Mesoscopic analysis of Gibbs’ criterion for sessile nanodroplets on trapezoidal substrates

Mesoscopic analysis of Gibbs’ criterion for sessile nanodroplets on trapezoidal substrates

Abstract

By taking into account precursor films accompanying nanodroplets on trapezoidal substrates we show that on a mesoscopic level of description one does not observe the phenomenon of liquid-gas-substrate contact line pinning at substrate edges. This phenomenon is present in a macroscopic description and leads to non-unique contact angles which can take values within a range determined by the so-called Gibbs’ criterion. Upon increasing the volume of the nanodroplet the apparent contact angle evaluated within the mesoscopic approach changes continuously between two limiting values fulfilling Gibbs’ criterion while the contact line moves smoothly across the edge of the trapezoidal substrate. The spatial extent of the range of positions of the contact line, corresponding to the variations of the contact angle between the values given by Gibbs’ criterion, is of the order of ten fluid particle diameters.

Gibbs’ criterion, contact line pinning, contact angles, nanodroplets
pacs:
68.03.Cd, 47.55.D-, 47.55.np, 68.15.+e

I Introduction

Recent progress in device miniaturization has led to an increased interest in adsorption of liquids on substrates structured topographically on the micron- and nanoscale Herminghaus, Brinkmann, and Seemann (2008); Rauscher and Dietrich (2008); Quere (2008); Seemann et al. (2005, 2011). The influence of the substrate structure on the morphology and on the location of interfaces and three-phase contact lines present in such systems is of particular interest. Already a hundred years ago Gibbs pointed out that an apex-shaped substrate can pin the solid-liquid-gas contact line Gibbs (1906). If a sessile droplet of fixed volume is placed on a planar substrate it forms a spherical cap with a contact angle given by the modified Young’s equation Swain and Lipowsky (1998)

(1)

where , , and denote the substrate-gas, substrate-liquid, and liquid-gas surface tension coefficients, respectively; is the line tension coefficient connected with the occurrence of the circular three-phase solid-liquid-gas contact line of radius . For macroscopic droplets or if the droplet is invariant in one spatial direction and forms a ridge, one has and the modified Young’s equation reduces to the original Young’s equation Rowlinson and Widom (1989); de Gennes, Brochard-Wyart, and Quere (2004). For a detailed account of the subtleties associated with the line tension see Refs. Schimmele, Napiórkowski, and Dietrich, 2007; Schimmele and Dietrich, 2009. If the substrate surface forms a sharp corner and the three-phase solid-liquid-gas contact line is located at its apex the modified Young’s equation is no longer valid. The corresponding local contact angle can take any value within the range Gibbs (1906); Oliver, Huh, and Mason (1977); Quere (2005); Rauscher and Dietrich (2008); Herminghaus, Brinkmann, and Seemann (2008); Langbein (2002)

(2)

where is the angle of the apex formed by the substrate faces (see Fig. 1). This ambiguity of the local contact angle at the apex is called Gibbs’ condition or Gibbs’ criterion Kusumaatmaja et al. (2008); Blow, Kusumaatmaja, and Yeomans (2009); Toth et al. (2011).

Figure 1: Illustration of Gibbs’ criterion for a liquid ridge which is translationally invariant in the direction normal to the plane of the cut shown here. The substrate forms an edge such that its surfaces meet at an angle . Far from the edge and in thermal equilibrium the liquid wedge forms Young’s contact angle with the local substrate surface. If the liquid wedge on the right is pushed left it maintains its contact angle until its three-phase contact line coincides with the edge of the substrate. If it is pushed further, the three-phase contact line remains pinned and the liquid wedge increases its angle with the horizontal substrate surface to a value until this angle reaches the value , which coincides with the case of the liquid wedge being far to the left from the edge of the substrate. The angle with the horizontal corresponds to Young’s angle relative to the tilted substrate surface on the left. Accordingly, if the liquid wedge is pushed further to the left it slides down the tilted substrate surface keeping its local contact angle . The same conclusions are reached if the liquid wedge on the far left recedes to the right passing the edge of the substrate. The ambiguity of the local contact angle if the three-phase contact line coincides with the edge of substrate vanishes in the limit of a planar surface . This macroscopic picture assumes that (in this cut) all interfaces are straight, geometric lines.

For liquid droplets deposited on conical Chang et al. (2010); Extrand (2005); Extrand and Moon (2008) or cylindrical Du, Michielsen, and Lee (2010); Mayama and Nonomura (2011); Wong, Huang, and Ho (2009); Toth et al. (2011) pillars the solid-liquid-gas contact line can remain pinned at the corresponding sharp edges of the substrates for a range of volumes of the droplets provided the contact angle fulfills Gibbs’ criterion. This fact is widely exploited in the so-called Vapor-Liquid-Solid growth process of nanowires made of semiconductors such as silicon (Si) or germanium (Ge) Wagner and Ellis (1964); Schmidt, Senz, and Gösele (2005); Li, Tan, and Gösele (2007); Roper et al. (2010); Ross, Tersoff, and Reuter (2005); Schwarz and Tersoff (2009); Oh et al. (2010); Algra et al. (2011); Dubrovskii et al. (2011). In this process a metal sessile droplet is deposited on a substrate exposed to the vapor phase of silicon or germanium. The semiconductor atoms are absorbed by the metal droplet which becomes supersaturated by them. The ensuing excess semiconductor material precipitates at the boundary of the metal droplet with the substrate, activating the growth of a semiconducting nanowire. Quite often gold droplets are used as a catalyst.

Three-phase contact line pinning and Gibbs’ criterion are also crucial for capillary filling in microchannels patterned by posts Kusumaatmaja et al. (2008); Blow, Kusumaatmaja, and Yeomans (2009); Mognetti and Yeomans (2010); Chibbaro et al. (2009); Mognetti and Yeomans (2009); Berthier et al. (2009) and for dewetting phenomena on geometrically corrugated substrates Ondarçuhu and Piednoir (2005). In the former case, depending on the shape and the height of the posts, and on Young’s contact angle , the liquid front can be pinned by the posts so that capillary filling of the microchannel might stop. In the dewetting case, the morphology of the emerging holes is modified by the height and the structure of the steps, also due to three-phase contact line pinning.

In addition to the surface and line tension coefficients present in Eq. (1), the mesoscopic description of sessile nanodroplets takes into account the effective interface potential acting between the substrate-liquid and the liquid-gas interface. In thermal equilibrium a nanodroplet with a contact angle less than is connected with the wetting layer of the liquid phase adsorbed at the substrate Brochard-Wyart et al. (1991); de Gennes (1985); Dietrich (1998); Bonn et al. (2009). The shape and stability of such effectively two-dimensional ridges or three-dimensional droplets have been discussed in the literature Yeh, Newman, and Radke (1999a, b); Bertozzi, Grun, and Witelski (2001); Gomba and Homsy (2009); Weijs et al. (2011); Dörfler (2011); Mechkov et al. (2007). Also the dynamics of nanodroplets was examined on substrates structured geometrically by rectangular steps Moosavi, Rauscher, and Dietrich (2006, 2009).

These kinds of mesoscopic studies for planar substrates have not yet been extended to the aforementioned apex-shaped substrate with an arbitrary angle. For such systems, here we focus on how the contact angle changes when the three-phase contact line crosses the edge of the substrate, and whether on the mesoscale the three-phase contact line remains pinned to the edge, as it is the case in the macroscopic description (see Fig. 1).

In Sec. II we describe the density functional based effective interface Hamiltonian which enables us to calculate the equilibrium shapes of the liquid-gas interface in the presence of geometrically structured substrates. In Sec. III we analyze how the contact angle of the liquid-gas interface separating the coexisting liquid and gas phases varies upon moving across an apex-shaped substrate. It turns out that on the mesoscale the three-phase contact line is not pinned to the edge of the substrate and the contact angle varies in agreement with Gibbs’ criterion. The shape of a liquid nanodroplet deposited on a trapezoidal substrate is examined in Sec. IV. Upon increasing the volume of this nanodroplet the three-phase contact line moves smoothly across the edge of the trapezoidal substrate while the apparent contact angle changes continuously between two limiting values fulfilling a modified Gibbs’ criterion. The modification stems from the fact that one has to take into account the change of the contact angle of the nanodroplet with its volume. We show that the spatial extent of the region within which the apparent contact angle changes significantly is of the order of ten fluid particle diameters and thus remains mesoscopic. We summarize and discuss our results in Sec. V.

Ii Model

In order to determine the effective interface Hamiltonian for an interface separating a liquid-like layer adsorbed on a substrate from the bulk gas phase we employ classical density functional theory (DFT). The corresponding grand canonical density functional is a function of the temperature and the chemical potential , and it is a functional of the the spherically symmetric interparticle pair potential and of the external potential encoding the influence of the substrate. The interparticle potential is split into a short-ranged repulsive part and an attractive part :

(3)

Two models of the attractive part will be discussed: a short-ranged Yukawa-type potential and a long-ranged van der Waals potential.

We adopt a simple random phase approximation for the density functional Evans (1979); Napiórkowski and Dietrich (1993); Napiórkowski (1994); Napiórkowski and Dietrich (1995):

(4)

The equilibrium number density profile minimizes . The first term on the rhs represents the free energy in the local density approximation of the reference fluid interacting via the short-ranged repulsive potential . The external potential acting on a fluid particle located at position stems from its interactions with all particles forming the substrate,

(5)

where denotes the spatial region occupied by the substrate with homogeneous number density . As an approximation we take in that spatial region where is repulsive; in the remaining part of space is determined by the attractive fluid-substrate interaction .

The thermodynamic state of the fluid is taken to be at the bulk liquid-gas coexistence line and sufficiently below the critical point. This implies that the bulk correlation length is comparable with the diameter of the fluid particle. Under these conditions the nonuniform number density profile can be described within the so-called sharp-kink approximation

(6)

where is the Heaviside function while and denote the bulk number densities of the coexisting liquid and gas phase, respectively. The local position of the liquid-gas interface is described in terms of the Monge parametrization . Thus by invoking the sharp-kink approximation we disregard the actual smooth variation of the density profile due to thermal fluctuations and due to the long range of the interactions governing the system which give rise to so-called van der Waals tails Napiórkowski and Dietrich (1989); Dietrich and Napiórkowski (1991). For a finite system the density functional in Eq. (4), evaluated at , can be systematically decomposed into a sum of bulk, surface, line, etc. contributions Bauer, Bieker, and Dietrich (2000); Getta and Dietrich (1998); Merath (2008).

For a system which is translationally invariant along, say, the -direction, the -dependent surface contribution to the density functional is the sum of two terms:

(7)

where is the system size in the invariant direction. The first term in the bracket corresponds to the free energy functional per length of a free, fluctuating liquid-gas interface:

(8)

The second term describes the effective interaction per length of the liquid-gas interface with the surface of the substrate:

(9)

The function is the surface density of the interaction functional and it is called effective interface potential. The limit is taken after the appropriate leading terms proportional to are extracted from the above expressions (see for example, c.f., Eq. (12)).

For small undulations the expression in the bracket in Eq. (7) can be approximated by its local form which is called the local effective interface Hamiltonian of the system:

(10)

Within the present approximation the surface tension coefficient of the liquid-gas interface is given by

(11)

Iii Liquid-gas interface close to an apex-shaped substrate

In this section we aim at finding the equilibrium shape of the liquid-gas interface which minimizes the effective Hamiltonian (Eq. (10)) for an apex-shaped substrate (see Fig. 2).

Figure 2: Schematic profile of a liquid-gas interface at an apex-shaped substrate with a characteristic angle . The liquid layer thickness on the left hand side of system is finite and the interface detaches from the substrate with an angle on the right hand side of the system. The shapes of the interface and of the substrate surface, and , respectively, are measured relative to the plane running through the apex horizontally. The system is translationally invariant in -direction.

The adsorption of a liquid phase at this kind of a substrate has been investigated in the context of wetting phenomena Parry, Greenall, and Romero-Enrique (2003). We shall focus on the thermodynamic states below the wetting temperature at the bulk liquid-gas phase coexistence line. We consider configurations which attain a finite width at the far left hand side of the apex and detach from the substrate with an angle on the right hand side of the apex (see Fig. 2). We aim at determining the range of accessible angles for such configurations. As a first step, we recall the results for a planar substrate Indekeu (1992); Dobbs and Indekeu (1993); Getta and Dietrich (1998); Merath (2008); Indekeu (2010); Indekeu, Koga, and Widom (2011), for which – for specific choices of the fluid-fluid and the substrate-fluid intermolecular pair potentials – one is able to carry out the whole analysis analytically.

iii.1 Planar substrate

For a planar substrate the effective interface potential does not depend explicitly on . Accordingly the effective interface Hamiltonian of the system (Eq. (10)) is given by

(12)

where on the rhs of Eq. (12) the free energy per length corresponding to the asymptotic configuration (i.e., for )

(13)

is subtracted so that is finite for . The contact angle (unknown a priori) is formed by the asymptotes in the limits and (see Fig. 3):

(14)

The parameter determines the position of the intersection of the asymptotes, which is also unknown a priori.

The equilibrium shape of the liquid-gas interface minimizing this functional fulfills the equation

(15)

which after one integration leads to

(16)

where is an integration constant. Demanding implies , , and . According to Eq. (16) the integration constant equals

(17)

For a profile diverging linearly for one obtains Dietrich (1998)

(18)
Figure 3: Schematic plot of the equilibrium liquid-gas interface which attains a constant value for and forms a contact angle with the substrate covered by the wetting film of equilibrium thickness . The parameter is the lateral position of the intersection of the asymptotes and . There is a family of solutions with the same free energy which follow from shifting the profile laterally by a constant .

For a planar liquid-gas interface corresponding to a wetting film on a flat substrate, the substrate-gas surface tension coefficient equals the equilibrium surface free energy density

(19)

where denotes the substrate-liquid surface tension. Together with Eq. (18) this renders Young’s law

(20)

Short-ranged forces

In order to find from Eq. (15) the explicit expression for the equilibrium shape of the liquid-gas interface, we choose

(21)

as a specific model for the effective interface potential, where is the film thickness divided by the bulk correlation length in the wetting phase. We take the dimensionless amplitude within the range . Within this model the effective interface potential attains its minimum at with . The amplitude is a unique function of temperature and corresponds to the transition temperature of continuous wetting at which the equilibrium film thickness diverges and the contact angle vanishes ( due to Eq. (18)). The value corresponds to . In Subsec. III.1.1 all lengths (e.g., and ) are measured in units of .

Deriving the above expression (Eq. (21)) for short-ranged intermolecular forces requires to go beyond the sharp-kink approximation (Eq. (6)), which corresponds to setting the bulk correlation length equal to zero (see Refs. Aukrust and Hauge, 1985, 1987; Dietrich, 1998). On the other hand, in the case of long-ranged intermolecular forces the sharp-kink approximation turns out to be not a severe one. Using this approximation in the latter case one obtains the exact expressions for the coefficients multiplying the two leading-order terms in the expansion of the corresponding effective interface potential in terms of powers of (see Ref. Dietrich and Napiórkowski, 1991).

For weakly varying interfaces, i.e., , one can expand the effective Hamiltonian:

(22)

The equilibrium shape of the interface minimizes the above functional and satisfies the equation

(23)

which upon integration yields

(24)

Here and in the following we omit the overbar indicating the equilibrium shape of the interface. The parameter is the first integration constant. (Note that the first term on the rhs of Eq. (24) can be negative (compare Eq. (18)); therefore the second term must be positive because the lhs is positive.) A second integration renders the general solution of Eq. (23):

(25)

where is the second integration constant which shifts the position of the liquid-gas interface in the horizontal direction (see Fig. 3). We put on note the property

(26)

which allows one to focus on the case . The derivative can be rewritten in the form

(27)

According to Eq. (25) for finite and nonzero the interface profile diverges for :

(28)

On the other hand

(29)

(which implies ), provided

(30)

Otherwise is either infinite or undetermined. For the liquid-gas equilibrium interface profile takes the form

(31)

and the contact angle fulfills the equation

(32)

Equation (31) implies (compare Eq. (18)). Thus for a given the contact angle predicted by Eq. (22) is smaller than the one predicted by the full model in Eq. (12). Accordingly, for Eq. (22) the range of angles accessible upon changing the parameter is with .


The other solution of Eq. (23) is trivial. It corresponds to and in Eq. (25).

Boundary conditions at a finite lateral position of the three-phase contact line

One way to determine the parameters and of the equilibrium liquid-gas interface profile in Eq. (25) is to fix the value of the function and of its derivative at a finite position . This implies

(33)

and

(34)

In the previous subsection we checked that only for (Eq. (30)) the interface attains a finite value for . Due to Eq. (33) this leads to the relation

(35)

which is depicted in Fig. 4.

Figure 4: Dependence of the derivative on the height of the interface in cases for which the interface attains a finite value for . Note that , , and so that . The plot corresponds to Eq. (21) with rendering . The inset shows the liquid-gas configuration for the specific choice and the height of the interface . Note that all lengths are measured in units of (see Eq. (21)).

According to Eq. (31) the allowed values of are bounded from below by , i.e., .

With and Eqs. (34) and (35) the second parameter is given by the equation

(36)

For fixed and the shape of the interface which attains a finite value for is determined uniquely. For the function is not defined in the whole range and thus physically not acceptable.

Boundary conditions at the flat asymptote

If one requires a finite value for , this implies (Eq. (29)), (Eq. (35)), (Eq. (30)), and with Eq. (34)

(37)

The value of the parameter follows from Eq. (37), but it depends on the way in which vanishes and approaches minus infinity. In contrast to the case of fixing the height of the liquid-gas interface at a finite lateral position , in the present case one obtains a whole family of solutions, which is given by Eq. (31) and parametrized by . The members of this family of solutions differ only by a constant lateral shift.

iii.2 Apex-shaped substrate

In this section we investigate the equilibrium shape of the liquid-gas interface at an apex-shaped substrate (Fig. 2). The substrate is translationally invariant in the direction perpendicular to the plane of the figure and the surface of the substrate is described by the function . We assume that the adsorbed liquid layer attains a finite width for and the liquid-gas interface detaches from the substrate with an angle on the right hand side of the apex. The value of the angle is not known a priori. If the detachment occurs far to the right of the apex, for the geometry in Fig. 2 one expects . For this shape of the substrate even for the short-ranged intermolecular pair potentials the effective interface potential cannot be obtained in an analytical form so that the equilibrium shape of the liquid-gas interface has to be determined numerically.

In view of this loss of analytic advantage we now consider long-ranged interactions as they are realistic for actual fluid systems. For the attractive parts of the fluid-fluid and substrate-fluid pair potentials we take Tasinkevych and Dietrich (2006, 2007); Hofmann et al. (2010)

(38)

where and are the amplitudes of the interactions while and are related to the molecular sizes of the fluid and substrate particles. For this model the surface tension coefficient in Eq. (11) takes the form

(39)

In the case of an apex shaped substrate and for the above interparticle potentials the effective interface potential reads:

(40)

This leads to the disjoining pressure

(41)

where

(42)

Here and in the following we use the fluid-fluid interaction parameter as the unit of length, thus setting . Upon introducing dimensionless quantities

(43)

Eq. (41) reduces to

(44)

Equilibrium shape of the liquid-gas interface

The equilibrium profile minimizes the effective Hamiltonian

(45)

of the system where the contributions from the asymptote (see, c.f., Fig. 6)

(46)

have been subtracted which renders the integral in Eq. (45) finite. The profile satisfies the Euler-Lagrange equation

(47)

Here and in the following we again omit the overbar indicating the equilibrium shape of the interface.

Equation (47) is integrated numerically. This is a second-order differential equation so that fixing the value of the function and its derivative at a certain point, say , leads to a unique solution, similarly as discussed for the flat substrate (see Sec. III.1.2). We search for solutions which attain for . There it corresponds to the same thickness of the liquid layer as for the corresponding wetting film on a planar substrate (see Fig. 2 and, c.f., Fig. 6). The solutions of Eq. (47), which satisfy this boundary condition, will be called . In order to find them we proceed as follows:

  1. fix and at certain values ;

  2. integrate Eq. (47) numerically within the range , where and are the numerically imposed limits of the system size on the left and the right hand side, respectively;

  3. compare with and with ;

  4. if the differences and are not small enough we return to step 1 with a different choice of , but the same choice of .

Typical values of are in the range of tens of whereas can be very large, e.g., . The necessity to consider only relatively small values of (as compared with ) is related to the fact that a significantly higher accuracy is needed to solve the differential equation in the region where the liquid-gas interface is close to the substrate. Due to limited numerical accuracy and due to finite system sizes we are not able to find (for a fixed value ) the value of the derivative which renders exactly the function with and . What can be achieved numerically is to find values and rendering solutions which for sufficiently large and negative follow the asymptote and differ only slightly from it in the vicinity of , as depicted in Fig. 5. These functions are called and , respectively. The values of and change continuously with so that the contact angle is bounded from below by and from above by , i.e., .

Figure 5: Schematic plots of the liquid-gas interface at an apex-shaped substrate for fixed and with (a) and (b). In the former case , and in the latter case . The dashed line indicates the asymptote (Eq. (46)). These two plots correspond to case (b) in Fig. 6.

Gibbs’ criterion

In order to access Gibbs’ criterion we analyze the dependence of the results of the procedure described in the previous subsection on the choice of the value of . As in the case of a planar substrate (Sec. III.1) there exists a minimal value of , denoted as , such that solutions of Eq. (47) exist for . The solution corresponding to is symmetric: . For solutions corresponding to we define the contact angle and the parameter , where fulfills the equation . The parameter characterizes the position at which the liquid-gas interface detaches from the substrate. This is defined as the intersection of the corresponding asymptotes (see Fig. 6).

Figure 6: Schematic shapes of liquid-gas interfaces at an apex-shaped substrate. The parameters and characterize the equilibrium liquid-gas interface and are defined in the main text. Various choices for lead to the cases , , and . For the local contact angle on the far right side approaches the contact angle on a planar substrate.

Upon increasing the contact angle increases and the parameter decreases, i.e., the three-phase contact line approaches the apex. Changing the value of enables one to plot the dependence of the contact angle on the parameter (Fig. 7). The difference between and is so small that the error bars of are not visible on the present scale.

In the case of a planar substrate the free energies corresponding to the asymptotic configurations (Eq. (13)) are the same for each interface profile. Therefore the task of finding the equilibrium configuration, i.e., the profile with the lowest free energy (Eq. (12)), is posed well. In the case of the apex-shaped substrate, for liquid-gas configurations fulfilling Eq. (47) with appropriate boundary conditions, the free energies of the corresponding asymptotic configurations, which represent different constraints, are different (Eqs. (45) and (46)). Thus, comparing free energies corresponding to different configurations amounts to compare free energies characterizing different constraints. These free energies as function of can be interpreted as the potential of the effective interaction between the three-phase contact line and the apex.

For the local contact angle tends to its limiting values from below, which are those expected from Gibbs’ criterion for this geometry (compare Eq. (2) which holds for the geometry shown in Fig. 1). For the local contact angles are, slightly, smaller then (Fig. 7). The spatial extent of the region within which the contact angle changes significantly can be chosen as the region of where . For and for parameters of the effective interface potential rendering and , this width equals and thus it is mesoscopic.

Figure 7: Dependence of the contact angle on the parameter characterizing the position of the liquid-gas interface detachment from the substrate (see Fig. 6). The horizontal dashed lines from bottom to top indicate the angles , , , and , respectively, where is the contact angle on a planar substrate and is the opening angle of the apex. The quantity measures the width of the region within which the contact angle changes between the values and and is the position above which . These data correspond to , , , , , and such that the effective interface potential renders and . All lengths are measured in units of . The break in slope of at is caused by the discontinuity in the derivative of the function (Eq.(46)) which enters the definition of the point .

Iv Sessile droplets on trapezoidal substrates

iv.1 Planar substrate

As preparatory work, in this subsection we discuss the shapes of interfaces characterizing sessile droplets on planar substrates (see Fig. 8).

Figure 8: Schematic equilibrium shape of a ridgelike liquid nanodroplet deposited on a planar substrate. The system is translationally invariant in -direction and has a finite lateral extent .

We assume that the system under consideration has a finite extent in -direction, , and is translationally invariant in the -direction. The shape of the interface is described by a function . The total volume of liquid in the system per length is fixed. denotes the size of the system in direction.

Within our mesoscopic description for weakly varying liquid-gas interfaces the corresponding effective Hamiltonian is given by (compare Eq. (22))

(48)

where the limits of the -integration reflect the finite extent of the system. The equilibrium shape of the interface minimizes the functional

(49)

where is a Lagrange multiplier, and

(50)

The equilibrium profile satisfies the differential equation

(51)

In the following we again omit the overbar denoting the equilibrium configuration. In -direction we impose Neumann and periodic boundary conditions:

(52)

with the thickness not fixed a priori. The conditions can be realized by vertical sidewalls at exhibiting a contact angle of .

Integrating Eq. (51) renders

(53)

where the integration constant is determined by the boundary conditions:

(54)

which leads to

(55)

We examine effective interface potentials with a minimum at , , and one inflection point at (see Fig. 9).

We search for solutions , such that and that there is one for which ; we denote the maximum value of the function as . As a result one obtains from Eq. (51) the relation and from Eq. (55) one has

(56)

From the structure of one infers . Thus the slope of the line connecting the points and is equal to the Lagrange multiplier . In order to fulfill the condition this line segment must be located below the graph of the effective interface potential (see Fig. 9). This implies two restrictions: one for the thickness of the liquid layer at the boundaries , and the other for the maximum value of the function .

Figure 9: Schematic plot of the effective interface potential considered here in the context of discussing the shape of small droplets (Eqs. (55) and (57)). The slope of the dotted line connecting the points and equals the Lagrange multiplier (see Eq. (56)).

Integrating Eq. (55) one obtains

(57)

On the other hand, by rearranging the limits of integration on the rhs of Eq. (57) it can be expressed as which renders . In addition, the function also fulfills Eq. (55) so that in the following we consider functions which are symmetric with respect to . The excess volume

(58)

of the adsorbed liquid is given by (see Eq. (55))

(59)

By combining Eqs. (56) – (59) with a given volume and a size one determines the quantities , , ; integrating Eq. (55) gives the shape of the equilibrium liquid-gas interface in the form (for )

(60)