Maximum Spreading of Liquid Drops Impacting on Groove-Textured

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Maximum spreading of liquid drops impacting on groove-textured surfaces: Effect of surface texture Visakh Vaikuntanathan, and D. Sivakumar Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.5b04639 • Publication Date (Web): 17 Feb 2016 Downloaded from http://pubs.acs.org on February 18, 2016

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Maximum spreading of liquid drops impacting on groove-textured surfaces: Effect of surface texture Visakh Vaikuntanathan Research scholar Department of Aerospace Engineering Indian Institute of Science Bangalore 560 012 INDIA D. Sivakumar* Associate Professor Department of Aerospace Engineering Indian Institute of Science Bangalore 560 012 INDIA

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2 ABSTRACT

Maximum spreading of liquid drops impacting on solid surfaces textured with

unidirectional parallel grooves is studied for drop Weber number in the range 1–100 focusing on the role of texture geometry and wettability. The maximum spread factor of impacting drops measured perpendicular to grooves, m, is seen to be less than that measured parallel to grooves, m,||. The difference between m, and m,|| increases with drop impact velocity. This deviation of m, from βm,|| is analyzed by considering the possible mechanisms, corresponding to experimental observations, – (1) impregnation of drop into the grooves, (2) convex shape of liquid-vapor interface near contact line at maximum spreading, and (3) contact line pinning of spreading drop at the pillar edges – by incorporating them into an energy conservation-based model. The analysis reveals that contact line pinning offers a physically meaningful justification of the observed deviation of m, from βm,|| compared to other possible candidates. A unified model, incorporating all the above-mentioned mechanisms, is formulated which predicts m, on several groove-textured surfaces made of intrinsically hydrophilic and hydrophobic materials with an average error of 8.3%. The effect of groove-texture geometrical parameters on maximum drop spreading is explained using this unified model. A special case of the unified model, with contact line pinning absent, predicts βm,|| with an average error of 6.3%.

Keywords: drop impact; groove-textured surface; maximum spreading; contact line pinning


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3 INTRODUCTION The maximum spreading of a liquid drop impacting on a solid surface is a direct measure of the maximum solid-liquid contact area and is a significant parameter in applications involving interfacial fluid flow and heat transfer such as spray cooling. It is generally quantified in terms of the maximum drop contact diameter on the solid surface, Dm and is often normalized with the drop diameter prior to impact, Do to give the maximum spread factor, βm. Many research groups have reported experimental studies of maximum spread factor of liquid drops impacting on smooth and randomly rough solid surfaces for a range of impact parameters (see the review articles by Yarin1 and Marengo et al.2 and references therein). In general, βm depends on the liquid drop properties such as surface tension, σ, dynamic viscosity, μ and density, ρ, physical characteristics (surface roughness, Ra) and chemical nature (intrinsic wetting contact angle/Young’s contact angle,

Y  cos1   SV   SL    where SV and SL are the solid-vapor and solid-liquid interfacial tensions, respectively) of solid surface often quantified in terms of a static wetting contact angle, θe of liquid drop on the surface, and drop impact velocity, Uo. Often the liquid drop parameters are grouped into dimensionless parameters such as Reynolds number, Re = ρUoDo/μ, Weber number, We = ρUo2Do/σ, and Ohnesorge number, Oh = μ/(ρσDo)0.5. Semi-empirical correlations3-5 and theoretical models6-11 have been proposed to offer a quantitative prediction of βm as a function of the above-mentioned drop impact parameters. Table I summarizes some of the relevant and widely-used models for predicting βm on smooth surfaces and their range of applicability in terms of impact parameters. It is clear from Table I that the effect of surface wetting characteristics is not explicitly taken into account in the semi-empirical correlations3-5. Current literature lacks studies which consider the effect of physical characteristics of solid surface such as surface roughness and texture geometry details on these models of βm during drop impact on rough/textured surfaces.


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4 A liquid drop impacting on a textured surface with well-defined surface features can reside in one of the following states: Cassie, Wenzel, or intermediate state in which respectively none, all, or some of the roughness valleys underneath the drop are impregnated by the impacting drop.12 An irreversible transition from Cassie to Wenzel or intermediate state caused by drop impact velocity can be detrimental in applications where superhydrophobicity is desired.13,14 Interesting instances of intermediate impregnation states in which the impregnated area forms a ring-like structure have been recently reported for water drop impact on micro-textured surfaces.13-15 Although it is known that such impregnated states could affect the receding dynamics13,14 and outcomes15 of impacting drops, their effect on maximum drop spread has not been explored. Textured solid surfaces comprising uni-directional parallel grooves, referred to as groove-textured surfaces, have the potential to be employed in practical applications involving liquid drainage from condenser surfaces, drag reduction, etc. due to their anisotropic wettability and are used to understand the effect of surface roughness geometry on drop-surface interaction process.16-21 In a previous study of drop impact on groove-textured surfaces at low to moderate range of We, it was observed that both βm,|| and βm, increase with We, with βm,|| closely following the quantitative trend of βm on smooth surface of the same material and βm,|| > βm,.22 The pinning of drop front spreading perpendicular to grooves at the edges of pillars and the absence of such pinning for drop front spreading parallel to grooves was hypothesized to be the cause of this difference between βm,|| and βm,. The difference between βm,|| and βm, was also seen to increase with We. A similar observation was also reported in the experimental study of drop impact on groove-textured surfaces by Pearson et al.23. In the present study, we focus on understanding the role of surface texture on the maximum spreading of liquid drop on groove-textured surfaces during impact. Results are presented to


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5 highlight the various possible physical events and mechanisms during drop impact on groovetextured surfaces by carrying out experiments of drop impact at different impact conditions on groove-textured surfaces varying in their texture geometry. A unified model incorporating the various physical mechanisms encountered by the impacting drop during spreading process is proposed for the determination of βm (βm, and βm,||) on groove-textured surfaces. The model is sufficiently checked with the experimental measurements of βm on both groove-textured and isotropic textured surfaces reported in literature. Further the observed trends on the effect of groovetexture on βm, are explained using the unified model. EXPERIMENTAL DETAILS Liquid drop impact experiments were conducted on groove-textured surfaces using an experimental set-up comprising of a micro-meter-syringe-needle system to dispense liquid drops of a given diameter, Do, a high speed video acquisition system comprising a high speed digital video camera (Redlake Y4) with a strobe lamp in backlighting mode, and a target surface holder to keep the target surfaces flat and directly below the drop dispenser. The inner and outer diameters of the flat-tipped hypodermic needle were 0.25 mm and 0.37 mm, respectively. The height of the needle tip from the target surface could be adjusted using a vertical traverse system so as to simulate different impact velocities, Uo of the liquid drop. Distilled water (ρ = 997 kg/m3, σ = 72.8 mN/m, and μ = 0.89 mPa-s) and ethanol-water mixture in the mass ratio 5:95 (ρ = 987 kg/m3, σ = 55.7 mN/m, and μ = 0.90 mPa-s)24 were used as liquids. The average drop diameters of distilled water and ethanol-water mixture delivered by the drop dispenser were 2.60 mm and 2.40 mm, respectively. The target surfaces comprised of groove-textured surfaces with uni-directional parallel grooves of well-defined geometry and spatial uniformity. Figure 1 shows a schematic of groove-textured surface highlighting the relevant geometrical parameters. Different groove-textured surfaces were


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6 made using stainless steel (intrinsically hydrophilic) and PDMS (intrinsically hydrophobic) materials differing in the geometrical parameters of the groove-texture. The morphological and geometrical parameters of the surface texture were extracted from scanning electron microscopy (SEM, Quanta), optical profilometry (Wyko), and video microscopy (Keyence) measurements. The geometrical parameters of the groove-texture, as defined in Fig. 1, extracted from these measurements are also given in Fig. 1 along with secondary geometrical parameters calculated from these measurements: solid fraction,  and roughness factor, r defined as follows.



b wb

(1)

 2d  1  cos   r  1     w  b  sin  

(2)

Here b is the pillar top width, w is the groove top width, d is the groove depth, and  is the pillar side angle as defined in Fig. 1. The wetting characteristics of groove-textured surfaces relevant in the current study, quantified by Young’s contact angle, θY, equilibrium contact angle, θe and advancing contact angle, θa of drop liquids on the target surfaces, were measured from static wetting experiments. Details of the contact angle measurement methodology can be found in Ref. 25. These values, relevant in theoretical model predictions, are given in Fig. 1. Drop impact experiments were conducted for We in the range 1-100. High speed videos of the entire drop impact dynamics perpendicular and parallel to the grooves were captured from separate side views of drop impact under identical conditions with three experimental trials each. The maximum drop spreading on target surfaces were measured from images extracted from such high speed videos for all drop impact conditions studied here. EXPERIMENTAL OBSERVATIONS


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7 Figure 2 shows the image sequences, obtained from high speed video, highlighting the spreading process of distilled water drop with DO = 2.60 mm and UO = 0.44 m/s (first row) and DO = 2.59 mm and UO = 0.88 m/s (second row) perpendicular to the grooves on TS220. The first row of Fig. 2 shows that the impacting drop spreads perpendicular to the grooves without any drop impregnation into the grooves of the target surface (unfilled grooves seen as bright portions at the bottom of drop). As the impact velocity is increased (second row), the drop is seen to impregnate the grooves in the vicinity of impact location which remain filled during the entire spreading process. Hence the drop configuration at maximum spreading on the target groove-textured surface could be one of the following – Cassie-Baxter (all grooves underneath the drop remain unfilled by drop), Wenzel (all grooves underneath the drop are filled by drop), or intermediate (some of the grooves underneath the drop near the impact location are filled by drop) – depending on whether the impact velocity is less or more than the critical impact velocity of transition corresponding to the ‘liquid drop–target groove-texture’ combination.12,26-28 During the process of drop spreading perpendicular to grooves, the drop front gets pinned at the edge of solid pillars which prevent the advancing motion of three phase contact line (TPCL).22,25,29 This causes the drop contact angle to increase which is seen as an increase in the convexity (bulging) of the advancing drop front (see 3rd, 4th and 5th images in top row of Fig. 2). Until the drop contact angle reaches a particular value (given by the advancing contact angle, θa), the TPCL remains pinned at pillar edges squeezing the drop front perpendicular to the grooves. This results in a reduced maximum drop spreading perpendicular to the grooves compared to that parallel to grooves (see Fig. 3 (a) and (b)) since pinning edges are absent. Such anisotropy in drop spreading resulting from the pinning of drop front spreading perpendicular to grooves is also reported in the case of static drop wetting (no impact velocity) on groove-textured surfaces.16,19,20 The conventional energy


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8 conservation models for predicting βm does not take this effect of the TPCL pinning on maximum spreading perpendicular to grooves into account. Further, it is evident that the drop front at maximum spreading perpendicular to grooves is convex in shape (last image in Fig. 2) which deviates from the assumption of a cylindrical disc shape (with flat drop front) in conventional energy conservation models for predicting βm.7-9 THEORETICAL ANALYSIS The spreading of impacting liquid drops perpendicular to the grooves on groove-textured surfaces undergo physical events in addition to those encountered on a smooth surface. These events include, as drawn from the experimental observations summarized in the previous section, (A) drop impregnation into the grooves leading to the modification in drop configuration (Cassie/Wenzel), (B) convexity of the spreading drop front, and (C) contact line pinning of drop at the asperity edges. In what follows, we attempt to predict the trend of m, with We, as shown in Fig. 3(b), by including the abovementioned observations in an analysis based on the general framework of energy conservation.7-9 The assumption of axisymmetry in drop configuration at maximum spread, as made in conventional energy conservation models, is retained here due to the limitation in simultaneous prediction of βm, and βm,|| using the current approach. The deviation in model predictions due to this assumption is not significant for the current impact conditions (see Sec. S1 of Supporting Information). A. Effect of drop impregnation into surface texture Consider a liquid drop of diameter Do impacting a textured solid surface with a velocity Uo. It is known that on textured solid surfaces which support a Cassie drop state at static wetting condition an impact velocity driven impregnation of drop into the valleys of roughness asperities occurs.12,26-28 Here the drop state at maximum spreading on textured surfaces is taken into account through the


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9 area fraction underneath the drop which is impregnated by liquid. The conventional energy conservation model, Ei = Em + Wv (where Ei and Em are the total energies of ‘liquid drop-solid surface’ system just prior to impact and at maximum drop spreading, respectively and Wv is the viscous energy dissipation during the drop spreading process modeled as given in Pasandideh-Fard et al.8 given by Eqs. (3)–(5)), is modified to take into account the impregnated area fraction underneath the drop base at maximum spreading. 1  3  2   Do U o 2 6 

(3)

      Em   Dm2 ,  Dm, h   Dm2 , SL   A  Dm2 ,  SV 4 4 4   

(4)

Ei  Do2   SV A 

Wv 

 3 Re

U o2 Do Dm2 ,

(5)

In the above equations A is the total surface area of the target solid surface and h is the thickness of liquid drop at maximum spreading, assumed to be a circular cylindrical disc of diameter Dm,, given from mass conservation between states i and m as h = 2Do3/3Dm2. Ei remains the same as given by Eq. (3) whereas Em, taking into account the impregnated area fraction, becomes  E m   Dm2 , 1  1  f 4

 

2

1     D

m,

  h  Dm2 , 1  f 2   f 2 r  SL 4 







     A  Dm2 , 1  f 2   f 2 r  SV 4  







(6)

Here f is the fraction of maximum drop spread diameter, Dm, which is impregnated by drop (or, f2 is the area fraction beneath the drop at maximum spreading which is impregnated by drop). When f becomes zero, the drop state corresponds to Cassie and when it becomes 1, the drop state is Wenzel (see Fig. 4(a)). Assuming the viscous dissipation term (Eq. (5)) remains unchanged due to the impregnation process, Eq. (3), Eq. (5), and Eq. (6) when applied in the energy conservation


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10 framework, Ei = Em + Wv gives a modified equation to predict the maximum spread factor as follows: We  3  2 2 3 2  1  f  1  cos Y   f 1  r cos Y   4   m,  We  12 m,  8  0 Re  

 





(7)

Figure 5(a) shows the variation of βm, with We for water drop impact on TS220 as predicted by Eq. (7) for f = 0 and f = 1. Figure 5(a) also shows the current experimental measurements of βm, on TS220. It is clear that the impregnation model alone as given by Eq. (7) does not capture the experimental measurements well for f = 0 and f = 1even though it predicts the qualitative trend of βm, with We. However, there seems to be an apparent matching of predictions from Eq. (7) with f = 0 and experimental trend of βm,|| and βm on smooth, un-textured surface with We as seen in Fig. 5(a). B. Effect of convex drop front at maximum spread The liquid-vapor interfacial energy due to convex drop front can be calculated as follows (see Fig. 4(b)):   Econvex  Dm,  hconvex  2 

(8)

The drop thickness at maximum spread, hconvex can be calculated from mass conservation between states i and m as follows.



 2   Do3    Dm2 , hconvex  hconvex Dm,   6 8 4 

(9)

Solving the above quadratic equation in hconvex, we get

hconvex 

Dm , 



  4   1  1  3   3 m, 

1  2       

Substituting Eq. (10) in Eq. (8) and on simplification we get


(10)

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11

Econvex

  4Do3 2   Dm,  1  1   3D 3 2 m,   



1  2       

(11)

The effect of convex drop front at maximum drop spreading state is modeled using conventional energy conservation principle, Ei = Em + Wv by incorporating the modified surface energy, Econvex, as given by Eq. (11), in the energy of system at maximum drop spreading state as given by Eq. (6). We consider the extreme states of Cassie and Wenzel corresponding to f = 0 and 1, respectively in Eq. (6). On simplification, this yields the following. For drop in Cassie state at maximum spreading: 2    We  We  2   4  36 3 1  cos  Y   4  m,  2We  123 1  cos  Y   4    m, Re  Re      

(12)

 48 m,  We  12  0 2

For drop in Wenzel state at maximum spreading: 2    We  We  2   4   3 1  r cos  Y  4   36 m,  2We  1231  r cos  Y   4   m, Re  Re      

(13)

 48 m,  We  12  0 2

Figure 5(b) shows the comparison of predictions from Eq. (12) and Eq. (13) with the experimental data of βm, on TS220. It is clear that even though the model captures the qualitative trend of increasing βm, with We, quantitative match with the experimental data is rather poor. However, there seems to be an apparent matching of predictions from Eq. (12) and experimental trend of βm,|| and βm on smooth, un-textured surface with We as seen in Fig. 5(b). C. Effect of contact line pinning Here, the effect of contact line pinning encountered by spreading drop front at the pillar edges of the textured surface is taken into account in the form of an apparent energy loss, Wp to overcome contact angle hysteresis due to contact line pinning. The contact angle hysteresis force per unit


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12 length of the drop TPCL is σ(cosθr – cosθa) where θr and θa are the receding and advancing contact angles of drop perpendicular to the grooves on groove-textured surface measured from static wetting experiments. Since the drop spreading does not involve a receding motion of drop TPCL, it is physically more meaningful to use σ(cosθe – cosθa) as a measure of contact angle hysteresis force per unit length of drop TPCL (here, θe is the equilibrium/static/‘as-placed’ contact angle of drop perpendicular to the grooves on groove-textured surface measured from static wetting experiments). The energy lost to overcome this pinning force at Np = Dm,/(w+b) number of pillar edges during spreading can be estimated as25

Wp 



4w  b 

 cos  e  cos  a Dm3 ,

(14)

Equation (14) together with Eq. (3), Eq. (5), and Eq. (6) when applied in the framework of energy conservation, Ei = Em + Wv + Wp gives the model equation for predicting maximum drop spreading perpendicular to grooves on groove-textured surfaces as follows. Here we consider the two extreme wetting states of drop at maximum spreading – Cassie and Wenzel – corresponding to the extreme values of f (0 and 1, respectively). For drop in Cassie state at maximum spreading:  We  3  3Do   m,  We  12 m,  8  0  cos  e  cos  a  m4 ,   32   1  cos  Y   4 Re  wb 

(15)

For drop in Wenzel state at maximum spreading:  We  3  3Do   m,  We  12 m,  8  0  cos  e  cos  a  m4 ,   31  r cos  Y   4 Re  wb 

(16)

In obtaining Eq. (15) and Eq. (16) f values of 0 and 1, respectively, are used in Eq. (6). In Eq. (15) and Eq. (16), the contribution of an apparent energy loss, due to contact line pinning and the


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13 associated contact angle hysteresis, in maximum spread factor prediction comes from the coefficient of (m,)4. Figure 5(c) shows the comparison of experimental measurements of βm, on TS220 with the predictions from Eq. (15) and Eq. (16). It is clear that Eq. (15) and Eq. (16) capture the experimental trend of βm, on TS220 quite well. The experimental observation that the deviation of βm, on TS220 from βm,|| increases with increase in We is predicted well by the model. This could be physically explained as follows. As We increases, βm, increases by covering more number of pillar edges during spreading. This leads to a higher energy loss to overcome pinning at pillar edges and, hence, a higher deviation of βm, from the maximum spread measured parallel to grooves on groovetextured surface where pinning is absent. D. A unified model: drop impregnation, drop convexity, and TPCL pinning effects A unified model which combines the effects of drop impregnation into surface texture, convexity of drop front shape at maximum spread, and pinning of drop TPCL during spreading across the pillar edges can be formulated under the energy conservation approach, Ei = Em + Wv + Wp using Eq. (3), Eq. (5), Eq. (6) modified using Eq. (11), and Eq. (14) to give equations for predicting βm, during liquid drop impact on groove-textured surfaces as follows. For this the extreme cases of drop liquid impregnation corresponding to Cassie (f = 0) and Wenzel (f = 1) states of liquid drop at its maximum spread configuration are considered. For drop in Cassie state at maximum spread: 2

6 Do We  5   3Do  6  w  b cos  e  cos  a   m,  w  b cos  e  cos  a 3 1  cos  Y   4 Re   m ,     2   4 6 Do We  cos  e  cos  a We  12 m3 ,  3 1  cos  Y   4  36  m ,   w  b Re    We  2  2  2We  123 1  cos  Y   4   m,  48 m ,  We  12  0 Re  


(17)

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14 For drop in Wenzel state at maximum spread: 2

6 Do We  5   3Do  6  w  b cos  e  cos  a   m,  w  b cos  e  cos  a 31  r cos  Y   4 Re   m ,     2   4 6 Do We  cos  e  cos  a We  12 m3 ,  31  r cos  Y   4  36  m ,   w  b Re    We  2  2  2We  1231  r cos  Y   4   m,  48 m ,  We  12  0 Re  

(18)

Figure 5(d) shows the comparison of experimental measurements of βm, on TS220 with the predictions from Eq. (17) and Eq. (18). It is clear that unified model Eq. (17) and Eq. (18) capture the experimental trend of βm, on TS220 quite well. The experimental observation that the deviation of βm, on TS220 from βm,|| increases with increase in We is predicted well by the model. COMPARISON OF EXPERIMENTAL MEASUREMENTS OF MAXIMUM DROP SPREADING WITH MODEL PREDICTIONS A. Comparison of m, from current experiments with model predictions Comparison of the current experimental measurements of βm, for liquid drop impact on all the target groove-textured surfaces explored here with the predictions from unified model (Eq. (17) and Eq. (18)) are provided here to confirm the applicability of the unified model to target groovetextured surfaces with a wide range of texture geometry (comparison of the predictions from models considering the effects of impregnation, convex drop front, and contact line pinning individually with experimental measurements of βm, for all the drop impact conditions studied here is available in Sec. S4 of Supporting Information). In addition to this, the current experimental measurements of βm, are also compared with the predictions from a recently reported model, based on energy conservation with a modified expression for viscous dissipation, for maximum drop spreading during impact on surfaces decorated with circular cylindrical posts.30 The viscous dissipation term


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15 was modified by Li et al.,30 adopting from models reported by Ishino et al.,31 yielding two expressions for the cases of short posts and long posts (details of this model are given in Sec. S5 of Supporting Information; comparison of experimental measurements of βm, with predictions from other theoretical models listed in Table I are given in Sec. S2 and Sec. S3). Figure 6 (a) and (b) shows the comparison of experimental measurements of βm, on all the groove-textured surfaces (see Table II) with the predictions from the unified model (Eq. (17) and Eq. (18)), and Li et al.’s model30, respectively. In the case of Li et al.’s model30, the values reported here correspond to the predictions which show lower deviation from current experimental measurements (for example, long post model for TS220 and TS140 which have higher groove depths of 220 μm and 140 μm, respectively and short post model for TS11 and TS12 which have relatively lower groove depths of 54 μm and 53 μm, respectively – refer to Sec. S5 of Supporting Information for more details on this). It is evident that the unified model gives better quantitative predictions of βm, on all the groove-textured surfaces explored here. B. Effect of surface texture on m, This section deals with the effect of surface texture through pinning term in the contact line pinning model (Eq. (15) and Eq. (16)) on βm, for drop impact on groove-textured surfaces. In the following analysis, the drop is considered to be in Cassie state at maximum spreading and Eq. (15) is taken for further discussion (the same approach can be used with Eq. (16) for drop in Wenzel state also). Rewrite the form of Eq. (15) as

A4  m4 ,  A3  m3 ,  A1  m,  A0  0

(19)

where  We   3Do  A4    cos  e  cos  a ; A3   32   1  cos  Y   4 Re  wb  A2  0; A1  We  12; A0  8


(20)

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16 The coefficient A4 will be referred to as the pinning coefficient and is a function of the texture geometry of target surface as well as wetting nature of target surface by drop liquid. Differentiating Eq. (19) with respect to A4 and rearranging the terms,

4 A  4

3 m,

 3 A3  m2 ,  A1

  ddA

m, 4

 dA dA     m4 ,   m3 , 3   m, 1   0 dA4 dA4  

(21)

Note that A1 is purely a function of drop impact velocity and drop liquid properties and, hence, independent of target surface nature whereas A4, as mentioned above, is a function of the surface texture characteristics and, hence, is independent of drop impact velocity. Keeping the drop liquid properties and impact velocity constant gives

dA1  0 . Now, A3 is a function of both the texture dA4

geometry and material of target surface (first term) as well as drop impact velocity (second term). In addition to keeping drop liquid properties and impact velocity constant, if the solid fraction,  of textured surface is fixed, then

d m, dA4



dA3  0 and Eq. (21) reduces to the following. dA4

 m4 ,

(22)

4 A4  m3 ,  3 A3  m2 ,  A1

Rearranging the denominator of RHS in Eq. (22) as follows, 4 A4  m3 ,  3 A3  m2 ,  A1  

1

 m, 1

 m,

3 A  4

4 m,

3 A  4

4 m,

 2 A3  m3 ,  A4  m4 ,  A3  m3 ,  A1  m,   2 A3  m3 ,  A0 

Substituting this rearranged denominator in Eq. (22) with A0 = 8,

d m, dA4



 m5 ,

(23)

3 A4  m4 ,  2 A3  m3 ,  8


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17 Figure 7(a) shows the experimental values of the denominator of RHS in Eq. (23) corresponding to all the impact conditions of the current study. It is clear that (3 A4  m4 ,  2 A3  m3 ,  8)  0 , at least for the experimental drop impact conditions explored here. This leads to the conclusion that d m, dA4

 0 from Eq. (23). Therefore, as the pinning coefficient increases the maximum drop

spreading decreases when the drop liquid properties, drop impact velocity, and the solid fraction of the textured surface are kept constant. Further, rewriting the pinning coefficient as  3D  A4   o  cos e  cos  a  , we find that A4 can be increased by decreasing the pillar top width, b  b 

and/or increasing the contact angle hysteresis, cos e  cos a (assuming both can be independently varied through a combination of variation in surface texture and surface material). Hence, decreasing the pillar width of target groove-textured surface leads to an increased pinning coefficient and a decrease in maximum drop spreading on the target textured surface. This implies that reducing the size of roughness asperities on groove-textured surfaces results in a reduction in maximum spreading achieved by a liquid drop impacting on the surface if the solid fraction is kept constant. It is clear that for textured surfaces with the same solid fraction but different contact angle hysteresis, the one with higher contact angle hysteresis will have a larger pinning coefficient and hence smaller maximum spreading perpendicular to grooves. In order to understand this, we consider groove-textured surfaces made of the same material and having the same solid fraction but different contact angle hysteresis. From Fig. 1, it can be seen that the target surfaces TS11, TS140, and TS220 satisfy these criteria. Figure 7(b) shows the variation of βm, with We for water drop impacted on these surfaces. It is evident from the figure that, for high We, βm, is higher for the surface with less contact angle hysteresis perpendicular to grooves, cos e  cos a (TS11: 0.097;


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18 TS140: 0.107; TS220: 0.238) as shown by the guiding lines. In other words, there is no significant difference between the experimental measurements of βm, on these surfaces for low We and the effect of target surface geometry is seen only at high We regime. Consider the terms highlighting the effect

of

surface

texture

and

wetting

in

Eq.

(15):

 3Do   cos  e  cos  a  m4 , and w  b  

32   1  cosY  m3 , . To find out at what impact condition the pinning term governed by contact angle hysteresis becomes more dominant than 32   1  cos  Y  m3 , , consider the following.  3Do  4 3  cos  e  cos  a  m,  32   1  cos Y  m,  wb

This implies that for  m, 

32   1  cos  Y   3Do   cos  e  cos  a  wb

(24)

, the pinning term will be more dominant and

lead to a difference in the experimentally measured βm, between TS11, TS140, and TS220. The maximum value of  m, 

32   1  cos  Y   3Do   cos  e  cos  a  wb

among the target surfaces considered is 1.78. In

Fig. 7(b), βm, ~ 1.78 corresponds to We ~ 33. Hence for We > 33, the maximum spreading perpendicular to grooves on these target surfaces having similar solid fraction will show a difference with the least maximum spreading being on the surface with the highest contact angle hysteresis; for We < 33, no effect of target surface geometry/wetting is seen in Fig. 7(b) because the dominant term

32   1  cos Y  m3 , is similar on these target surfaces. C. Prediction of m,|| as a special case of the unified model for m,


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19 The theoretical model developed for prediction of m,, as outlined and discussed above, could be used for the prediction of maximum drop spreading parallel to grooves, m,|| by considering the special case of absence of any pillar edges causing contact line pinning for liquid drop front spreading parallel to grooves. A model for prediction of m,|| could be obtained from Eq. (17) and Eq. (18) by eliminating the terms containing the pinning coefficient as a factor. Together with this, if r and ϕ are set to 1 (since drop spreading parallel to grooves is seen to be similar to that on smooth surface)29, we get a special case of Eq. (17) and Eq. (18), which could be used to predict m,||, as follows. 2   We  We  2    4  36 31  cos Y   4  m,||  2We  1231  cos Y   4    m,|| Re  Re      

(25)

 48 m,||  We  12  0 2

Ukiwe and Kwok9 argued that using the advancing contact angle, θa obtained from static wetting experiments instead of θY is more suitable for the case of an advancing drop front on a smooth (untextured) surface. However, in a more recent study, Vadillo et al.11 reported the significance of using dynamic advancing contact angle, θa,d measured during spreading of drops impacted on smooth (untextured) surfaces, in predicting m on smooth un-textured surfaces. Here, we adopt the same reasoning for maximum spreading parallel to the grooves on groove-textured surfaces since it is very similar to maximum spreading on smooth surfaces29. Hence for predicting m,|| using Eq. (25) or one of the theoretical models in Table I, θa,d is a more physically meaningful input than θa or θY (see Sec. S7 of Supporting Information for comparison of experimental measurements of m,|| with predictions from these models using static advancing contact angle, θa; in the case of perpendicular to grooves, experimental measurements show there is no significant difference between θa,d and θa). Figure 8 shows comparison of experimentally measured values of m,|| with that predicted by Eq.


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20 (25) and theoretical models reported in literature (given in Table I) using the dynamic advancing contact angle. Equation (25) from current study as well as the models by Mao et al.7 and Ukiwe and Kwok9 are seen to offer good predictions of m,|| at all the drop impact conditions explored here. Since the unified model is formulated under the assumption of axisymmetric drop configuration at its maximum spread, it should also be ideally suitable for predicting maximum drop spread during impact on isotropic textured surfaces (such as surfaces decorated with microcircular/square cylindrical posts in a square lattice) (see Sec. S8 of Supporting Information). CONCLUSIONS Maximum drop spreading during impact on groove-textured surfaces was studied experimentally and theoretically. From experiments, it was observed that the maximum drop spreading perpendicular to grooves, βm, is always less than that parallel to grooves, βm,|| which, in turn, followed the trend of βm on smooth un-textured surface. The difference between βm, and βm,|| was seen to increase with increase in We. Different physical mechanisms that could be responsible for the experimentally observed trend of βm, with We were explored in the framework of conventional energy conservation based models: (i) impact velocity driven drop liquid impregnation into grooves, (ii) convex shape of drop front at maximum spreading, and (iii) contact line pinning of spreading drop front at the pillar edges. The theoretical analysis led to the conclusion that, among the potential candidates, the contact line pinning modeled as an apparent energy loss due to contact angle hysteresis of drop liquid best explains the experimental trend of βm, with We. A unified model taking into account the effects of drop liquid impregnation, convex drop front at maximum spreading, and contact line pinning at pillar edges during drop spreading perpendicular to grooves was formulated. The increase in the deviation of βm, from βm,|| with We could be explained using this model as due to the increase in the number of pinning pillar edges that the spreading drop front


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21 has to cross and hence an increased ‘energy loss’ with increase in We. On surfaces with similar solid fraction, the one with larger contact angle hysteresis was seen to show a smaller βm, beyond We of 33 which is in line with the predictions of the theoretical model. A special case of the unified model, with no pinning, was able to predict the experimental measurements of βm,|| provided the average dynamic advancing contact angle was used in the model. Supporting Information Available: Effect of anisotropy in maximum drop spread on the model predictions; Comparison of maximum drop spreading perpendicular and parallel to grooves with

predictions from semi-empirical correlations and theoretical models reported in literature; Comparison of maximum drop spreading perpendicular to grooves with predictions from theoretical models considering various mechanisms (drop liquid impregnation, convex drop front, TPCL pinning) individually; Details on the model reported by Li et al. (2013); Comparison of experimental measurements of average dynamic advancing contact angle, θa,d during drop impact on groovetextured surfaces with the corresponding static advancing contact angle, θa measured from static wetting experiments; Comparison of experimental data on maximum drop spreading during impact on groove- and post-textured surfaces reported in literature with unified model predictions. This material is available free of charge via the internet at http://pubs.acs.org. Corresponding Author * E-mail: [email protected]; Phone: +91-80-2293-3022 ACKNOWLEDGMENTS The financial support from Department of Science & Technology (DST), India through FIST program is gratefully acknowledged. REFERENCES


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22 (1) Yarin, A. L. Drop impact dynamics: Splashing, spreading, receding, bouncing…….. Annu. Rev. Fluid Mech. 2006, 38, 159–192. (2) Marengo, M.; Antonini, C.; Roisman, I. V.; Tropea, C. Drop collisions with simple and complex surfaces. Curr. Opin. Colloid Interface Sci. 2011, 16, 292–302. (3) Scheller, B. L.; Bousfield, D. W. Newtonian drop impact with a solid surface. AIChE J. 1995, 41, 1357–1367. (4) Roisman, I. V. Inertia dominated drop collisions. II. An analytical solution of the Navier-Stokes equations for a spreading viscous film. Phys. Fluids 2009, 21, 052104. (5) Bayer, I. S.; Megaridis, C. M. Contact angle dynamics of droplets impacting on flat surfaces with different wetting characteristics. J. Fluid Mech. 2006, 558, 415–449. (6) Roisman, I. V.; Rioboo, R.; Tropea, C. Normal impact of a liquid drop on a dry surface: Model for spreading and receding. Proc. R. Soc. London Ser. A 2002, 458, 1411–1430. (7) Mao, T.; Kuhn, D. C. S.; Tran, H. Spread and rebound of liquid droplets upon impact on flat surfaces. AIChE J. 1997, 43, 2169–2179. (8) Pasandideh-Fard, M.; Qiao, Y. M.; Chandra, S.; Mostaghimi, J. Capillary effects during droplet impact on a solid surface. Phys. Fluids 1996, 8, 650–659. (9) Ukiwe, C.; Kwok, D.Y. On the maximum spreading diameter of impacting droplets on wellprepared solid surfaces. Langmuir 2005, 21, 666–673. (10) Clanet, C.; Béguin, C.; Richard, D.; Quéré, D. Maximal deformation of an impacting drop. J. Fluid Mech. 2004, 517, 199–208. (11) Vadillo, D. C.; Soucemarianadin, A.; Delattre, C.; Roux, D. C. D. Dynamic contact angle effects onto the maximum drop impact spreading on solid surfaces. Phys. Fluids 2009, 21, 122002.


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23 (12) Deng, T.; Varanasi, K. K.; Hsu, M.; Bhate, N.; Keimel, C.; Stein, J.; Blohm, M. Nonwetting of impinging droplets on textured surfaces. Appl. Phys. Lett. 2009, 94, 133109. (13) Maitra, T.; Tiwari, M. K.; Antonini, C.; Schoch, P.; Jung, S.; Eberle, P.; Poulikakos, D. On the nanoengineering of superhydrophobic and impalement resistant surface textures below the freezing temperature. Nano Lett. 2014, 14, 172-182. (14) Maitra, T.; Antonini, C.; Tiwari, M. K.; Mularczyk, A.; Imeri, Z.; Schoch, P.; Poulikakos, D. Supercooled Water Drops Impacting Superhydrophobic Textures. Langmuir 2014, 30, 1085510861. (15) Tsai, P. C.; Hendrix, M. H. W.; Dijkstra, R. R. M.; Shui, L. L.; Lohse, D. Microscopic structure influencing macroscopic splash at high Weber number. Soft Matter 2011, 7, 11325-11333. (16) Rahman, M. A.; Jacobi, A. M. Wetting behaviour and drainage of water droplets on microgrooved brass surfaces. Langmuir 2012, 28, 13441–13451. (17) McHale, G.; Shirtcliffe, N. J.; Aqil, S.; Perry, C. C.; Newton, M. I. Topography driven spreading. Phys. Rev. Lett. 2004, 93, 036102. (18) Quere, D. Wetting and roughness. Annu. Rev. Mater. Res. 2008, 38, 71–99. (19) Chen, Y.; He, B.; Lee, J.; Patankar, N. A. Anisotropy in the wetting of rough surfaces. J. Colloid Interface Sci. 2005, 281, 458–464. (20) Kusumaatmaja, H.; Vrancken, R. J.; Bastiaansen, C. W. M.; Yeomans, J. M. Anisotropic drop morphologies on corrugated surfaces. Langmuir 2008, 24, 7299–7308. (21) Bixler, G. D.; Bhushan, B. Bioinspired rice leaf and butterfly wing surface structures combining shark skin and lotus effects. Soft Matter 2012, 8, 11271–11284. (22) Kannan, R.; Sivakumar, D. Drop impact process on a hydrophobic grooved surface. Colloids Surf. A 2008, 317, 694–704.


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24 (23) Pearson, J. T.; Maynes, D.; Webb, B. W. Droplet impact dynamics for two liquids impinging on anisotropic superhydrophobic surfaces. Exp. Fluids 2012, 53, 603–618. (24) Vazquez, G.; Alvarez, E.; Navaza, J. M. Surface tension of alcohol water + water from 20 to 50 degree C. J. Chem. Eng. Data 1995, 40, 611. (25) Vaikuntanathan, V.; Sivakumar, D. Directional motion of impacting drops on dual-textured surfaces. Phys. Rev. E 2012, 86, 036315. (26) Jung, Y. C.; Bhushan, B. Dynamic effects of bouncing water droplets on superhydrophobic surfaces. Langmuir 2008, 24, 6262–6269. (27) Bartolo, D.; Bouamrirene, F.; Verneuil, E.; Buguin, A.; Silberzan, P.; Moulinet, S. Bouncing or sticky droplets: Impalement transitions on superhydrophobic micropatterned surfaces. Europhys. Lett. 2006, 74, 299–305. (28) Vaikuntanathan, V.; Sivakumar, D. Transition from Cassie to impaled state during drop impact on groove-textured solid surfaces. Soft Matter 2014, 10, 2991–3002. (29) Kannan, R. Experimental study on the impact of water drops on groove-textured surfaces. PhD Thesis 2011, Indian Institute of Science. (30) Li, X.; Mao, L.; Ma, X. Dynamic behavior of water droplet impact on microtextured surfaces: The effect of geometrical parameters on anisotropic wetting and the maximum spreading diameter. Langmuir 2013, 29, 1129–1138. (31) Ishino, C.; Reyssat, M.; Reyssat, E.; Okumura, K.; Quere, D. Wicking within forests of micropillars. Europhys. Lett. 2007, 79, 56005.


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25 LIST OF FIGURES

Figure 1. Schematic of a typical groove-textured surface along with the geometrical details and relevant wetting characteristics perpendicular to grooves of groove-textured surfaces used in the present study. TXi where T refers to ‘Textured’, X indicates the surface material (S for Stainless Steel and P for

PDMS), and i is a number assigned as per our internal reference.

Figure 2. (a) Image sequences of water drop impact on TS220 showing the events during drop spreading process perpendicular to the grooves at two different values of impact velocity. (First row) Do = 2.60 mm, Uo = 0.44 m/s. (Second row) Do = 2.59 mm, Uo = 0.88 m/s. Figure 3. (a) Ethanol-water (E05) drop at its maximum spread during impact on TS220 at three different values of We. Corresponding to each We both perpendicular () and parallel (||) to grooves cases are given. (b) The variation of maximum drop spreading measured perpendicular, βm, and parallel to grooves, βm,|| with We during drop impact on groove-textured surfaces TS11, TS140, and TS220. The continuous and dashed lines are guides to eye for differentiating the trends of βm,|| and βm,, respectively. Figure 4. (a) Schematic sketches of drop states prior to impact and at maximum spreading with the extreme cases of impregnated area fraction, f = 0 and 1, underneath the drop on textured surface. (b) Schematic sketch showing the geometry of the convex drop front as modeled to calculate the difference in liquid-vapor interfacial energy due to the assumption of flat drop front at maximum spreading in the conventional energy conservation framework. Figure 5. Comparison of current experimental trend of maximum drop spreading perpendicular to grooves on TS220 with We with the predictions of (a) impregnation model (Eq. (7)), (b) convex drop front model (Eq. (12) and Eq. (13)), (c) contact line pinning model (Eq. (15) and Eq. (16)), and (d) unified model (Eq. (17) and Eq. (18)). Experimental measurements of maximum drop spreading parallel to grooves on TS220 and on smooth un-textured surface are also shown for comparison.


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26 Figure 6. Comparison of the current experimental measurements of maximum drop spreading perpendicular to grooves on groove-textured surfaces, given in Fig. 1, with the theoretical predictions from (a) unified model (Eq. (17) and Eq. (18)), and (b) model reported by Li et al. (2013). Figure 7. (a) Experimental values of the denominator of RHS in Eq. (23) plotted for all the impact conditions studied here. (b) Plot showing the variation of βm, with We for water drop impact on TS11, TS140, and TS220 to highlight the effect of contact angle hysteresis on the observed trend. The vertical and horizontal dashed lines respectively indicate the values of We = 33 and βm, = 1.78 beyond which the effect of contact angle hysteresis becomes dominant as per Eq. (24). The lines shown for We > 33 are guide to eyes to highlight the difference in βm, among the three groovetextured surfaces. (Inset) Cross-sectional profiles of TS11, TS140, and TS220 are shown. Figure 8. Comparison of the current experimental measurements of maximum drop spreading parallel to grooves on groove-textured surfaces with the predictions from Eq. (25) (with dynamic advancing contact angle, θa,d) as well as theoretical models reported in literature as given in Table I: (a) Eq. (25), (b) Mao et al. (1997), (c) Pasandideh-Fard et al. (1996), and (d) Ukiwe and Kwok (2005).


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27 Table I. Semi-empirical correlations and theoretical models for prediction of maximum drop spreading during impact on smooth solid surfaces

Reference

Modeling approach

Range of parameters

Equation

We

Re

SCA*/Surface

Scheller & Bousfield3

Semi-empirical

We: [47,1826]

Re: [19,16400]

Polystyrene film, glass

βm = 0.61(ReWe0.5)0.166

Bayer & Megaridis5

Semi-empirical

We: [0.1,120]

Re: (140,2100)

[20o,157o]

βm = 0.72(ReWe0.5)0.14

Roisman et al.4

Semi-empirical

High We >> 10

High Re >> 100

Hydrophilic surfaces

βm = 0.87Re1/5–0.40Re2/5We-1/2

Mao et al.7

Energy conservation

σ: 25-73 mN/m

μ: 1-100 mPa.s

[30o,120o]

Uo: 1-6 m/s

Uo: 1-6 m/s

 We 0.83  3 31  cos  Y   2.4 0.33   m  We  12 m  8  0 Re  

Do: 1.5-3.5 mm

Do: 1.5-3.5 mm Re: [213, 35339]

PasanidehFard et al.8

Energy conservation

We: [26, 641]

Clanet et al.10

Mass conservation with an effective gravity

We: [2,900]

Ukiwe & Kwok9

Energy conservation

We: [18,370]

Re: [10, 10000]

Re: [1866, 9735]

[20o,140o]

m 

We  12

31  cos  a   4We

Re

Superhydroph obic surfaces, plastic

P = WeRe-4/5 < 1: βm = We0.25

[66.3o, 90.7o]

 We  3 31  cos  Y   4   m  We  12 m  8  0 Re  

* Static Contact Angle


P > 1: βm = Re0.2

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28

b

w d

α

Solid pillars separating parallel grooves Surface texture geometry

Sl. No.

Target surface

Experimental liquid

1

TS11

2





We range

w (m)

b d (m) (m)

Distilled water (W)

4 to 74

180

116

54

42

0.39

TS140

Distilled water (W)

1 to 74

173

126

140

80

3

TS220

Distilled water (W)

4 to 73

173

126

220

4

TS12

Distilled water (W)

4 to 84

177

22

5

TP300

Distilled water (W)

3 to 55

125

6

TS220

Ethanol-water (E05)

8 to 83

7

TP300

Ethanol-water (E05)

8 to 84

Surface wetting

r

(deg.)

Y

e

a

(deg.)

(deg.)

(deg.)

1.14

76

124

131

0.42

1.79

76

137

147

89

0.42

2.45

76

135

161

53

43

0.11

1.21

76

121

134

300

300

90

0.71

2.41

105

125

135

173

126

220

89

0.42

2.45

71

136

160

125

300

300

90

0.71

2.41

100

125

135

Figure 1.


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29

Drop spreading process Prior to impact

t increases

2 mm

Figure 2.


Maximum spread

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30

(a) We ~ 2.0 

We ~ 21.0 ||



We ~ 43.0 ||



3 mm 3.0

(b) 2.5

m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

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2.0

1.5

1.0

TS11 TS140 TS220

0

20

40

60

We

Figure 3.


  

  

80

||

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31

System configurations in state m System configuration in state i Liquid

Liquid Textured surface f = 0: Cassie

drop

hconvex

Textured surface bw α d

(a)

Semi-circular cylinder of diameter hconvex and height Central cylindrical disc with πDm, wrapped around the diameter Dm, and height central cylindrical disc hconvex

Liquid Dm,

Liquid Textured surface f = 1: Wenzel

Solid surface (b)

Figure 4.


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32 4

4

Impregnation model, Eq. (7): f=0 f=1

3

2

2

m

3

Convex drop front model: Eq. (12) Eq. (13)

1

(a) 4



Experiments: TS220: Smooth:

0

20

40

We

60

80

1 100 0

20

40

(b)

Experiments: TS220:  Smooth: Pinning model: Eq. (15)

4

|| Eq. (16)

3

3

2

2



Experiments: TS220: Smooth:

||

We

60

Experiments: TS220:  Smooth: Unified model: Eq. (17)

80

||

100

|| Eq. (18)

m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

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1

(c)

0

20

40

We

60

80

1 100 0

20

(d)

Figure 5.


40

We

60

80

100

Page 33 of 36

33

4

Unified model Average error ~ 8%

0

1

TS11 TS140 TS220 TS12 TP300

3

1

0

(a)

4

Eq. 17 Eq. 18 TS11: TS140: 3 TS220: TS12: TP300: 2

m, (model)

m, (model)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

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2

m, (experiment)

3

2

1

0

4

(b)

Li et al. (2013) Average error ~ 33%

0

1

2

m, (experiment)

Figure 6.


3

4

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34

1000

2.5

TS11 TS12 TS140 TS220 TP300

750

3

2.0

m,

500

4

1.5

250

0

(a)

TS11 TS140 TS220

Groove depth (mm)

3A4m, +2A3m, -8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

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1.0 0

20

40

We

60

80

100

0.32

TS11

TS140

TS220

0.24 0.16 0.08 0.00 0.00

0.25

0.50

0.75

1.00

Length  to grooves (mm)

0

(b)

Figure 7.


20

40

We

60

80

Page 35 of 36

35 4

TS11 TS140 TS220 TS12

3

m, (model)

m, (model)

4

TS11 TS140 TS220 TS12

3

2

1

2

1

Eq. (25) using a,d

(a)

Mao et al. (1997) using a,d

Average error ~ 6%

0

4

1

2

m, (experiment)

3

4

(b)

Average error ~ 6%

0

4

TS11 TS140 TS220 TS12

3

0

2

1

1


(c) Figure 8.

1

2


3

4

1 Ukiwe & Kwok (2005) using a,d

Average error ~ 16%

0

3

2

Pasandideh-Fard et al. (1996) using a,d

0

2

TS11 TS140 TS220 TS12

3

m, (model)

0

m, (model)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

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0

4

Average error ~ 8%

0

(d)


1

2


3

4

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36

TOC Graphic


Maximum Spreading of Liquid Drops Impacting on Groove-Textured

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