# Static and Dynamic Optical Analysis of Micro Wrinkle Formation on a Liquid Surface

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^{2}

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## Abstract

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## 1. Introduction

^{1/2}, where Δρ is the difference in density between the liquid and air phases [12]. Distortions in which Lc > p are dominated by surface tension driven levelling and distortions for which Lc < p are primarily levelled by gravitational effects [11].

_{ave}. Since the magnitude of the electric field is at a maximum above the gaps between the electrode fingers, this also results in the liquid preferentially collecting within these regions and so the liquid-air interface develops a periodic “wrinkle” deformation, i.e., a corrugation, with pitch determined by the underlying electrode structure [16].

_{ave}< p < L

_{C}. We demonstrate that Optical Coherence Tomography (OCT) imaging can be used to make precise measurements of a static wrinkle deformation profile on the liquid surface in equilibrium, with sub-micrometer axial resolution.

## 2. Materials and Methods

^{−2}, a surface tension of $\gamma $ = 0.043 ± 0.001 N m

^{−1}, a refractive index of ${n}_{TMP-TG-E}=$ 1.477 at a wavelength of 532 nm, and a mass density of 1157 ± 10 kg m

^{−3}[34]. With no applied voltage, TMP-TG-E formed a spherical cap shaped droplet on the SU-8 coated substrates with an equilibrium contact angle of 30 ± 2°. All experiments were performed on the open bench in a temperature-controlled laboratory (21 ± 1 °C).

_{C}≈ 1950 μm >> p, indicating that when we subsequently significantly reduce the voltage, the levelling of these induced surface wrinkle distortions will be dominated by capillary forces. Dielectrophoresis forces are independent of polarity and so, in theory, either D.C. or A.C. voltages can be used create the wrinkle deformation. However, in practice A.C. voltages are used to the avoid the D.C. and low frequency shielding effects of the field induced migration of free charges that are inevitably present even in the low electrically conductivity liquid TPM-TG-E. We used voltages of 1 kHz (Section 3.1) and 2.5 kHz (Section 3.2) which provides a suitable compromise that avoids low frequencies that would have caused charge migration and would also have significantly modulated the wrinkle amplitude during each half period, and also avoids high frequencies where the finite slew rate of the amplifier and the finite conductivity of the indium tin oxide electrodes would have caused signal losses.

## 3. Results

#### 3.1. Static Equilibrium Profile of the Liquid Surface Wrinkle

_{ave}that have been defined in the Experimental section and in Figure 1 of this paper. This analytical expression was derived by assuming a spatially periodic potential at z = 0 that is a sinusoidal function of x and that $h\left(x\right)={h}_{ave}+\left(A/2\right)\mathrm{cos}\left(kx\right)$, with the condition that $2\pi A/p\ll 1$. This expression predicts that the voltage scaled wrinkle peak to trough amplitude will be given by $A/{V}_{o}^{2}$ = 0.86 × 10

^{−10}mV

^{−2}for the data shown in Figure 4a, for which the experimentally measured amplitudes are in the range (0.91–1.06) × 10

^{−10}mV

^{−2}, and by $A/{V}_{o}^{2}$ = 1.87 × 10

^{−10}mV

^{−2}for the data shown in Figure 4b, for which the experimentally measured amplitudes are in the range (1.85–2.17) × 10

^{−10}mV

^{−2}. Whilst this analytical expression should not be expected to provide precise quantitative predictions, due to the simplified model of the non-uniform electric fields in the system, the sinusoidal profile approximation, and due to the presence of a thin solid dielectric protective film over-coating the electrodes, we do find very good correspondence between the predicted voltage-scaled wrinkle amplitude values and the values measured from our OCT results. The results from OCT imaging shown in Figure 4 illustrate how OCT techniques can be used to obtain high axial resolution profiles of micrometer scale features on a liquid-air surface, across a range of liquid-air wrinkle amplitudes spanning from $A$ = 0.24 μm (100 V, Ω = 1.48 ± 0.05 μL) up to $A$ = 4.42 μm (300 V, Ω = 0.85 ± 0.05 μL).

#### 3.2. Dynamic Growth and Decay of the Liquid Surface Wrinkle

_{ave}during the measurements. In addition, the selected parameters allow sufficient time for the measurement of growth to saturation and decay to negligible peak to trough amplitude of the surface corrugation during each 5 ms time period. At the 46 V minimum voltage, the equilibrium value of the wrinkle amplitude was sufficiently negligible to enable study of levelling, whilst this value also acted effectively to prevent the liquid film from beginning to de-wet and hence avoided any significant changes in the overall shape of the liquid film during the measurements. The efficacy of this approach was confirmed during the measurements using a CMOS camera (DCC1545M, Thorlabs, Ely, UK) fitted with a 5× objective lens and a 150 mm tube to image the edge of liquid film in the y-direction to monitor the overall liquid film shape and the value of ${h}_{ave}$ at the position on the liquid film at which the collimated incident laser light was transmitted and diffracted. This imaging set-up is shown in Figure 1.

^{−7}m(t) metres, for transmitted laser light of wavelength ${\lambda}_{air}$ = 532 nm, with $\Delta n=\left({n}_{TMP-TG-E}-{n}_{air}\right)=$ 0.477, and assuming a spatially periodic wrinkle profile and commensurate optical path variation as given in Equation (1). Taking the measured 0th order diffraction data ${I}_{o}\left(t\right)$ shown by the open circles in Figure 5a as an example, the solid line through the data shows the fit to the square of the Bessel function of first kind, ${J}_{0}^{2}\left(m\left(t\right)\right)$, to the normalised data, ${I}_{o}\left(t\right)$, which yielded the time-dependent function $m\left(t\right)$, and hence the plot of $A\left(t\right)$ against time $t$ shown by the solid line in Figure 6a. Extracting $m\left(t\right)$ from fitting ${J}_{0}^{2}\left(m\left(t\right)\right)$ to the experimental data is an inverse problem, and the fitting function has an oscillatory dependence on its argument, $m\left(t\right)$. We obtained the fit in a straightforward manner by considering each time interval between adjacent maxima and minima in turn.

^{−4}s, 3.496 × 10

^{−4}s, and 3.456 × 10

^{−4}s (±0.002 × 10

^{−4}s) for the curves ${A}_{\mathrm{on},i=0}\left(t\right)$, ${A}_{\mathrm{on},i=1}\left(t\right)$ and ${A}_{\mathrm{on},i=2}\left(t\right)$, respectively. The voltage was then abruptly decreased from 293 V to 46 V, and Figure 6b shows the wrinkle amplitude $A\left(t\right)$ responding by decaying steeply from initial equilibrium amplitude $A\left(t=0\right)\approx $ 3.6 μm, at $t=$ 0 s towards the equilibrium asymptote of $A\left(t\to \infty \right)\approx $ 0.1 μm. Here capillary driven levelling of the surface of the liquid film, driven by the Laplace pressure associated with surface tension forces, is resisted by the liquid viscosity. Figure 6b shows very similar shapes, as should be expected, for the three curves $A\left(t\right)$ derived from the separate fits to the 0th order diffracted order (${A}_{\mathrm{off},i=0}\left(t\right)$, solid line), the 1st order (${A}_{\mathrm{off},i=1}\left(t\right)$, dashed line), and the 2nd order (${A}_{\mathrm{off},i=2}\left(t\right)$, dotted line). The inset graph in Figure 6b compares these three curves with the exponential decay function,$A\left(t\right)=A\left(t\to \infty \right){e}^{-t/{\tau}_{\mathrm{off}}}$. The linear dependence demonstrates that the exponential decay provides an excellent description of the time dependence of the wrinkle levelling. Linear regression analysis gave ${\tau}_{\mathrm{off}}$ = 4.410 × 10

^{−4}s, 4.438 × 10

^{−4}s, and 4.394 × 10

^{−4}s (±0.003 × 10

^{−4}s) for ${A}_{\mathrm{off},i=0}\left(t\right)$,${A}_{\mathrm{off},i=1}\left(t\right)$ and ${A}_{\mathrm{off},i=2}\left(t\right)$, respectively.

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic diagram of the experimental system and the device substrate geometry. The Optical Coherence Tomography (OCT) apparatus is shown in detail in the schematic diagram in subsequent Figure 2. The dynamic optical diffraction measurements with auxiliary side-view CMOS camera imaging were performed separately to the OCT measurements.

**Figure 2.**(

**a**) Schematic diagram of the Optical Coherence Tomography (OCT) imaging apparatus that was used to measure the equilibrium surface height profile h(x) of thin films of the liquid TMP-TG-E on the device substrate; (

**b**) A raw “BScan” image of the TMP-TG-E film obtained using the OCT apparatus with a constant static A.C. voltage ${V}_{R.M.S.}$ = 250 V (1 kHz) applied to the electrodes of the device.

**Figure 3.**Plot of the static equilibrium surface height profile h(x) of two different spread thin films of the liquid TMP-TG-E, measured using Optical Coherence Tomography imaging. Two different volumes of liquid were dispensed onto the substrate, 1.48 ± 0.05 μL (upper plot) and 0.85 ± 0.05 μL (lower plot). In each case, a constant A.C. voltage ${V}_{R.M.S.}$ = 200 V (1 kHz) applied to the electrodes resulted in a spread liquid film that exhibited a static periodic surface wrinkle deformation.

**Figure 4.**Plot of the static equilibrium surface height profile h(x) of two different spread thin films of the liquid TMP-TG-E measured using optical coherence tomography imaging, zoomed in from Figure 3. Profiles are shown on films formed from two different dispensed volumes of liquid: (

**a**) 1.48 ± 0.05 μL, and (

**b**) 0.85 ± 0.05 μL. The equilibrium periodic wrinkle deformation profiles are shown for 5 different voltages (R.M.S. 1 kHz) applied to the electrodes under the liquid, ${V}_{R.M.S.}$ = 100, 150, 200, 250, 300 V. The average vertical positions of the profiles are offset for clarity.

**Figure 5.**The solid lines show the measured time dependent intensity of laser light at 532 nm diffracted into the 0th, 1st and 2nd orders, by transmission through a dielectrophoresis induced periodic wrinkle deformation on a TMP-TG-E liquid film of thickness ${h}_{ave}$ = 40 ± 1 µm. The 1st and 2nd order plots use the same vertical scale as the 0th order, but are shifted upwards by 0.35 and 0.70 units, respectively, for clarity. In plot (

**a**) the 2.5 kHz A.C. sinewave driving voltage had been increased abruptly from 46 V to 293 V at time $t=0$, and in plot (

**b**) the voltage value had been decreased abruptly from 293 V to 46 V at time $t=0$. The open symbols show the fits to squares of Bessel functions of first kind, ${J}_{i}^{2}\left(m\right)$, with $i=0$ (circles), $i=1$ (squares), and $i=2$ (diamonds).

**Figure 6.**The measured time dependence of the peak to trough amplitude $A\left(t\right)$ of a growing (

**a**) and decaying (

**b**) periodic wrinkle deformation on a TMP-TG-E liquid film of thickness ${h}_{ave}$ = 40 ± 1 µm. $A\left(t\right)$ was extracted from fitting to the optical diffraction data shown in Figure 5, with plot (

**a**) derived from the fits shown in Figure 5a, and plot (

**b**) derived from the fits shown in Figure 5b, where the solid line shows the fit to the 0th order, the dashed line the 1st order, and the dotted line the 2nd order. The inset graphs show fits of the $A\left(t\right)$ results (same time axis) to, (

**a**) an exponential rise to equilibrium function, and (

**b**) to an exponential decay function, with $A\left(t\right)$ functions from the 1st and 2nd orders displaced vertically by 0.1 and 0.2 units, respectively, for clarity.

**Figure 7.**Measured time constant values for the exponential rise of the amplitude $A\left(t\right)$ of a liquid surface wrinkle towards equilibrium, ${\tau}_{\mathrm{on}}$ (filled diamonds), and for the exponential decay of the wrinkle amplitude, ${\tau}_{\mathrm{off}}$ (open diamonds), for periodic wrinkle deformations of pitch $p$ = 160 μm on different thickness (${h}_{ave}$) TMP-TG-E liquid films. The solid line shows the theoretical prediction by Orchard [11], ${\tau}_{\mathrm{off}}\left(k{h}_{ave}\right)$, for the surface tension driven levelling of a liquid surface corrugation, where $k=2\pi /p$ is a constant. The dashed line shows our theoretical prediction of the wrinkle growth time constant, ${\tau}_{\mathrm{on}}\left(k{h}_{ave}\right)$, from numerical solution of the equations of Stokes flow with electrostatic forcing.

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**MDPI and ACS Style**

Saxena, A.; Tsakonas, C.; Chappell, D.; Cheung, C.S.; Edwards, A.M.J.; Liang, H.; Sage, I.C.; Brown, C.V.
Static and Dynamic Optical Analysis of Micro Wrinkle Formation on a Liquid Surface. *Micromachines* **2021**, *12*, 1583.
https://doi.org/10.3390/mi12121583

**AMA Style**

Saxena A, Tsakonas C, Chappell D, Cheung CS, Edwards AMJ, Liang H, Sage IC, Brown CV.
Static and Dynamic Optical Analysis of Micro Wrinkle Formation on a Liquid Surface. *Micromachines*. 2021; 12(12):1583.
https://doi.org/10.3390/mi12121583

**Chicago/Turabian Style**

Saxena, Antariksh, Costas Tsakonas, David Chappell, Chi Shing Cheung, Andrew Michael John Edwards, Haida Liang, Ian Charles Sage, and Carl Vernon Brown.
2021. "Static and Dynamic Optical Analysis of Micro Wrinkle Formation on a Liquid Surface" *Micromachines* 12, no. 12: 1583.
https://doi.org/10.3390/mi12121583