Mechanical derivation of the Wenzel and Cassie equations using a statistical interpretation of drop dispensation

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Mechanical derivation of the Wenzel and Cassie equations using a statistical interpretation of drop dispensation
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   Journal of Colloid and Interface Science 327 (2008) 477–479 Contents lists available at ScienceDirect  Journal of Colloid and Interface Science www.elsevier.com/locate/jcis Short Communication Mechanical derivation of the Wenzel and Cassie equations using a statisticalinterpretation of drop dispensation Miguel A. Rodríguez-Valverde ∗ Biocolloid and Fluid Physics Group, Department of Applied Physics, University of Granada, Campus of Fuentenueva, E-18071 Granada, Spain a r t i c l e i n f o a b s t r a c t  Article history: Received 16 May 2008Accepted 15 August 2008Available online 3 September 2008 Keywords: Contact anglesRoughnessHeterogeneityMechanical equilibriumWenzel–Cassie equation The global mechanical equilibrium condition of a liquid on a rough and chemically heterogeneous surfacewas derived for three-dimensional situations from a statistical outlook of dispensation of many dropsand the assumption of local mechanical equilibrium. Unlike the conventional thermodynamic derivationsfrom variational methods, the current proof is based on vector algebra rather than differential geometry.The mechanics-based derivation becomes less intricate although the minimum energy condition is notestablished. An effective contact angle is computed from the directional sampling of three-phase linesafter local drop dispensations. The final expression is a combined mechanical version of the Wenzel andCassie equations. ©  2008 Elsevier Inc. All rights reserved. 1. Introduction One of the main challenges in surface thermodynamics is toquantify meaningfully surface energetics of solids from multiplemeasured contact angles [1]. This variability in contact angle arises from the non-ideal features of surfaces. There is just one measur-able contact angle closely related to the concerning solid interfacialenergies (material properties), at least on average. However, unlessvery particular conditions, this angle is hardly experimentally ac-cessible, not even recognizable if the contact line is corrugated.Instead, different effective contact angles have been proposed inthe literature [2] in order to estimate thermodynamically mean-ingful contact angles. In this paper, inspired in the work of Swainand Lipowsky [3], a new effective contact angle is formulated and further related to the Wenzel and Cassie angles, i.e. the equilibriumcontact angles on rough-homogeneous and smooth-heterogeneoussurfaces [4,5], respectively. Swain and Lipowsky [3] derived the thermodynamic equilib-rium condition for rough and chemically heterogeneous surfacesusing an elegant formalism based on energy minimization, differ-ential geometry and spatial averaging. However, the Wenzel andCassie equations can be intuitively derived [6], although less rigor-ously. Variational calculus has been frequently used to obtain thethermodynamic equilibrium conditions involved in a solid–liquid–vapour system. In this framework, the Young–Laplace equation de-scribes the shape of the liquid–vapour interface and the Youngequation [7] rises as its “natural” boundary condition due to the *  Fax: +34 958 24 32 14. E-mail address:  marodri@ugr.es. solid phase contact. The Young equation is locally fulfilled pro-vided that the triple contact line is unconstrained, i.e. when theYoung equation becomes a Neumann-type boundary condition.The Young–Laplace equation is indistinctly interpreted either asthe mechanical or thermodynamic equilibrium condition for mostliquid–fluid interfaces (isotropic phases) due to the fast moleculesdiffusion between phases (i.e. chemical equilibrium). However, al-though Thomas Young srcinally formulated his famous equationfrom mechanistic arguments, the Young equation is not indeedthe mechanical equilibrium condition of a solid–liquid–vapour sys-tem because of the absence of diffusion equilibrium at commonsolid–liquid interfaces [8,9]. Both thermodynamic and mechanical equilibrium conditions will coincide just on an  ideal  solid surface:smooth, chemically homogeneous, chemically inert and rigid sur-face; where the solid interfacial tensions numerically coincide withthe respective specific interfacial energies. Hence, on moderatelyrigid solids and at experimental time scales, a metastable mechani-cal configuration is usually visualized at the three-phase line ratherthan a global thermodynamic equilibrium state [10]. Thermody-namic equilibrium state will be truly attained when the wholesolid–liquid interface is accordingly deformed or molecularly ar-ranged, at very long times. This picture stimulates the currentcontroversy about the applicability of the Wenzel and Cassie equa-tions: does the attainable global equilibrium state depend on thesurface conditions underneath the drop or just at its three-phasecontact line?From the local mechanical equilibrium condition of a solid–liquid–vapour system, i.e. the local balance of interfacial forces,the current note provides a combined mechanical version of theWenzel and Cassie equations, if there is no trapped gas inside theconcave grooves of solid surface. In order to probe all surface fea- 0021-9797/$ – see front matter  © 2008 Elsevier Inc. All rights reserved.doi:10.1016/j.jcis.2008.08.048  478  M.A. Rodríguez-Valverde / Journal of Colloid and Interface Science 327 (2008) 477–479 tures over the entire solid area, a gedanken dispensation of manydrops allowed applying the concerning spatial averaging. 2. Definitions of contact angle In the context of Gibbs’ dividing surface, “interfaces” are treatedas mathematical surfaces, continuous and differentiable; thereforetheir normal unit vectors can be defined analytically. In order toavoid misleading interpretations, every angle involved in the three-phase contact line of a solid–liquid–vapour system must be conve-niently defined (Fig. 1): (1)  Actual  contact angle,  θ  , is the local angle formed by the normalunit vectors to the solid surface   N   and to the liquid–vapourinterface   N  lv  at a point  s  of the three-phase line:cos θ( s )  ≡   N  lv ( s ) ·   N  ( s ). This angle is also referred to as microscopic contact angle.(2)  Apparent   contact angle,  θ  app , is the local angle formed by theapparent unit normal of the solid surface (i.e. the vertical di-rection   e  z  ) and the unit normal of the liquid–vapour interface  N  lv  at a point  s  of the three-phase line:cos θ  app ( s )  ≡   N  lv ( s ) ·  e  z  . The apparent or macroscopic contact angle is experimentallyaccessible by optical techniques rather than the actual contactangle.(3) Roughness angle or surface pitch,  α , is the local angle formedby the vertical direction   e  z   and the unit normal to the solidsurface   N  ( s )  at a point  s  of the three-phase line:cos α ( s )  ≡   N  ( s ) ·  e  z  . (4) Angle  β  is the planar angle that points out about the absenceof co-planarity of the triplet   e  z  ,   N   and   N  lv :cos β( s )  ≡  (  ˆ N  lv ( s ) × ˆ e  z  ) · (  ˆ N  ( s ) × ˆ e  z  ) | ˆ N  lv ( s ) × ˆ e  z  || ˆ N  ( s ) × ˆ e  z  | . The mentioned triplet will be coplanar when  β  =  0 , π .Applying vector algebra [11], the following identities can be readily proved: ˆ N  lv  · ˆ N   − ( ˆ e  z   · ˆ N  lv )( ˆ e  z   · ˆ N  )  =  (  ˆ N  lv  × ˆ e  z  ) · (  ˆ N   × ˆ e  z  ),   1 − ( ˆ e  z   · ˆ N  lv ) 2   1 − ( ˆ e  z   · ˆ N  ) 2 = | ˆ N  lv  × ˆ e  z  || ˆ N   × ˆ e  z  | , and substituting the above-mentioned definitions into them, thelocal angles involved in the three-phase contact line are relatedamong themselves as follows:cos θ( s )  =  cos θ  app ( s ) cos α ( s ) + sin θ  app ( s ) sin α ( s ) cos β( s ).  (1)This expression can be also derived from spherical trigonometrybecause the vectorial triplet  e  z  ,   N   and   N  lv  (Fig. 1) draws a sphericaltriangle. Indeed, Eq. (1) is the identity between the surface tensionforce directly projected on the solid surface and the addition of the projections of the horizontal and vertical components of thesurface tension force on the solid surface. It is worth to emphasizethat this equality is not a condition of mechanical equilibrium. Thisexpression can be rewritten as:cos θ( s )  =  cos  θ  app ( s ) − α ( s )  cos 2  β( s ) 2 + cos  θ  app ( s ) + α ( s )  sin 2  β( s ) 2 Fig.1.  Layout of unit vectors (spherical triangle) and local angles involved in a pointof three-phase contact line over a rough heterogeneous surface. in order to gather the geometrical findings of Shuttleworth andBailey [12]:  θ   =  θ  app  ∓ α  for cylindrical interfaces  (β  =  0 , π ) .In addition to the mentioned angles, there is another “angle”known as  intrinsic   contact angle. In fact, this angle is an abstract,unlocated angle. The intrinsic contact angle at a point  (  x ,  y )  of thesolid surface is defined by the following expression:cos θ  i (  x ,  y )  ≡  γ  sv (  x ,  y ) − γ  sl (  x ,  y ) γ  lv ,  (2)where  γ  lv ,  γ  sv  and  γ  sl  symbolize the interfacial tensions of eachinterface, respectively. Strictly,  θ  i  is not a geometrical angle be-cause it is defined on the entire solid surface, regardless of theexistence of a three-phase contact line. In this paper, the varia-tions in  θ  i (  x ,  y )  are considered smooth with a macroscopic lengthscale and so, the long-range interactions acting in the core re-gion are negligible. The assumptions of weak chemical contrast anddifferentiable surface profiles (i.e. no sharp edges) allow avoidingcontact line pinning. Otherwise, the contribution from line tensionwas neglected.According to the local mechanical equilibrium condition for un-pinned contact lines, once a liquid drop placed on a real surfaceattains a stable configuration (not necessarily static), the actualcontact angle  θ   at a point  s  of the contact line  (  x c ( s ),  y c ( s ))  mustbe equal to the concerning intrinsic contact angle:cos θ( s )  =  cos θ  i   x c ( s ),  y c ( s )  .  (3)This equality would be the precursor one to the Young equa-tion before the thermodynamic equilibrium conditions were estab-lished. Although the interfacial tensions involving the solid phase γ  si  ( i  =  l , v )  do not generally coincide with the respective specificinterfacial energies  σ  si , the local difference  γ  sv (  x ,  y ) − γ  sl (  x ,  y )  canbe roughly approximated to  σ  sv (  x ,  y )  − σ  sl (  x ,  y )  [9], provided thatthe chemical equilibrium is guaranteed at the liquid–vapour inter-face [8]. Consequently, Eq. (3) becomes  numerically  equivalent tothe Young equation [7].Substituting Eq. (3) into Eq. (1), the following local equality is hold over all contact line:cos θ  i   x c ( s ),  y c ( s )   =  cos θ  app ( s ) cos α ( s ) + sin θ  app ( s ) sin α ( s ) cos β( s ).  (4)Marmur derived the two-dimensional version of Eq. (4) for roughhomogeneous surfaces by variational methods [13] and from theYoung equation (i.e. using energies instead of tensions). Alike, us-ing their generalized Young equation, Swain and Lipowsky reportedEq. (4) although in differential form (see Eq. (9.1) in [3]). 3. Statistical interpretation of drop dispensation Experimentally it is not feasible to scan all possible stableconfigurations (i.e. metastates) of a system in a given pseudo-  M.A. Rodríguez-Valverde / Journal of Colloid and Interface Science 327 (2008) 477–479  479 Fig. 2.  Geometrical interpretation of the statistical dispensation of drops around afixed surface point. thermodynamic state (e.g. temperature, volume and chemical po-tential). Instead, in principle, we might dispense a great number of drops, with fixed volume, whose stable contact lines pass over agiven surface point in any direction. The dispensation probabilitywould be the same one because the surface is considered uni-formly rough and heterogeneous, everywhere. In practice, a scalelength of inhomogeneities much lower than the drop size is equiv-alent to evenly distributed inhomogeneities. It is worthy to men-tion that Swain and Lipowsky used the same statistical approachin their working definition of effective angle (Eq. (6.8) in [3]). Geo-metrically, the multiple dispensation of drops (or scanning of dropconfigurations) involves the rotation of the unit normal   N  lv  aroundthe vector   N   fixed at a surface point  (  x c ( s ),  y c ( s )) , for a given  θ  i value (Fig. 2). Since the local vector   N  lv  will depend on the localsurface point,  s , and the rotating angle,  ψ , the apparent contactangle,  θ  app , and the  β  angle will also depend on them. However,these two angles are mutually independent.Averaging over all possible configurations in the considered“ensemble,” i.e. all drops with constant volume having locally anactual angle equal to  θ  i  and which contact lines pass over the lo-cal surface point  s , the mean cosine value of the local apparentcontact angles will be given by:cos θ  app ( s )  ≡  12 π 2 π   0 cos θ  app ( s ,ψ) d ψ. Otherwise, the mean value of the product sin θ  app ( s ,ψ) cos β( s ,ψ) will be proportional to the product of each mean value:sin θ  app ( s ) cos β( s )  because both variables are uncorrelated. Sincethe  β( s ,ψ)  angle scans as the rotating angle,  ψ , the entire range [ 0 , π ]  then cos β( s )  will be practically zero. Applying these resultsto Eq. (4), the following expression is derived: cos θ  app ( s )  =  sec α ( s ) cos θ  i   x c ( s ),  y c ( s )  ,  (5)which might be interpreted as the local Wenzel equation, i.e.the macroscopic condition of local mechanical equilibrium at thethree-phase line regardless of the drop shape.Since a rough heterogeneous surface locally manifests differentchemical and topographic features, the drop dispensation shouldbe extended over the entire surface. Thus, after the concerningspatial averaging, 1 Eq. (5) is rewritten in terms of surface-averagedvalues as follows:  cos θ  app   xy  =  r   cos θ  i   A ,  (6)where  r   ≡  sec α   xy  is the Wenzel factor and the following prop-erty was applied:   f   sec α   xy  =  sec α   xy   f     A . 1 Surface average over the nominal area:  ·  xy  ≡  1  A  z     · dxdy  and surface averageover the actual area:  ·  A  ≡  1  A    · dA . As Swain and Lipowsky suggested [3], an effective measurable an- gle  θ  eff   might be defined as:cos θ  eff   ≡  cos θ  app   xy that substituting it into Eq. (6) provides:cos θ  eff   =  r   cos θ  i   A .  (7)The effective measurable angle  θ  eff   represents, of all possible con-figurations of a drop on a rough heterogeneous surface, the config-uration of global mechanical equilibrium. In fact, Eq. (7) is a unifiedmechanical version of the Wenzel and Cassie equations [4,5] forthree-dimensional situations. Hence, accordingly, for a smooth het-erogeneous surface  ( α  =  0 ) , Eq. (7) is reduced to:cos θ  eff   =  cos θ  i   xy and for a rough homogeneous surface  (θ  i      =  θ  i (  x ,  y )) , Eq. (7) drawsto:cos θ  eff   =  r  cos θ  i . There is no extra term in Eq. (7) related to gravity and root meansquare roughness, as found in Eq. (9.2) of the paper of Swain andLipowsky [3]. Furthermore, this term has been no more reported in the rest of literature. This term indeed depends on the  β  angleand it does not vanish if a single spatial average  ·  xy  is applied. 4. Summary  In this note, the global mechanical equilibrium condition of aliquid on a rough heterogeneous surface was derived for three-dimensional situations from a statistical view of dispensation of many drops and the assumption of local mechanical equilibrium.Unlike the derivation of Swain and Lipowsky [3], based on the en- ergy minimization but the same statistical arguments and effectiveangle than the current proof, this is based on vector algebra in-stead of differential geometry. Moreover, the concerning surfacesaverages were correctly applied from the directional sampling of three-phase lines after local drop dispensations. The mechanicalderivation becomes less intricate although the minimum energycondition is not established. The expression (7) is a combined me-chanical version of the Wenzel and Cassie equations.  Acknowledgments This work was supported by the “Ministerio Español de Edu-cación y Ciencia” (contract “Ramón y Cajal” 18-08-463B-750), theEuropean Social Fund (ESF) and the “Junta de Andalucia” (projectFQM-02517). References [1] A. Marmur, Soft Matter 2 (1) (2006) 12–17.[2] S.D. Iliev, N.C. Pesheva, Langmuir 19 (23) (2003) 9923–9931.[3] P.S. Swain, R. Lipowsky, Langmuir 14 (1998) 6772–6780.[4] R. Wenzel, J. Phys. Chem. 53 (1949) 1466–1467.[5] A. Cassie, Discuss. Faraday Soc. 3 (1948) 11–16.[6] G. McHale, M.I. Newton, Colloids Surf. A Physicochem. Eng. Aspects 206 (2)(2002) 193–201.[7] T. Young, Philos. Trans. R. Soc. 95 (1805) 65–87.[8] A.I. Rusanov, Pure Appl. Chem. 64 (1) (1992) 111–124.[9] P. Roura, J. Fort, J. Colloid Interface Sci. 272 (2) (2004) 420–429.[10] N. Eustathopolous, M. Nicholas, B. Drevet, Fundamental Equations of Wetting,Elsevier Science, Oxford, 1999, Chap. 1, pp. 16–17.[11] J.W. Gibbs, E.B. Wilson, Vector Analysis: A Text-Book for the Use of Students of Mathematics and Physics, Founded upon the Lectures of J. Willard Gibbs, YaleUniversity Press, New Haven, 1929.[12] R. Shuttleworth, C. Bailey, Discuss. Faraday Soc. 3 (1948) 16–22.[13] A. Marmur, Adv. Colloid Interface Sci. 50 (1994) 121–141.
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