Electric Field Induced Patterning of Thin Coatings on Fiber Surfaces

Article pubs.acs.org/JPCC

Electric Field Induced Patterning of Thin Coatings on Fiber Surfaces V. Anoop Kishore† and Dipankar Bandyopadhyay*,†,‡ †

Department of Chemical Engineering, and ‡Centre for Nanotechnology, Indian Institute of Technology, Guwahati, India S Supporting Information *

ABSTRACT: We explore the electric field induced instabilities of a thin liquid film coated on a fiber surface. Thin liquid films on the curved surfaces spontaneously self-organize into interesting patterns such as a string of beads/droplets when the destabilizing radial curvature force dominate over the stabilizing in plane curvature. Application of an external electrostatic field in such a situation opens up the added possibility of fabricating ridge like structures issuing outward from the film surface when electric field dominates over the curvature forces. With the help of a general linear stability analysis and long-wave nonlinear simulations, the study uncovers the conditions under which the ridges and beads form on the fiber surface. In particular, we show that the ridges are favored morphologies when electric field is stronger because of higher film to air filling ratio in between the electrodes, higher applied voltage, and larger fiber radius. The beads and the ridges are found to coexist when the destabilizing forces are equally dominant whereas only beads are formed when the radial curvature force is the dominant destabilizing force. The analysis also shows that tuning the voltage, fiber radius, and film thickness, length scales ranging from a few hundred nanometers to a few micrometers can be achieved for all these morphologies. The results reported can potentially be exploited in the applications related to the field-induced microfabrication on the curved surfaces. overcomes the stabilizing in-plane curvature.3,4 Interestingly, when such a configuration is exposed to a strong electrostatic field the electrical stress at the film surface can also deform into columns, ridges, holes, or channel like structures, as observed for thin films over the flat surfaces.36,37 Clearly, a thin film over a curved surface can lead to a host of interesting morphologies such as beads, columns, ridges, droplets and channels on fibers when subjected to an external electric field. Electric field adds more versatility to this fabrication process because tuning the external voltage bias, (i) a wide range of soft films of thickness ranging from a few nanometers to a few micrometers can be destabilized on thick to thin fibers, (ii) the length scale of the patterns can be controlled by tuning the electric field from a few hundred nanometers to a few micrometers, and (iii) a transition from bead to columnar patterns can be achieved. In the present study, we illustrate the instabilities of a thin dielectric liquid film on a stationary cylindrical fiber exposed to an external electrostatic field as schematically shown in Figure 1. A few recent studies have shown some very interesting aspects of these instabilities, (i) a thin viscoelastic film resting on a fiber and under the influence of an electrostatic field arising from a bounding conducting gas19 and (ii) a thin film resting on a curved surface with very small slope and under the influence of an external electric field.22 However, the salient features of electric field induced instabilities of a thin film resting on a fiber with a complete electrohydrodynamic (EHD) description is yet to be pursued. We perform a general linear stability analysis

I. INTRODUCTION Instabilities of liquid threads1,2 and thin coatings on fibers3−5 have been investigated extensively in the past owing to their scientific and technological importance in the flows inside micropores, liquid jets, protective fiber coatings, electro-spinning devices, heat exchangers, flexible tubes and spider silks.6−13 A thin coating spontaneously undergoes Plateau-Rayleigh instability and breaks down to a string of beads/droplets on a motionless fiber when the destabilizing radial curvature force dominates over the stabilizing in plane curvature force.3−5 The perturbations with wavelengths larger than the combined circumference [2π(r + h)] of film (thickness h) and fiber (radius r) are found to be unconditionally unstable in such configurations. The meandistance between the beads follow a simple relation, 2π √2r when h ≪ r. Previous studies14−18 show the importance of the motion of the fiber, viscosity of the film, ratio of film thickness to fiber radius, and the strength of the intermolecular to capillary forces on the length and time scales of the instabilities. Recent advances in fabrication and characterization of micro/ nano devices have renewed the interest to study the instabilities of thin films on fibers,19−22 especially when they are subjected to an external electric field.19,22 The instabilities of a thin film on a curved surface is rather different from the same when the film rests on a flat surface and under the influence of destabilizing intermolecular force23−32 or electrostatic field.33−54 On the flat surfaces the electric field overcomes the in-plane curvature to engender columns, holes, or channel like structures on the films.36,37 In contrast, on the curved surfaces, the radial curvature force acts as a destabilizing influence even when the film is at undeformed state. This enables the films to spontaneously disintegrate into bead morphologies when the radial curvature © 2012 American Chemical Society

Received: October 29, 2011 Revised: January 29, 2012 Published: January 31, 2012 6215

dx.doi.org/10.1021/jp210402j | J. Phys. Chem. C 2012, 116, 6215−6221

The Journal of Physical Chemistry C

Article

To obtain the expressions for the potentials the following boundary conditions are enforced: (i) at the fiber-liquid interface (r = R), ψ1 = 0; (ii) at the electrode−air (r = d) interface, ψ2 = ψ; (iii) at the liquid−air interface (r = R + h), the balance of the normal [ε0 E2·n = ε0εE1·n] and tangential [E1·t = E2·t] components of the electric field. Here ε0 = 8.85 × 10−12 C2/N m2 is the dielectric permittivity of free space, ε is the dielectric constant of the liquid film of thickness h resting on a fiber of radius R. The notations n[(− rz2/(rz2 + 1)3/2, 1/r(rz2 + 1))], and t [(1/r(rz2 + 1)1/2, rz2/(rz2 + 1)3/2)] represent the unit outward normal and tangent vectors. The base state governing equations for the electric field potentials, (1/r) d[r(ψi0/dr)]/dr = 0, have the solutions, ψi0 = A1i ln(r) + A2i. The constants (A1i and A2i) are evaluated employing the following boundary conditions: (i) at the fiber− liquid interface (r = R), ψ10 = 0; (ii) at the liquid−air interface (r = R + h0), ψ10 = ψ20and ε(∂ψ10/∂r) = (∂ψ20/∂r); (iii) at the electrode-air interface (r = d), ψ20 = ψ. Solving the ODEs with the help of the boundary conditions the following base state potentials are obtained:

Figure 1. Schematic diagram of a liquid film coated on a cylindrical fiber of radius R. An electric field of voltage ψ is applied through the electrodes separated by a distance, d. The mean and local film thicknesses are denoted by h0 and h (z, t), respectively. The symbols γ, ε, and μ represent surface tension, dielectric constant, and viscosity of the film, respectively.

(GLSA) of a thin film resting on a fiber (Figure 1) considering the Maxwell stresses for the electric field in the governing equations and boundary conditions. An axisymmetric evolution equation under the lubrication approximation is derived for the film surface to perform a long-wave LSA (LWLSA) and nonlinear simulations. The results for LWLSA and GLSA are compared to highlight the accuracy of the GLSA in predicting the length and time scales of the instabilities especially when the instability is in the shorter wavelength regime. GLSA identifies two distinct regimes of instabilities dominated by the radial curvature and electrostatic force when the ratio of the film thickness to the fiber radius is varied. The length and time scales of the instabilities are also found to vary significantly with the air to liquid filling ratio in between the electrodes, dielectric permittivity of the liquid film, and the applied voltage. The long-wave nonlinear simulations show morphologies such as a string of beads and an array of ridges in the regimes where the radial curvature and the electric field are the dominant destabilizing influences, respectively. The simulations also show the possibility of a mixed hierarchical morphology composed of beads in ridges or ridges in beads when the destabilizing forces are of similar strength. Concisely, the study highlights that the electric field induced instabilities of thin films on curved surface can lead interesting ridges, beads, and mixed morphologies over curved surface. The results shown here can be of significant importance in micro/nano fabrication on the curved surfaces.

(1)

∇ × Ei = 0

(2)

(4)

ψ02 =

ψε ln[r /d] ln[R /R C0] + (ε ln[R C0/d])

(5)

d2ψ̃i

1 dψ̃i + − k2 ψ̃i = 0. 2 r d r dr

(6)

Here ψ̃ i is the perturbed potential and ω and k are the linear growth coefficient and the wavenumber of the disturbance, respectively. The general solution of eq 6 is ψ̃i = Bij I0(kr ) + Bij K 0(kr )

(7)

where I0 and K0 are the zeroth order modified Bessel functions of first and second kind, respectively. The constants Bij (i = 1 and 2; j = 1 and 2) in eq 7 are evaluated enforcing the following boundary conditions to the total potential (ψi = ψi0 + ψ′i = ψi0 + ψ̃ i eωt+ikz): (i) at the fiber-liquid interface (r = R), ψ1 = 0; (ii) at the liquid−air interface (r = R + h), ψ1 = ψ2 and ε(∂ψ1/∂r) = (∂ψ2/∂r); (iii) at the electrode−air interface (r = d), ψ2 = ψ. The expressions for the perturbed potentials are cumbersome and not provided along with the text. The excess pressures at the interfaces because of electric field is obtained from the total electric field potential 2 ⎡ ⎛ ∂ψ ⎞2 ⎤ ε ⎛ ∂ψ ⎞ π = − 0 ⎢ε⎜ 1 ⎟ − ⎜ 2 ⎟ ⎥ 2 ⎢⎣ ⎝ ∂r ⎠ ⎝ ∂r ⎠ ⎥⎦

(8)

It may be noted here that the formalism employed here to derive the excess pressure because of electric field has been employed previously to model a completely different configuration composed of a thin film confined by a conducting gas.19 A recent study22 also employs a similar potential in the long-wave

The electric fields can be also be expressed in terms of potential functions ψi as, Ei = −∇ψi, leading to the Laplace equations ∇2 ψi = 0

ψ(ln[r / R C0] + ε ln[R C0/ d]) (ln[R / R C0] + ε ln[R C0/ d])

Here RC0 is the combined fiber and film thickness (R + h0) at the base state. In the perturbed state, the governing equations and the boundary conditions for electric field are linearized by employing the normal linear modes, ψi = ψi 0 + ψ̃ i eωt+ikz, with respect to the base state and the following linear form of the perturbed electric field potential is obtained:

II. PROBLEM FORMULATION A. Electric Field. In this section, initially the governing equations and the boundary conditions for the electric field are discussed. Following this, the methodologies to obtain the base state and perturbed state potentials are described. Figure 1 shows a schematic diagram of a dielectric liquid film resting on a solid cylindrical fiber. We assume that the electrostatic field is irrotational, a nonconducting liquid−air interface, and the characteristic flow time is much larger than the time scales for the electrically induced magnetic effects.33,34 Thus, the following governing equations are appropriate for the electric field, Ei, in the liquid film (i = 1) and the air gap (i = 2) ∇·Ei = 0

ψ10 =

(3) 6216



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to the GLSA dispersion relation, ω = f(k). A concise derivation of the GLSA and the analytical expression of the dispersion relation are provided in the Supporting material available on the web. The dominant growth coefficient (ωm) and the corresponding wavelength (λm) are obtained by finding the maximum of ω and the corresponding wavelength from the dispersion relation.

limit to describe the deformation of a thin film on a curved surface having a very small slope. B. Electrohydrodynamic Field. In this section, the electrohydrodynamic (EHD) governing equations and the boundary conditions are shown for a thin film resting over a fiber and under the influence of a destabilizing homogeneous electric field (Figure 1). The liquid film is assumed incompressible, isothermal, Newtonian and perfectly dielectric. Inertia is neglected from the equations of motion owing to small thickness of the film. In the following sections, the notations vr and vz in the equations represent the r and z components of velocities, respectively; the symbols μ and γ represent the viscosity and surface tension, respectively; and the notations p and π denote the static pressure in the liquid and excess pressure from the applied electric field. The following continuity equation and the equations of motion in the Stokes flow regime describe the EHD of a thin viscous film resting on a cylindrical fiber

∇·v = 0 −∇p + ∇·(τ + M) = 0

IV. NONLINEAR SIMULATIONS In order to perform nonlinear simulations the evolution equation for liquid−air interface is derived under the lubrication approximation. The continuity equation, the long-wave z and r components of the equations of motion for the film, −(∂p/∂z) + (μ/r)[∂(r(∂vz/∂r))/∂r] = 0 and −∂p/∂r = 0, together with kinematic equation for the interface, ∂h/∂t + vz∂h/∂z = vr leads to the following evolution equation for the liquid−air free surface ⎤ ⎡ 3 R R2 R 3 ⎡ R ⎤⎞ ∂ ⎢ Pz ⎛⎜ 5R C ∂h − C − C ln⎢ C ⎥⎟⎟⎥ + ⎜ ⎣ R ⎦⎠⎥⎦ 2 3 ∂z ⎢⎣ μ ⎝ 18 ∂t

(9)

P ⎛ 13R C3 R 3 ⎡ R ⎤⎞ R4 + zz ⎜⎜ − + R CR2 + + C ln⎢ C ⎥⎟⎟ = 0 ⎣ R ⎦⎠ 4μ ⎝ 36 4R C 3

(10)

Here the constitutive relation,τ = μ(∇v + ∇vT), represents a Newtonian liquid film where v[vr, vz] is the velocity vector. The Maxwell’s stresses because of the external electric field is expressed as, Mi = ε0ε[Ei·Ei − 0.5(Ei·Ei)I], where i = 1 and 2 represent the liquid and air. Since we assume the liquid to be perfectly dielectric, electro neutrality prevails within the bulk (∇·M = 0) of the film. The viscous film is assumed to be perfectly bonded (v = 0) to the rigid fiber surface (r = R). The normal [−p0 + n·M2·n + p − n·(τ + M1)·n = γκ] and the tangential [t·M2·n = t·(τ + M1)·n] stress balances, and the kinematic condition [∂h/∂t + vz∂h/∂z = vr] are enforced as boundary conditions at the liquid−air (r = R + h) interface. Here p0 is the ambient pressure and κ (=∇·n) is the curvature of film surface.

(13)

Here RC = R + h is the variable combined thickness of the film and fiber. The boundary conditions employed to derive the eq 13 are, at the solid−liquid interface (r = R) no-slip and impermeability (vz = vr = 0); at liquid−air interface (r = RC) the shear, μ(∂vz/∂r) = 0, and normal, P + π + γκ = 0, stress balances. In eq 13, the expression for pressure is obtained from the normal stress balance, ⎛ ∂ 2h 1 ⎞ ⎟⎟ P = p0 − γ⎜⎜ 2 − RC ⎠ ⎝ ∂z +

III. LINEAR STABILITY ANALYSIS In the general linear stability analysis (GLSA), the EHD governing equations and the boundary conditions are linearized by the normal linear modes, vr = ṽreωt+ikz, vz = ṽzeωt+ikz, p = p0 + p̃eωt+ikz, and h = h0 + δ̃eωt+ikz, where the perturbed variables ṽr, ṽz, and p̃ are f(r). The symbols δ̃ denote infinitesimal perturbation of height at the interface. Eliminating the perturbed pressure p̃ from the resulting linear governing equations the following fourth order ODE is obtained

+

2

2

2(R C) (ln[R C/R ] + ε ln[d /R C]) 3B

+

A 6πh3

(d − R C)4

(14)

where A and B denotes Hamaker constant and Born Repulsion coefficient. In eq 14, the terms from the left represent ambient pressure, curvature, electric field, van der Waals force, and Born repulsion, respectively. The radial curvature force is expected to break the liquid film into droplets, which can lead to the contact line singularities (h → 0) on the fiber surface. A repulsive van der Waals term in eq 14 ensures the removal of contact line singularity by means of an ultrathin precursor at the zones where film dewets the fiber surface. The Born repulsion term in eq 14 regularizes the contact line singularity that can appear when the ridges touch the outer electrode (d − RC → 0). The expression for B is obtained from the condition π = 0 when d − RC = l0, where l0 is the equilibrium cutoff thickness.43 Equations 13 and 14 is linearized employing the normal linear mode, h = h0 + δeωt+ikz, to obtain the following long-wave (LWLSA) dispersion relation

⎤ ⎡ d 1 d ⎛ d ⎛ 1 d(rv ̃ ) ⎞⎞ ⎛ d 1 d(rvr̃ ) ⎞ r ⎟ ⎢ ⎜r ⎜ ⎟ + k 4vr̃ ⎥ = 0 ⎟ − 2k2⎜ ⎢⎣ dr r dr ⎝ dr ⎝ r dr ⎠⎠ ⎝ dr r dr ⎠ ⎦⎥ (11)

The general solution for eq 11 is vr̃ = C1rK 0[kr ] + C2K1[kr ] + C3rI0[kr ] + C4I1[kr ]

ε0ε(ε − 1)ψ 2

(12)

where I1 and K1 (I0 and K0) are the first order (0th order) modified Bessel functions of the first and second kind. As a consequence of the linearization of the governing equations, now all the perturbed variables can be expressed in terms ṽr. Expressing the perturbed variables (ṽz and p̃) in terms of ṽr in the linearized boundary conditions leads to a set of four homogeneous linear algebraic equations involving four unknown constants Ci (i = 1 to 4). Equating the determinant of the coefficient matrix of the algebraic equations to zero leads

⎛ ⎛ ⎞ γk2 1 2 ⎟ k ω = − ⎜⎜ ⎜⎜−γk 4 + − π | h h 0⎟ R C02 ⎠ ⎝μ ⎝ ⎞ ⎛ 3R 3 R R2 R 3 ⎡R ⎤ R4 ⎞⎟⎟ × ⎜⎜ C0 − C0 − C0 ln⎢ C0 ⎥ + ⎟ ⎣ R ⎦ 16R C0 ⎠⎟ 4 4 ⎝ 16 ⎠ (15) 6217



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The dimensional eqs 13 and 14 are made dimensionless with the help of following dimensionless variables to perform the nonlinear simulations ⎛ ε ε(ε − 1)ψ 2 ⎞1/2 h R ⎟⎟ z , , R̅ = , Z = ⎜⎜ 0 H= h0 h0 2γh03 ⎝ ⎠ Π=

⎛ ε ε(ε − 1)ψ 2 ⎞2 t ⎜⎜ 0 ⎟⎟ , T π = 2 2γh03 0.5ε0ε(ε − 1)ψ ⎝ ⎠ μ/γ h02

The resulting nondimensional evolution equation is ⎤ ⎡ 3 R̅ CR̅ 2 R̅ C3 ⎡ R̅ C ⎤⎞ ∂M̅ ⎥ ∂ ⎢⎛⎜ 5R̅ C ∂H ⎟ ln − − = ⎢ ⎥ ⎣ R̅ ⎦⎟⎠ ∂Z ⎥⎦ 2 3 ∂Z ⎢⎣⎜⎝ 18 ∂T ⎛ 13R̅ 3 R̅ R̅ 2 R̅ 3 ⎡ R̅ ⎤⎞ ∂ 2M̅ R̅ 4 C + ⎜⎜ − + C + + C ln⎢ C ⎥⎟⎟ 2 ⎣ R̅ ⎦⎠ ∂Z 4 12 16R̂ ⎝ 144 (16)

where R̅ C = R̅ + H, M̅ = (∂ H/∂Z ) − (1/R̅ C) + Π. Equation 16 is discretized using a central difference scheme in space with half node interpolation. The resulting set of coupled stiff ordinary differential equations is time marched employing Gear’s algorithm from the NAG libraries. The simulations are initialized with a volume preserving small amplitude random perturbation at the liquid−air interface. The domain size is chosen as the multiples of the dimensionless dominant linear length scale (Λ) and periodic boundary conditions are enforced at the spatial boundaries. The numerical accuracy and convergence is ensured by varying the number of grid points. In most of the cases, ∼200 grid points per Λ is found to provide satisfactory results. 2

2

Figure 2. Results from the general (GLSA, solid lines) and long-wave (LWLSA, broken lines). The curves 1 (1a) and 2 (2a) correspond to ψ = 20 and 70 V, respectively. Plots (a) and (b) show variation ω with k. In plot (a) h0 = 100 nm. In plot (b) the curves 1−3 correspond to R = 1000 nm, 750 and 250 nm at h0 = 100 nm, respectively. The plots (c) and (d) show the variations of ωm and λm with h0/R where h0 = 100 nm. The plots (e) and (f) show the variations of ωm and λm with h0/(d − h0) where R = 1000 nm. The other necessary parameters for plots considered as, d = 1150 nm, γ = 0.015 N/m, μ = 1 Pa s, and ε = 2.5.

wavelength (λm)] with the ratio of the film thickness to the fiber radius (h0/R) and liquid to air filling ratio [F = h0/(d − h0)], respectively. We start the discussion with a comparison between LWLSA and GLSA. Although we do not report, to judge the accuracy of the LWLSA, the results are validated against the results obtained in the ref 22. The broken (LWLSA) and the solid (GLSA) lines in Figure 2a shows that results from LWLSA matches well with the GLSA at lower voltages, when the instability is indeed in the long-wave regime. However, they vary significantly at higher voltages where the electric field induced instabilities show a shorter length scale. The LWLSA fails to predict the large wavenumber unstable modes when the instability is in the smaller wavelength regime whereas the smaller wavenumber modes are predicted with similar accuracy by both GLSA and LWLSA. For example, the solid (curves 1 and 2) and broken (curves 1a and 2a) lines in Figure 2c−f depict that the LWLSA from the GLSA differ significantly especially in the domains where the instability shows a smaller wavelength (large wavenumber) under a stronger electric field. The figure clearly depicts the limitations of the LWLSA in predicting the length scales of this type of instabilities especially when the wavelength to film thickness ratio is small. The figure also highlights the importance of the GLSA in this type of analysis. Figure 2c−f also shows the sensitivity of length and time scales of the instabilities with h0/R and F. The solid lines 1−3 in Figure 2b and the curves 1 and 2 in Figure 2c together show that with increase in h0/R, the magnitude of ωm initially reduces

V. RESULTS AND DISCUSSION Figure 1 shows a schematic diagram of a thin film coated on a cylindrical fiber and under the influence of an external electric field. A finite air gap between the film and the outer confining electrode ensures that the free surface of the thin film deforms to generate patterns. Under the exposure of a weaker electrostatic field, the radial curvature force initially deforms the film surface and then causes a flow from thinner to thicker regions of the film. In consequence, the liquid film dewets the fiber surface to form drop or bead like structures. However, when the film is exposed to a strong electrostatic field the induced dipoles present in the liquid film accumulate near the free surface to generate an additional electrical stress at the liquid−air interface. Again, the electrical stress causes flow from thinner to the thicker regions of the films to form ridge like structures issuing outward from the fiber surface toward the confining electrode. When the air gap is large, the axial symmetry of the ridges can be broken to produce columnar structures. The relative strengths of the electric field and the radial curvature force can be modulated by changing the ratio of the film thickness to the fiber radius, the liquid to air filling ratio, applied voltage, and the distance between the electrodes. In what follows, we first discuss the LSA results followed by the results obtained from the nonlinear simulations. Figure 2 shows the results obtained from the long-wave (LWLSA) and general (GLSA) linear stability analysis. Figure 2a,b show the variation of linear growth coefficient (ω) with the wavenumber (k). Figure 2c,e [Figure 2d,f] show the variation in the dominant linear growth coefficient (ωm) [the corresponding 6218



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GLSA show a length scale of 1.9 μm at high h0/R. In comparison, when the voltage is increased to ∼150 V the LWLSA predicts almost double (1.1 μm) the length scale predicted by the GLSA. The plot emphasizes the importance of GLSA in predicting the length scales of the instabilities, which are not necessary long-wave under all conditions. The plots in this figure also highlight that patterns with wide variety of length scales starting from a few micrometers to a few hundred nanometers can be obtained by changing the process parameters h0/R, ψ, F, and ε. In addition, the plots confirm that miniaturization of patterns on the fiber surface is a possibility by reducing the wavelength to smaller values at larger electric field strength (high h0/R, ψ, F, and ε) or radial curvature force (high h0/R). Figure 4 shows the morphological evolution of the interface under varied conditions. We ensure that the simulations are

to reach a minimum value and then progressively increases. A thin film on a thick fiber (low h0/R) is expected deform under a dominant destabilizing electric field and a comparatively weaker radial curvature force. With progressive increase in h0/R, ωm progressively reduces because the strength of the electric field reduces. However, the reduction in fiber radius leads to an increase in the strength of the radial curvature force. Beyond a threshold value of h0/R, ωm again progressively increases under the influence of a strong radial curvature force. Curves 1 and 2 in Figure 2c clearly show that there are two distinct regimes of instabilities. In the left-hand side of the minimum, the electric field dominates and the films are expected to deform into ridges. In contrast, the films are expected to form beads under the dominant radial curvature force when the system parameters are similar to the right-hand side of the minimum. The curves 1 and 2 in Figure 2d show that the change in the strength of the electric field and the radial curvature force also changes the length scale significantly. The curves show that λm passes through maxima as the transition from electric field induced to radial curvature dominated instability takes place. The strength of the destabilizing field can also be tuned by changing the air to liquid filling ratio F. Figure 2e,f show that with increase in F as the strength of the destabilizing electric field progressively increases, the time (increasing ωm) and length scales (decreasing λm) of instability reduce. Figure 2d,f highlight that even under moderately strong destabilizing field the thin films can deform into submicrometer morphologies on a fiber surface. Figure 3 shows contours λm in the h0/R − ψ plane (Figure 3a,b) and F−ε plane (Figure 3c,d). Figure 3a,c correspond to the

Figure 4. Nonlinear simulations for a 100 nm thick film when (a) R = 1000 nm and d = 1300 nm, (b) R = 500 nm and d = 800 nm, and (c) R = 100 nm and d = 550 nm. The domain size chosen is 3Λ and the applied voltage is ψ = 20 V. The dimensionless times for the curves 1−4 in the plot (a) T = 0.00732, 0.00855, 0.00953, and 106; plot (b) T = 0.496, 0.588, 0.652, and 0.698; plot (c) T = 125, 154, 169, and 178. The other parameters for plots are γ = 0.015 N/m, μ = 1 Pa s, and ε = 2.5.

carried out under the conditions where the instability is indeed long-wave. Figure 4a shows that in response to a random initial perturbation at the liquid−air interface, a thin film on a thick fiber (lower filling ratio, F) can deform into an array of ridges when exposed to a strong destabilizing electric field. The spacing between the ridges follows exactly the linear length scale. The figure also shows that in conjunction to the ridge formation, the regions with thinner films adjacent to the ridges can simultaneously dewet the fiber under the weak influence of the radial curvature force. Thus, the final morphology composed of an array of microridges issuing out of the fiber surface. At late time the symmetry broken ridges are expected to form columns under this condition. Figure 4a shows an example where electric field induced ridge formation and radial curvature induced dewetting can be simultaneously observed. The ridges develop at the primary phase of instability whereas at the late stage dewetting takes place as a secondary instability. It may be noted here that the term dewetting corresponds to the liquid−air interface touching the fiber surface because of

Figure 3. Plots show the contours of λm with various parameters. Plots (a) and (b) correspond to h0/R vs ψ plane. Plots (c) and (d) show the h0/(d − h0) vs ε plane. Plots (a) and (c) correspond to LWLSA whereas plots (b) and (d) depict GLSA. The other necessary parameters for plots considered as, h = 100 nm, R = 1000 nm, d = 1150 nm, γ = 0.015 N/m, and μ = 1 Pa s.

LWLSA whereas Figure 3b,d show the results from GLSA. The figure shows the LWLSA and GLSA predict similar length scales for instability when the strength of the electric field is low. However, they differ significantly when the applied voltage is on the higher side. For example, the Figure 3a,b show that when the applied electric field is of 10 V both the LWLSA and 6219



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electric field. A general linear stability analysis has been performed to predict accurately the length scale of these instabilities than the available long-wave analysis.22 The linear analysis complemented by the nonlinear simulations shows two different regimes of instabilities dominated by the radial curvature and electrostatic forces in which the ridge like (bead) morphologies are favored when the electric field (radial curvature) is dominant. Possibility of a mixed hierarchical morphology composed of both beads and ridges are also shown when these two forces are of similar strength. A parametric analysis uncovers that the ridges (beads) are preferred morphology at high (low) filling ratio, higher (lower) applied voltage, and larger (smaller) fiber radius. The mixed morphology is found to possess a larger length scale compared to the beads and the ridges. The study highlights that controlling the strength of these destabilizing forces a wide range of patterns with mean spacing from a few hundred nanometers to a few micrometers can be decorated on the fiber surface. The results shown here can be useful in developing superhydrophobic/selfcleaning threads, micro/nano channels on curved surfaces, fibers with rough surfaces, and micro/nano rotors/impellers. The analysis shown here can be extended to predict the length and time scales of bead formation on fibers during electrospinning or spider silk formation by considering the threedimensional governing equation and including symmetry breaking azimuthal linear perturbations. However, this is beyond the scope of the present study and left as a future scope of research work.

the rapid thinning of the liquid film. The simulations performed here are expected to have an ultrathin film precursor layer at the dewetted zones because of the van der Waals repulsion in the potential (eq 14). However, the larger ordinate ranges make the precursor invisible in the figures. Figure 4b shows that when the electric field and the radial curvature forces are equally strong a part of the film can dewet the fiber under the destabilizing radial curvature force whereas the other part simultaneously can form ridges under the destabilizing electric field. Thus, a mixed morphology composed of both beads and ridges can be observed under this condition. The patterns shown in Figure 4b are very similar to hierarchical patterns with beads between the ridges or ridges in between the beads over fiber surface. As observed for many biological surfaces, hierarchical patterns on curved surfaces can be useful in the fabrication of superhydrophobic or self-cleaning surfaces. Figure 4c shows that when the destabilizing curvature force is the dominant the liquid film dewets the fiber surface to form a periodic string of beads on the fiber surface. The bead structures closely resemble the patterns on the fiber surface during electro spinning, spider silk formation, or when morning dew deposits on the spider net. These periodic columnar, bead or mixed hierarchical morphologies protruding outward from the fiber surface can also be a useful in fabricating micro/nano fluidic mixers where the fibers with these patterns can be employed as surfaces with baffles or rotors inside micro/nano channels. Figure 5 shows a phase diagram where the color map shows the conditions under which the beads, ridges, and the mixed

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ASSOCIATED CONTENT

S Supporting Information *

Steps for the linear stability analysis have been provided in the text. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +91 361 258 2254. Fax: +91 361 258 2291. Notes

The authors declare no competing financial interest.

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Figure 5. Phase diagram shows the variations of λm with ψ and R keeping h0 = 100 nm from GLSA. The color map from nonlinear simulations identifies the zones of morphologies with beads, ridges and a mixed morphology. The other parameters for plots are h = 100 nm, d = 1150 nm, γ = 0.015 N/m, μ = 1 Pa s, and ε = 2.5.

ACKNOWLEDGMENTS D.B. acknowledges initiation grant from IIT Guwahati and FastTrack research grant SR/FTP/ETA-091/2009 from DST India. Discussions with Prof. Ashutosh Sharma, Prof. V. Shankar, and Dr. Gaurav Tomar are gratefully acknowledged.

morphologies can be expected. A series of nonlinear simulations have been performed in order to obtain this parametric color map. The map is then superimposed on the surface contour, which shows that variation of λm with R and ψ, obtained from the GLSA. The plot clearly shows that at lower (higher) voltages and smaller (larger) fiber radius beads (ridges) can be formed on the fiber surface whereas a mixed morphology can be observed at the intermediate voltages and fiber radius. The wavelength of the structures for the mixed morphologies is mostly found to be larger than when only beads or ridges are formed.

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REFERENCES

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VI. CONCLUSIONS We discuss an interesting transition from ridges to beads for a thin film resting on a cylindrical fiber and under the influence of 6220



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