Supplementary Material on Electrically modulated dynamic spreading of drops on soft surfaces Ranabir Dey 1, Ashish Daga 1, Sunando DasGupta 2,3, Suman Chakraborty 1,3 1 Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur- 721 32, West Bengal, India. 2 Department of Chemical Engineering, Indian Institute of Technology Kharagpur, Kharagpur- 721 32, West Bengal, India. 3 Advanced Technology Development Centre, Indian Institute of Technology Kharagpur, Kharagpur- 721 32, West Bengal, India Corresponding author, E-mail: suman@mech.iitkgp.ernet.in 1
Table S1: Young s modulus ( E ), thickness of differently cross-linked Sylgard 184 dielectric films ( h ); equilibrium contact angles ( θ eq ), macroscopic advancing and receding contact angles ( θa θ r ) of 1 mm aqueous KCl solution drops, on the different dielectric films, without any electrical effects. Sylgard 184 (1:1) Sylgard 184 (3:1) Sylgard 184 (5:1) E (MPa) (refs. 33, 34) 1.5.6.2 h (μm)* 9.65 11.34 7.2 θ eq ( ) 19.24 ±.5 11.7 ± 3 113.44 ± 2.25 θ / θ ( ) 115.6/12.7 121.6/39.13 126.95/29.6 a r Note: *All films are spin-coated at- 5 rpm for 3s and then at 5 rpm for 7s (intermediate acceleration 4 rpm/s 2 ). The root mean square roughness of the different film surfaces is in the order of.33 nm. Initial (equilibrium) droplet contact radius: r c = 1.28 mm, is independent of substrate elasticity; only 2.45% deviation is observed across different substrates. Table S2: Measured (see the discussion about the results shown in Fig. 2a and Fig. 2b in the main paper) non-dimensional values of the final, macroscopic equilibrium contact angle [ θeq = θeq ( η; E) θ measured eq ] on the dielectrics of different elasticity, under different values of the applied electrical voltage; fitted (by a robust, nonlinear least-squares method) values of the same, on the soft dielectrics, as consistent with Eq. (1). The experimental and fitted values show good agreement. E (MPa) η θ ( η; E) θ θ ( η; E) θ 1.5.6.2 eq measured.25.91 NA.34.86 NA.44.81 NA.25.92.9.34.89.88.44.85.84.25.96.93.34.91.9.44.87.86 eq eq fitted eq Table S3: Physically consistent values of the different parameters obtained from the classical description (Eq. (S1) and Eq. (S2)) of the electrospreading behaviour on the apparently rigid film, forη =.44. E=1.5 MPa; η =.44 ζ (Pa s) θ dm, ( ) L ln φ l 1.64 91.34 19.99 ( L ~1.28mm) 2
FIG. S1. Detailed schematic of the experimental setup. 3
E=1.5 MPa E=.2 MPa t= t= t=2.5 ms t=25 ms t=3.75 ms t=525 ms t=6.25 ms t=2 s t=8.75 ms t=4 s t=25 ms t=6 s t=2 ms t=9 s FIG. S2. Image-sequences exhibiting the transient drop deformation, during electrospreading (at 16 V) on dielectric films having different values of the Young s modulus. 4
FIG. S3. Blow-ups (of the initial transience) of the temporal variations of (a) the nondimensional macroscopic dynamic contact angle ( θ d ), and (b) the non-dimensional contact radius ( r c ), during electrospreading at different magnitudes of the applied electrical voltage- 14 V (blue) and 16 V (black), as shown in Fig. 2(a) and Fig. 2(b) in the main paper. These blow-ups clearly demonstrate the electrospreading characteristics on the dielectric film with E = 15. MPa (1:1 Sylgard 184 film). The error-bars are shown here, as well as in Fig. 2(a) and Fig. 2(b) in the main paper, only for the data corresponding to 16 V to maintain clarity. The error-bars for the data corresponding to other magnitudes of the applied electrical voltage are comparable to this. 5
FIG. S4. The standard error of estimate of the power-law fit lies well within the experimental error (shown here forη =.44 ). This further substantiates the fact that the evolution of the non-dimensional contact radius, during electrospreading on soft dielectric films, conforms to a universal power-law in a non-dimensional time, based on the surface elasticity. 6
3:1 (E=.6 MPa) 3:1 (E=.6 MPa) 5:1 (E=.2 MPa) 5:1 (E=.2 MPa) 3:1 (E=.6 MPa) 5:1 (E=.2 MPa) FIG. S5. Bar graphs showing the values (evaluated by a robust, nonlinear least-squares method) of the non-dimensional co-efficients- (a)( ζv ) γ, (b) ( 3γ) ( 2πEδ ), and (c) the power-law exponent for viscoelastic dissipation, n, as obtained from Eq. (1), for electrospreading on the soft dielectrics, under different strengths of the electrical actuation (represented by η ). These bar graphs indicate that the non-dimensional co-efficients and n, as evaluated for the general electrospreading behaviour on soft substrates ( Ο( γ Er c ) 4 1 ) represented by Eq. (1), are approximately constants, independent of the Young s modulus and the electrical voltage. 7
3:1 (E=.6 MPa) 5:1 (E=.2 MPa) 3:1 (E=.6 MPa) 5:1 (E=.2 MPa) FIG. S6. Bar graphs showing the values of the (a) contact line friction co-efficient, ζ and (b) radius (δ ) of the region, about the TPCL, within which the linear elasticity theory is no longer applicable, as obtained from the constant co-efficients in Eq. (1), for electrospreading on the soft dielectrics ( ( Ο γ Er ) 1 4 c ), under different strengths of the electrical actuation (represented by η ). These bar graphs indicate that ζ and δ increase with increasing substrate softness (decreasing E ), while being approximately independent of the applied electrical voltage. The invariance of ζ over a wide range of applied electrical voltage is also evidenced in the existing literature (see ref. 12) 8
Image processing for measurement of contact angle and contact radius: FIG. S7. (I) Images of the droplet on a dielectric film, with E=1.5 MPa, grabbed by the high speed camera (a) before the application of the electrical voltage, and (b) during electrospreading after the application of the electrical voltage. (II) Images of the droplet on a dielectric film with E=.2 MPa grabbed by the high speed camera (a) before the application of the electrical voltage, and (b) during electrospreading after the application of the electrical voltage. The blue knotted lines, shown in these images, represent the fit of the droplet contour, as generated by ImageJ. The corresponding contact angles ( θ eq or θ d ) and the contact radii ( r c ), as subsequently measured by ImageJ, are also shown here. The blow-up in Ib shows that the droplet and the dielectric film with E=1.5 MPa form an apparently sharp (undeformed) solid-liquid-air interface (TPCL; red dashed line). However, the blow-up in IIb depicts a capillarity-induced deformation of the film surface (with E=.2 MPa) about the TPCL, under identical conditions. 9
High speed imaging and image processing: The images of a sessile droplet undergoing electrospreading on a dielectric film, with a definite value of E, are recorded by a high speed camera in a setup identical to that shown in Fig. S1. It must be noted here that the dropletand-dielectric system, in between the light source and the camera lens, is oriented in such a manner that there exists a vertical plane passing though the sessile droplet centre, which is almost parallel to the camera lens (a small tilt is however maintained). Fig. S7(I) and Fig. S7(II) show sample images captured during droplet electrospreading on the dielectric film with E=1.5 MPa, and on the dielectric film E=.2 MPa respectively. The image sequence obtained for each experiment, for each substrate-voltage combination, is then processed using ImageJ (see ref. 36) to evaluate the macroscopic dynamic contact angle ( θ d ) and the contact radius ( r c ). ImageJ uses a plugin to measure θ d by using a piecewise polynomial fit of the droplet contour (see the ImageJ generated fit (blue knotted line), and the corresponding measured contact angle in each of Fig. S7(I) and Fig. S7(II)). For measuring r c, ImageJ identifies the solid liquid interface, in the captured image, by fitting the droplet contour and by identifying the beginning of the droplet reflection on the substrate (see the yellow line marking the solid-liquid interface in each of Fig. S7(Ib) and Fig. S7(IIb)). Any inadvertent variation in the initial droplet condition (due to droplet placement or slight change in droplet volume) does not affect the final results/conclusions, as the data are all presented in nondimenisonalized form as d = d eq θ θ θ and c c c r = r r. The aforementioned methodology for contact angle and contact radius measurements, by using ImageJ, is devoid of any general assumption, and hence, is applicable for all types of sessile droplets (even non-axisymmetric ones). Moreover, this method for contact angle and contact radius measurements is prevalent in the existing literature (see refs. 14 and 16). For further details on the ImageJ measurement of the contact angle see ref. 36. It is instructive to note here that the contact line velocity is evaluated by subsequent numerical differentiation of the contact radius data with respect to time. High speed imaging with microscopy lens: The blow-ups in Fig. S7(Ib) and Fig. S7(IIb) (which are the same images shown in Fig. 1(b) in the main paper) are recorded with the high speed camera fitted with a microscopy lens (see Fig. S1 for the lens details). During this recording, the microscopy lens is focused on a small portion of the droplet near the three phase contact line (TPCL). It must be noted here that for acquiring these images the dropletand-substrate system is tilted towards the lens, so that the TPCL (red dashed line), the 1
dielectric surface, and the air-liquid interface (blue dashed line) are clearly visible. These microscopy images are shown here (and in the main paper) solely to give a qualitative idea about the difference in behaviours of the dielectric film surfaces about the TPCL, during electrospreading, with decreasing value of E. These images are not used for the quantitative measurement of θ d. The blow-up in Fig. S7(Ib) shows that the droplet and the dielectric film with E = 1.5 MPa form an apparently sharp (undeformed) solid-liquid-air interface (TPCL; red dashed line). Alternatively, it can be said that the air-liquid interface (blue dashed line) and the TPCL (red dashed line) form an apparent wedge-shaped structure in the image. On the other hand, Fig. S7(IIb) shows that under identical conditions, there is a local deformation of the film surface, with E =.2 MPa, about the TPCL. The interplay of such substrate elasticity dependent film surface deformation and the interfacial electrostatic forces alters the electrospreading characteristics from that established for rigid films. This is substantiated by the difference between the average θd and r c characteristics on the soft films, as measured by using ImageJ from images such as those shown in Fig. S7(IIb), and the same on the apparently rigid film. 11
Classical description of electrospreading dynamics on rigid dielectric films: During electrospreading process on non-deformable substrates, the spreading dynamics in the regime of high contact line velocity ( v cl ) is predominantly dictated by the dynamic energy balance between the excess free energy and the dissipation due to the threephase contact line (TPCL) friction (see refs. 12-16). Consequently, this dynamic energy 2 balance can be written as: γ cosθeq ( V ) cosθd vcl = ζvcl, where θd () t is the macroscopic dynamic contact angle, θ ( V ) is the electrostatic energy-dependent, final macroscopic, eq equilibrium contact angle (see ref. 4), γ is the liquid surface tension andζ is the co-efficient of wetting line friction (see refs. 17, 18). On rearrangement, the ensuing electrospreading dynamics can be described by the following θd vs. vcl relationship (see ref. 12-16): ζ cosθd = cos θeq(v ) vcl (S1) γ On the other hand, during such electrically-mediated advancement of the liquid meniscus, the variation inθ d with lower values of v cl conforms to a hydrodynamic description. The mathematical form of this spreading dynamics physically stems from the matching of the internal Stokes flow solution in the quasi-static macroscopic region to that in the mesoscopic viscocapillary region of the electrically-mediated advancing droplet meniscus (see refs. 14, 16-18). The culminating θd vs. vcl relationship, referred to in the literature as the Cox-Voinov model, can be written as (see refs. 14, 16-18): 3 3 L θd = θd,m + 9Caln φ l μvcl Here, θd,mis the microscopic dynamic contact angle, Ca = is the capillary number, μ is γ the liquid viscosity, L is a macroscopic length comparable to the droplet size,l is a microscopic cut-off length beyond which the hydrodynamic framework is inapplicable, and φ is a problem specific numerical constant (see ref. 18). It must be noted here that the electrospreading behaviour of a millimetre-sized sessile droplet on the apparently rigid dielectric film (E=1.5 MPa) adheres to these classical descriptions (see Fig. 3(b) and its inset in the main paper), as portrayed in the literature so far (see refs. 12-16), which do not take in purview the deformability of the substrate. However, (S2) 12
Eq. (S1) and Eq. (S2) fail to describe the electrospreading behaviour on the soft dielectric films ( Ο( γ Er c ) 4 1 ). 13