Cassie and Wenzel_ Were They Really So Wrong_

Cassie and Wenzel:Were They Really So Wrong?G.McHale*School of Biomedical &Natural Sciences,Nottingham Trent Uni V ersity,Clifton Lane,Nottingham NG118NS,UKRecei V ed April 17,2007.In Final Form:May 18,2007The properties of superhydrophobic surfaces are often understood by reference to the Cassie -Baxter and Wenzel equations.Recently,in a paper deliberately entitled to be provocative,it has been suggested that these equations are wrong;a suggestion said to be justified using experimental data.In this paper,we review the theoretical basis of the equations.We argue that these models are not so much wrong as have assumptions that define the limitations on their applicability and that with suitable generalization they can be used with surfaces possessing some types of spatially varying defect distributions.We discuss the relationship of the models to the previously published experiments and using minimum energy considerations review the derivations of the equations for surfaces with defect distributions.We argue that this means the roughness parameter and surface area fractions are quantities local to the droplet perimeter and that the published data can be interpreted within the models.We derive versions of the Cassie -Baxter and Wenzel equations involving roughness and Cassie -Baxter solid fraction functions local to the three-phase contact line on the assumption that the droplet retains an average axisymmetry shape.Moreover,we indicate that,for superhydrophobic surfaces,the definition of droplet perimeter does not necessarily coincide with the three-phase contact line.As a consequence,the three-phase contact lines within the contact perimeter beneath the droplet can be important in determining the observed contact angle on superhydrophobic surfaces.IntroductionGao and McCarthy have recently argued that experimental data demonstrates “how Wenzel and Cassie were wrong”.1-5They take the view that thinking of the interfacial tensions as forces per unit length is more intuitive than thinking in terms of energy per unit area.Their objective is to educate modern surface and materials scientists working with superhydrophobic surfaces.They do not advocate never using the Wenzel and Cassie -Baxter’s equations but that “they should be used with knowledge of their faults ”.The aim of our paper is not to refute everything they claim but to show how the Cassie -Baxter and Wenzel equations can be used to explain the presented data and to provide an intuitive understanding of when it might be reasonable to use the local forms of these equations.It is our concern that these equations are used not so much with “knowledge of their faults ”but within their assumptions and range of applicability and with appropriate generalization.In reviewing the derivations of the Wenzel and Cassie -Baxter equations we show how the Wenzel roughness and the Cassie -Baxter surface fractions are either global constants,if the surfaces satisfy certain assumptions,or local functions,if the surface is of the type presented by Gao and McCarthy.For more randomly structured surfaces,we discuss the conditions under which a droplet might be expected to average out the surface structure and for Cassie -Baxter and Wenzel type of equations to then apply.Wenzel and Cassie -Baxter RestatedTo summarize the key results of this paper,our statements of the Wenzel and Cassie -Baxter equations areandwhere the contact angle,θi s ,on a smooth flat surface is given byYoung’s equationand f 1(x )+f 2(x ))1.These equations differ from those given in ref 1by explicitly defining the roughness,r (x ),and the Cassie -Baxter surface fractions,f 1(x )and f 2(x ),as functions taken in the region of the droplet perimeter indicated by the radial coordinate x .The roughness,r (x ),is the ratio of the true area of the solid to its planar projection for a small displacement of the droplet perimeter at location x ,i.e.r (x ))∆A True (x )/∆A (x )evaluated at x .The Cassie -Baxter surface fraction,f i (x ),is the ratio of area of type i to the total area in a small representative region around the droplet perimeter at location x .In so much as these are local parameters in the vicinity of the droplet perimeter,we are in agreement with Gao and McCarthy that only a small fraction of the surface probed by the droplet perimeter is important in affecting the contact angle behavior.However,we will later argue that for a superhydrophobic surface the equivalence of the contact line at the perimeter of a droplet and the three-phase contact line needs to be considered carefully.As a consequence,the contact area within the droplet contact perimeter can be important for superhydrophobic surfaces.We recognize that eqs 1and 2require droplets with experimentally confirmed axi-symmetric droplet shapes for the assumptions underlying them to be regarded as reasonable,and we emphasize that this paper does not address advancing or receding contact angles.*E-mail:glen.mchale@.(1)Gao,L.;McCarthy,ngmuir 2007,23,3762-3765.(2)Cassie,A.B.D.;Baxter,S.Trans.Faraday Soc.1944,40,546-551.(3)Wenzel,R.N.Ind.Eng.Chem .1936,28,988-994.(4)Wenzel,R.N.J.Phys.Colloid Chem.1949,53,1466-1467.(5)Johnson,R.E.;Dettre,R.H.Contact Angle,Wettability and Adhesion .Ad V ances in Chemistry Series;American Chemical Society:Washington,DC,1964;Vol.43,pp 112-135.cos θe W (x ))r (x )cos θes(1)cos θe CB (x ))f 1(x )cos θ1s +f 2(x )cos θ2s(2)cosθis )γSV i -γSLi γLV(3)8200Langmuir 2007,23,8200-8205We now focus on the suggestion that experiments in ref1 show that the Wenzel and Cassie equations are wrong.In the case that a surface is everywhere similar and isotropic,the roughness and Cassie-Baxter fractions in eqs1and2become global constants.By“everywhere similar and isotropic”we mean that one could randomly choose a small part of the surface and would not be able to identify where on the surface the location was or identify an orientation simply by looking at the structure of the chosen small surface area.A surface with small randomly placed defects,chemical or topographic,satisfies this assumption, but one with a single defect does not.The surfaces used in ref 1involve a single small circular area(referred to in this article as a defect)within a larger area and so do not satisfy this assumption and therefore the local form of the equations(eqs 1and2)should be used.The Original Wenzel and Cassie-Baxter Equations In the original article by Wenzel,he stated that“the forces that oppose each other along a gi V en length of the ad V ancing periphery of the wetted area are proportional in magnitude,not to the tensions of the respecti V e interfaces but to their energies per unit of geometric surface.”3To further clarify,his Figure1showed a section of area at the periphery of the droplet and the discussion concerned surfaces characterized by a single,constant value of roughness r.His subsequent communication(ref4)further refers to properties of the roughness of the solid surface itself rather than of the wetted portion of the surface.A potential confusion of language is that his summary in the original article states“the wetting properties of a solid substance should be directly proportional to the roughness of the surface wetted”.3However, given his previous discussion,we believe the statement“surface wetted”refers to the substrate surface as a whole and not simply the portion of the substrate surface on which a droplet might have been deposited.Moreover,if the roughness of a substrate surface is constant,then there is no difference in the value obtained by calculating the roughness parameter using a small part of the surface at the periphery of the droplet,the whole of the substrate surface,or the specific area of surface wetted under a droplet. In this case,all of these calculational methods are equivalent. Johnson and Dettre5also discussed the Wenzel equation.They noted that the basic assumption of Wenzel was that“a rough surface can be treated as a smooth surface of surface energy rγSV or rγSL,depending on the interface.With this assumption Wenzel’s equation can be deri V ed using the same techniques as in the deri V ation of Young’s equation itself.”These authors went on to list four basic assumptions that were needed to derive Wenzel’s equation via a variational technique requiring the first-order change in surface free energy to vanish.They therefore demonstrated that Wenzel’s equation is compatible with a derivation based upon surface free energy changes at the droplet perimeter in the same manner as in the derivation of Young’s law.The second of the assumptions they listed required the surface free energy change associated with the solid-liquid interfacial area change,∆F SL,to equal r∆A,where∆A is one period of the surface.They did explicitly state in their work that r was defined in terms of total solid-liquid interfacial area under the drop divided by the horizontal projection of contact area.However, this was in the context of a surface with a constant roughness, so that the roughness parameter was assumed independent of droplet contact area.Thus,the basis of Wenzel’s equation is of a constant roughness defined as a global property of the surface itself and not of the specific portion of the surface under the configuration,the Johnson and Dettre derivation limits its applicability to those whereby a small change in area of the droplet would encompass one period of the surface roughness. The derivation by Cassie and Baxter of their equation in ref 2considered a liquid advancing on a regular(periodic)grid of cylindrical fibers.They provided two equationswhereθD andθW were the apparent advancing and receding contact angles andθA andθR were the advancing and receding contact angles for the solid-liquid interface.They also noted that the factors f1and f2in the equation for the apparent receding contact angle are determined by the advancing contact angle. Their paper states“Let f1be the total area of solid-liquid interface and f2be the total area of liquid-air interface in a plane geometrical area of unity parallel to the rough surface”.The energy they considered is that expended in forming unit geometrical area of the interface for“water ad V ancing onto a dry surface”.The experiments reported by Cassie and Baxter in ref2did not use droplets but used a plate method.In discussing this method they stated that the“present analysis is inapplicable when the wires are parallel to the surface,since it assumes that for an increase in areaδA of the porous surface water interface there is formed an area of solid-water interface,f1δA,and of water-air interface,f2δA,for any infinitesimal V alue ofδA.”Thus,the original Cassie-Baxter concept involved an apparent contact angle determined using an infinitesimal change,δA,that covered a typical portion of the surface containing both types of surface fraction f1and f2.It is a key aspect of their derivation that they considered the energy per unit geometrical area for a liquid advancing on a periodic substrate.This means there is an implicit assumption that the f1and f2should not depend on the location on the surface where a contact angle is measured.For a droplet experiment,the surface fractions f1and f2should have the same constant values characterizing the substrate-liquid contact area no matter where the droplet is deposited. Johnson and Dettre5also demonstrated how the Cassie-Baxter equation could be derived using the condition that the first-order change in surface free energy should vanish.Similar to their treatment of the Wenzel equation,their rederivations of the Cassie-Baxter equations again explicitly listed four assumptions, three of which relate to small changes in area at the three-phase contact line;the surface was assumed to be periodic.Johnson and Dettre define the solid surface fractions as total areas of solid-liquid and liquid-air interfaces under the drop.However, the implicit assumption arising from the periodicity is that the precise location at which a droplet is deposited does not alter the values of f1and f2.Within the context of spreading on a periodic substrate and with a smallest change of one period,there can be no dependence of the surface fractions f1and f2on the precise location at which a droplet is deposited.In our opinion,the original Wenzel and Cassie-Baxter equations interpreted with r and f i from formulas based on surface areas under a droplet are restricted in applicability to quite specific types of surfaces.These should be surfaces for which the roughness and surface fraction parameters are everywhere constant and which do not have values dependent on the location of the droplet or the size of the droplet contact area.We also believe that the original papers imply that it is the surface cosθD)f1cosθA-f2and cosθW)f1cosθR-f2(4)Cassie and Wenzel:Were They Really So Wrong?Langmuir,Vol.23,No.15,20078201Interpretation of the Single Defect Experiment If we now consider a simple circular defect area of one hydrophobicity within a flat sample of another hydrophobicity (e.g.,Figure 1a),we cannot define Cassie -Baxter surface fractions f 1and f 2for the surface that are independent of where the droplet is placed.Similarly,a roughness factor for a surface with a single rough defect within a smooth area of a substrate would not be well-defined as a global surface property,since the overall substrate size is arbitrary.In the experiments reported by Gao and McCarthy,the comparisons to the models define the Cassie -Baxter fractions and the roughness parameter with reference to the contact area of the droplet.This may seem reasonable in following some of the literature language describing r and f i ,but the fact that for these substrates the values of r and f i depend on the specific location for the droplet deposition and its size does not conform to the assumptions in the original models.The roughness parameter r is no longer a property of the substrate alone,and the surface fractions f 1and f 2do not describe a substrate for which a small differential change of droplet area for the given droplet configuration encompasses one period of the substrate.The interpretation used by Gao and McCarthy 1of these parameters in the Cassie -Baxter and Wenzel equations then leads to the conclusion that the equations themselves are wrong.In our opinion,the experiments in Gao and McCarthy 1and those of Extrand 6are entirely consistent with our statements of the Wenzel and Cassie -Baxter equations with roughness of surface fractions that are locally defined at the droplet perimeter (i.e.,eqs 1and 2).By redefining the roughness and surface fractions as functions local to the droplet perimeter,we also agree with these authors that it is interactions at the three-phase contact lines that determine the contact angle.If a droplet sits only on a defect area of constant roughness r def ,then only the surface properties of the defect area matters and the local roughness parameter at the droplet perimeter is the roughness of the defect area,i.e.,r (x ))r def ,and Wenzel’s equation becomescos θe W (x ))r def cos θdef s .If the droplet fully encompasses thedefect and extends into an exterior region of constant roughness r ext ,then only the properties of the exterior region matter and the local roughness parameter at the droplet perimeter is the roughness of the exterior area,i.e.,r )r ext ,so Wenzel’s equation becomescos θe W (x ))r ext cos θext s .This is entirely consistent with all ofthe data in refs 1and 6and with our eqs 1and 2.We would not claim the original Wenzel and Cassie -Baxter equations arewrong,but we would say the experimental situation does not match the assumptions of the models.Moreover,with a reintepretation of the meaning of some of the symbols to match the experimental situation,the Wenzel and Cassie -Baxter equations can be applied to explain the observed data,subject to the usual warnings that real surfaces display advancing and receding contact angles rather than simply a single equilibrium contact angle.Minimum Energy PrinciplesWe believe that the surface free energy per unit area view of interfacial tensions can be an extremely intuitive way of understanding contact angles.Our preference for an energy per unit area approach is because superhydrophobic surfaces are often continuous,but nondifferentiable.Due to sharp asperities,it is not possible to define the physical slope at all points across such a surface,and as a consequence,a force based approach can become difficult.An energy-based approach inherently averages over some microscopic area.In the following we examine this approach for a solid surface composed of patches with two types of surface chemistry (i.e.,the Cassie -Baxter case).Generalization to multiple patches and to a rough surface follows the same principles,so we have simply cited the results for the Wenzel case.When a droplet is placed on a solid substrate possessing two types of surface chemistry,and the droplet is assumed small enough that only surface energies need be considered,the total surface free energy,F ,is,In this equation,A i is the total solid area of type i covered by the droplet and A i ∞is the total area of patches of type i across the solid surface.The notation of eq 5uses an infinity symbol in the area superscript precisely because in principle the surface can be of infinite area;this is the equivalent of saying a given substrate can be of arbitrary size.We are therefore forced by this formulation to recognize at the outset that the total energy itself is only determined to within an overall constant.This is not a problem,since an equilibrium state is defined by a droplet configuration giving a minimum in the energy function;changing the zero of the energy by an overall constant does not alter the configuration giving the minimum.Thus,the derivation of the equilibrium state relies on identifying the configuration of the droplet from which a change in configuration satisfies the condition ∆F )0;this is independent of any overall constant in eq 5.It should be clear that for a solid surface continuously contacted by a droplet,making a small change in surface free energy by making a small change in contact area of a droplet relies on changes of the three-phase contact line at the droplet perimeter and not of the area interior to the droplet perimeter.The situation for a droplet in a Cassie -Baxter state on a superhydrophobic surface is different because any possibility of penetration into features within the droplet contact area within the perimeter becomes important for minimizing the surface free energy;changes can occur at any three-phase contact line.In the (non-superhydrophobic)Cassie -Baxter case of a continuously contacted solid surface possessing two different surface chem-istries,a change at the droplet perimeter transfers area from just outside the perimeter to just inside the perimeter or vice versa.Hence,the Cassie -Baxter concept of using surface fractionsFigure 1.Isolated circular hydrophobic defect of radius r p characterized by a Young’s law contact angle θ1s within a hydro-philic region characterized by a Young’s law contact angle θ2s :(a)planar view of the substrate and (b)two droplet configurations with minima in their local surface free energy corresponding to the same droplet volumes.F )γSV 1(A 1∞-A 1)+γSV 2(A 2∞-A 2)+γSL 1A 1+γSL 2A 2+γLV A LV (5)8202Langmuir,Vol.23,No.15,2007McHaleshould be defined locally at the droplet perimeter.More generally,it is changes at the three-phase contact line that matter,which for a superhydrophobic surface is not limited to the droplet perimeter but includes locations within the droplet perimeter.The importance of three-phase contact lines,which for the (non-superhydrophobic)Cassie -Baxter case of a smooth surface composed of multiple solids with different surface chemistries is equivalent to the droplet perimeter,is a direct consequence of the minimum energy principle.Now consider the case of a hydrophobic circular defect patch of radius r p with a Young’s law contact angle θ1s surrounded by a hydrophilic area with a Young’s law contact angle θ2s (Figure 1a).If a small droplet is deposited on this surface it will form a spherical cap shape described by a spherical radius R ,contact radius r ,cap height h ,and volume V .These parameters are related by the equations 7where (θ)is a function of the contact angle and is defined asThe solid -liquid and liquid -vapor interfacial areas are given byThe total surface free energy,F ,is given by eq 5.In this case,it is possible for a suitable volume droplet to have two equilibrium states on the surface with contact radii r 1<r p and r 2>r p related byThese two states are illustrated schematically in Figure 1b.The surface free energies,F 1and F 2,for two droplets of the same volume,but with one entirely contained within the patch and the other encompassing the patch,are given by eq 5aswhere Young’s law (eq 3)has been used.Here A p )πr p 2is the area of the patch and F o is an overall constant given bywhere A ∞)A 2∞+A p is the total surface areas of both chemical types.In these equations,the droplet volume and contact angle,θ1,determines r 1,and the droplet volume and contact angle,θ2,determines r 2.Equations 10and 11are illustrated numerically as a function of contact angle (at constant droplet volume)inFigure 2for the case of θ1s )70°,θ2s )110°,V )1×10-9m 3,and r p )0.782mm.In each case,the minimum energy of the curve (indicated by the dots in Figure 2)is the local Young’slaw contact angle;the corresponding equilibrium contact radiiare equal to 0.921and 0.643mm,respectively.This illustrates that local equilibrium corresponds to the vanishing of the first-order change in surface free energy and that the change is taken at the three-phase contact line,which for a chemically hetero-geneous solid surface is the same as the droplet perimeter.Local Cassie -Baxter and Wenzel Equations To be more general than allowed by the assumption of a small droplet of spherical cap shape,it is usual to consider surface free energy changes at the three-phase contact line;this is the approach we now adopt.Figure 3b shows a two-dimensional model in which a displacement of the liquid -vapor interface along a one-dimensional line surface has an accompanying small change in length,∆A (x ),at the three-phase contact point at location x ;the one-dimensional line surface is assumed to be of the Cassie -Baxter type with alternate solid patches of different wettability characterized by different Young’s law contact angles.We use the terminology of area rather than length because the one-R )(3V π)1/3r )R sin θh )R (1-cos θ)(6)(θ))2-3cos θ+cos 3θ)(1-cos θ)2(2+cos θ)(7)A SL )πr 2and A LV )2πR 2(1-cos θ)(8)r 2)sin θ2s sin θ1s ( (θ1s) (θ2s ))1/3r 1(9)F (θ1)γLV )F o γLV+A LV (θ1)-A SL (r 1)cos θ1s (10)F (θ2)γLV )F o γLV+(cos θ2s -cos θ1s )A p +A LV (θ2)-A SL (r 2)cos θ2s (11)F o )A ∞γSV 2+A p (γSV 1-γSV 2)(12)Figure 2.Surface free energy curves for two spherical cap droplets of volume V )1×10-9m 3on a surface of type Figure 1a possessingθ1s )70°,θ2s)110°,and r p )0.782mm.In the first case,the equilibrium contact radius r )0.921mm is larger than r p ,and in the second case,the equilibrium contact radius r )0.643mm is smaller than r p .In both cases,the minima in energy correspond to the local Young’s law value of contact angle at the dropletperimeter.Figure 3.(a)An axially symmetric concentric ring type substrategeneralized from a one-dimensional line substrate consisting of two types of surface chemistries and (b)a small local displacement of a three-phase contact point on the one-dimensional Cassie -Baxter surface.Cassie and Wenzel:Were They Really So Wrong?Langmuir,Vol.23,No.15,200782033a).For a three-phase contact on a surface,which is locally of type i ,the change in surface free energy is then (Figure 3b)where ∆A (x )is a small local change at point x .The assumption in eq 13that makes the contact angle independent of droplet size or shape is that changes in the contact angle are second-order;sharp transitions in droplet configuration may break this assumption.Setting the first-order change in surface free energy to zero gives eq 3and shows that the local equilibrium contact angle is given by the Young’s law contact angle for the local surface of type i .Using the axial symmetry assumption generalizes the one-dimensional line surface to the two-dimensional surface case of multiple consecutive rings (Figure 3a);Figure 1a is a simple version of this type of surface.It is self-evident that this one-dimensional line surface approach to deriving an equilibrium contact angle does not give the Cassie -Baxter equation for the two-dimensional surface situation.However,the difficulty with this argument against the Cassie -Baxter equation is that the assumption of an axially symmetric surface can only result in surfaces of the concentric ring type shown in Figure 3a.In contrast,experiments are often interested in axially symmetric droplets on more complex surfaces.The contact perimeter of a three-dimensional droplet on a two-dimensional surface does not simply involve a set of independent three phase contact points arising from consideration of the problem as separate two-dimensional profile sections of the droplet on one-dimensional line surfaces.The key point is that the three-phase contact points along the perimeter of a droplet are connected to form a contact line and so are not independent.Thus,motion of one point on the droplet perimeter influences its neighbors along the contact line.For a surface with regular defects arranged in a lattice,Cubaud et al.have reported transitions between circular and faceted contact area shapes due to competition between the tension of the drop edge and the local force exerted on the contact line by the defects.8-10They also reported coordinated depinning of the contact line along a facet.Very recently,Kusumaatmaja and Yeomans have shown from lattice Boltzmann simulations the extent of these effects for the specific cases of droplets whose contact areas are on arrays of up to 6×6posts;11they note that larger scale simulations are in progress.As they indicate,the macroscopic contact angle is difficult to define uniquely because it then depends on the profile view direction used for the cross-section.However,even on these types of surfaces,droplets can often display approximately circular contact areas.Consistency of multiple profile views should therefore allow an assessment experimentally of whether the contact perimeter is approximately circular.Many droplet-surface problems of interest will either involve droplets encom-passing regularly arranged defects,but nonetheless displaying approximately circular contact perimeters,or the surfaces involved will be irregular composites (e.g.,Figure 4a).In such cases,droplets will certainly have local deformations from a circular contact area,but,if on average,the contact perimeter can be approximated by a circular shape,a Cassie -Baxter equation approach and an axially symmetric droplet should be valid.Surfaces with strong symmetry or directionality of features canstill be handled by minimizing the surface free energy,but we would not expect this to result in an axially symmetric droplet.In our approach,we suggest that provided a circular contact perimeter approximation is a reasonable assumption,then the displacement ∆A (x )can be viewed as an average along the whole length of the droplet perimeter;such an assumption will not apply to all surfaces.Under this assumption,the displacement ∆A (x )should,on average,encompass a representative set of points along the droplet perimeter.This is equivalent schematically to taking the two-dimensional view shown in Figure 4b,but with the displacement ∆A (x )(on average)fully encompassing both a type 1and a type 2patch;in this manner,the displacement should be representative of the surface along the entire droplet perimeter.This is a generalization of the condition stated by Johnson and Dettre 5that the small differential change of area should correspond to exactly one period of the surface.We therefore define Cassie -Baxter type surface fractions,f i (x ))∆A i (x )/∆A T (x ),where ∆A i (x )is the area of a patch of chemical properties of type i at location x ,and ∆A T (x ))∆A 1(x )+∆A 2(x ).The equivalent surface free energy change then becomesSetting this change in surface free energy to zero and usingYoung’s law for each type of surface then gives eq 2for the equilibrium contact angle θe CB (x ).Strictly this could be a local minimum rather than a global minimum in the energy.Similar considerations can be used to derive eq 1for the Wenzel case.Superhydrophobic Surfaces and the Droplet Perimeter In the previous section we considered a Cassie -Baxter surface composed of solids of two types of surface chemistries,but smooth so that a droplet contacted the surface at all points within its contact perimeter.A common approach to modeling the contact angle on a superhydrophobic surface is to imagine a droplet bridging between the peaks of the (rough or topographically (8)Cubaud,T.;Fermigier,M.;Jenffer,P.Oil Gas Sci.Technol.s Rev.IFP 2001,56,23-31.(9)Cubaud,T.;Fermigier,A.Europhys.Letts.2001,55,239-245.∆F (x ))(γSL i -γSV i )∆A i (x )+γLV cos θ(x )∆A (x )(13)Figure 4.(a)A random type Cassie -Baxter substrate consisting of two types of surface chemistries (the size of features is exaggerated)and (b)a small local displacement of a three-phase contact point on the equivalent one-dimensional Cassie -Baxter surface representing multiple one-dimensional sections through the surface in part a.∆F (x ))(γSL 1-γSV 1)f 1(x)∆A T (x )+(γSL 2-γSV 2)f 2(x )∆A T (x )+γLV cos θ(x )∆A T (x )(14)8204Langmuir,Vol.23,No.15,2007McHale。