 7 and 10 kV revealed a slight increase in the number of fingers formed after drop impact. Also, longer fingers are formed when the pulling electric force is acting on the spreading drop compared to the case without voltage at 0 kV in both Figs. 8 and 13.

Note also that the experimental evidence provides a strong indication that grooves at the wood surface affect the rim evolution during the spreading stage. On a rougher surface, such as balsa wood, longer and thinner fingers might protrude out over the rim compared to smooth surfaces without grooves, such as polypropylene. Moreover, the front of the spreading drop tends to come to rest earlier when it encounters grooves during its motion, whereas the already formed fingers keep moving radially due to a residual kinetic energy from the initial impact onto the balsa substrate.

An additional set of experiments at a lower acquisition frame rate (100 fps) revealed the long-term evolution of drops on balsa wood substrates II (cf. Fig. 14). It should be emphasized that here, once again, t = 0 s corresponds to the time moment when the short-term evolution of Fig. 13 is already over.

FIG. 14.

Long-term evolution of drops of water–gold oak aniline dye on balsa wood substrate II. Drops impacted onto the dielectric balsa wood substrate II without an applied voltage at 0 kV (rows 1 and 3), and with the applied voltage of 10 kV (the electric field strength of 6.6 × 105 V/m; rows 2 and 4). The Weber number We = 77.3, the Ohnesorge number Oh = 0.002, and the composite parameter K = 928.5. The electric capillary number is CaE = 0.847 for the applied voltage of 10 kV. The left-hand-side electrode is the anode, and the right-hand-side one is grounded (the cathode). The time moments are listed on top of the columns. Top views are presented in rows 1 and 2, whereas the side views are shown in rows 3 and 4; the snapshots corresponding to the side views (rows 3 and 4) were taken in different trials from the top-view snapshots shown in rows 1 and 2. The average diameter of each drop before the impact was D 2.50 mm, and the impact velocity was V0 1.50 m/s. Scale bar is 3 mm.

FIG. 14.

Long-term evolution of drops of water–gold oak aniline dye on balsa wood substrate II. Drops impacted onto the dielectric balsa wood substrate II without an applied voltage at 0 kV (rows 1 and 3), and with the applied voltage of 10 kV (the electric field strength of 6.6 × 105 V/m; rows 2 and 4). The Weber number We = 77.3, the Ohnesorge number Oh = 0.002, and the composite parameter K = 928.5. The electric capillary number is CaE = 0.847 for the applied voltage of 10 kV. The left-hand-side electrode is the anode, and the right-hand-side one is grounded (the cathode). The time moments are listed on top of the columns. Top views are presented in rows 1 and 2, whereas the side views are shown in rows 3 and 4; the snapshots corresponding to the side views (rows 3 and 4) were taken in different trials from the top-view snapshots shown in rows 1 and 2. The average diameter of each drop before the impact was D 2.50 mm, and the impact velocity was V0 1.50 m/s. Scale bar is 3 mm.

Close modal

An interesting observation regarding the long-term evolution of drops at 0 kV (rows 1 and 3) and 10 kV (rows 2 and 4) in Fig. 14 is that the effect of the electric field progressively increases the visible wetted area starting at t = 4 s. It should be emphasized that the snapshots in Fig. 14 are fully representative of many trials, because similar long-term patterns were also observed in the other trials conducted. Water with aniline die imbibing into balsa wood substrates II reveals in Fig. 14 a fractal-like spreading, which seems to be enhanced and accelerated when a 10 kV voltage is applied. Moreover, it should be emphasized that in Fig. 9 (wood substrate I), the observations revealed a slight effect of the 10 kV applied voltage, which resulted in an increase in the footprint area, especially in the orthogonal direction, visible at t = 12 s and t = 16 s in the overall long-term patterns. Note that the snapshots up to t = 16 s presented in Fig. 14 do not end up in a steady-state (as they did not end in a steady-state in Fig. 9), because drop spreading could continue beyond the frame limit of the top-view camera. To confirm the effect of the electric field on drop spreading and imbibition on balsa wood substrate II, the recording continued for another 40 s until most of the liquid remaining above the surface had been imbibed.

It should be emphasized that the side-view snapshots for balsa substrate I (Fig. 9) reveal that there was almost no liquid remaining over the wood surface at t = 12 s when the 10 kV voltage was applied compared to the same time moment for the case without voltage applied (at 0 kV). In the case without voltage applied, there was some liquid leftover seen over the surface until the latest image taken at t = 16 s. This pattern was also observed for wood substrate II in Fig. 14. Here, the side-view recordings reveal that the application of an external DC voltage of 10 kV enhanced liquid stretching over the surface, which facilitated its imbibition into balsa wood grooves. As a result, with the applied voltage of 10 kV at t 16 s, one could see a tiny amount of liquid remaining over the surface, whereas in the case without voltage (0 kV), a still larger puddle of liquid is remaining on the balsa substrate II.

The traces of the wetted area corresponding to the top-view observations in Fig. 14 are depicted in Fig. 15. The superposition of the contours at 0 kV [Fig. 15(a)] with those corresponding to 10 kV DC voltage applied (the electric field strength of 6.6 × 105 V/m) [Fig. 15(b)] reveals the effect of the electric force on the wetted area on the balsa wood substrate II. An increase in the wetted area in the case of 10 kV is clearly seen. Thus, the applied electric field facilitates liquid stretching over the wood surface, which in turn facilitates the wettability-driven imbibition of liquid into porous balsa substrate.

FIG. 15.

Contours of the wetted area of a single drop of water with aniline dye observed in the long-term experiments on balsa wood substrate II: (a) 0 and (b) 10 kV. The left-hand-side electrode is the anode, and the right-hand-side electrode is grounded (the cathode). The Cartesian coordinates on the substrate are denoted as x (in the field direction) and y.

FIG. 15.

Contours of the wetted area of a single drop of water with aniline dye observed in the long-term experiments on balsa wood substrate II: (a) 0 and (b) 10 kV. The left-hand-side electrode is the anode, and the right-hand-side electrode is grounded (the cathode). The Cartesian coordinates on the substrate are denoted as x (in the field direction) and y.

Close modal

The wetted areas can be converted in the area-equivalent radii a. For all the trials conducted at 0 and 10 kV (six and eight trials, respectively), the values of a at different time moments are presented in Fig. 16. It should be emphasized that each data point represents here an individual trial and their corresponding values are listed in Table III.

FIG. 16.

Area-equivalent radius of wetted area of drops of water with aniline dye on balsa wood substrate II as a function of time without and with the applied voltage of 10 kV. The experimental data are shown by symbols and the fitting by curves corresponding to the equations a=0.6212t+0.4726 and a=1.0071t+0.4159 for 0 and 10 kV, respectively.

FIG. 16.

Area-equivalent radius of wetted area of drops of water with aniline dye on balsa wood substrate II as a function of time without and with the applied voltage of 10 kV. The experimental data are shown by symbols and the fitting by curves corresponding to the equations a=0.6212t+0.4726 and a=1.0071t+0.4159 for 0 and 10 kV, respectively.

Close modal
TABLE III.

Area-equivalent radius of water–aniline dye drops recorded on balsa wood substrate II without and with the applied voltage.

Time (s)a (mm), for 0 kVa (mm), for 10 kVPercentage increase (%) due to the applied voltage
1.35 ± 0.06 1.84 ± 0.13 36.3 ± 3.4 
1.54 ± 0.15 2.43 ± 0.12 57.8 ± 6.9 
1.86 ± 0.11 3.16 ± 0.14 69.9 ± 2.4 
2.26 ± 0.26 3.68 ± 0.12 62.8 ± 12.1 
10 2.61 ± 0.15 3.77 ± 0.19 44.4 ± 0.9 
12 2.82 ± 0.16 4.09 ± 0.18 45.1 ± 1.7 
14 2.89 ± 0.10 4.28 ± 0.16 48.1 ± 0.4 
16 2.91 ± 0.11 4.49 ± 0.18 54.3 ± 0.3 
18 3.11 ± 0.82 4.69 ± 0.13 50.8 ± 28.2 
Time (s)a (mm), for 0 kVa (mm), for 10 kVPercentage increase (%) due to the applied voltage
1.35 ± 0.06 1.84 ± 0.13 36.3 ± 3.4 
1.54 ± 0.15 2.43 ± 0.12 57.8 ± 6.9 
1.86 ± 0.11 3.16 ± 0.14 69.9 ± 2.4 
2.26 ± 0.26 3.68 ± 0.12 62.8 ± 12.1 
10 2.61 ± 0.15 3.77 ± 0.19 44.4 ± 0.9 
12 2.82 ± 0.16 4.09 ± 0.18 45.1 ± 1.7 
14 2.89 ± 0.10 4.28 ± 0.16 48.1 ± 0.4 
16 2.91 ± 0.11 4.49 ± 0.18 54.3 ± 0.3 
18 3.11 ± 0.82 4.69 ± 0.13 50.8 ± 28.2 

It should be emphasized that the fitting curves in Fig. 11 (balsa wood substrate I) a=0.8348t+0.5477 and a=0.8897t+1.0656 corresponding to the applied voltages of 0 and 10 kV, respectively, and Fig. 16 (balsa wood substrate II) a=0.6212t+0.4726 and a=1.0071t+0.4159 for 0 and 10 kV, respectively, reveal that the square root of time dependence relatively accurately fits the experimental data. It should be emphasized that the data points from all the trials conducted (six and eight for 0 and 10 kV, respectively) in Figs. 11 and 16 were used to find the fitting lines through the root mean square procedure. Moreover, it is worth mentioning that the coefficient near the square root C1 = 0.8348 for 0 kV in Fig. 11 (wood substrate I) is larger compared to the coefficient C1 = 0.6212 for wood substrate II at 0 kV (Fig. 16). Such a difference is due to the fact that in the observations with balsa wood substrates I, the area-equivalent radius values are slightly greater over time than the values obtained for balsa wood substrates II at the same time moments when there is no voltage applied. On the other hand, the multiplier of the square root C1 = 0.8897 for 10 kV in Fig. 11 (wood substrate I) is smaller than the multiplier C1 = 1.0071 for wood substrate II at the applied voltage of 10 kV (Fig. 16). The latter implies that the effect on wet area spreading of the 10 kV voltage acting on the balsa wood substrate II with grooves across the field lines is more pronounced than in balsa wood substrate I.

An additional characterization of the wetted contours was conducted to establish the characteristic dimensions of the wetted area Δx and Δy in the case of the balsa wood substrate II; cf. Fig. 17. The results in Fig. 17 demonstrate that spreading across the field lines is much more significant in this case, that is, Δy > Δx, which means that grooves facilitate significantly liquid imbibition. This is dramatically different from the results in Fig. 12 for balsa wood substrate I, where Δx > Δy. One can conclude that the direction of grooves determines the strength of the imbibition process. The effect of the electric field on the long-term liquid spreading is rather secondary, as the comparison of Figs. 12(a) and 12(b), or 17(a) and 17(b), shows.

FIG. 17.

Characteristic dimensions of the wetted area of drops of water with aniline dye on balsa wood substrate II: (a) 0 and (b) 10 kV.

FIG. 17.

Characteristic dimensions of the wetted area of drops of water with aniline dye on balsa wood substrate II: (a) 0 and (b) 10 kV.

Close modal

Unlike the characteristic length Δy, the characteristic length in the field direction Δx reveals an increase at the applied voltage of 10 kV compared to the case of 0 kV [cf. Figs. 17(a) and 17(b)]. Note also that in Fig. 17(a) at 0 kV, the experimental data for Δx and Δy closely follow the square-root-of-time dependence. On the other hand, in Fig. 17(b) for 10 kV both Δx and Δy deviate quite significantly from this dependence.

The developed PFM algorithm is used to simulate numerically the situations characteristic of the experiments described in the preceding Secs. II and V. First, a drop shape over a porous substrate is tackled, which depends on the contact angle. For this case, a specific contact angle value is imposed as the equilibrium wetting boundary condition at all walls of the porous substrate. Then, the model is applied to simulate drop impact on porous balsa substrate.

All the required physical properties and special (numerical, for PFM) parameters used in the simulations are listed in Table IV.

TABLE IV.

List of parameters used in numerical simulations.

ParameterNotationValueUnit
Density of liquid ρ 997 kgm3 
Density of air ρair 1.25 kgm3 
Viscosity of liquid μ 1×103 Pas 
Viscosity of air μair 2×105 Pas 
Surface tension of liquid γ 72.8 m Nm1 
Dielectric permittivity of liquid εl 80 ⋯ 
Dielectric permittivity of air εair ⋯ 
Drop interface thickness εint 2.5×105 
Volume-equivalent drop radius r0 2.5 mm 
Initial drop velocity  1.5 ms1 
Mobility tuning parameter χ mskg1 
Equilibrium contact angle θs 50 (Hydrophilic) ° 
130 (Hydrophobic) ° 
ParameterNotationValueUnit
Density of liquid ρ 997 kgm3 
Density of air ρair 1.25 kgm3 
Viscosity of liquid μ 1×103 Pas 
Viscosity of air μair 2×105 Pas 
Surface tension of liquid γ 72.8 m Nm1 
Dielectric permittivity of liquid εl 80 ⋯ 
Dielectric permittivity of air εair ⋯ 
Drop interface thickness εint 2.5×105 
Volume-equivalent drop radius r0 2.5 mm 
Initial drop velocity  1.5 ms1 
Mobility tuning parameter χ mskg1 
Equilibrium contact angle θs 50 (Hydrophilic) ° 
130 (Hydrophobic) ° 

The predictions of the numerical model for the shape of a drop on a porous phase are depicted in Fig. 18. Initially, half of a 2.5 mm (in diameter) water drop was placed on top of a porous balsa substrate creating a 90° angle at the contact line, with an initial zero velocity. Figure 18 demonstrates that first under the action of the gravity force, the drop begins to spread on top of the porous surface accompanied by imbibition into the porous substrate. These results can be compared to the experimental results depicted in Fig. 4. In particular, analyzing the evolution of the drop shape in time, one can observe that at ∼23 s most of the drop has already been imbibed into the porous substrate, similarly to the last side-view image recorded in experiments in Fig. 4. It is impossible to compare the exact shape with finger forming and liquid transport through the balsa wood grooves, as observed experimentally, because the model is axisymmetric. Despite all our efforts, it was impossible to run a fully three-dimensional model due to the needed extra-fine mesh in the porous substrate, as well as for the resolution of the drop interface. Thus, for the equilibrium contact angle, as well as for the impact modeling, the comparison between the experimental and the modeling results is qualitative.

FIG. 18.

Predicted evolution of a water drop over porous balsa substrate. The initial geometry is a half of the spherical drop with diameter of 2.5 mm and zero initial velocity. The geometrical details of the porous substrate are sketched in Fig. 3.

FIG. 18.

Predicted evolution of a water drop over porous balsa substrate. The initial geometry is a half of the spherical drop with diameter of 2.5 mm and zero initial velocity. The geometrical details of the porous substrate are sketched in Fig. 3.

Close modal

Next, the PFM calculations were performed for water drop impact onto hydrophilic (θs = 50°) and hydrophobic (θs = 130°) porous surfaces. Drop impact onto a dry solid porous substrate assumed to be hydrophilic was simulated first, with the corresponding results presented in Fig. 19. The upper row in Fig. 19 depicts the evolution of the drop shape in time, with the red color representing the water drop, blue color—the gas-phase (air), and brown color—the porous substrate. The second row in Fig. 19 depicts the velocity magnitude together with the velocity vectors color-marked for different phases. The third row in Fig. 19 illustrates the pressure field during the impact, spreading, and imbibition into the porous substrate. The predicted trends were qualitatively compared to the experimentally observed trends corresponding to the data depicted in Figs. 8 and 9 (the first row, no voltage applied). In particular, it is predicted numerically that the drop first spreads over the surface and in the subsurface layer of the porous substrate, leaving a substantial part visible over the solid surface similarly to the experiment. After the initial spreading at ∼2 ms, the predicted drop begins imbibing deeper into the porous substrate and loses its integrity. Obviously, the present axisymmetric simulation's prediction of fingering formation is impossible. Analyzing the drop evolution and the corresponding pressure contours in Fig. 19, it could be concluded that liquid is imbibed into the subsurface layer of the porous phase driven by hydrophilicity (corroborated by the comparison with the hydrophobic case). However, once the drop is slightly imbibed, at ∼0.1 mm from the top porous surface, the pressure difference is responsible for its further imbibition into the porous medium (based on the comparison with the hydrophobic case).

FIG. 19.

Predicted drop evolution (the upper row), with red color being the water drop, blue color being air, and brown color being the porous balsa substrate. The situation is assumed to be hydrophilic (with the contact angle of water on balsa assumed to be 50°). Velocity magnitude together with the corresponding velocity vectors is depicted in the second row. The pressure field is presented in the third row. The drop diameter is 2.5 mm, and the initial drop velocity is 1.5 m/s.

FIG. 19.

Predicted drop evolution (the upper row), with red color being the water drop, blue color being air, and brown color being the porous balsa substrate. The situation is assumed to be hydrophilic (with the contact angle of water on balsa assumed to be 50°). Velocity magnitude together with the corresponding velocity vectors is depicted in the second row. The pressure field is presented in the third row. The drop diameter is 2.5 mm, and the initial drop velocity is 1.5 m/s.

Close modal

In the simulations, the air presence in pores is accounted for. The gas phase (air) is over and inside the pore space. It can be seen in Fig. 19, second row, magenta arrows describe the airflow during the impact. Eventually, drop forces air out during the spreading and imbibition. There are air pockets in the dead zones (Fig. 19, 2 ms), where air is entrapped as a result of drop spreading through the pores. However, it does not influence much liquid penetration, because these air pockets are extremely small and located in dead zones.

Then, the balsa substrate in the simulations was artificially made hydrophobic (θs = 130°). Figure 20 presents the predicted results for the drop impact, spreading and rebounding over such an artificially hydrophobic porous balsa surface with the contact angle of 130°. Similarly to Fig. 19, in Fig. 20 the first row displays the predicted evolution of the drop shape, the second row shows the velocity magnitude and vectors, and the third row depicts the pressure field. The numerical predictions revealed that the spreading lamella bumps up as a rim, which is typical of the experimental observations on intact surfaces (without liquid penetration). Analyzing the velocity magnitude and velocity vectors presented in the second row in Fig. 20, one observes the highest values in liquid corresponding to the initial moment of the impact, while in air this happens during the drop rebound stage. In the present case, the drop spreads and rebounds off the surface, practically without any imbibition into the pores, as the surface would be Teflon.12 

FIG. 20.

Predicted drop evolution (the upper row), with red color being the water drop, blue color being air, and brown color being the substrate artificially made hydrophobic. Velocity magnitude together with the velocity vectors is displayed in the second row. The pressure field is depicted in the third row. The drop diameter is 2.5 mm, and the initial drop velocity is 1.5 ms1.

FIG. 20.

Predicted drop evolution (the upper row), with red color being the water drop, blue color being air, and brown color being the substrate artificially made hydrophobic. Velocity magnitude together with the velocity vectors is displayed in the second row. The pressure field is depicted in the third row. The drop diameter is 2.5 mm, and the initial drop velocity is 1.5 ms1.

Close modal

The experimental, theoretical, and numerical studies conducted in this work elucidated the effect of an in-plane electric field on a diffusion-like spreading of paint drops resulting in a significant increase in the painted area of balsa wood substrates. Drops of paint (aqueous suspensions of pigment particles) impacted onto the porous surface of balsa wood.

The short-term recordings (on the scale of a few ms) demonstrated a slight effect of the applied in-plane electric field (at the applied voltage of 10 kV). Here, some trials conducted revealed a greater number of finger-like structures developing along the drop rim compared to the trials without any voltage applied. It should be emphasized that these fingers appeared during the advancing motion of the drop rim.

Additionally, the experimental observations of drops of water with aniline dye impacting onto balsa wood substrates revealed a slight effect of an in-plane electric field (only at a high voltage of 10 kV) on drop spreading on such dielectric substrates. The long-term recording (on the scale of ∼10 s) revealed a wettability-driven increase in the area-equivalent radius of the wetted area at 18 s after drop impact by 18.6% ± 0.4% and 50.8% ± 28.2% for balsa wood substrates I and II, respectively. These substrates had groves in the inter-electrode direction, and the cross-field direction, respectively. The wettability-driven spreading in the long term was a much more significant effect than the effect of the electric field. The electric field presumably stretched the diminishing upper part of the drop, which was still above the porous surface, that is, still was not imbibed.

The analysis of the experimental data revealed an approximately square-root dependence on time of the area-equivalent radius of the wetted area on both balsa wood substrates I and II in cases without voltage applied, which attests to the diffusion-like moisture transport during imbibition of porous balsa. However, this dependence was inapplicable in the cases with the electric field applied. The side-view observations of water drops with aniline dye softly located on balsa surface confirmed wettability-driven imbibition hypothesis, and together with a novel analytical model developed here allowed measurements of the moisture transfer coefficient. The results for the latter were in good agreement with those reported in the literature.

Furthermore, the PFM results built upon experimentally observed balsa geometrical features revealed that a change in the contact angle from hydrophobic to hydrophilic leads to the drop imbibition into the balsa porous substrate.

The authors acknowledge the financial support from the National Science Foundation through award CBET-1906497.

The authors have no conflicts to disclose.

Rafael Granda: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Vitaliy Yurkiv: Funding acquisition (equal); Investigation (equal); Methodology (equal); Software (equal); Writing – original draft (equal); Writing – review & editing (equal). Farzad Mashayek: Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal). Alexander L. Yarin: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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