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Wetting on axially-patterned heterogeneous surfaces

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Wetting on axially-patterned heterogeneous surfaces
M.A. Rodríguez-Valverde
⁎
, F.J. Montes Ruiz-Cabello, M.A. Cabrerizo-Vilchez
Biocolloid and Fluid Physics Group, Department of Applied Physics, University of Granada, Campus of Fuentenueva; E-18071 Granada, Spain
Available online 27 December 2007
Abstract
Contact angle variability, leading to errors in interpretation, arises from various sources. Contact angle hysteresis (history-dependent wetting)and contact angle multiplicity (corrugation of three-phase contact line) are irrespectively the most frequent causes of this uncertainty. Secondaryeffects also derived from the distribution of chemical defects on solid surfaces, and so due to the existence of boundaries, are the known
“
stick/ jump-slip
”
phenomena. Currently, the underlying mechanisms in contact angle hysteresis and their connection to
“
stick/jump-slip
”
effects and the prediction of thermodynamic contact angle are not fully understood. In this study, axial models of smooth heterogeneous surface were chosen inorder to mitigate contact angle multiplicity. For each axial pattern, advancing, receding and equilibrium contact angles were predicted from thelocal minima location of the system free energy. A heuristic model, based on the local Young equation for spherical drops on patch-wise axial patterns, was fruitfully tested from the results of free-energy minimization. Despite the very simplistic surface model chosen in this study, it allowed clarifying concepts usually misleading in wetting phenomena.© 2007 Elsevier B.V. All rights reserved.
Keywords:
Contact angle; Hysteresis; Stick-jump-slip; Heterogeneity; Axial pattern
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852. Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852.1. Spherical sessile drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862.2. Axially-patterned heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862.3.
“
Stick/jump-slip
”
phenomena. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 882.4. The local Young equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892.5. Equilibrium contact angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893. Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.1. Free-energy minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.2. Heuristic model for patch-wise patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.2.1. Advancing mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.2.2. Receding mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924. Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.1. Step pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.2. Square pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.3. Periodic patch-wise pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.4. Periodic continuous pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955. Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Advances in Colloid and Interface Science 138 (2008) 84
–
100www.elsevier.com/locate/cis
⁎
Corresponding author. Tel.: +34 958 24 00 25; fax: +34 958 24 32 14.
E-mail address:
marodri@ugr.es (M.A. Rodríguez-Valverde).0001-8686/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.cis.2007.12.002
Notation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
1. Introduction
There are manyapplicationswhere wetting plays animportant role, such as coatings (paints, inks
…
), waterproofing, detergencyand adhesive industries [1]. Contact angle is used as measure of the wettability of a solid surface by a liquid [2] and to obtaininformation about the surface energy of materials [3]. However,therearestillmanyopenquestionsaboutthemeasurementandtheinterpretation of contact angle [4,5] due to phenomena such as
Contact Angle Multiplicity (CAM) [6
–
8] and Contact AngleHysteresis (CAH) [9
–
11].Variability in contact angle is mainly caused by CAM andCAH,regardless.Experimentally,forasolid
–
liquid
–
vapor(SLV)system,alimitedrangeofcontactanglescanbeobservedeitherasalongthethree-phasecontactline(CAM)oraccordingtohowthesystem reached the
“
equilibrium
”
state (CAH). The main dif-ference between both phenomena is that the first one appears in a particular
configuration
of the system whereas the second one, indifferent configurations of the same thermodynamic state (e.g.temperature, volume and chemical potential), i.e. wetting is ahistory-dependent phenomenon. In addition, CAM provides a bounded, uncountable set of contact angles due to the continuityofdistortedcontactlinesandCAHinvolvesafinitecountable set,instead. Owing to these ever-present phenomena, the contact angle values frequently become meaningless concerning surfaceenergetics [4,5].
The identification of the thermodynamic contact angle (anglecorresponding to the global energy minimum of the SLV system)within the range of observable contact angles, is the main chal-lenge for Surface Thermodynamics [12
–
14]. Thereby, experi-mentallyaccessiblecontact anglesare also referredtoas apparent contact angles (
θ
app
). The existence of more than one apparent contact angle can be caused either as by the corrugation of three- phase contact lines (CAM) or by their irreversible movement during liquid spreading (CAH) [12]. Both phenomena becomeobservable at different scales. Frequently, CAH and CAM areobservedatonce(e.g.droponatiltedplate[15])andconsequentlythey are not clearly distinguished in literature, so that many morestudies have been mainly addressed to understand CAH [16
–
18].The presence of topographic and/or chemical defects onsolid surfaces is assumed to be the srcin of CAH [19
–
21].Hence, CAH is an indicator of the imperfection degree of thesubstrates and it is a characteristic of all real materials. Attemptsto understand CAH began many years ago [22
–
25], although it is not fully understood as yet.Some authors reported that CAH actually depends on thespecific distribution of surface defects as regards to wetting lines[26]. Furthermore, boundaries between surface domains of different chemistry or elevation can cause several observableeffectsonthethree-phasecontactlinesuchascorrugation(CAM)[6,7,20] and
“
pinning
”
[27,28]. Consequently, the orientation of the three-phase contact line regarding the heterogeneity bound-aries is critical for the reinforcement or the mitigation of CAH,CAM and pinning as well [26,29], according to the measuring
scale used. Indeed, analysis of CAH should take into account thedetails of heterogeneity pattern and the shape of liquid
–
vapor interface. Generally, CAH is examined from the minimization of system free energy in order to determine the different (meso)-equilibrium states [14]. However, global-system free energyminimizations are mathematically difficult because of the problem ofanalyticalrepresentationofthe liquid
–
vapor interfaceof deformed menisci or non-axisymmetric drops [30
–
32].The main motivation of this theoretical study is to examineexclusively CAH and pinning effects on particular heteroge-neous pattern and liquid
–
vapor interface shape, which jointlymagnify CAH and eliminate CAM. The chosen pattern consistsin an axial distribution (alternating concentric circular bands) of surface energy and as a result, the
“
natural
”
model of liquid
–
vaporinterface in absence ofgravity isthesphericalcap [24,33].
This free surface model is more realistic and experimentallyaccomplishable than those ones used by other authors [34
–
37].Quantitative measures of CAH and pinning/depinning featurescan be extracted from a suitable minimization of system freeenergy, known the surface defect pattern and drop volume [10].
2. Theory
Fromanenergydescription,CAHrequirestwoconditions:theexistence of multiple local minima in the system free energy,referred to as metastable equilibrium states or
metastates
[23,38];and the dependence of the location and the number of thesemetastates on liquid volume [36]. In absence of CAM, a liquidmeniscus in contact with a heterogeneous substrate will reach avalue of contact angle within of a finite discrete range. Hence, asan example, values of apparent contact angle measured duringspreading or condensation of drops differ from those of eva- porating drops, even for identical liquid volumes [39]. Further-more, in the case offinite-size menisci (drops), the amplitude anddensity of the mentioned spectrum of contact angles will changeastheliquidvolume.Therefore,fluctuationsintheexperimentallyaccessible metastates, as the liquid volume varies, will illustratethe existence of CAH.For a given SLV system and a fixed volume, three distinctivemetastatescanbe identifiedinthefree energycurve(Fig.1).Thesemetastates correspond to the maximum contact angle
θ
adv
(
advancing contact angle
, ACA), the minimum contact angle
θ
rec
(
receding contact angle
, RCA) and the contact angle of globalminimum energy
θ
eq
(
equilibrium contact angle
, ECA). Measure-ments of ACA and RCA are accomplished from the incipient relative movement of advancing and receding contact lines,respectively [40
–
42]. Otherwise, ECA or thermodynamic contact angle,ishardlyobservableincontrolledexperiments.Theoretically,the apparentcontactangles associatedtoeachmentionedmetastatecan be accordingly estimated from the free-energy minimization.
85
M.A. Rodríguez-Valverde et al. / Advances in Colloid and Interface Science 138 (2008) 84
–
100
Various experimental approaches allow examining the dropsize effect on apparent contact angle and in consequence, tomonitor CAH. This is the case of growing/shrinking sessiledrops or captive bubbles [40], which are usually employed tomeasure ACA and RCA. Otherwise, due to the different horizontal spacing of metastates depending on the slope of thefree energy curve (Fig. 1) for low energy surfaces, ACA isusually located in a well-defined metastate while RCA canattain one of many metastates separated by small energy barriers[43]. This explains the relative experimental reproducibilityon low energy surfaces of ACA compared to RCA, which isstrongly susceptible to environmental vibrations [17,35,39].
Just the opposite case is found for predominantly high energysurfaces. The mentioned experiments can be virtually repro-duced by the identification of the singular metastates through a befitting free-energy minimization for different drop volumes.
2.1. Spherical sessile drop
Liquid
–
vapor interfaces in wetting phenomena can bemodeled analytically by two-dimensional solutions of theYoung
–
Laplace equation [44] in order to understand CAH. Asolid surface with small randomly placed defects, chemical or topographies, produces CAH although no visible CAM[27,45,46], whereas both are often observed in grossly hetero-
geneous surfaces [47]. Since CAH and CAM come about indifferent situations, they might be separately analyzed at least from a conceptual point of view. Therefore, when CAM is totallyabsent,three differentmodelsofliquid
–
vapor interface incontact to a solid surface can be used in analytical form: semi-infinitewedge, cylindrical drop and spherical drop. These two-dimen-sional models, which all hold the Young
–
Laplace equation,reduce significantly the mathematical complexity associated tothe three-dimensional shape of free capillary surfaces [30,31].
Moreover, the free energy of SLV system can be expressed inclosed form using analytical models of liquid
–
vapor interface.Spherical cap corresponds well with the symmetry of sessiledrops and is widely used as experimental model for contact anglemeasuring. Although the semi-infinite meniscus is the simplest geometry (zero apex curvature and one of the principal radii of curvature equal to zero), this model necessarily considers thegravity effect [34,35,48]. Furthermore, only finite-size interfaces
allow examining system size effects for a given solid surface.Although cylindrical drops (non-zero apex curvature and one of the principal radii of curvature equal to zero) are regularly proposed in literature [14,36,37,49,50], their experimental
accomplishment becomes an intricate task. Instead, sphericaldrops become more realistic physically and moreover, none of their curvatures is zero. In addition, the model might be readilyextended to
“
pancake
”
drops (zero apex curvature), in order toinclude gravity effects [47].In practice, spherical cap is employed when the drop size issmall enough and so the gravity effects can be negligible. Interms of characteristic lengths, this happens when the capillarylength (
l
0
¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
g
lv
=
q
g
ð Þ
p
) is much larger than the apex curvatureradius of drop,
R
ap
. Here,
γ
lv
is the surface tension at liquid
–
vapor interface,
ρ
the liquid density and
g
the constant of localgravitational acceleration. Indeed, sessile drops are nearlyspherical when their volume
V
is not greater than 0.02
l
03
, re-gardless of contact angle value (see Appendix A).Several authors [51,52] have reported that, even though the
drop shape may be significantly altered, the effect of gravityon thermodynamic contact angle is not relevant within theclassical theories of capillarity [53]. The energy contributionof those body forces, such as gravity (not dependent on thederivative of the drop profile), to the SLV system does not alter to the boundary condition, namely the local Young'sequation (see Section 2.4). In conclusion, the analysis of CAHcan be performed upon gravity-free conditions.
2.2. Axially-patterned heterogeneity
Once the model of liquid
–
vapor interface is selected, theheterogeneity pattern should be conveniently described to
Fig. 1. Dimensionless free energy as function of cosine of apparent contact anglefor sessile drops of different dimensionless volume: (a)
V
˜
1/3
=10.0 (b)
V
˜
1/3
=25.2; placed on a patch-wise heterogeneous surface. The pattern is axially distributedwith sharp wettability transitions (
ω
˜
=0.5,
θ
1
=128.0° and
θ
2
=57.5°) and a centrallyophilicpatch.Seetextforthepatterndescription.Sincethereisafinitenumberof allowed values of
θ
app
, in fact the free energy curve should be drawn as a finiteseries of points. In case of continuous heterogeneity patterns, the sawtooth-structured curve would be softened.86
M.A. Rodríguez-Valverde et al. / Advances in Colloid and Interface Science 138 (2008) 84
–
100
analyzing CAH without CAM. Whereas the most consistent heterogeneitypatternaccordingtothesymmetryofsemi-infinitemenisci and cylindrical drops is the strip-wise distribution,axially-patterned heterogeneity would be the
“
natural
”
distribu-tion of surface energy for spherical drops. Unlike to strip-wisegeometry, axial distribution of surface energy allows exploringdynamics of
curved
contact lines. Although well-structured patterns are unrealistic, the main results derived from themmight be qualitatively usable for random heterogeneity patterns[27,45,46].
Generally, two types of heterogeneities distribution might beconsidered according to the transition between surface domains: patch-wise and continuous patterns. Patch-wise patterns are thoseones with infinitely sharp transitions (
“
mesa
”
defects) in thesurface energy distribution. Precisely, piece-wise heterogeneous patterns appear in open micro- and nanofluidic systems wherefluids are transported on chemical channels, i.e. lyophilic stripesembeddedinlyophobicsubstrates[54].Otherwise,heterogeneous patterns in which surface energy varies continuously (regular defects) might be applicable to more systems. Additionally,heterogeneity distribution can be arranged as isolated spots on ahomogeneous field (e.g. impurities) or periodic array of defects.As with the liquid
–
vapor interface, the heterogeneity patternmust be expressed analytically for free energy calculations.Hence, local
wettability
of a SLV system can be defined by:
μ
slv
x
;
y
ð Þ ¼
g
sv
x
;
y
ð Þ
g
sl
x
;
y
ð Þ
g
lv
ð
1
Þ
where
γ
sv
and
γ
sl
stand for the surface energy and solid
–
liquidinterfacial energy, respectively; and
x
and
y
are the coordinatesof a surface point. This function is defined on the entire solidsurface, regardless of the existence of a three-phase contact line.Indeed, wettability at a given position on the heterogeneoussurface is interpreted as the cosine of the local intrinsic contact angle (ICA), which is essentially a material property.In this study, inspired by the highly cited work of Johnson andDettre [24], the patch-wise axial pattern consists of two smoothand homogeneous patches of different lyophilicity/lyophobicity(high/low-surface energy), distributed in alternating concentriccircularbands.Forthispattern,therespectiveICA'sofeachpatchare
θ
1
and
θ
2
. The orientation ofboundariesasregards the patterncentre is symbolized by internal patch's ICA: external patch'sICA (e.g.
θ
2
:
θ
1
). Equally, two different characteristic lengths can bedefined(seeFig.2):thesemi-widthofthecentralpatch(
ω
)andthe spatial period or representative radius of pattern (
λ
). Alllengths involved in the SLV system are normalized by this last term (see Notation). Three types of patch-wise axial patterns
µ
slv
(
r
˜
;
θ
1
,
θ
2
,
ω
˜
) can be analytically defined as follows:Step pattern: cos
θ
2
+(cos
θ
1
–
cos
θ
2
)
U
(
r
˜
,
ω
˜
)Square pattern: cos
θ
2
+(cos
θ
1
–
cos
θ
2
)(
U
(
r
˜
,
ω
˜
)
–
U
(
r
˜
,
ω
˜
+1))Periodic patch-wise pattern: cos
θ
2
þ ð
cos
θ
1
cos
θ
2
Þ
P
ni
¼
1
ð
U
ð
r
˜
;
i
x
˜
þ
i
1
Þ
U
ð
r
˜
;
i
x
˜
þ
i
ÞÞ
where the central patch is arbitrarily chosen as
θ
2
-patch,
U
(
x
,
x
0
) is the unit step function (0 for
x
b
x
0
and 1 for
x
≥
x
0
) and
n
isan integer number greater than 1 (number of bands). Except for step pattern, 1
–
ω
˜
is the dimensionless width of the
θ
1
-patch.Using patch-wise axial patterns, the effect of boundaries onapparent contact angle can be directly examined with different
ω
˜
values.
Unlike the patch-wise pattern, the continuous axial pattern ischaracterized by concentric circular bands in which the ICAvaries continuously between two extreme values (
θ
1
and
θ
2
). Inthis study, two parameters might define the details of surfaceenergy distribution: the spatial period (
λ
) and a form coefficient
ν
(
N
1), which provides information about the surface proportionoflyophilic/lyophobiccharacter,accordingly.Hence,thewettabi-lity of the continuous axial pattern can be expressed by:
μ
slv
r
˜
;
θ
1
;
θ
2
;
m
ð Þ ¼
cos
θ
2
þ
cos
θ
1
cos
θ
2
ð Þ
f r
˜
;
m
ð Þ
where
f
is the following piece-wise polynomial (see Fig. 3):
f r
˜
;
m
ð Þ ¼
4
r
˜
1
r
˜
ð Þ½
m
;
r
˜
V
1
f r
˜
1
;
m
ð Þ
;
r
˜
N
1
:
The
f
-function gives information about the relative concen-tration of each domain. Once
ν
reaches the value 2.384, the pattern is sinusoidal and thereby, the surface proportion of lyophilic/lyophobic character is 0.5. For
ν
values greater than2.384, the
θ
2
-domain will be prevailing on the surface. Con-tinuous axial patterns allow relaxing the effects of boundarieson apparent contact angle with different
ν
values.Axial patterns can be described, on average, by a meanwettability
b
µ
slv
N
xy
as regards their wetted area (surface or
Fig. 2. Radial sections of patch-wise axial patterns: (a) 1 transition (step), (b) 2transitions (square), (c) 3 transitions and (d)
i
+2 transitions (wave). Thelyophobic patch is labeled by 1, the semi-width of the central patch (1 in the picture) is symbolized by
ω
and the spatial period by
λ
.87
M.A. Rodríguez-Valverde et al. / Advances in Colloid and Interface Science 138 (2008) 84
–
100
areal average). This parameter will depend on both ICA's, thecharacteristic lengths of the pattern (
ω
˜
or
ν
) and the droprelative dimensions (
V
˜
1/3
). Consequently, the mean wettabilityof axially-patterned heterogeneous surfaces will not remainconstant during the drop motion because of the pattern features.
2.3.
“
Stick/jump-slip
”
phenomena
Wetting can be interpreted from a mechanistic view [29],
i.e.
in terms of external and internal forces. The external forces arethose ones caused by an external agent. In contrast, two types of internal forces can be involved in a SLV system, apart from the bulk forces occurring in each phase: capillary force (exerted bythe liquid) and adhesion force (exerted by the solid surface).While the purpose of external forces is the deliberate relativemovement of the three-phase contact line, the capillary andadhesion forces can be driving or resistive, accordingly. Drivingforces induce liquid spreading or dewetting, while resistiveforces oppose movement of the contact line like a frictionalforce [55]. If there is none external force, wetting is consideredas spontaneous (known as spreading) and otherwise, wetting isforced [56].Forced wetting can be accomplished by dipping/risingvertically the solid in/from the liquid, tilting the solid surfaceor changing the drop volume by direct liquid addition/ withdrawal [15,40,41]. All these experimental approaches
purposely produce a variation in solid
–
liquid area (wettedarea) through the contact line movement in quasi-staticconditions [42], driven by vertical push/pull forces, gravity or volume variations, accordingly. However, the movement of triple line is not always a steady, continuous process. In fact, a
‘‘
jerky
’’
movement is observed with the drop or liquid front pinned at a position for a time and then jumping to a new position. This phenomenon is often referred to as
‘‘
stick/jump-slip
’’
(SJS) process, in analogy to dry friction. The underlyingfriction in SJS phenomena is different to the internal friction produced by liquid viscosity, although both are dissipative.Although SJS phenomena are equally visualized during spon-taneous wetting, forced wetting allows attaining on purpose areproducible sequence of metastates of the SLV system. Inliterature, certain complex contact angle patterns have beenreported in relation to SJS behavior [40,57]. Such measure-
ments can be misleading and are not used for the interpretationin terms of surface energetics.If the external force and the driving internal force cannot overcome the resistive internal force (critical threshold force)[58], the contact line is pinned:
“
stick
”
stage. This
“
staticfriction
”
or canthotaxis [59] is caused by the presence of
“
mesa
”
Fig. 3. Polynomial
f
in function of dimensionless radial coordinate, used to generate sinusoid-based axial patterns with different forms: (a)
ν
=1.10, (b)
ν
=2.53 and(c)
ν
=10.0.88
M.A. Rodríguez-Valverde et al. / Advances in Colloid and Interface Science 138 (2008) 84
–
100

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