Surface forces and wetting features in drops and capillaries

Advances in Fluid Mechanics VIII 345 Surface forces and wetting features in drops and capillaries M. G. Velarde1 & V. M. Starov2 1 2 Instituto Pluridisciplinar, Universidad Complutense de Madrid, Spain Department of Cmical Engineering, Loughborough University, UK Abstract Using t DLVO (Derjaguin, Landau, Verwey, Overbeek) tory, which accounts for quantum mechanics and electrostatics at t macroscopic level, t trmodynamic expressions for (trmodynamic) equilibrium contact angles of drops on solid substrates and menisci in solid wall capillaries are, operationally and unambiguously, expressed in terms of t corresponding Derjaguin pressure. T latter s S-shape is responsible for microdrops and otr pnomena appearing on flat solid substrates. Keywords: DLVO, Derjaguin pressure, surface forces, wetting, drops, capillaries. 1 Introduction: mechanics versus trmodynamics In t study of t equilibrium of a sessile drop partially wetting a smooth, homogeneous, flat solid substrate, tre have been numerous attempts to deduce t Young equation for t equilibrium contact angle,, [1-3] 0 sl cos sv, (1) wre, sl, are solid-vapor (of a bare solid surface), solid-liquid and liquid-vapor interfacial tensions, respectively. As reflected in most textbooks and scientific publications, traditional analyses use arguments eitr from trmodynamics or from mechanics, always insisting on eqn. (1) defining an equilibrium condition. As stated, eqn. (1) defines a contact angle,, that does 0 sv doi:10.2495/afm100301

346 Advances in Fluid Mechanics VIII Figure 1: A sessile drop on a solid surface in apparent mechanical equilibrium. As shown, t angle defines, following Young, t mechanical equilibrium of tangential forces. Figure 2: Wetting (left figure) and mostly non-wetting (center figure) liquid films and t ideal complete wetting case (figure on t right). not depend on t vapor pressure in t ambient air or on t volume of t drop or on its radius of curvature,. To be also noted is that neitr sv0 nor sl can be measured [2]. However, to a useful first approximation, we have t Kelvin equation [2] Pe 2 RT p, ln vm ps (2) wre m is t molar volume of t liquid, p s is t pressure of t saturated vapor at t temperature T, R is t universal gas constant, p is t vapor pressure that is at equilibrium with t liquid in t drop, and Pe is t excess pressure inside t drop. As t right hand side of eqn. (2) must be positive t vapor pressure, p, must be above t pressure of t saturated vapor p s (corresponding to a flat surface, R ). Hence t drop can be at equilibrium with oversaturated vapor pressure only. Thus, eqn. (2) determines a single radius of curvature of t liquid drop for each given value of oversaturated vapor pressure, p. Accordingly, t contact angle determined by eqn. (1) does not correspond to trmodynamic equilibrium. Note that, with oversaturation, t equilibration process demands quite long time intervals (on occasion up to hours). Surely most of t publisd data on measured contact angles actually

Advances in Fluid Mechanics VIII Figure 3: 347 Evaporation flux along a sessile drop surface. correspond to so-called (static) advancing contact angles, different from t true equilibrium value [2, 4, 5]. If eqn. (1) refers to mechanical equilibrium tn, disregarding gravity effects, t drop is bound to retain a sprical shape down to t contact with t solid substrate, that is, down to t three-phase contact line wre t thickness of t liquid vaniss. To be recalled is that a displacement of t three-phase contact line through t solid substrate as t liquid wets it is hard to explain. Indeed, according to standard hydrodynamics, t tangential friction force at t threephase contact line diverges to infinite value. Anotr difficulty with eqn. (1), as it stands, is that evaporation seems to take place most vigorously in t vicinity of such three-phase contact line (fig. 2). This means that trmodynamic equilibrium is violated mostly just in t vicinity of t three-phase contact line [6, 7]. 2 DLVO tory drops and capillaries Starting with t work of Frumkin, Derjaguin and Landau, and Verwey and Overbeek, and Casimir, in t mid-xxth century, a useful approach (DLVO tory) to t problems raised by eqn. (1) [8-10] has been developed, thus leading to a sound trmodynamic derivation of t true equilibrium contact angle [4]. From t work of those authors it became clear that as t three-phase contact line belongs to t hundred nano-meter range tn at such distances between two surfaces quantum mechanics and electrostatics intervene. Wn all tir integrated molecular interactions are considered, this led to t introduction of so-called surface forces and t definition of a new form of pressure of significance in t study of thin liquid films and, in general, wn two surfaces approach each otr in vacuum or otrwise, as t ideal case illustrated in fig. 2. Originally, it was denoted as disjoining pressure though depending on t confronted surfaces and t medium separating tm it could be eitr disjoining or conjoining. We shall call it Derjaguin pressure. How much t action of Derjaguin pressure affects t liquid profile eventually deviating it from a sprical shape wn approaching t three-phase

348 Advances in Fluid Mechanics VIII Figure 4: Measuring Derjaguin pressure (as proposed by Derjaguin, Churaev and Sludko) using a porous container A: h gh (H may be positive or negative). Figure 5: Exaggerated view (as H ) of a liquid drop partially wetting a solid substrate. is t radius of t sprical cap. In t left figure t angle e can be considered as t contact angle at (trmodynamic) equilibrium corresponding (following Gibbs) to t minimum excess free energy of t system relative to t homogeneous flat thin liquid film. T right figure illustrates t sprical macro-region, t adsorbed nano-metric liquid film and t transition region. contact line? How t (trmodynamic) equilibrium contact angle can be defined? Tse questions can be answered by considering t combined action of Laplace capillary ( surface tension ) pressure and Derjaguin ( surface forces ) pressure acting along t overall drop profile. T Laplace pressure is [2-5] pl p a 2 or otrwise h 1 h 2 32 Pe, (3) wre for simplicity in two-dimensional geometry (a cylindrical drop) we have provided t value of in terms of t function h defining t drop profile. Pe p a pl is negative for a drop and positive for a capillary. To carry on t proposed task we must consider that at trmodynamic equilibrium t liquid in t drop must be at equilibrium with its own vapor (recall Kelvin equation), and also at equilibrium with t solid, both under t drop and around it, and t vapor must be at equilibrium with t solid substrate. T latter demands t existence of an adsorbed liquid film (of thickness ) on t solid surface as illustrated in fig. 5. A similar figure can be provided for a capillary with, however, opposite curvature [2, 4, 5].

Advances in Fluid Mechanics VIII 349 Let us assume that we can maintain t oversaturated vapor condition over t solid substrate long enough until trmodynamic equilibrium is reacd. Tn t liquid molecules in t vapor are in equilibrium with t liquid molecules in t drop and with those in t adsorbed film on t solid surface. With eitr partial or complete wetting conditions, t presence of an adsorbed liquid layer on t solid substrate demands consideration of a new surface tension sv of lower values than t surface tension of t bare solid surface, sv0. In view of t above, t expression for t excess free energy of t overall liquid layer over t solid substrate contains t free energies due to t sprical part of t drop and to t adsorbed liquid film with t transition zone: f Pe h dh sl sv0. (4) As earlier noted, Pe is a given Laplace excess capillary pressure and h is t Derjaguin pressure for a given temperature. We can rewrite t expression (4) as f sv sv0, by introducing sv Pe h dh sl, (5) as t interfacial tension (which is an excess free energy) of t solid substrate covered with t adsorbed liquid film. Tn, inserting sv ratr than sv0 in Young equation (1) we get cos e sv sl, (6) thus defining a contact angle at trmodynamic equilibrium. Consequently, cos e Pe h dh 1 1 h dh, (7) if recalling t smallness of H we neglect t term Pe. T latter equation is t Derjaguin-Frumkin equation for t (trmodynamic) equilibrium contact angle. It provides an operational definition relative to eqn. (1) which is conceptually unacceptable. T argument is formally valid for both a drop and a capillary. As 1 cos e 1, in t case of partial wetting (water and aqueous solutions over most solid substrates) t integral in t right hand side of eqn. (7) must be negative h dh 0, and nce for partial wetting S S (actually, this condition defines partial wetting). Note that tre are an infinite number of equilibrium contact angles for a drop on a solid substrate (each of tm corresponds to a different value of t oversaturated vapor pressure in t

350 Advances in Fluid Mechanics VIII Figure 6: Derjaguin pressures, h, interpreted as isotrms corresponding to t cases of complete wetting (1) and partial wetting (2). T left figure is for drops and t right figure is for capillaries. In t latter t liquid film thicknesses,, h and h, are thin flat films that for a given non-vanishing value of Pe could coexist at equilibrium with t central part of t meniscus in a capillary wn Pe max. However, only t first two are trmodynamically stable as d h 0 for eitr or h. Hence h is unstable. For a drop dh as Pe is negative t left figure shows that tre is only and h, stable and unstable thicknesses, respectively. T values h0 and h1 denote t zeros of h with no otr meaning. For hmin and hmax see t main text. surrounding air) but tre is a single, uniquely defined equilibrium contact angle of t liquid meniscus in a partially wetted solid capillary for any undersaturated vapor pressure. For completeness let us now focus on t capillary case. Tre are three solutions of equation Pe 0 (fig. 6, right part, curve 2). Only those d 0 are trmodynamically stable. T thinnest film dh belongs to t so-called -branch of t Derjaguin isotrm and it is absolutely stable. T thickest film, h, is a metastable solution, with a barrier hmax, and satisfying belongs to t so-called -branch. Tn h corresponds to an unstable equilibrium flat film. On t surface of thin capillaries only equilibrium -films are observed if Pe max wre max is t stability limit of -films (fig. 6). This is t case for narrow enough capillaries. However, if Pe max, t formation of metastable -films is possible, and it has been observed experimentally.

Advances in Fluid Mechanics VIII 351 T -film is like an adsorbing film while t -film is called a wetting film as it can be obtained by draining a thicker liquid layer. Generally, t free energy of -films is lower than t free energy of -films, ruling out spontaneous transitions. Typical -films are t equilibrium films occurring wn a hydrophilic surface is wetted with water. On t otr hand, if more of t same liquid is added once t -film is formed, disregarding gravity, it will not spread but ratr it will form drops over t -film (t fact that a liquid does not spread on itself is called autophobicity). Note that or -films may not exhibit interference colors wn irradiated with white light and wn ty appear black ty are usually called black films. In fact ty do have destructive interference as two light rays/waves one reflected from t upper surface of t film and t otr from t opposite surface, travel about t same path length but t second comes from t liquid film phase-shifted half a wavelength. Tse black liquid films were studied by Newton and it is customary to call t -black film, a Newton black film, wreas t -black film is called common black film. Aqueous soap solutions easily permit visualizing this pnomenon. Indeed, t transition from a thick film to a black film is easily demonstrated by allowing a vertical soap film supported on a horizontal frame to drain. Initially t whole film exhibits interference colors wn illuminated, and tn a dark boundary, separated from t colored area by a silvery band appears at t top and moves downward. This tn divides into two areas occupied by black films. Note that a bi-layer of soap molecules is just a few nano-meters thick. 3 Non-flat equilibrium liquid films on flat surfaces: general trmodynamic approach In a thick liquid film if its surface is curved or uneven, t Derjaguin and t Laplace (capillary) pressures acting togetr permit stable non-flat equilibrium shapes. Dictated by t S-shaped Derjaguin pressure, in t case of partial wetting, micro-drops, micro-depressions and spatially periodic (wavy) films could exist on homogeneous, flat solid substrates [11]. Let us insist in that eqn. (7) also shows that t equilibrium contact angle is completely determined by t shape of t disjoining pressure in t case of molecularly smooth solid substrates. Tre is no doubt that t surface roughness influences t apparent value of t contact angle. However, t roughness cannot result in a transition from non-wetting to partial wetting, or from partial wetting to complete wetting. T possible existence of non-flat equilibrium liquid films points to a scenario of rupture of thick metastable -films and tir transition to absolutely stable -films. Neglecting t action of gravity and, for simplicity, restricting again consideration to a two-dimensional capillary, t excess free energy of t system can be written as

352 Advances in Fluid Mechanics VIII 1 h 2 1 Pe h h dh h dx, h (8) wre t coordinate x is taken along t flat solid surface of t wetted capillary. Such excess free energy is t excess over t energy of t same flat surface covered by a trmodynamically stable equilibrium -film. Although such reference state results in an additive constant in expression (8), it is needed to study t liquid profile in a vicinity of t apparent three-phase contact line. Any liquid profile, h x, which gives t minimum value to t excess free energy,, describes an equilibrium configuration. For a minimum we must have: (i) 0 ; (ii) if f 1 h 2 1 Pe h h dh h, h 2 f 0 ; (iii) as t solution is a curve, h x, t problem is in functional h 2 space and a secondary variational principle is needed [12]. This leads to a Jacobi u d d h u 0 whose solution cannot vanish wn t equation dx 1 h 2 3 2 dh tn unknown, x, is inside t region under consideration, except at t boundaries of t integration; and (iv) at t three-phase contact line, a smooth transition must occur from a non-flat liquid profile to a flat liquid film (transversality condition): f 0. T actual three-phase contact line, virtually located at t f h h x border of in fig. 5, corresponds to t intersection of t evolving liquid profile with t equilibrium liquid film of thickness,, and not to t h 2 intersection with t solid substrate. Tn 1 h 2 1 0, or 1 h 2 x 1 1. T latter and nce t mentioned smooth transition occurs 2 1 h x wn h x 0, or h 0, nce h 0, x, wre x means that it tends to t end of t transition zone. T first condition (i) yields h 1 h 2 32 h Pe, (9) which is t Laplace-Derjaguin equation. Indeed, t first term in t left hand side of eqn. (9) corresponds to t Laplace pressure and t second term to t Derjaguin pressure. Note that for a flat film t first term of t left hand side vaniss and nce h Pe. If t thickness of t liquid is beyond t range of t surface forces action (above hundred nm) tn eqn. (9) describes eitr a

Advances in Fluid Mechanics VIII 353 flat liquid surface Pe 0, or a sprical drop profile Pe 0, or a curved meniscus profile Pe 0. T second condition (ii) is always satisfied. h d d h dh Direct differentiation of eqn. (9) yields 0. dx 1 h 2 3 2 dh dx Accordingly, t solution of t Jacobi equation is u const h. Hence, tre is stability if h x0 0 with x0 different from t boundary points; otrwise t system is unstable. 1 C Pe h h dh h, wre C is 1 h 2 an integration constant. Needless to say t right hand side must be positive. In t case of a meniscus in a flat capillary t integration constant, C, is determined from t condition that at t capillary center: h H, which Integrating equation (9) we have: gives C Pe H, wre re H is t half-width of t capillary. In t case of equilibrium drops t constant should be chosen using t condition that at t drop apex, h H : h H 0, which results in C Pe H. An alternative way of getting t integration constant C is using t transversality condition. A profile of t transition zone between a non-flat meniscus in a flat capillary and a thin -film aad of it in t case of partial wetting is depicted in fig. 7 (to be compared with fig. 6, right part, curve 2, for a flat meniscus). In t transition zone both Laplace and Derjaguin pressures act simultaneously. It appears that t liquid profile is not of uniform concavity from t vapor side, but changes curvature inside t transition zone. In t transition zone (fig. 7, right part) all thicknesses appear from t very thick, outside t range of t Derjaguin pressure action, to t thin -films. Figure 7: Partial wetting of a capillary. Enlarged view of t liquid profile inside t transition zone in thick capillaries. S-shaped disjoining pressure isotrm (left side) and t liquid profile in t transition zone (right side). Here corresponds to t stable thin -film, and h to thick t -films as defined by h Pe 0.

354 Advances in Fluid Mechanics VIII Tn t Jacobi equation condition (iii) above must be used ratr than t stability condition (4) used in t case of flat films. Indeed, t transition zone is stable if h x does not vanish inside it. Such peculiar shape of t transition zone, wre both t capillary pressure and t disjoining pressure are equally important, suggests looking for stable solutions otr than flat liquid films of constant thickness. 4 Non-flat equilibrium films. Microdrops: t case Pe 0 (undersaturation conditions) Let us consider t possible existence of microdrops, i.e., drops with an apex ight in t range of influence of t Derjaguin pressure [4, 11]. In this case t drop does not have a sprical part even at its apex, because its shape is everywre distorted by t action of t surface forces. T transversality condition at h gives h 0, and nce t drop edge approacs t equilibrium film of thickness on t solid surface at vanishing microscopic contact angle. This permits to fix t earlier mentioned integration constant C Pe h dh. Tn, t microdrop profile (fig. 8; to be compared with fig. 6, left part, curve 2) follows from imposing h Figure 8: 2 L h 2 1, Microdrops. Left figure, upper part: S-shaped Derjaguin (pressure) isotrm; bottom part: L h (see main text). T value L h 0 determines t drop apex, h. Right figure: profile of an equilibrium microdrop. T apex of t microdrop is in t range of t Derjaguin pressure action, and nce t drop is not sprical even at t drop apex.

Advances in Fluid Mechanics VIII with h L h Pe h h dh 0 L h. and T first 355 equality corresponds to vanishing derivative, h, while t second one corresponds to t infinite derivative. Let h define t apex of t microdrop. T top part of t left fig. 8, shows t S-shaped dependence of t Derjaguin isotrm, h, while t bottom part shows t curve, L h, which is maximum or minimum wn Pe h. At t apex of t drop, wn h h, t first derivative must be zero, h h 0 and nce L h 0. Take t origin placed at t center of t microdrop which corresponds to its formation at undersaturation, Pe 0. In t range 0 Pe max t equation Pe h has three roots (fig. 8, left part); t smallest corresponds to t equilibrium flat -film of thickness,. For t existence of microdrops we must have: h 0 at h h and h 0 as h, h nce h h h. Tn Pe h h dh, whose solution, h, must satisfy t inequality hu h h as illustrated in t bottom part of fig. 8, left part. As t left hand side is positive, so must be its right hand site. Hence, a sufficient condition for t existence of equilibrium microdrops is that S S. This implies that t S-shaped Derjaguin (pressure) isotrm leads to an equilibrium contact angle of zero value. Consequently, microdrops at trmodynamic equilibrium are not possible if tre is partial wetting, wn S S or with complete wetting, wn t Derjaguin pressure decreases in a monotonic way, as for example with h A h 3. 5 Concluding remarks Summarizing, we have shown that t trmodynamics of drops and liquid films with curved menisci involves, for a given temperature, both t Laplace (capillary) pressure and t Derjaguin (surface forces) pressure. Once t Derjaguin (pressure) isotrm is known we can define, operationally and unambiguously, what do we mean by complete wetting, partial wetting and nonwetting. For t partial wetting case, t S-shaped Derjaguin isotrm, determines true trmodynamic equilibrium contact angles for drops of capillaries. Such S-shaped form is also responsible of t possible existence of microdrops, and even for micro-depressions and equilibrium periodic films on flat solid substrates. Acknowledgement This research was supported by t EU under Grant MULTIFLOW, FP7-ITN2008-214919.

356 Advances in Fluid Mechanics VIII References [1] Young, T., An essay on t cosion of fluids. Phil. Trans. Royal Soc. London, 95, pp. 65-87, 1805. [2] Adamson, A.W. & Gast, A.P., Physical Cmistry of Surfaces, 6th ed., Willey: New York, 1997 (and references trein). [3] de Gennes, P.G., Brochard-Wyart, F. & Quéré, D., Capillarity and Wetting Pnomena. Drops, Bubbles, Pearls, Waves, Springer-Verlag: New York, 2004. [4] Starov, V.M., Velarde, M.G. & Radke, C.J., Wetting and Spreading Dynamics, Taylor & Francis /CRC: New York, 2007. [5] Starov, V.M. & Velarde, M.G., Surface forces and wetting pnomena. J. Phys. Cond. Matter, 21, pp. 464121-1-11, 2009. [6] Starov, V.M. & Sefiane, K., On evaporation rate and interfacial temperature of volatile sessile drops. Colloids & Surfaces A. Physicocm. Engng. Aspects, 333, pp. 170-174, 2009. [7] Ajaev, V.S. Gambaryan-Roisman, T. & Stephan, P., Static and dynamic contact angles of evaporating liquids on ated surfaces. J. Colloid Interface Sci., 342, pp. 550-558, 2010. [8] Derjaguin, B.V., Churaev, N.V. & Muller, V.M., Surface Forces, Consultants Bureau-Plenum Press: New York, 1987 (and references trein). [9] Verwey, E.J.W. & Overbeek, J.Th.G., Tory of t Stability of Lyophobic Colloids, Elsevier: Amsterdam, 1948. [10] Israelachvili, J.N., Intermolecular and Surface Forces, 2nd ed., Academic Press: London, 1991. [11] Starov, V.M., Nonflat equilibrium liquid shapes on flat surfaces. J. Colloid Interface Sci., 269, pp. 432-441, 2004. [12] Sagan, A., Introduction to t Calculus of Variations, Dover reprint: New York, chap. 7, 1992.