ISSN 1744-683X

www.rsc.org/softmatter Volume 9 | Number 31 | 21 August 2013 | Pages 7391–7658

1744-683X(2013)9:31;1-4

PAPERSushanta K. Mitra et al.Drop deposition on under-liquid low energy surfaces

Soft Matter

PAPER

Publ

ishe

d on

24

May

201

3. D

ownl

oade

d by

Uni

vers

ity o

f N

ewca

stle

on

24/0

8/20

14 2

2:54

:51.

View Article OnlineView Journal | View Issue

Micro and Nanoscale Transport Laboratory

University of Alberta, Edmonton, Alberta

mitra@ualberta.ca

† Electronic supplementary informa10.1039/c3sm50981h

Cite this: Soft Matter, 2013, 9, 7437

Received 10th April 2013Accepted 24th May 2013

DOI: 10.1039/c3sm50981h

www.rsc.org/softmatter

This journal is ª The Royal Society of

Drop deposition on under-liquid low energy surfaces†

Prashant R. Waghmare, Siddhartha Das and Sushanta K. Mitra*

In this paper, we propose a new technique for non-intrusive drop deposition on under-liquid low energy

surfaces. This technique addresses the limitations of the conventional drop deposition method, in which a

needle holding the drop is brought in close proximity to the solid substrate, and the drop gets deposited on

the solid by detaching from the needle spontaneously owing to the favorable drop-substrate surface

energy. Therefore, for low energy surfaces, irrespective of whether such surfaces are in air or under

liquid, the fact that the drop-substrate surface energy is much smaller than the drop-needle surface

energy, there are extreme difficulties in depositing the drop by getting it removed from the needle. In

our proposed method, we address this limitation for the special case of under-liquid low energy

surfaces. Under-liquid systems provide two distinct interfaces: first the surface–liquid interface where the

drop is to be deposited and the second the liquid–fluid interface at the location where the liquid

column ends. We achieve this non-intrusive, substrate-independent drop deposition method by

depositing the drop from this liquid–fluid interface, by using the (un)favorable spreading behavior of

the drop at this interface. Massive upsurge in use of low-energy under-liquid surfaces, requiring a

correct estimation of the corresponding surface energy, makes our proposed technique extremely

important and technologically significant.

1 Introduction

The most common technique to characterize the wettability ofsolid surfaces is to deposit a drop of liquid of known surfacetension on the surface and measure the corresponding contactangle. Therefore, one of the most important steps intrinsic topreparing and characterizing solid surfaces of desirable wetta-bility is to ensure a non-intrusive drop deposition on the solidsurface. In this paper, we shall propose a new technique of dropdeposition that will improve the most widely practiced tech-nique of drop deposition, namely pendant drop deposition tech-nique1–4 (henceforth this technique will be referred to as theClassical or the Conventional drop deposition technique),particularly in the context of depositing a drop on a low energysurface. Preparing low energy water-repelling surfaces, oenmotivated by the abundant natural examples,5–8 as well as theirremarkable practical applications, have been widely practicedover the past few decades.9–39 While preparing surfaces that aresuperhydrophobic in air remains a strong research focus, therehave been a massive upsurge in efforts in preparing amphi-phobic surfaces (or surfaces that repel multiple liquids)2,40–43

and surfaces that repel one/multiple liquids when placed insideanother liquid.1,44–47 The new drop deposition technique that we

, Department of Mechanical Engineering,

, Canada T6G 2G8. E-mail: sushanta.

tion (ESI) available. See DOI:

Chemistry 2013

shall discuss here is specially relevant for such under-liquid lowenergy surfaces. Please note that when we state that one surfaceS1 has lower energy than another surface S2 corresponding to aliquid L in a medium M, we effectively imply that contact angleof a drop of the liquid L (in surrounding medium M) on surfaceS2(q2) is smaller than that on surface S1(q1), i.e., q2 < q1, whichyields (from Young's Law) (gS2M � gS2L) > (gS1M � gS1L) (where gij

is the surface tension between components i and j), i.e., surfaceS1 is more repelling to liquid L than surface S2.

To elucidate the relevance of our proposed drop depositiontechnique, we rst need to highlight the drawbacks of theclassical pendant drop deposition technique,1–4 which is themost widely practiced drop deposition technique for the surfacewettability characterization. In this classical technique, a needlecontaining the drop at its end is brought near the surface. Ontouching the surface the drop spreads and attains an equilib-rium and in the process the drop gets detached from the needlespontaneously (see Fig. 1a–c). This apparently simple andwidely practiced mechanism inevitably requires the conditionthat the drop-substrate surface energy is larger than the drop-needle surface energy. Hence for surfaces for which the corre-sponding drop-substrate surface energy is much smaller thanthe drop-needle surface energy (these are the surface that wedescribe as low energy surfaces with respect to the concerneddrop), such simple steps, as discussed above, do not occur.Rather, depending on the extent of smallness of the drop-substrate surface energy, we can have two distinct situations.First, when this energy is substantially smaller than the drop-needle surface energy (oen quantied by an equilibrium

Soft Matter, 2013, 9, 7437–7447 | 7437

Fig. 1 (a–c) Drop generation and deposition on high energy surfaces, clearly showing a spontaneous deposition with the drop getting detached from the needlespontaneously. We have reported experiments with such high energy surfaces elsewhere.51 (d–f) Challenges associated with drop deposition on very low energysurfaces (q $ 170�), referred to as Case 1 in the manuscript (Waghmare et al.52 has recently discussed creation of such low energy surfaces). (d) The drop comes incontact with the surface, but does not spread. (e) The drop moves up with the needle. (f) The drop is made to spread on the surface by applying a force on the needle,which remains adhered to the drop. (g–i) Challenges associated with the drop deposition on low energy surfaces (not as low as the case studied in (d–f), referred to asCase 2 in the manuscript (experiments on such surfaces have been reported by Waghmare et al.52 and Qian et al.53). (g) The drop spreads on the surface but does notcome out of the needle spontaneously. (h) A pulling force is applied on the needle to bring it out from the drop, and in the process the drop deforms. (i) The needlefinally comes out of the drop.

Soft Matter Paper

Publ

ishe

d on

24

May

201

3. D

ownl

oade

d by

Uni

vers

ity o

f N

ewca

stle

on

24/0

8/20

14 2

2:54

:51.

View Article Online

contact angle >170�; henceforth we shall refer to this situationas Case 1), the drop on the needle virtually does not spread at allon the solid. Hence effectively when the drop touches thesubstrate it rolls on the substrate, so that when the needle islied the drop moves up with the needle with virtually noresistance from the substrate (see Fig. 1d and e). Under suchcircumstances, very oen an external force is applied on theneedle to ensure that this force is transmitted to the drop andthe drop spreads on the solid. As a result, the needle penetratesinsides the drop, oen making the drop spread, and contactangles are obtained under such conguration (see Fig. 1f andESI movie 1,† as well as recent papers published else-where2,4,48,49), which may not be accurate. The second case(henceforth we shall refer to this situation as Case 2) is the onewhere the drop-substrate has relatively lower energy (ascompared to the drop-needle surface energy), but not as smallas Case 1. In this case, the drop does spread on the substrate,although unlike the classical case, the drop does not getdetached from the needle spontaneously. Hence, we encountera situation, where the drop is adhered simultaneously to thesubstrate at its bottom and the needle at its top (see Fig. 1g).Therefore, to ensure that the needle comes out of the drop, one

7438 | Soft Matter, 2013, 9, 7437–7447

needs to apply sufficient pulling force on the needle, which rstdeforms the drop (see Fig. 1h), eventually leading to a “needle-free” drop state (see Fig. 1i). Therefore, in both these cases ofdrop deposition on low energy surfaces, we need to apply anite force on the needle in contact with the drop, which will gettransmitted as a nite force on the solid. The intrinsically smallcontact area of the drop on the solid, on account of large contactangles characterizing the low energy surfaces, will ensure thatthe resulting stress on the solid will be substantially large.Consequently, there may be a possible deformation of thecarefully-engineered solid surfaces demonstrating the neces-sary low energy behavior.2 Also, for Case 1, there is a possibilitythat a nite horizontal force is triggered at the three phasecontact line or TPCL (previous studies illustrate that in absenceof any external inuence such horizontal force does not exist50),which may eventually lead to a drop contact angle that isdifferent from the Young's angle, thereby enforcing an erro-neous characterization of the solid.

From the above discussion, it is evident that the conven-tional drop deposition technique suffers from the differentlimitations, in particular for depositing drops on low energysurfaces (irrespective of whether such surfaces are in air or

This journal is ª The Royal Society of Chemistry 2013

Paper Soft Matter

Publ

ishe

d on

24

May

201

3. D

ownl

oade

d by

Uni

vers

ity o

f N

ewca

stle

on

24/0

8/20

14 2

2:54

:51.

View Article Online

inside any liquid), since this deposition technique is not a non-intrusive and substrate-independent technique. In fact, in orderto address these limitations, there have been some efforts indevising new drop deposition techniques on low energysurfaces.54,55 However, such methods are very scarce and mostlyrely on application of several additional complexities that arehighly unlikely for simple contact angle measurement set up.On the other hand, in this paper, we propose a simple noveldrop deposition technique for under-liquid low energy surfaces,where unlike these sophisticated techniques, the drop deposi-tionmechanism is based on imbalance of surface tension forcescreated at the liquid–uid interface inextricably present fordrop deposition on under-liquid surfaces. Our mechanism isdescribed in Fig. 2. We rst generate a drop at the tip of a needleinserted inside the liquid bath, where the low energy substrateis present (see Fig. 2a). Subsequently, instead of bringing thedrop closer to the substrate, as is done in the conventionaltechnique, we take it away from the substrate so that the drop-needle entity hits the liquid–uid interface (see Fig. 2b). Herethis uid can be air, or some other liquid whose lm has beendeliberately created on the liquid surface. The moment thedrop-needle combination hits the liquid–uid interface, thedrop and the needle are subjected to different surface tensionforces, thereby allowing them to get detached from each other(see Fig. 2b). Consequently, the needle goes out of the liquid

Fig. 2 Schematic of our proposed drop deposition technique. (a) The needle isinserted in the liquid ‘2 andadrop (of liquid ‘1) is generatedat the tip of the needle.The solid substrate (wherewedesire to deposit the drop) is placed at the bottomofthe liquid bath. A thin film of another liquid (liquid ‘3, lighter than liquid ‘2 andimmiscible in ‘2) is dispersed on top of liquid ‘2; else liquid ‘2 can be in contact withthe ambient air (henceforth this liquid/air on topof ‘2 is referredgenerically asfluid‘3). (b): The needle containing the drop is pulled up till the needle-drop arrange-ment hits the liquid–fluid interface. (c) Thedropgets detached fromtheneedle and(being heavier than the surrounding liquid ‘2) falls through the liquid bath. (d) Thedescending drop is deposited on the low energy surface.

This journal is ª The Royal Society of Chemistry 2013

bath, whereas the drop remains at the interface. Depending onthe combinations of the surface tension values, this separateddrop may either form a Neumann condition,56 or form a lm atthe interface, or may actually get repelled from the interface. Forthe last case, if the density of the liquid drop is more than thedensity of the surrounding liquid (e.g., if the drop is of siliconoil and the surrounding liquid is water), the drop descendswithin the liquid bath (see Fig. 2c) and gets deposited on theunder-liquid low energy substrate (see Fig. 2d). Therefore, ourproposed mechanism is based on three founding ideas. Firstly,the procedure is applicable for under-liquid surfaces, sincesuch a condition introduces a new liquid–uid interface thatensures a differential force on the needle and the drop thatmakes it detach from each other. Secondly, we need to choosethe liquid–uid combination which ensures that the drop getsrepelled from the liquid–uid interface. Thirdly, we need toselect the drop liquid and surrounding liquid combination in amanner which allows the drop to fall down. With a control onthese three factors, we can get rid of the fundamental cause thatis responsible for difficulty in drop deposition on low energysurfaces, namely the presence of unfavorable drop-substratesurface energy in comparison to the favorable drop-needlesurface energy. Our proposed mechanism is not affected by theheight of the liquid column: this is extremely signicant sinceby adjusting the liquid column we can ensure that the Webernumber of the impacting drop (on the low energy under-liquidsubstrate) is sufficiently small so as to prevent any post-impactdynamics of the drop.57

The structure of the paper is as follows: rst we quantify thekey challenges associated with the drop deposition on lowenergy surfaces by conventional drop deposition technique.Second, we describe in detail our experiments with differentcombinations of liquids to validate the proposed drop deposi-tion mechanism. Third, we provide the necessary theoreticalbackground to analyze this new technique of drop deposition.Fourth, we discuss the key issues that govern our proposedtechnique. Finally, we end with concluding remarks.

2 Challenges associated with dropdeposition on low energy surfaces usingclassical drop deposition technique

There are two major challenges associated with the dropdeposition on low energy surfaces using classical drop deposi-tion technique. First, which primarily occurs for Case 1, is thepossible alteration of the drop contact angle on account of anadditional force triggered at the three phase contact line (TPCL).Second, which occurs for both the Cases 1 and 2, is the largestress exerted on the low energy surface. Both these problems,as discussed above, are caused by the unfavorable drop-substrate surface energy as compared to the favorable drop-needle surface energy, leading to a preferential adhesion of thedrop on the needle. Below, we quantify this surface energydifference driven preferential adhesion, as well as the twoproblems associated with the drop deposition. Please note thatalthough the central theme of the paper is to illustrate a newtechnique of drop deposition, we would like to place substantial

Soft Matter, 2013, 9, 7437–7447 | 7439

Soft Matter Paper

Publ

ishe

d on

24

May

201

3. D

ownl

oade

d by

Uni

vers

ity o

f N

ewca

stle

on

24/0

8/20

14 2

2:54

:51.

View Article Online

emphasis on the major limitations associated with the dropdeposition with conventional technique – this would ramify thesignicance of the proposed method. In this light, therefore,issues such as possible alteration of the drop contact angle, orthe possible deformation of the low energy surface, both ofwhich are associated with drop deposition on low energysurfaces by conventional technique, form an integral part of thepresent study.

2.1 Surface energy driven preferential drop adhesion on theneedle

The large difference in the drop-substrate and the drop-needlesurface energies (or equivalently the difference in the contactangles, as described above) leads to large difference in thecorresponding adhesive forces, and the drop shows a prefer-ence to remain adhered to the needle, and not get depositedon the substrate. We can obtain this difference in the drop-needle and the drop-substrate adhesive forces by noting thatthe adhesion force on the liquid drop exerted by a solid, incontact with the drop over a contour of length scale d, can beexpressed as:58,59

Fadh z gd(cos qr � cos qa) (1)

where g is the liquid–vapor or the liquid–liquid [when we studythe dynamics of a liquid (oil) drop in a surrounding liquid(water) medium] surface tension and qa and qr are the corre-sponding advancing and receding contact angles.

Therefore, the ratio of the drop-needle adhesive force (Fadh,n)to the drop-surface adhesive force (Fadh,s) can be expressed as:

Fadh;n

Fadh;s

zdnðcos qr;n � cos qa;nÞdsðcos qr;s � cos qa;sÞ ; (2)

where all quantities with subscript “n” and “s” refer to thequantities corresponding to drop-needle and the drop-substrate interfaces. Here dn is the length of the contour overwhich the drop is in contact with the needle, and hence it isequal to pd0, where d0 is the needle diameter. On the contrary,ds is the contact circumference of the drop with the substrate.For Case 1, since the drop contacts the low energy surface overextremely small region, we have dn [ ds. Also for suchextremely low energy surfaces (as considered in Case 1), theequilibrium drop contact angle is extremely large, therebyensuring that the contact angle hysteresis is extremely small(2), making (cos qr,s � cos qa,s) � (cos qr,n � cos qa,n). There-fore, we get from eqn (2):�

Fadh;n

Fadh;s

�Case 1

[1: (3)

This large difference in the adhesion force [eqn (3)] estab-lishes the reason for which the drop under conventionaldeposition condition does not deposit on the surface for Case 1;rather it remains adhered to the needle and goes up with theneedle as the needle is retracted (see Fig. 1d and e).

On the contrary, for Case 2, where the difference in the drop-needle and the drop-substrate surface energies (and hence the

7440 | Soft Matter, 2013, 9, 7437–7447

contact angles) are less severe, we get (cos qr,s � cos qa,s) z(cos qr,n � cos qa,n). Also, there is a nite spread of the drop onthe substrate, so that ds z dn. Therefore, we get from eqn (2):�

Fadh;n

Fadh;s

�Case 2

z1: (4)

From eqn (4), we see that for Case 2, the drop-needle and thedrop-substrate adhesion forces are pretty similar, therebyestablishing the rationale behind the drop conguration withthe drop adhered simultaneously to the substrate at its bottomand the needle at its top (see Fig. 1g).

2.2 Alteration of the drop contact angle

The large disparity in the adhesion forces for Case 1 ensures thatthe drop does not deposit and spread on the solid. Hence itbecomes extremely difficult to quantify thewetting characteristicof the concerned solid by measuring the corresponding contactangle. In such a scenario, it is difficult to ensure that there is anite contact between the drop and the solid substrate. Conse-quently, oen an external force is applied to establish the nitecontact. In the process, the needle oen penetrates the drop,with the drop spreading and attaining a nite contact with thesubstrate. Contact angles are then measured in this congura-tion2,4,48,49 (see ESI movie 1† for illustration). The minimum ofthis external force is theminimum force needed to overcome theresistance on theneedle impartedby thedrop tomake theneedlepenetrate inside the drop, expressed as:

Fext,min ¼ pgd0(cos qr,n � cos qa,n). (5)

Such external forcing on the drop through the needle mayactually change the drop contact angle, as discussed below.Fig. 3 illustrates the resulting force picture.

In presence of the external forcing Fext acting on the dropthrough the needle, we can actually get a force distribution (perunitwidth of the contact line) of f 0

x and f0y at theTPCL (see Fig. 3b),

which may lead to a drop contact angle q0 that is different fromYoung's angle of q. The force balance (per unit length) on thesection cut off at the contact line (see Fig. 3b) reads:

f 0y ¼ g cos�q0 � p

2

�¼ g sin q0 ðvertical equilibriumÞ;

f 0x ¼ gsv � gsl þ g sin�q0 � p

2

�

¼ g�cos q� cos q0

� ðhorizontal equilibriumÞ:

(6)

Similarly, the force balance on the rest of the drop (excludingthe two symmetrically cut out sections at the contact line) willyield:

Nsolid ¼ Fext þ 2prg cos�q0 � p

2

�

¼ Fext þ 2prg sin q0 ðvertical equilibriumÞ;symmetry conditions ðhorizontal equilibriumÞ;

(7)where Nsolid is the normal reaction force exerted by the solid onthe liquid drop. We shall discuss the signicance of eqn (7) andNsolid in the following section.

This journal is ª The Royal Society of Chemistry 2013

Fig. 3 (a) Schematic of the deposited dropwith the needle in contact. (b) Force balance on the liquid section at the three phase contact line highlighted in (a). (c) Forcebalance in the rest of the drop barring two symmetrically cut out liquid sections at the three phase contact line [one such section is highlighted in (a)]. In (c) nsolid ¼Nsolid/2pr and fext ¼ Fext/2pr, where r is the drop contact radius at the solid and Nsolid and Fext are expressed in the main text.

Paper Soft Matter

Publ

ishe

d on

24

May

201

3. D

ownl

oade

d by

Uni

vers

ity o

f N

ewca

stle

on

24/0

8/20

14 2

2:54

:51.

View Article Online

Eqn (6) clearly establishes that in presence of an externallyimposed force there is the possibility of an additional hori-zontal force at the TPCL, which may lead to an equilibriumangle q0 other than the Young's angle q. Generation of thisadditional horizontal force f 0

x is rather non-intuitive, since weare applying a vertical force Fext on the drop. Below, we shallexplain the physical mechanism that triggers this horizontalforce on the drop. As has been already mentioned, this force isoriginated as the needle presses the drop against the solidsubstrate. As long as the needle remains at the topmost point onthe drop, or in other words, the three points, namely the pointof contact of the needle with the drop, the drop center and thepoint at which the drop contacts the surface are along onesingle straight line, the force exerted on the drop by the needleis strictly vertical. As a consequence, therefore, there is noadditional horizontal force on the contact line, and the dropnever spreads. On the contrary, in such a conguration the dropexperiences a vertical compressive force, which forces it todeform and bulge, while maintaining a point contact on thesolid substrate. Things change when the needle gets displacedhorizontally from the top of the drop. In such a situation, theabove referred three points are not on a single straight line,which would mean that the vertical force exerted by the needleon the drop and the normal reaction of the substrate on thedrop are not aligned. Consequently, these two forces wouldgenerate a torque on the drop making it move towards theneedle. In the process, the needle penetrates the drop at itshorizontally displaced location. The resistance of this penetra-tion by the drop would trigger a force on the drop, that isdirected along the normal to the drop at this point of penetra-tion. This point being such that the corresponding normal has ahorizontal component, this would trigger a horizontal force onthe drop. This horizontal force needs to be balanced by anequivalent horizontal force on the drop, which can only arise atthe drop-substrate contact. This explains the origin of the (perunit length) horizontal force f 0

x at the drop substrate contactline, leading to a contact angle q0 s q. Please note that in casethere is f 0

x ¼ 0, one gets q0 ¼ q from eqn (6) and we recover theforce picture illustrated in Marchand et al.50

From eqn (6), we can easily nd that the variation between q

and q0 is a direct function of the force (per unit length) f 0x. Using

eqn (5) and the fact that f 0x is triggered due to the penetration of

the needle inside the drop, we can write f 0x � gnH, where

This journal is ª The Royal Society of Chemistry 2013

nH ¼ dH/R0 (with dH being the horizontal displacement of theneedle) is the horizontal component of the normal vector at thepoint on the drop where the needle penetrates. Hence, from eqn(6) we get cos q0 � cos q � dH/R0: hence if dH � R0 there issubstantial variation between q0 and q.

2.3 Stress on the solid during drop deposition

Presence of an external force on the drop, on account ofapplication of an external force on the needle holding the drop,is relevant for both Cases 1 and 2. For Case 1, as has beendemonstrated in the previous subsection, this force on theneedle is required make the drop spread on the substrate,whereas for Case 2, as has been discussed earlier, this force isneeded to detach the needle from the drop. In either of thecases, there is an equivalent stress exerted on the solid. Thisstress is different for Cases 1 and 2 and is also different fromthat predicted for the case of a simple drop sitting on thesubstrate.50,60

In eqn (7), we have expressed the normal force exerted by thesolid on the drop for Case 1. Please note that even for Case 2, thisgeneral expression for the normal force (exerted by the solid onthe drop) remains unchanged (things that get changed will beexplicated later). Following the analysis by Marchand et al.,50 wecan state Nsolid, expressed in eqn (7), is also the normal forceexerted by the drop on the solid. For Case 1, we take Fext[appearing in eqn (7)] as the force expressed in eqn (5), andtherefore, the corresponding normal stress (snsolid,1) on thesolid is:

snsolid;1 ¼

Nsolid

pr2¼ 2g

R0

�1þ d0R0

2r2ðcos qr;n � cos qa;nÞ

; (8)

where R0 is the radius of curvature of the drop and r is the dropcontact radius (r¼ R0sin q0). As we are considering Case 1, wherethe drop-solid surface energy is very small, r is very small, i.e., R0

[ r and d0 [ r, ensuring that the relative contribution of thestress can be substantially larger than the corresponding Lap-lace pressure (2g/R0) contribution.

For Case 2, where the needle must be pulled out of the drop,Fext is the drop-needle adhesion force Fadh,n [see eqn (1) and (2)]and is exerted vertically upwards (i.e., fext in Fig. 3c will bedirected away from the drop). Therefore, the normal stress(snsolid,2) on the solid can be expressed as:

Soft Matter, 2013, 9, 7437–7447 | 7441

Soft Matter Paper

Publ

ishe

d on

24

May

201

3. D

ownl

oade

d by

Uni

vers

ity o

f N

ewca

stle

on

24/0

8/20

14 2

2:54

:51.

View Article Online

snsolid;2 ¼

2g

R0

�� 1þ d0 knR0

2r2ðcos qr;n � cos qa;nÞ

: (9)

Here d0 � r, but R0 [ r, so that snsolid,2 is enhanced, althoughthe extent is less severe as compared to snsolid,1. Also note thatsnsolid,1 represents a compressive stress on the solid, whereassnsolid,2 represents an tensile stress. Presence of such stresses inboth Case 1 and 2 may potentially damage the surface, whichis oen very delicately engineered to ensure large contactangles.

Through eqn (8) and (9) we get the idea of the additionalstresses on the solid that occurs during drop deposition usingclassical techniques. The exact impact of these additionalstresses in deforming the solid is dictated by the elasticity of thesolid, which may vary drastically depending on the desiredapplications. On the other hand, a better pinpointing of theeffect of these additional stresses can be obtained by comparingthem against the Laplace pressure 2g/R0. For example, for Case1, the ratio of this additional stress to the Laplace pressure isd0R0(cos qr,n � cos qa,n)/2r

2. We can provide an estimate of thisratio by considering a practical example. Let us consider that forCase 1, qr,n ¼ 179� and qa,n ¼ 180�, which would lead to a valueof this ratio (with drop volume of 2 mL and needle diameter d0¼

Fig. 4 Experiments for validating our proposed drop deposition technique. (a) Creainterface. (b) Pulling the needle and the drop-needle combination hitting this watercondition at the interface (also see the inset for the corresponding force balance). Ththe interface and (being heavier than water) descends in the liquid column. This is

7442 | Soft Matter, 2013, 9, 7437–7447

0.8 mm) as 1 (indicating that the stress on the solid is doublethe Laplace pressure). With a different set of parameters (e.g.,qr,n ¼ 170� and qa,n ¼ 171�), we would get this ratio as 0.026.From the above two examples, it can be clearly seen that thepresence of this additional stress is primarily important forextremely low energy surfaces, where the drop-surface contact isvirtually negligible. In this light, we can infer that for Case 2,where there is indeed a nite drop-surface contact, the magni-tude of this additional stress, as has been mentioned earlier,will be substantially smaller.

3 Experiments

In the present study, we use the DSA 100 setup with congu-rations similar to our previous work.61 A distortion-free glasscuvette (Kruss Germany, SC-02) of dimension 40 mm � 40 mm� 30 mm is used as a liquid container. We rst generate theliquid drop (e.g. silicon oil drop; henceforth we call this liquidas ‘1) of known volume at the tip of a stainless steel needle(diameter d0 ¼ 1.8 mm) inside another liquid (e.g. water;henceforth we call this liquid ‘2) (see Fig. 4a). The solid onwhich the drop is to be deposited is positioned at the bottom ofthe liquid-lled cuvette. The drop-needle combination is

tion of a silicon oil drop inside a column of water. The interface is the water–fluid–fluid interface. (c) In case this fluid is air, the drop forms a Neumann equilibriumis is the Case B in Table 1. (d) In case this fluid is canola oil, the drop is repelled fromour proposed drop deposition technique. This is the Case A in Table 1.

This journal is ª The Royal Society of Chemistry 2013

Paper Soft Matter

Publ

ishe

d on

24

May

201

3. D

ownl

oade

d by

Uni

vers

ity o

f N

ewca

stle

on

24/0

8/20

14 2

2:54

:51.

View Article Online

withdrawn away from the solid. Subsequently, the drop-needlecombination hits the liquid–uid interface once this combi-nation reaches the top of the liquid level in the cuvette (seeFig. 4b). This uid (henceforth we call this uid ‘3) can be air, ora carefully prepared lm of an appropriate liquid (e.g., canolaoil). The moment the needle-drop combination hits this liquid–uid interface, the needle and the drop are subjected todifferent forces, which ensure that they get detached from eachother. Therefore, we now have a needle-free drop at the liquid–uid interface. The ensuing drop dynamics (quantitativelyexplained in details in the following section) holds the key toour technique. Depending on the combinations of the surfacetensions (of the liquids–uids ‘1, ‘2 and ‘3), the drop at theinterface either attains a Neumann state (see Fig. 4c and ESImovie 3†), or spreads at the liquid–uid interface, or getsrepelled from the liquid–uid interface and falls inside theliquid bath (see Fig. 4d and ESI movie 2†). For this third case tohappen, we need to ensure that density of ‘1 is more than thedensity of ‘2 (e.g., silicon drop in water medium), and thismechanism is our proposed technique of drop deposition.Obviously, this technique essentially eliminates the role of thesubstrate on drop deposition, thereby ensuring, unlike theclassical drop deposition technique, a substrate-independent,non-intrusive drop deposition on low energy surfaces. Pleasenote that by drop “dynamics” at the liquid–uid interface, weimplied only the dynamics triggered by the imbalance of thesurface tension forces, and did not consider (we shall laterdiscuss the detailed rationale) the dynamics caused by the dropimpact at nite Weber number at the liquid–uid interface.62–68

Table 1 Different fluid combinations considered for drop deposition from theinterface and corresponding interfacial tensions between different phases. In thistable we characterize the drop dynamics at the liquid–fluid (‘2� ‘3) interface, andhence the onset of drop deposition, through parameters such as SNm,1, SNm,2 anddf,c/R. Also we always use a drop (silicon oil) and surrounding liquid (water)combination, such that the drop is heavier than the surrounding liquid

Phase Case-A Case-B

1 Silicon oil Silicon oil2 Water Water3 Canola oil Airg12 (mN m�1) 33.33 33.33g23 (mN m�1) 18.01 72.2g13 (mN m�1) 2.5 (see Appendix) 24SNm,1 1.19 1.194SNm,2 �8.66 1.34df,c/R 0.18 �0.15Neumann equilibriumCondition No NoDrop spreading at interface No YesDrop deposition Yes No

4 Drop dynamics at the liquid–fluidinterface: principle of proposed dropdeposition technique

The surface energy imbalance driven dynamics of the oil drop,detached from the needle, at the liquid–uid (‘2 � ‘3) interfaceholds the key to our proposed deposition mechanism. At thisinterface, the drop may either attain an equilibrium Neumannstate (see Fig. 4c and the ESI movie 3†),56 or may exhibit adynamic condition, dictated by the corresponding free energychange. In case, the drop attains the equilibrium Neumannstate (see the inset in Fig. 4c), we get from the force balance (weuse gij to represent the interfacial tensions between liquid–uidcomponents i and j and q1j to denote the contact angle made bythe oil drop in liquid–uid component j; see inset of Fig. 4c):

g23 ¼ g12cos q12 + g13cos q13, (10)

g12sin q12 ¼ g13sin q13. (11)

Solving eqn (10) and (11), we get:

cos q12 ¼ g122 þ g23

2 � g132

2g12g23

; (12)

cos q13 ¼ g132 þ g23

2 � g122

2g13g23

: (13)

This journal is ª The Royal Society of Chemistry 2013

Therefore, to form the equilibrium Neumann state, one

needs to simultaneously satisfy�1#g12

2 þ g232 � g13

2

2g12g23# 1 and

�1#g13

2 þ g232 � g12

2

2g13g23# 1. In case this condition is not satis-

ed, then when the liquid drop hits the liquid–uid interface, itwill try to spread at the interface or may get repelled from theinterface, with both these phenomena being dictated by thecorresponding free energy change. Let us assume that the dropforms a lm of thickness df and surface area A at the ‘2 � ‘3interface, so that using volume conservation, we get:

A ¼ 4pR3

3df; (14)

where R is the radius of the drop. By spreading as a lm of areaA, there is an increase in surface energy (g12 + g13)A and there isa decrease in the surface energy g23A. Also the initial surfaceenergy of the impacting drop (neglecting its kinetic energy, seelater for detail rationale) is 4pR2g12. Hence the net change ofenergy will be:

DE ¼ 4pR2

�ðg12 þ g13 � g23Þ

R

3df� g12

: (15)

If DE < 0, then the spreading will be favored, whereas if DE >0, then the spreading is not favored and the drop will fall backinside the liquid medium (provided gravity does not oppose it).

We analyze the experimental results demonstrating the dropdeposition or no-deposition as a function of gij. The results ofthe drop (of liquid ‘1) dynamics at the liquid–uid (‘2 � ‘3)interface has been summarized in Table 1. We dene

Neumann parameters as SNm;1 ¼ g122 þ g23

2 � g132

2g12g23and

SNm;2 ¼ g132 þ g23

2 � g122

2g13g23. As discussed above, only when�1#

SNm,1 # 1 and �1 # SNm,2 # 1 hold simultaneously, weencounter the equilibrium Neumann state. Once this conditiondoes not hold, the resulting dynamics of the drop (which can be

Soft Matter, 2013, 9, 7437–7447 | 7443

Soft Matter Paper

Publ

ishe

d on

24

May

201

3. D

ownl

oade

d by

Uni

vers

ity o

f N

ewca

stle

on

24/0

8/20

14 2

2:54

:51.

View Article Online

either spreading at the interface or getting repelled from theinterface) can be quantied by noting the value of the critical

lm thickness df;c ¼�R3

�g12 þ g13 � g23

g12, which is the lm

thickness for which DE ¼ 0 [see eqn (15)]. A nite positive valueof df,c, (which is equivalent to a negative value of the spreadingparameter S(¼ g23 � g12 � g13)), indicates that the drop cannotcontinue to spread at the interface (since for df < df,c, DE > 0),and it will be repelled from the interface, making it fall downthe bulk liquid (if the drop density is more than the density ofthe bulk liquid). On the other hand, if df,c is negative (which isequivalent to a positive value of the spreading parameterS(¼ g23 � g12 � g13)), that will suggest that irrespective of thelm thickness one always encounters DE < 0, which indicates anuninterrupted spreading of the drop at the liquid–uid (‘2 � ‘3)interface. In this light, we can analyze the results summarizedin Table 1. It is clearly seen and explained above that for Case A,where canola oil is present as the uid on top of the liquid bath,the drop dynamics at the liquid–uid (‘2 � ‘3) interface is suchthat the drop gets repelled from the interface (this is dictated bythe parameters SNm,1, SNm,2 and df,c), and by virtue of the factthat the silicon oil drop is heavier than the surrounding liquidwater, we attain the drop deposition (see Fig. 4a, b and d andESI movie 2†). On the other hand, for Case B, where it is simplyliquid–air interface, the condition dictates that the silicon dropwill keep on spreading at the water–air interface. However, asseen from Fig. 4a–c and ESI movie 3,† the silicon drop in Case B,actually forms an equilibrium Neumann state. This is ratherpuzzling, given that both the parameters SNm,1 and SNm,2 aregreater than unity (see Table 1) for Case B. This can be explainedfollowing an analysis similar to that done elsewhere.69,70 As thesilicon oil drop spreads on the water–air interface some siliconoil molecules may actually get trapped at the water–air inter-face. These entrapped oil molecules behave similar to surfac-tant molecules adsorbed at an water–air interface, and mayactually reduce the water–air surface tension (g23) signicantly.For example, with g23 � 40 mN m�1, we get SNm,1 ¼ 0.8 andSNm,2 ¼ 0.55, whereas with g23 � 30 mN m�1, we get SNm,1 ¼0.538 and SNm,2 ¼ 0.25. Hence, with such oil-molecule-entrap-ment-induced reduction of water–air interfacial tension, weindeed get�1# SNm,1 # 1 and�1# SNm,2# 1 simultaneously –this explains the attainment of the Neumann equilibrium stateby the oil drop at the liquid–uid (‘2 � ‘3) interface (see Fig. 4cand ESI movie 3†).

5 Discussions

In this section, we shall discuss certain key issues that dictatethis unique drop deposition technique and other possibledifficulties associated with drop deposition on low energysurfaces.

5.1 Role of the needle velocity

Wedenote the velocity of the needle (with the drop) at the time ofimpact of the drop at the liquid–uid (‘2 � ‘3) interface as vn.Therefore, the oil drop impacts the interface ‘2� ‘3 with a speed

7444 | Soft Matter, 2013, 9, 7437–7447

vn. Theanalysis developed so far is basedon thewettability issuesof this drop at this interface, i.e., we have implicitly neglected therole of this impacting velocity vn. However as has been wellestablished the drop impacting speed plays a major role indictating the dynamics of the drop on both solid–liquid57 andliquid–liquid interfaces.62–68 For small drops, neglecting gravity,there are two crucial dimensionless parameters that dictate thepost-impact drop dynamics, namely the Weber number (We ¼rRv2/g, where r, R and v are the density, diameter and velocity,respectively of the impacting drop and g is the surface tensionbetween the drop and the surrounding phase) and the Ohne-sorge number (Oh ¼ h=

ffiffiffiffiffiffiffiffiffirRg

p, whereh is thedynamic viscosity of

the liquid inside the drop).57 We dictates whether on impactthe drop will exhibit a deformation beyond that dictated by thesurface-tension driven dynamics: for We > 1, i.e., when thekinetic energy exceeds the surface energy (typically for the case oflow energy surfaces), we get a plethora of drop dynamics anddrop shapes.57Theexactnature of such shapes aredictatedby theOh number. Key to note is that whether or not a drop impact willyield a situation that is distinct from simple drop deposition andspreading is solely dictated by the value of We. For We� 1, thedrop impact is as good as simple drop deposition. In our study,the typical needle velocity is v ¼ vn z 1 mm s�1 and the dropradius is R z 7 mm; therefore with g ¼ g12 ¼ 33.33 mN m�1

(silicon-oil–water surface tension) and r ¼ rso z 1100 kg m�3

(silicon oil density), we get Wez 2.3 � 10�4. Such an extremelysmall impactWe of the oil drop at the ‘2� ‘3 interfacewill ensurethat this drop impact is as good as drop deposition, and the dropdynamics is dictated solely by the surface tension effects.

There is another important effect connected to the speed ofthe drop deposition at this ‘2 � ‘3 interface. The drop on hittingthis interface, will deform the interface, as is typical for dropimpact on liquid–uid interface.71 Whether, such interfacedeformation is important is dictated by the time scale (sid) ofthe interface deformation relative to the time scale (ss) of theeffects (e.g., drop spreading) triggered by the surface tensionforces. sid can be identied as the contact time72 of the drop withthe ‘2 � ‘3 interface. For low We, we can writesidz lnð1=WeÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

rR3=gp

,73,74 so that for the present case, we getsid z 1 s. On the other hand, ss will scale as the time requiredfor the spreading, driven by the imbalance of the surfaceenergies. Please note all the three possible cases, namely, dropspreading, attainment of Neumann condition and drop depo-sition will follow this time scale, since all these processes areeffectively triggered by the imbalance of the surface energies.Following the analysis by Bergeron and Langevin,75 we canestimate ss � R4/3(r1/2h1/2/S0)2/3, where S0 ¼ g23 � g12 � g13 is thespreading parameter, which considering an water–air (‘2 � ‘3)interface, yields S0 z 15 mN m�1. Therefore, with h ¼ 200 mPa� s (silicon oil drop viscosity), we obtain ss z 0.1 s. Conse-quently, for our case, we get ss � sid, and hence the interfacedeformation, on account of drop impact, is not signicant.

5.2 Role of the height of the liquid column

One of the key advantages of the proposed technique is thatthe feasibility of this technique is independent of the height of

This journal is ª The Royal Society of Chemistry 2013

Paper Soft Matter

Publ

ishe

d on

24

May

201

3. D

ownl

oade

d by

Uni

vers

ity o

f N

ewca

stle

on

24/0

8/20

14 2

2:54

:51.

View Article Online

the water column separating the liquid–uid (‘2 � ‘3) and thesolid–liquid interfaces. In other words, our proposed tech-nique, being only dependent on the presence of the liquid–uid interface becomes feasible for any height of the liquid(water) column. This is important as this allows us to controlthe speed of drop impact on the solid–liquid interface (pleasenote, we are dealing with the impact of the oil drop at twolocations: rst is the liquid–uid interface and second is thesolid–liquid interface). Unlike the case of conventional dropdeposition technique, our present case gives rise to a situationwhere the oil drop will hit the solid–liquid interface with anite velocity (vd), caused by the fact that it has attained anite velocity while getting repelled from the liquid–uidinterface (case denoted in Fig. 2b and c). We need to ensure,by adjusting the height of the liquid column, that vd is suffi-ciently small so that the corresponding Weber number is smallenough not to cause any drop deformation at the solid–liquidinterface. This is specially relevant given the fact that we areusing this technique to deposit drops on low energy surfaces,on which the drops demonstrate pronounced shape variationsand effects such as bouncing and splashing even for low andmoderate We values.

Silicon oil being heavier than water, the oil drop aer beingrepelled from the liquid–uid interface will undergo a verticaldescend under the inuence of its own weight, the buoyantforce from the surrounding water and the drag force. Therefore,we can express the instantaneous oil drop velocity v (in thedownward direction) at time t and position x as:

mo

dv

dt¼ mog � FB � FD0mov

dv

dx¼ mog � FB � FD (16)

where mo ¼ 4/3pR3rso is the mass of the oil drop, g is theacceleration due to gravity, FB ¼ 4/3pR3rwg (rw is the density ofwater) is the buoyant force and FD ¼ 6phRv is the drag force.Therefore, eqn (16) reduces to

vdv

dx¼ k2 � k1v; (17)

where k2 ¼ g(1� rw/rso) and k1 ¼ 9h/2R2rso. Integrating eqn (17)in presence of the condition v(t ¼ 0) ¼ v0 and x(t ¼ 0) ¼ 0 (weconsider t ¼ 0 as the time when the drop starts its verticaldescend from the liquid–uid interface), we get:

k2

k22ln

�k1v0 � k2

k1v� k2

�þ 1

k1ðv0 � vÞ ¼ x; (18)

which can be used to express the dimensionless velocityimplicitly as:

v ¼ aþ ð1� aÞ exp�� vþ x� 1

a

�; (19)

where a¼ k2/k1v0, �v¼ v/v0 and �x¼ x/(v0/k1). For sufficiently largeliquid column height, we get �x + �v [ 1, so that eqn (19)simplies to

v ¼ a0v ¼ k2

k1¼ 2R2g ðro � rwÞ

9h¼ vT; (20)

where vT is the terminal velocity. Depending on the ratio k2/k1,we may have vd/v0 < 1, or vd/v0 > 1. The main motivation being to

This journal is ª The Royal Society of Chemistry 2013

ensure a minimum possible value of vd, for the former case weshould attempt to make the liquid column large enough toensure that vd ¼ vT, whereas for the latter case the liquidcolumn should be as small as possible to ensure vd z v0.

On a different note, such a drop deposition technique,where the drop can impact an under-liquid low energy surface(with the ability to control the impact speed by regulating theliquid column height) will allow, possibly for the rst time,investigations on drop impact on under-liquid low energysurfaces and the resulting variations in drop shape anddynamics. Such analysis of post-impact (on low energysurfaces) drop shape and dynamics have been very wellstudied in air medium; however, in context of under-liquid lowenergy surfaces such studies are very rare. This is speciallysurprising given the rich physics expected in the dropdynamics in a medium that has a nite viscosity. One canimagine the only reason for this absence being the lack ofknowledge to deposit drops on under-liquid low energysurfaces, with a controllable impact speed. Our proposedtechnique successfully lls this void.

5.3 Other challenges associated with drop deposition on lowenergy surfaces

We end this paper by discussing few more challenges, otherthan those explicated in Sections 2.2 and 2.3, associated withthe drop deposition on low energy surfaces. One such chal-lenge, particularly relevant for Case 1, is the possible torque onthe drop generated during the time of deposition. In Case 1,there is very little contact of the drop with the substrate, andtherefore a force on the needle is required to make the dropspread on the substrate. However, if during the time of appli-cation of the force the needle is displaced slightly horizontally,it will generate a torque on the drop, thereby making the droproll up the needle (see ESI movie 4†). Once the drop attains thisposition it is virtually impossible to deposit this drop, and oneneeds to perform the deposition afresh.

In Case 2 (explained earlier), the drop is under comparableadhesive forces from the needle at its top and the substrate at itsbottom. In such a scenario, as demonstrated by Qian et al.,53

withdrawal speed of the needle dictates the size of the drop onthe substrate – this is not always a desired situation in case oneneeds to deposit a drop of specied volume.76 In an analogousproblem in Case 1, during the retraction of the needle, anunknown portion of the drop volume will invariably remainattached to the needle; as a consequence, during subsequentdeposition it will be difficult to deposit a drop of desiredvolume.

6 Conclusions

In this paper, we have proposed a new drop deposition tech-nique for under-liquid low energy surfaces. The methodreplaces the conventional drop deposition technique, thatexhibits several limitations in context of deposition on lowenergy surfaces (such surfaces may be in air or inside a liquid).The limitation of this classical technique stems from the

Soft Matter, 2013, 9, 7437–7447 | 7445

Soft Matter Paper

Publ

ishe

d on

24

May

201

3. D

ownl

oade

d by

Uni

vers

ity o

f N

ewca

stle

on

24/0

8/20

14 2

2:54

:51.

View Article Online

intrinsically intrusive nature of the technique that may poten-tially damage the carefully engineered low energy surface, oreven worse, may alter the contact angle of the deposited drop.Therefore, in essence, the key step involved in characterizingthe wettability of low energy surfaces – there has been massiveupsurge in fabricating such surfaces owing to their applicationin multiple disciplines – has remained riddled with funda-mental errors. In our proposed method, we ensure that we havesuccessfully removed the “intrusive” aspect of the drop depo-sition – through this method the researchers will now be able tosuccessfully deposit drop on under-liquid surfaces in a mannerthat is completely independent on the substrate wettability, andin the process will be able to successfully by-pass all the majorlimitations associated with the conventional drop depositiontechnique.

Appendix: measurement of silicon-oil-canola-oil surface tension

A part of the analysis requires the estimation of the surfacetension (gso,co) between the silicon oil and the canola oil (i.e.,surface tension g13 in Case A in Table 1). This value being noteasily available in the literature, we perform our own experi-ment to determine gso,co. The experiment is described in Fig. 5.We insert the drop-generating needle inside the bath of canolaoil, and attempt to generate a drop of the silicon oil. The resultis not a drop of silicon oil; rather a vertical column of silicon oilterminated by a spherical interface. Applying Bernoulli'stheorem between points A (the corresponding pressure is P1)and B (the corresponding pressure is P3) inside the silicon oilcolumn (see Fig. 5), we get (neglecting the velocity of the siliconoil column), P3 ¼ P1 + rsogh (where rso is the density of thesilicon oil and h is the height of the silicon oil column, seeFig. 5). Also P3 ¼ P4 + 2gso,co/Rso (where Rso is the radius ofcurvature of the curved interface and P4 is the pressure at thesame horizontal level as point B but inside the canola oil, seeFig. 5). Further, P2 ¼ P1 (where P2 is the pressure at the samehorizontal level as point A but inside the canola oil) and P4 ¼P2 + rcogh (where rco is the density of the canola oil). From theserelations, we nally obtain gso,co ¼ Rsogh(rso � rco)/2, whichyields (using Rso ¼ 0.75 mm, h ¼ 3.75 mm, rso ¼ 1100 kg m�3

and rco ¼ 920 kg m�3) gso,co z 2.5 mN m�1.

Fig. 5 Procedure to obtain silicon-oil-canola-oil surface tension gso,co.

7446 | Soft Matter, 2013, 9, 7437–7447

Acknowledgements

The authors gratefully acknowledge the Natural Sciences andEngineering Research Council of Canada (NSERC) for providingnancial support to S. D. in form of the Banting PostdoctoralFellowship. The authors also acknowledge Aleksey Baldygin forhis help during the preparation of the graphical abstract.

References

1 L. Zhang, Z. Zhang and P. Wang, NPG Asia Mater., 2012, 4, e8.2 X. Deng, L. Mammen, H.-J. Butt and D. Vollmer, Science,2012, 335, 67.

3 M. Liu, S. Wang, Z. Wei, Y. Song and L. Jiang, Adv. Mater.,2009, 21, 665.

4 A. Tuteja, W. Choi, M. Ma, J. M. Mabry, S. A. Mazzella,G. C. Rutledge, G. H. McKinley and R. E. Cohen, Science,2007, 318, 1618.

5 C. Neinhuis and W. Barthlott, Ann. Bot., 1997, 79, 667.6 X. Gao and L. Jiang, Nature, 2004, 432, 36.7 E. Bormashenko, Y. Bormashenko, T. Stein, G. Whyman andE. Bormashenko, J. Colloid Interface Sci., 2007, 311, 212.

8 B. Bhushan, Philos. Trans. R. Soc., A, 2009, 367, 1445.9 P. Roach, N. J. Shirtcliffe and M. I. Newton, SoMatter, 2008,4, 224.

10 T. Onda, S. Shibuichi, N. Satoh and K. Tsujii, Langmuir, 1996,12, 2125.

11 M. Lundgren, N. L. Allan, T. Cosgrove and N. George,Langmuir, 2003, 19, 7127.

12 A. Marmur, Langmuir, 2003, 19, 8343.13 N. A. Patankar, Langmuir, 2003, 19, 1249.14 B. He, N. A. Patankar and J. Lee, Langmuir, 2003, 19, 4999.15 H. Y. Erbil, A. L. Demirel, Y. Avci and O. Mert, Science, 2003,

299, 1377.16 R. Blossey, Nat. Mater., 2003, 2, 301.17 A. Lafuma and D. Quere, Nat. Mater., 2003, 2, 457.18 N. A. Patankar, Langmuir, 2004, 20, 7097.19 J. Jopp, H. Grull and R. Yerushalmi-Rozen, Langmuir, 2004,

20, 10015.20 R. D. Narhe and D. A. Beysens, Phys. Rev. Lett., 2004, 93,

076103.21 C. Ishino, K. Okumura and D. Quere, Europhys. Lett., 2004,

68, 419.22 Y.-T. Cheng and D. E. Rodak, Appl. Phys. Lett., 2005, 86,

144101.23 Y.-L. Zhang, H. Xia, E. Kim and H.-B. Sun, So Matter, 2012,

8, 11217.24 M. Wolfs, T. Darmanin and F. Guittard, So Matter, 2012, 8,

9110.25 L. Li, S. Pan, X. Pang, H. Chen, D. Hu, L. Ke, Y. Xiong and

W. Xu, So Matter, 2012, 8, 7357.26 E. Hosono, S. Fujihara, I. Honma and H. Zhou, J. Am. Chem.

Soc., 2005, 127, 13458.27 Z. Guo, F. Zhou, J. Hao and W. Liu, J. Am. Chem. Soc., 2005,

127, 15670.28 G. Carbone and L. Mangialardi, Eur. Phys. J. E: So Matter

Biol. Phys., 2005, 16, 67.

This journal is ª The Royal Society of Chemistry 2013

Paper Soft Matter

Publ

ishe

d on

24

May

201

3. D

ownl

oade

d by

Uni

vers

ity o

f N

ewca

stle

on

24/0

8/20

14 2

2:54

:51.

View Article Online

29 M. Reyssat, A. Pepin, F. Marty, Y. Chen and D. Quere,Europhys. Lett., 2006, 74, 306.

30 C. Yang, U. Tartaglino and B. J. Persson, Phys. Rev. Lett.,2006, 97, 116103.

31 M. Lundgren, N. L. Allan and T. Cosgrove, Langmuir, 2007,23, 1187.

32 L. Barbieri, E. Wagner and P. Hoffmann, Langmuir, 2007, 23,1723.

33 S. Moulinet and D. Bartolo, Eur. Phys. J. E: So Matter Biol.Phys., 2007, 24, 241.

34 M. Reyssat, J. M. Yeomans and D. Quere, Europhys. Lett.,2008, 81, 26006.

35 H. Kusumaatmaja, M. L. Blow, A. Dupuis and J. M. Yeomans,Europhys. Lett., 2008, 81, 36003.

36 T. An, S. J. Cho, W. Choi, J. H. Kim, S. T. Lim and G. Lim, SoMatter, 2011, 7, 9867.

37 L. Zhang, N. Zhao, X. Li, Y. Long, X. Zhang and J. Xu, SoMatter, 2011, 7, 4050.

38 M. Xu, N. Lu, H. Xu, D. Qi, Y. Wang, S. Shi and L. Chi, SoMatter, 2010, 6, 1438.

39 J. S. Lee, J. Ryu and C. B. Park, So Matter, 2009, 5, 2717.40 M. Im, H. Im, J.-H. Lee, J.-B. Yoon and Y.-K. Choi, SoMatter,

2010, 6, 1401.41 S. H. Wong, T.-S. Kang, S. K. Y. Tang, E. J. Smythe,

B. D. Hatton, A. Grinthal and J. Aizenberg, Nature, 2011,477, 443.

42 T. Darmanin, F. Guittard, S. Amigoni, E. T. de Givenchy,X. Noblin, R. Kofman and F. Celestini, So Matter, 2011, 7,1053.

43 H.-J. Butt, C. Semprebon, P. Papadopoulos, D. Vollmer,M. Brinkmann and M. Ciccotti, So Matter, 2013, 9, 418.

44 Z. Xue, S. Wang, L. Lin, L. Chen, M. Liu, L. Feng and L. Jiang,Adv. Mater., 2011, 23, 4270.

45 B. Su, S. Wang, Y. Songa and L. Jiang, So Matter, 2011, 7,5144.

46 Q. Cheng, M. Li, F. Yang, M. Liu, L. Li, S. Wang and L. Jiang,So Matter, 2012, 8, 6740.

47 Q. Cheng, M. Li, Y. Zheng, B. Su, S. Wang and L. Jiang, SoMatter, 2011, 7, 5948.

48 H.-T. Kwon, A. T. Paxson, K. K. Varanasi and N. A. Patankar,Phys. Rev. Lett., 2011, 106, 036102.

49 X. Liu, J. Gao, Z. Xue, L. Chen, L. Ling, L. Jiang and S. Wang,ACS Nano, 2012, 6, 5614.

50 A. Marchand, J. Weijs, J. H. Snoeijer and B. Andreotti, Am. J.Phys., 2011, 79, 999.

51 S. Das, P. R. Waghmare, M. Fan, N. S. K. Gunda, S. S. Roy andS. K. Mitra, RSC Adv., 2012, 2, 8390.

This journal is ª The Royal Society of Chemistry 2013

52 P. R. Waghmare, S. Das and S. K. Mitra, Sci. Rep., 2013, 3,1862.

53 B. Qian, M. Loureiro, D. A. Gagnon, A. Tripathi andK. S. Breuer, Phys. Rev. Lett., 2009, 102, 164502.

54 T. Leıchle, L. Tanguy and L. Nicu, Appl. Phys. Lett., 2007, 91,224012.

55 T. Vuong, A. Qi, M. Muradoglu, B. H.-P. Cheong, O. W. Liew,C. X. Ang, J. Fu, L. Yeo, J. Friend and T. W. Ng, So Matter,2013, 9, 3631.

56 P.-G. de Gennes, F. Brochart-Wyart and D. Quere, Capillarityand Wetting Phenomena: Drops, Bubbles, Pearls, Waves,Springer, New York, 2003.

57 A. L. Yarin, Annu. Rev. Fluid Mech., 2006, 38, 159.58 C. W. Extrand and Y. Kumagai, J. Colloid Interface Sci., 1995,

170, 515.59 C. Antonini, F. J. Carmona, E. Pierce, M. Marengo and

A. Amirfazli, Langmuir, 2009, 25, 6143.60 S. Das, A. Marchand, B. Andreotti and J. H. Snoeijer, Phys.

Fluids, 2011, 23, 072006.61 P. R. Waghmare and S. K. Mitra, Langmuir, 2010, 26,

17082.62 J. J. Thomson andH. F. Newall, Proc. R. Soc. London, 1885, 39,

417.63 O. W. Jayaratne and B. J. Mason, Proc. R. Soc. London, Ser. A,

1964, 280, 545.64 Y. Couder, E. Fort, C.-H. Gautier and A. Boudaoud, Phys. Rev.

Lett., 2005, 94, 177801.65 A. Prosperetti and H. N. Oguz, Annu. Rev. Fluid Mech., 1993,

25, 577.66 S. L. Manzello and J. C. Yang, Exp. Fluids, 2002, 32, 580.67 J. Zou, P. F. Wang, T. R. Zhang, X. Fu and X. Ruan, Phys.

Fluids, 2011, 23, 044101.68 E. Yakhshi-Tai, H. J. Cho and R. Kumar, J. Colloid Interface

Sci., 2010, 350, 373.69 W. D. Harkins, The Physical Chemistry of Surface Films,

Reinhord Publishing Corporation, New York, 1954.70 C. M. Phan, B. Allen, L. B. Peters, T. N. Le and M. O. Tade,

Langmuir, 2012, 28, 4609.71 Z. Mohamed-Kassim and E. K. Longmire, Phys. Fluids, 2003,

15, 3263.72 D. Richard, C. Clanet and D. Quere, Nature, 2002, 417, 811.73 G. Foote, J. Atmos. Sci., 1975, 32, 390.74 A. Gopinath and D. Koch, J. Fluid Mech., 2002, 454, 145.75 V. Bergeron and D. Langevin, Phys. Rev. Lett., 1996, 76,

3152.76 E. Bormashenko, R. Pogreb, T. Stein, G. Whyman and

M. Hakham-Itzhaq, Appl. Phys. Lett., 2009, 95, 264102.

Soft Matter, 2013, 9, 7437–7447 | 7447