Colloids and Surfaces A: Physicochem. Eng. Aspects 455 (2014) 19–27
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Colloids and Surfaces A: Physicochemical andEngineering Aspects
journa l h om epage: www.elsev ier .com/ locate /co lsur fa
xperimental and computational study of triple line shape andvolution on heterogeneous surfaces
eeharika Anantharajua, Mahesh V. Panchagnulab,∗
Department of Mechanical Engineering, Tennessee Tech University, Cookeville, TN 38505, USADepartment of Applied Mechanics, Indian Institute of Technology Madras, Chennai 600036, India
i g h l i g h t s
Current work studies the effect ofmacroscale chemical heterogeneitieson triple line.Triple line shape and kinetics arestudied, especially the nature of pin-ning.The triple line shape is shown todepend on the geometry of the chem-ical heterogeneity.The interfacial energy is shown toreach a maximum value in the pinnedstate.The system energy shows a gradientdescent phase prior to the suddenjump due to de-pinning.
g r a p h i c a l a b s t r a c t
r t i c l e i n f o
rticle history:eceived 27 March 2014eceived in revised form 13 April 2014ccepted 18 April 2014vailable online 26 April 2014
eywords:riple lineinetics of wettingontact angle hysteresis
a b s t r a c t
Wetting of smooth, chemically heterogeneous surfaces was studied experimentally and computationallyduring the advancing and receding processes. The motion of the triple line is known to play an impor-tant role in determining the macroscopic contact angle due to its ability to be pinned at various defectlocations on real surfaces. This effect is known to cause contact angle hysteresis. The shape of the tripleline during these pinning/de-pinning events on various chemically heterogeneous surfaces was capturedusing an experimental and a computational technique. The experimental study employed a ModifiedWilhelmy Plate Technique. The novelty in the current experimental setup lies in its ability to capture themicroscopic triple line shape and its evolution in addition to measuring the local contact angles, whichwere both studied. The triple line shape was observed to be very sensitive to minor imperfections of thesubstrate. In addition, Surface Evolver was used to study the triple line shape computationally. Evolver
was used to solve the complete three-dimensional problem by minimization of the total energy takinginto consideration, both gravity and contact angle hysteresis. The studies showed that the temporal evo-lution of the triple line was significantly different during the advancing and receding processes based onthe nature of the chemical heterogeneity. The results from the current work could be used in the designand fabrication of chemically heterogeneous surfaces for desired wetting applications.
Wetting of chemically heterogeneous surfaces, fabricated fromtwo or more component materials with different wetting proper-ties, has a wide range of practical applications [1]. For example,
0 N. Anantharaju, M.V. Panchagnula / Colloids and S
uch composite surfaces can be engineered with the objective ofmploying surface energy forces originating from the hydropho-ic/hydrophilic interactions, to control the motion of the liquid [2].
n most instances, however, the component materials, themselves,re not entirely defect free and could therefore exhibit intrinsic con-act angle hysteresis, apart from the hysteresis originating due tohe heterogeneity.
A fundamental property that characterizes wettability of aaterial is the contact angle, �, which a sessile drop manifests on
he solid substrate. However, this macroscopic contact angle mea-urement is observed to vastly vary with the triple line structure.1
n fact, for a real surface, it is not possible to identify a unique con-act angle for the sessile drop, as it varies locally around the tripleine [3]. The triple line is subject to local energy minima in the wet-ing process, owing to which it is likely to be pinned at varioushemical and/or topological defects on the real surface [4–7]. Thishenomenon is believed to be one of the causes of contact angleysteresis [8]. Thus, a study of the triple line shape and its tempo-al evolution is important to understand contact angle hysteresisn real surfaces as first pointed out by Shanahan [9]. For example,riple line pinning and de-pinning events in the case of evaporat-ng sessile drops has been shown to affect the rate of evaporation10,11].
The experimental studies of the wetting/de-wetting of liquidn a solid surface are broadly performed by either sessile droppreading on a flat surface or by plunging/withdrawing a substratento/from a liquid [12]. The studies using a sessile drop capturehe two-dimensional shape of the drop and measure the contactngle therein [13]. The traditional Wilhelmy technique, on the otherand, is used to measure the forces acting on the plate, in order toalculate the contact angle exhibited by the liquid meniscus on theolid surface [14]. Both these approaches are consistent as far ashe measurement of the macroscopic contact angles [15] are con-erned but fail to capture the local triple line behavior. For instance,n the sessile drop method, the triple line is curved and is difficulto image. Most of the recent research in understanding the tripleine motion was performed either numerically or computationally
ith very little validation against experiments [16–18]. The currentffort involves the use of a novel experimental setup to experi-entally capture the local triple line behavior and compare to the
esults from a computational study.The current experimental apparatus uses the setup of the tradi-
ional Wilhelmy plate [19], with two major modifications. Firstly,he substrate is inserted/withdrawn into/from the liquid bath at anngle and secondly, it includes the capability to capture the tripleine along the width of the surface. The triple line is captured bothuring the insertion and withdrawal of the surface from a waterath at very low speeds. This yields the shape of the triple line dur-
ng both the advancing and receding states. In this process, it isnsured that the liquid meniscus is allowed to reach equilibrium atach time instant; therefore, the process is quasistatic. The averageontact angle is measured using the height to which the free sur-ace rises/drops above the liquid meniscus surface. Thus the currentxperimental technique, which is referred to as the modified Wil-elmy plate technique is able to capture the microscopic triple linehape in addition to the local and average contact angles during theame experimental run, thus providing a more complete picture ofhe wetting process useful for validating computational studies.
The computational study on wetting was performed by employ-
ng the Surface Evolver (SE) [20], an interactive program for thetudy of surfaces shaped by surface tension, and gravity. The userpecifies an initial surface, the constraints that the surface should
1 The triple line in this context is the set of points where the solid, liquid and vaporhases meet.
Fig. 1. (a) Schematic of a modified Wilhelmy plate apparatus. (b) Shape of themeniscus for a hydrophobic surface submerged in a water bath.
satisfy throughout the evolution, and an energy function thatdepends on the surface. SE then modifies the surface, subject to thegiven constraints, so as to minimize the energy. This minimizationof energy is performed through a gradient descent method.
The strength of SE that makes it applicable to a breadth ofproblems lies in its capability to handle arbitrary topology, vol-ume constraints, boundary constraints, boundary contact angles,prescribed mean curvature, crystalline integrands, gravity, andconstraints expressed as surface integrals [21]. For this reason, SEproved to be a good tool for capturing the triple line shapes and theirtemporal evolution in the current computational study of chemi-cally heterogeneous surfaces. However, being a gradient descentalgorithm devised to operate on the energy landscape, there is nonatural source of directional information. Directionality is impor-tant to handling contact angle hysteresis, as the advancing andreceding angles are states, which are only accessible from specificdirections. If this information can be coupled into SE, it is likelyto become a tool with significantly expanded capabilities, espe-cially in the realm of modeling wetting of real surfaces. Recently,Santos and White [22,23] and Prabhala et al. [24] have indepen-dently demonstrated a methodology by which the Dussan [25] andHocking [26] model for contact angle hysteresis can be coupledinto SE. This approach allows for the directional information to beincorporated into SE. The mathematical description is briefly pre-sented in the Computational method section hereunder. The resultsfrom exercising this model on a wide range of model chemical het-erogeneity surfaces are presented and discussed. We also presentexperimental measurements of the triple line shape and contactangles for the same cases. Finally, the experimental and computa-tional results are compared and discussed. It must be mentionedthat the surfaces under study are smooth but chemically heteroge-neous. But the arguments presented herein could also be extendedto triple line pinning due to topological heterogeneities.
2. Methods and materials
2.1. Experimental method
The experimental set up (see Fig. 1) consists of a substrateoriented at angle, ̨ (60◦, in this case) with respect to the liquidinterface. The specimen under study was cleaned thoroughly and
urface
mTpacettcl
aorttctosrto
Wtmeocl[itc
�
�
g
k
wt
2
Pntciwmbt
N. Anantharaju, M.V. Panchagnula / Colloids and S
ounted on the plate and partially submerged in a bath of water.he specimen in the current work is a glass slide, as the trans-arency of the glass slide reveals two triple lines, one on the frontnd another on the rear. The reference height of the liquid menis-us was measured at this alignment position before starting thexperiment. The plate was then immersed into (withdrawn from)he water bath at very slow speeds using a precision stepper motoro study the advancing (receding) process. A high-resolution CCDamera SONY X-15 was used to capture the instantaneous tripleine shape during the advancing and receding processes.
As mentioned before, mounting the plate at an angle allows for clear image of the shape of the triple line, especially for the casef hydrophobic materials. This is because, for a hydrophobic mate-ial, the curvature of the liquid meniscus obscures the view of theriple line (see Fig. 1(b) for a schematic depiction). By aligning theransparent substrate at an angle, the front and the rear interfacesan be simultaneously observed. If the meniscus blocks a view ofhe triple line on the front surfaces of the glass slides, the interfacen the rear side is visible. Hence, the triple line shape was mea-ureable under all wetting situations. The choice of the front or theear surface in a given experiment was made apriori based on theriple line visibility. The series of images in the experiment werebtained by following the triple line on that face of the slide.
In addition to capturing the shape of the triple line, the modifiedilhelmy plate apparatus can also be used to measure the instan-
aneous contact angle. The liquid meniscus near the substrate isodified due to the local contact angle, which obeys the Young’s
quation [27]. For the modified Wilhelmy plate, the meniscus takesn a shape as shown in Fig. 1(b). The height, h, to which the menis-us rises near the substrate, is measured. For a vertical plate, theocal contact angle, �E, is related to h through an algebraic relation28] given by �E = sin−1(1 − (h2k2/2)) (when h > 0). When the plates inclined, this expression needs to be modified for the angle sub-ended by the plate with respect to the horizontal. Therefore, in theurrent configuration, �E is related to h as follows.
E = sin−1
(1 − h2k2
2
)+ �
2− ˛, (when the rear face is used)
(1a)
and
E = sin−1
(1 − h2k2
2
)+ ̨ − �
2, (when the front face is used)
(1b)
Fig. 1(b) shows a pictorial representation of the contact angleiven by equations (1). Here k−1 is the capillary length given by,
−1 =√
�
�g sin ˛(2)
here � is the liquid density, g is the gravity and � is the surfaceension.
.2. Computational method
As mentioned before, the computational model is similar torabhala et al. [24] and is reported herein briefly for complete-ess. The SE is used to predict the shape of the triple line fromhe modified Wilhelmy plate experiments. In order to begin theomputations, the initial and boundary conditions need to be spec-fied. Three edges of the surface that are set to meet the container
all are assumed not to contribute any energy (akin to the sym-etry condition). The contact angle for these edges is specified to
e 90◦ (boundary conditions). Further these edges are constrainedo remain on these walls. The fourth edge, i.e. the triple line itself,
s A: Physicochem. Eng. Aspects 455 (2014) 19–27 21
is constrained to lie on a vertical substrate. The position of the het-erogeneity and the properties of the materials defined by theiradvancing and receding contact angles are specified as knownmaterial properties of the substrate. The experiment is modeledby increasing (decreasing) the volume of water contained in thetank quasistatically to simulate an advancing (receding) process.At each volume, the meniscus is allowed to evolve by minimizationof its energy subject to the constraint of a fixed volume. Since thelocal material properties of substrate are specified, SE calculatesthe equilibrium shape of the meniscus, subject the constraints onthe motion of the vertices and the total volume of liquid. The dis-placement of each vertex on the triple line is only allowed to occuralong a normal to the triple line in the plane of the substrate. Themagnitude of this displacement is calculated by,
ˇ
�LV
∂n
∂t= �
�LV� + (cos �Y (s)) (3)
where ∂n/∂t is the rate of displacement along the unit normal vec-tor. In the above equation, the triple line is parameterized by itsarc co-ordinate position, s, in the undeformed configuration. Thefirst term on the right hand side represents the displacement ofthe vertex driven by curvature of the triple line. Here, � is the linetension and assumed to be equal to 10−5 N following Vedantamand Panchagnula [29,30], �LV is the surface tension and � is thecurvature calculated at every vertex. The second term representsthe imbalance in the capillary forces. Further, �(s) and �Y(s) are theinstantaneous local contact angle and instantaneous local Young’sangle at the arc coordinate, s. To include the effect of hysteresis, thekinetic coefficient, ˇ, is taken as a combination of rate dependent(ω) and rate independent hysteresis (). The reader is referred toreferences [24,29,30] for a detailed discussion of these parameters.The rate independent component of the hysteresis, is the imbal-ance in the Young’s force at which the drop remains pinned in theadvancing (�a) and receding angle (�r) state. This is given by,
= cos �r − cos �a
2(4)
The Young’s angle is estimated by equation,
�Y = cos−1
(cos �a + cos �r
2
)(5)
It has been shown that the actual definition of �Y such as the onein Eq. (5) is not crucial [24]. Thus the final evolution equation thatinvolves CAH and controls the shape of the triple line is given by,
ω∂n
∂t=
{�
�LV� + (cos �(s) − cos �Y (s)) if
∣∣∣ �
�LV� + (cos �(s) − cos �Y (s))
∣∣∣ >
0 otherwise
(6)
The physical meaning of Eq. (6) is as follows. The triple line islocally allowed to move if � is either greater than �a or less than�r. Otherwise, the triple line locally remains at rest. The new vertexpositions that form the shape of the triple line follow from Eq. (6).SE now calculates the meniscus shape (until convergence) assum-ing the triple line to be immovable in order to minimize any energygenerated during the motion of the triple line to satisfy the prop-erties of the local material. The process is repeated until the tripleline and meniscus no longer move. The volume is then incremented(decremented) for an advancing (receding) process.
2.3. Materials
The triple line shape was measured during advancing and reced-
ing events on smooth substrate and heterogeneous substrates withfour classes of heterogeneity. A description of the heterogene-ity arrangement is given in Table 1. Glass slides were silanizedusing dimethyldichlorosilane (DMDCS) following the procedure
22 N. Anantharaju, M.V. Panchagnula / Colloids and Surface
Table 1Description of the different chemical heterogeneity configurations tested.
Arrangement Description
I Plain DMDCS silanized glass slideII 4 mm DMDCS silanized stripe in a plain glass slideIII 4 mm plain glass stripe in a DMDCS silanized slide
owotg(mtmtesaTwTs
3
3
sldtncAbt
aacittTntrmStdW
stp(sdt
IV 4 mm diameter DMDCS silanized circular region in a plainglass slide
V 4 mm diameter glass circular region in a DMDCS silanized slide
utlined in Oner and McCarthy [31]. The first set of experimentsere performed on plain silanized glass slides. In the second set
f experiments (Arrangement II), heterogeneity was introduced inhe form of a 4 mm wide hydrophobic stripe along the length of thelass slide in the middle of the width of the slide. In the third caseArrangement III), a heterogeneous substrate similar to Arrange-
ent II was fabricated except with a hydrophilic stripe of glass inhe hydrophobized glass slide. Arrangement IV is a heterogeneous
aterial in the form of a 4 mm diameter hydrophobized circle inhe plain glass slide. Arrangement V is similar to Arrangement IVxcept with a hydrophilic circular defect in the hydrophobized glasslide. Arrangements II–V were fabricated by masking the appropri-te region using adhesive tape. This was followed by silanization.he masked region (which was left as plain glass) was then cleanedith ethanol and acetone to remove any remnants of the adhesive.
he computational simulations were also performed for similarurfaces for comparison.
. Results and discussion
.1. Validation of the surface evolver model
The triple line shape was initially captured for a plain DMDCSurface as a validation exercise. Fig. 2(a) and (b) shows the tripleine shape captured on a plain DMDCS glass slide (Arrangement I)uring the advancing and receding processes, respectively. For thisopologically smooth surface, the triple line can be observed to beearly a straight line. The minor contortions in the triple line, espe-ially in Fig. 2(b) (receding case), arise due to wettability defects.s an aside, the shape of the receding triple line was observed toe very sensitive to minor defects. It could be a sensitive methodo measure surface topology.
The meniscus shape, especially close to substrate is of interests it is determined by the local contact angle at the substrate. Theverage height, h, to which the triple line rises above the menis-us level, was measured during the advancing process from eachnstantaneous image. The contact angle at the substrate is relatedo this h through Eq. (1). This methodology was used to measurehe advancing contact angle of the pure surface (Arrangement I).his value was verified against the value measured from captiveeedle sessile drop infusion method. The advancing angle fromhe modified Wilhelmy plate method was found to be 115◦ with aepeatability error less than 0.7%. The two contact angle measure-ents were also observed to agree with each other within 0.8%.
imilarly, the receding angle of the DMDCS surface was estimatedo be 64◦ and consistent with the captive needle sessile drop with-rawal method. This serves as a first validation step for the modifiedilhelmy experiments.The Surface Evolver simulations were verified with an analytical
olution for the shape of the meniscus as well as with an experimen-al measurement for the microscopic triple line shape. This processrovides a complete three-dimensional validation of the SE model
meniscus as well as triple line shapes). The liquid meniscus in aolid vessel with vertical walls is horizontal away from the walls,ue to gravity, and undergoes a distortion closer to the walls dueo capillary effects [28]. The meniscus shape close to our substrate
s A: Physicochem. Eng. Aspects 455 (2014) 19–27
was computed from the SE simulations for receding process on asmooth glass surface. This was verified against an analytical solu-tion for the same case. The implicit relation for the meniscus shape,z(x) is given by,
x − x0 = k−1 cosh−1
(2k−1
z
)− 2k−1
(1 − z2
4k−2
)1/2
(7)
where, k−1 is the capillary length given by Eq. (2) and is calculated tobe 0.28 cm for water. In the above equation, x0 is the distance suchthat the equation limits to z = h at x = 0, i.e. at the wall. The height, hto which the meniscus rises, is related to the contact angle throughEq. (1).
Fig. 3(a) shows the meniscus shape on a plain DMDCS substrate(Arrangement I) obtained from SE simulations during the recedingprocess. Fig. 3(b) shows a comparison of this meniscus shape versusthe z(x) profile evaluated from Eq. (7) for the same conditions. It canbe observed from Fig. 3(b) that there is a good agreement betweenthe analytical solution and the Evolver result for the shape of themeniscus. Further, for the receding contact angle of 64◦ measuredwith the captive needle approach on plain DMDCS surface, theheight of the meniscus (from Eq. (1)) is calculated to be 0.12 cm.An agreement in this meniscus height can also be clearly observedfrom figure.
3.2. One-dimensional heterogeneity results
A one-dimensional heterogeneity is introduced to the substrate inthe form of a stripe of one material along the length of the substratecomprised of another material. Fig. 4(a) and (b) contain images ofthe triple line shape captured experimentally on heterogeneoussurface consisting of a hydrophilic stripe in hydrophobic material(Arrangement III) during advancing and receding processes respec-tively (red line). The SE calculation of the triple line shape is alsoshown as a blue line. Again, good agreement is observed betweenthe experiment and SE calculation of the triple line shape, even ininstance where the triple line is not a straight line. Of special inter-est is the ability to predict the triple line height at x = 0, which hasimplications to the increase in the liquid–vapor interfacial energycaused by the triple line pinning on the defect. This parameter iswell predicted in both the cases.
Fig. 5(a) and (b) shows the results from the advancing study onArrangement II substrate obtained from SE simulations. As the vol-ume of water in the both (indicated in the legend) is increased, themean triple line position increases in mean z position. Fig. 5(b) rep-resents the height of the meniscus from the free surface of the liquid(i.e. h in cm). The contiguous material to the hydrophobic stripefor Arrangement II is plain glass, which is innately hydrophilic innature. For this reason, it can be observed from Fig. 5 that thecontiguous material is readily wetted during the advancing pro-cess. The stripe on the other hand shows its hydrophobic natureby a lower triple line height in relation to the free surface. Thisshows that the triple line is pinned on the silanized stripe dur-ing the advancing process. It can be observed that the triple linecurvature at x = 0 is negative. This is because, during the advancingprocess, a relatively higher contact angle needs to be realized on thehydrophobic material before it can wet any additional surface. Thus,the triple line was observed to remain pinned on the hydrophobicstripe until the advancing angle of silane was achieved [1].
Fig. 6(a) and (b) show the Surface Evolver calculations of theshape of the triple line during the receding process for a hetero-geneous substrate of Arrangement III. Fig. 6(a) shows the triple
line shape and position as observed in the lab frame of reference.Fig. 6(b) shows the triple line shape referenced to the free meniscuslevel at the given volume. The initial shape of the meniscus for thestart of this process is the triple line shape obtained during steady
N. Anantharaju, M.V. Panchagnula / Colloids and Surfaces A: Physicochem. Eng. Aspects 455 (2014) 19–27 23
Fig. 2. Shape of the triple line during (a) advancing and (b) receding process on a plain, silanized glass sample.
re gla
acfianarfia1lfo
Fr
Fig. 3. (a) SE calculated meniscus shape for advancement process on pu
dvancing process. The ability of the current model to incorporateontact angle hysteresis is demonstrated through the results in thisgure. As the volume decrement (receding) process is initiated from
volume of 16.6 cm3, it can be observed that the triple line doesot change shape until a first critical volume (between 15.8 cm3
nd 16.0 cm3) is reached. At this point, the triple line begins toecede only on the (hydrophobic) contiguous part of the specimenrst (while remaining pinned on the stripe region). The triple linet x = 0 shows no motion until a second critical volume (between
5.6 cm3 and 15.8 cm3) is reached. Beyond this volume, the triple
ine begins to move and attains a steady state which can be observedor all volumes less than 15.8 cm3 overlapping within an rms errorf 2.3 × 10−3 cm in the meniscus height. Simulation of this process
ig. 4. Shape of the triple line for Arrangement I under (a) advancing and (b) receding
eferred to the web version of this article.)
ss surface, (b) comparison of meniscus shape versus analytical solution.
is shown to include the directional information for the motion ofthe triple line, which is captured mathematically in Eq. (6).
3.3. Two-dimensional heterogeneity results
The substrates considered for the two-dimensional variation ofthe chemical heterogeneity are in the form of a circle of one materialin a substrate comprised of another material. Both ArrangementsIV and V were investigated for the triple line shape, but only
experimental and computational results from Arrangement V arereported here for brevity in Figs. 7–10. Figs. 7 and 9 show asequence of images captured during the advancing and recedingprocesses, respectively, both on a specimen of Arrangement V. The
conditions. (For interpretation of the references to color in the text, the reader is
24 N. Anantharaju, M.V. Panchagnula / Colloids and Surfaces A: Physicochem. Eng. Aspects 455 (2014) 19–27
Fig. 5. (a) Sequence of triple lines (b) height of the meniscus from the free surface foradvancing process on a surface with hydrophobic stripe in a glass substrate obtainedfrom SE simulations (Arrangement II).
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
z, cm
x,cm
v_16.6 c m3
v_16.4 c m3
v_16.2 c m3
v_16.0 c m3
v_15.8 c m3
v_15.6 c m3
v_15.4 c m3
v_15.2 c m3
v_15.0 c m3
v=14.8 c m3
v_14.6 c m311 66
11.77
11.88
11.99
22.00
2222.111
222..2222
22.33
22.44
22.55
222.666z, cm
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
0.10
0.12
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
z, cm
x,cm
v_16.6 c m3
v_16.4 c m3
v_16.2 cm3
v_16.0 c m3
v_15.8 c m3
v_15.6 c m3
v_15.4 c m3
v_15.2 c m3
v_15.0 c m3
v=14.8 c m3
v_14.6 c m3000 111000
0.0 0088
0.0 0066
0.0.000444
0.00 0022
00.00000
00..000222
00.0044
00.000666
000.0000008888
00.1100
00.1122
2200 00 0..0 00.
zz, cm
(a)
(b)
Fig. 6. (a) sequence of triple lines (b) height of the meniscus from the free surfacefor receding process on a surface with hydrophilic stripe in a hydrophobized glasssubstrate obtained from SE simulations (Arrangement III).
Fig. 7. Shape of the triple line captured experimentally during advancing process
on a heterogeneous surface of a hydrophilic circular defect in hydrophobized plainglass material (Arrangement V). The circular defect has a diameter of 4 mm for scalereference.
corresponding results from the SE simulations are presented inFigs. 8 and 10 respectively. Videos of both the experiment (Sup-plementary Video 1) and simulation (Supplementary Video 2) areavailable as part of the Supplementary information.
Arrangement V is a substrate where the contiguous material isDimethyldichlorosilane (DMDCS) and the circular defect is plainglass. Fig. 7 is a sequence of images captured during the experi-mental advancing process. The corresponding numerical process isrepresented in Fig. 8. The advancing process can be divided intofour different stages of wetting by the triple line. The first and finalimages are captured with the triple line completely resident onthe contiguous hydrophobic material before and after wetting thecircular hydrophilic material, respectively. The triple line in thesestages is nearly a straight line. It can be observed from Fig. 7(b)that when the triple line just touches the hydrophobic defect, it
wets the circular defect almost instantaneously, as it lies at a higherenergy state. In other words, the triple line in this situation almostentirely lies on the contiguous hydrophobic material, as a result
0.2
0.4
0.6
0.8
1.0
1.2
1.4
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
z, cm
x, cm
v=5.4 c m3
v=6.0 c m3
v=6.4 c m3
v=7.0 cm3
v=7.8 cm3
v=8.6 c m3
v=9.2 c m3
Fig. 8. Sequence of triple lines for advancing process on a surface with hydrophiliccircle in a hydrophobized glass substrate obtained from SE simulations (Arrange-ment V).
N. Anantharaju, M.V. Panchagnula / Colloids and Surfaces A: Physicochem. Eng. Aspects 455 (2014) 19–27 25
Fig. 9. Shape of the triple line captured experimentally during receding processogr
occucctrimc
Aoiwhon
Fc
n a heterogeneous surface of a hydrophilic circular defect in hydrophobized plainlass material (Arrangement V). The circular defect has a diameter of 4 mm for scaleeference.
f which, it advances readily, attempting to wet the hydrophilicircular defect. This behavior of the triple line is observed in theomputational simulations as well, as can be seen in Fig. 8 at vol-me 8.6 cm3. On continuing the advancing process, the triple linean be observed to evolve further until it wets the circular defectompletely (as experimentally seen in Fig. 7(c)). The fact that theriple line is observed to readily wet the hydrophilic circular mate-ial almost instantaneously is a result of the spontaneous decreasen interfacial energy. The viscous dissipation owing to this sudden
ovement is purported to be one mechanism that gives rise toontact angle hysteresis.
Figs. 9 and 10 represent the receding process captured onrrangement V. It can be observed from the straight-line shapef the triple line in Fig. 9(a) that the circular hydrophilic defects completely underneath the free surface of the water. Fig. 9(b)
as captured when the triple line started to de-wet the circularydrophilic defect. It can be observed that the triple line gets pinnedn the hydrophilic circular defect as a result of the hydrophobicature of the contiguous material. In other words, the triple line
0.4
0.6
0.8
1.0
1.2
1.4
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
z,cm
x, cm
v=8.8 cm3
v=8.2 cm3
v=8.0 cm3
v=7.6 cm3
v=7.0 cm3
v=6.4 cm3
v=5.8 cm3
v=5.4 cm3
v=4.4 cm3
ig. 10. Sequence of triple lines for receding process on a surface with hydrophilicircle in a hydrophobized glass substrate.
Fig. 11. Plot of interfacial area during the receding process for Arrangement V spec-imen.
is observed to de-wet the contiguous material readily. As can beseen from Fig. 9(c), the triple line is observed to remain pinnedon the hydrophilic material until the receding angle of glass wasachieved, locally. At this point, the triple line pinned on the glassdefect could become unpinned. Fig. 9(d) is obtained in a state wherethe two materials are completely de-wet with the triple line com-pletely lying on the plain silanized glass, thus, nearly regaining theshape of a straight line. When the triple line is not resident on thedefect, it plays no role in determining the meniscus shape [32]. Sim-ilar pinning and de-pinning behavior in triple line motion can beobserved from the SE simulation in Fig. 10. Further, in Fig. 10, thetriple line is observed to exhibit sharp variations in the vicinity ofthe circular defect especially following the de-pinning event. Thesecontortions possibly arise from the numerical stiffness associatedwith the de-pinning event when modeled in Surface Evolver. Anaccurate model of the de-pinning process, especially when the pin-ning energy is large, may require a dynamical treatment. It may berecalled that Surface Evolver is only suited to study quasistatic evo-lution of the triple line. The wetting of a circular defect as observedin these four stages is similar to the study by Joanny and deGennes[33].
The process of pinning and de-pinning of the triple line is knownto play a role in manifesting contact angle hysteresis. Joanny anddeGennes [8] presented a theoretical model for contact angle hys-teresis based on pinning and de-pinning events occurring at variousdefect sites. Their model assumes that as the triple line is sweepingacross a single defect, it develops contortions. At a critical value ofthe triple line position, it instantaneously switches to a depinnedstate. Decker and Garoff [34] have attempted to validate this modelby making triple line measurements on microscale defects. Weherein presented an experimental and computational study ofthe triple line sweeping across a macroscale defect (∼4 mm). Thedetailed kinematics of the pinning and de-pinning processes is bestunderstood using macroscale defects, since making detailed tripleline measurements is possible.
Two cases from our current study exhibit pinning/de-pinningphenomenon – (i) advancing process on an Arrangement IV spec-imen and (ii) receding process on an Arrangement V specimen. Inboth these situations, the triple line encounters an energy barrierfollowed by de-pinning process. In the other two cases, viz. (iii)receding process on an Arrangement IV specimen and (iv) advanc-ing process on an Arrangement V specimen, one only notices a“run-ahead” of the triple line and hence is fundamentally differ-ent from a pinning/de-pinning process. We will focus next on cases(i) and (ii) described above to discuss the process in terms of surfaceenergy [35].
Fig. 11 is a plot of the liquid–vapor interfacial area versus vol-ume for the receding process on an Arrangement V specimen (case
(ii) above). These results are obtained from SE simulations andare for the case shown in Figs. 9 and 10. Since, the process isreceding (one of the decreasing volume), the abscissa needs to be
26 N. Anantharaju, M.V. Panchagnula / Colloids and Surface
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ig. 12. Plot of interfacial area during the advancing process for Arrangement IVpecimen.
ead in the decreasing co-ordinate direction. Four distinct featuresan be identified from this plot. During the first phase, where theriple line is a horizontal line, the meniscus does not undergo anyeformation. Hence the liquid–vapor interfacial area is constant.s the triple line approaches the defect (say at V = 8.6 cm3), the
iquid–vapor interfacial area begins to increase. This is observablen the Evolver simulations as an increasingly corrugated meniscus.he interfacial area now reaches a maximum at a volume of 7.2 cm3.rom this point, as the volume is further decreased, the interfacialrea decreases gradually until V = 5.2 cm3. At this critical volume,he triple line comes unpinned from the defect. The liquid–vapornterfacial energy then becomes equal to the initial value at thetart of the process. The nature of the plot (first increasing, thenecreasing followed by a jump) is a function of the geometry of theefect. In addition, all the energy associated with the jump duringhe de-pinning process is dissipated and contributes to hysteresis.herefore, it is conceivable that the geometry of the defect can bengineered to minimize the dissipation (jump) during de-pinning.
Fig. 12 is similar to Fig. 11, except that it is for the case of andvancing process on an Arrangement IV specimen. Since the pro-ess is one of increasing the volume (advancing), the abscissa forhis process can be read in the increasing order. The four distincteatures identified from the previous figure are also observable inhis plot. For example, the liquid–vapor interfacial area remainsonstant until V = 5.8 cm3, at which point it begins to increase grad-ally until a volume of 7.8 cm3 is reached. Further increase inolume causes a gradual decrease in the interfacial area until a crit-cal volume of 8.6 cm3 is reached. At this point, the triple line comesnpinned followed by a jump in interfacial energy to a value nearlyqual to the initial energy, indicating that the pinning/de-pinningrocess is complete.
Two points need to be noted from studying the data inigs. 11 and 12 along with the images in Figs. 9 and 10. Firstly, thenterfacial area reaches a maximum value followed by a region ofradual decrease in energy. In other words, it is not a sudden jumprom the maximum value to the de-pinned state. Even though thisbservation is being made for the case of macroscale defects, theame could be applicable to microscale defects as well. Secondly,he point of occurrence of the jump in relation to the point of max-mum energy is asymmetric (for advancing and receding process)nd is related to the specific geometric shape of the defect underonsideration.
. Conclusions
The triple line shapes and its temporal evolution on smooth andhemically heterogeneous surfaces with typical arrangements of
ts component materials have been studied experimentally, using
modified Wilhelmy plate apparatus, and computationally, usingurface Evolver. The triple line behavior was observed to be signifi-antly different during the advancing and receding processes based
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s A: Physicochem. Eng. Aspects 455 (2014) 19–27
on the arrangement of the two chemically heterogeneous compo-nent materials. This behavior was observed to be consistent for bothexperimental and computational techniques. The current studyprovides insight into the response of the triple line shape to thelocal surface wetting properties, thus providing an improved eval-uation of its effect on the macroscopic phenomenon. The modifiedWilhelmy plate apparatus proves to be a better suited experimen-tal technique than the traditional methods using a sessile drop,specifically in capturing the triple line motion and measuring thelocal contact angles of the triple line. This is because of the sen-sitivity of the one-dimensional nature of the triple line to veryminor chemical and/or topological imperfections on the surface.The energy jump during pinning/de-pinning process is studied fortwo cases and shown to be asymmetric. In addition, the de-pinningprocess for a macroscale defect is observed to be a relatively grad-ual change prior to the jump. Surface Evolver is shown to be a goodtool for computational studies on wetting due to its capability insolving the three-dimensional problem taking into considerationhysteresis and gravity.
Appendix A. Supplementary data
Supplementary data associated with this article can befound, in the online version, at http://dx.doi.org/10.1016/j.colsurfa.2014.04.052.
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