# Wetting on a spherical wall: influence of liquid-gas interfacial properties

## Abstract

We study the equilibrium of a liquid film on an attractive spherical substrate for an intermolecular interaction model exhibiting both fluid-fluid and fluid-wall long-range forces. We first reexamine the wetting properties of the model in the zero-curvature limit, i.e., for a planar wall, using an effective interfacial Hamiltonian approach in the framework of the well known sharp-kink approximation (SKA). We obtain very good agreement with a mean-field density functional theory (DFT), fully justifying the use of SKA in this limit. We then turn our attention to substrates of finite curvature and appropriately modify the so-called soft-interface approximation (SIA) originally formulated by Napiórkowski and Dietrich [Phys. Rev. B 34, 6469 (1986)] for critical wetting on a planar wall. A detailed asymptotic analysis of SIA confirms the SKA functional form for the film growth. However, it turns out that the agreement between SKA and our DFT is only qualitative. We then show that the quantitative discrepancy between the two is due to the overestimation of the liquid-gas surface tension within SKA. On the other hand, by relaxing the assumption of a sharp interface, with, e.g., a simple ÒsmoothingÓ of the density profile there, markedly improves the predictive capability of the theory, making it quantitative and showing that the liquid-gas surface tension plays a crucial role when describing wetting on a curved substrate. In addition, we show that in contrast to SKA, SIA predicts the expected mean-field critical exponent of the liquid-gas surface tension.

###### pacs:

05.20.Jj, 71.15.Mb, 68.08.Bc, 05.70.Np^{1}

## I Introduction

The behavior of fluids in confined geometries, in particular in the
vicinity of solid substrates, and associated wetting phenomena are
of paramount significance in numerous technological applications and
natural phenomena. Wetting is also central in several fields, from
engineering and materials science to chemistry and biology. As a
consequence, it has received considerable attention, both
experimentally and theoretically for several decades. Detailed and
comprehensive reviews are given in
Refs. Dietrich (1988); Bonn *et al.* (2009); Schick (1990); Sullivan and da Gama (1986).

Once a substrate (e.g. a solid wall) is brought into contact with a gas, the substrate-fluid attractive forces cause adsorption of some of the fluid molecules on the substrate surface, such that at least a microscopically thin liquid film forms on the surface. The interplay between the fluid-fluid interaction (cohesion) and the fluid-wall interaction (adhesion) then determines a particular wetting state of the system. This state can be quantified by the contact angle at which the liquid-gas interface meets the substrate. If the contact angle is non-zero, i.e. a spherical cap of the liquid is formed on the substrate, the surface is called partially wet. In the regime of partial wetting, the cap is surrounded by a thin layer of adsorbed fluid which is of molecular dimension. Upon approaching the critical temperature, the contact angle continuously decreases and eventually vanishes. Beyond this wetting temperature one speaks of complete wetting and the film thickness becomes of macroscopic dimension. The transition between the two regimes can be qualitatively distinguished by the rate of the disappearance of the contact angle, which is discontinuous in the case of a first-order transition or continuous for critical wetting.

From a theoretical point of view, it is much more convenient to take the adsorbed film thickness, , rather than the contact angle, as an order parameter for wetting transitions and related phenomena. An interfacial Hamiltonian is then minimized with respect to as is typically the case with the (mesoscopic) Landau-type field theories and (microscopic) density functional theory (DFT) – where can be easily determined from the Gibbs adsorption, a direct output of DFT.

In this study, we examine the wetting properties of a simple fluid
in contact with a spherical attractive wall by using an
intermolecular interaction model with fluid-fluid and fluid-wall
long-range forces. The curved geometry of the system prohibits a
macroscopic growth of the adsorbed layer (and thus complete
wetting), since the free energy contribution due to the liquid-gas
interface increases with the film thickness , and thus for a
given radius of a spherical substrate there must be a maximum finite
value of
Dietrich (1988); Bieker and Dietrich (1998); Hołyst and Poniewierski (1987).
For the mesoscopic approaches, the radius of the wall , is a new
field variable that introduces one additional -dependent term
to the effective interface Hamiltonian of the system, compared to
the planar geometry, where the only -dependent term is the
binding potential between the wall-liquid and liquid-gas interfaces.
Furthermore, for a fluid model exhibiting a gas-liquid phase
transition, such as ours, it has been found that two regimes of the
interfacial behavior should be distinguished: , in which
case the surface tension can be expanded in integer powers of
and , where the interfacial quantities exhibit a
non-analytic behavior Evans *et al.* (2004). Moreover, for an
intermolecular interaction model with fluid-fluid long-range
interactions, there is an additional contribution to
the surface tension in the regime Stewart and Evans (2005a). These
striking observations actually challenge all curvature expansion
approaches. In addition, a certain equivalence between a system of a
saturated fluid on a spherical wall and a system of an unsaturated
fluid on a planar wall above the wetting temperature has been
found Bieker and Dietrich (1998); Stewart and Evans (2005a).
Somewhat surprisingly, DFT computations confirmed this
correspondence at the level of the density profiles down to
unexpectedly small radii of the wall Stewart and Evans (2005a).

Most of these conjectures follow from the so-called sharp-kink approximation (SKA) Dietrich (1988), based on a simple piece-wise constant approximation of a one-body density distribution of the fluid, i.e. a coarse-grained approach providing a link between mesoscopic Hamiltonian theories and microscopic DFT. The simple mathematical form of SKA has motivated many theoretical investigations of wetting phenomena as it makes them analytically tractable. At the same time SKA appears to capture much of the underlying fundamental physics for planar substrates (often in conjugation with exact statistical mechanical sum rules Henderson (1992)).

However, as we show in this work, SKA is only qualitative for spherical substrates, even though the functional form of the film growth can still be successfully inferred from the theory Stewart and Evans (2005a). We attribute this to the particular approximation of the liquid-gas interface adapted by SKA. In particular, since the -dependent contribution to the interface Hamiltonian due to the curvature is proportional to the liquid-gas surface tension, the latter plays an important role compared to the planar geometry.

More specifically, the curved geometry induces a Laplace pressure whose value depends on both film thickness and the surface tension and so the two quantities are now coupled, in contrast with the planar geometry where a parallel shift of the liquid-gas dividing surface does not influence the surface contribution to the free energy of the system. We further employ an alternative coarse-grained approach, a modification of the one originally proposed by Napiórkowski and Dietrich Napiórkowski and Dietrich (1986) for the planar geometry, which replaces the jump in the density profile at the liquid-gas interface of SKA by a continuous function restricted by several reasonable constraints. We show that in this “soft-interface approximation” (SIA) the leading curvature correction to the liquid-gas surface tension is , rather than , in line with the Tolman theory. Once a particular approximation for the liquid-gas interface is taken, the corresponding Tolman length can be easily determined. Apart from this, we find that the finite width of the liquid-gas interface significantly improves the prediction of the corresponding surface tension when compared with the microscopic DFT computations, which consequently markedly improves the estimation of the film thickness in a spherical geometry.

In Sec. II we describe our microscopic model and the corresponding DFT formalism. In Sec. III we present results of wetting phenomena on a planar wall obtained from our DFT based on a continuation scheme that allows us to trace metastable and unstable solutions. The results are compared with the analytical prediction as given by a minimization of the interface Hamiltonian based on SKA. We also make a connection between the two approaches by introducing the microscopic model into the interfacial Hamiltonian. In Sec. IV we turn our attention to the main part of our study, a thin liquid film on a spherical wall. We show that the SKA does not perform as well as might be desired, in particular, it does not account for a quantitative description of the liquid-gas surface tension which plays a significant role when the substrate geometry is curved. We then introduce SIA and present an asymptotic analysis with the new approach. Comparison with DFT computations reveals a substantial improvement of the resulting interface Hamiltonian even for very simple approximations of the density distribution at the liquid-vapour interface, indicating the significance of a non-zero width of the interface. We conclude in Sec. V with a summary of our results and discussion. Appendix A describes the continuation method we developed for the numerical solution of DFT. In Appendix B we show derivations of the surface tension and the binding potential for both a planar and a spherical geometry within SKA. Finally, Appendix C shows derivations of the above quantities, including Tolman’s length, using SIA.

## Ii Dft

### ii.1 General formalism

DFT is based on Mermin’s proof Mermin (1965) that the free energy of an inhomogeneous system at equilibrium can be expressed as a functional of an ensemble averaged one-body density, (see e.g. Ref. Evans (1979) for more details). Thus, the free-energy functional contains all the equilibrium physics of the system under consideration. Clearly, for a 3D fluid model one has to resort to an approximative functional. Here we adopt a simple but rather well established local density approximation

(1) |

where is the free energy per particle of the hard-sphere fluid (accurately described by the Carnahan-Starling equation of state), including the ideal gas contribution. The contribution due to the long-range van der Waals forces is included in the mean-field manner. To be specific, we consider a full Lennard-Jones 12-6 (LJ) potential to model the fluid-fluid attraction according to the Barker-Henderson perturbative scheme

(2) |

where for the sake of simplicity the Lennard-Jones parameter is taken equal to the hard-sphere diameter.

The free-energy functional, , describes the intrinsic properties of a given fluid. The total free energy including also a contribution of the external field is related to the grand potential functional through the Legendre transform

(3) |

where is the chemical potential and is the external field due to the presence of a wall ,

(4) |

consisting of the atoms interacting with the fluid particles via the Lennard-Jones potential, , with the parameters and , and uniformly distributed throughout the wall with a density :

(5) |

Applying the variational principle to the grand potential functional, Eq. (3), we attain the Euler-Lagrange equation:

(6) |

where denotes the first term in the right-hand-side of (1). In general, the solution of (6) comprises all extremes of the grand potential as given by (3) and not just the global minimum corresponding to the equilibrium state. Here we develop a pseudo arc-length continuation scheme for the numerical computation of (6) that enables us to capture both locally stable and unstable solutions and thus to construct the entire bifurcation diagrams for the isotherms (details of the scheme are given in Appendix A).

The excess part of the grand potential functional (3) over the bulk may be expressed in the form

(7) |

where is the density of the bulk phase and

(8) |

is the negative pressure, or grand potential per unit volume, of a system with uniform density and . In particular, the equilibrium value of the excess grand potential (7) per unit area of a two-phase system of liquid and vapour in the absence of an external field, yields the surface tension between the coexisting phases, . The prediction of as given by minimization of (7) agrees fairly well with both computations and experimental data as shown in Fig. 1.

### ii.2 Translational symmetry: planar wall

If the general formalism outlined above is applied on a particular external field attaining a certain symmetry, it will adopt a significantly simpler form. In the next subsection we will formulate the basic equations resulting from the equilibrium conditions obtained from the minimization of (7), for a spherical model of the external field, i.e. a system with rotational symmetry. But prior to that, it is instructive to discuss the zero-curvature limit of the above model, corresponding to an adsorbed LJ fluid on a planar wall, a system with translational symmetry.

For a planar substrate in Cartesian coordinates, the density profile is only a function of , so that the Euler-Lagrange equation reads

(9) | ||||

where is the chemical potential of the hard-sphere system. A fluid particle at a distance from the wall experiences the wall potential:

(10) |

in Eq. (9) is the surface potential exerted by the fluid particles uniformly distributed (with a unit density) over the - plane at distance :

(11) | ||||

In the framework of DFT, the natural order parameter for wetting transitions is the Gibbs adsorption per unit area:

(12) |

### ii.3 Rotational symmetry: spherical wall

If the external field is induced by a spherical wall, , the variational principle yields

(13) | ||||

where is the surface interaction potential per unit density generated by fluid particles uniformly distributed on the surface of the sphere centered at the origin at distance ,

(14) | ||||

(see also Appendix B.1). The wall potential in Eq. (4) for the spherical wall is:

(15) |

Replacing the distance from the origin by the radial distance from the wall , one can easily see that the external potential (15) reduces to the planar wall potential (10), for . Analogously to the planar case, we define the adsorption as the excess number of particles of the system with respect to the surface of the wall:

(16) |

## Iii Wetting on a planar substrate

In this section we make a comparison between the numerical solution
of DFT and the prediction given by the effective interfacial
Hamiltonian according to SKA for the first-order wetting transition
on the planar substrate. We consider a planar semi-infinite wall
interacting with the fluid according to
(10) with the typical parameters
and that correspond to the class of intermediate-substrate
systems Pandit *et al.* (1982) for which prewetting phase
transitions can be observed. We note that wetting on planar and
spherical walls is a multiparametric problem and hence a full
parametric study of the global phase diagram is a difficult task,
beyond the scope of this paper.

### iii.1 Numerical DFT results of wetting on a planar wall

Figure 2 depicts the surface-phase diagram of the considered model in the plane, where is the departure of the chemical potential from its saturation value. The first-order wetting transition takes place at wetting temperature , well bellow the critical temperature of the bulk fluid for our model. The prewetting line connects the saturation line at the wetting temperature and terminates at the prewetting critical point, . The slope of the prewetting line is governed by a Clapeyron-type equation Hauge and Schick (1983), which, in particular, states that the prewetting line approaches the saturation line tangentially at with

(17) |

in line with our numerical computations. Schick and Taborek Schick and Taborek (1992) later showed that the prewetting line scales as . In Ref. Bonn and Ross (2001), this power law was confirmed experimentally, such that

(18) |

with Bonn and Ross (2001). A fit of our DFT results with (18) leads to a coefficient , in a reasonable agreement with the experimental data – see Fig. 2.

Figure 3 depicts the adsorption isotherm in terms of the thickness of the adsorbed liquid film as a function of for the temperature and in the interval between the wetting temperature and the prewetting critical temperature . can be associated with the Gibbs adsorption through

(19) |

for both finite and infinite , where is the difference between the liquid and gas densities at saturation.

The isotherm exhibits a van der Waals loop with two turning points depicted as B and C demarcating the unstable branch. Points A and D indicate the equilibrium between thin and thick layers, corresponding to a point on the prewetting line in Fig. 2. The location of the equilibrium points can be obtained from a Maxwell construction. Details of the numerical scheme we developed for tracing the adsorption isotherms are given in Appendix A.

### iii.2 SKA for a planar wall

For the sake of clarity and completeness we briefly review the main features of SKA for a planar geometry (details are given in Ref. Dietrich (1988)).

Let us consider a liquid film of a thickness adsorbed on a planar wall. According to the SKA the density distribution is approximated by a piecewise constant function

(20) |

where is the density of the gas reservoir and is the density of the metastable liquid at the same thermodynamic conditions stabilized by the presence of the planar wall, Eq. (10) and . The off-coexistence of the two phases induces the pressure difference

(21) |

where is the pressure of the metastable liquid and is the pressure of the gas reservoir, and where we assume that is small.

The excess grand potential per unit area of the system then can be expressed in terms of macroscopic quantities as a function of

(22) | |||

where and are the SKA to the wall-liquid and the liquid-gas surface tensions, respectively, and is the effective potential between the two interfaces (binding potential). In the following, we will suppress the explicit -dependence of these quantities.

The link with the microscopic theory can be made, if the contributions in the right-hand-side of Eq. (22) are expressed in terms of our molecular model, which, when summed up, give the excess grand potential (7) where we have substituted the ansatz (20):

(23) | ||||

(24) | ||||

(25) | ||||

where we considered the distinguished limit . is the Hamaker constant given by:

(26) |

We note that the Hamaker constant is implicitly temperature dependent and that the attractive contribution of the potential of the wall enables the Hamaker constant to change its sign. Hence, in contrast with the adsorption on a hard wall, where the Hamaker constant is always negative, there may be a temperature below which its sign is positive (large ) and negative above. Clearly, complete wetting is only possible for .

Making use of only the leading-order term in (25) the minimization of (22) with respect to gives:

(27) |

Hence, at this level of approximation the equilibrium thickness of the liquid film is:

(28) |

When substituted into (22), the wall-gas surface tension to leading order reads:

(29) |

Equation (28) can be confirmed by a comparison against the numerical DFT, see Fig. 4. We observe that the prediction of SKA becomes reliable for corresponding to a somewhat surprisingly small value of the liquid film, . Beyond this value, the coarse-grained approach looses its validity and also the prewetting transition is approached, both of which cause the curve in Fig. 4 to bend (see also Fig. 3). It is worth noting that the only term in (22) having an -dependence and thus governing the wetting behavior, is the term related to the undersaturation pressure and the binding potential, . Clearly, does not come into play in the planar case since the translation of the liquid-gas interface along the axis does not change the free energy of the system. The situation becomes qualitatively different if the substrate is curved. Nevertheless, at this stage we conclude in line with earlier studies, that SKA provides a fully satisfactory approach to the first-order wetting transition on a planar wall.

## Iv Wetting on a curved substrate

### iv.1 SKA for the spherical wall

For the spherical geometry, SKA adopts the following form:

(30) |

The corresponding excess grand potential now reads

(31) |

where . Within this approximation, the liquid-vapour surface tension becomes (see also Appendix B)

(32) |

and an analogous expansion holds for .
The correction to is due to the decay of our
model. We note that short-range potentials lead to different
curvature dependence of the surface tension, a point that has been
discussed in detail in Refs.
Stewart and Evans (2005a); Evans *et al.* (2004); Parry *et al.* (2006). Interestingly, the
correction to the surface tension, as one would expect
from the Tolman theory Tolman (1948), is missing. It corresponds to
a vanishing Tolman length within SKA, as we will explicitly show in
the following section.
Although the
value of the Tolman length is still a subject of some controversy, it is
most likely that its value is non-zero, unless the system is
symmetric under interchange between the two coexisting
phases Fisher and Wortis (1984). This observation has been confirmed
numerically in Ref. Stewart and Evans (2005a) from a fit of DFT results
for the wall-gas surface tension in a non-drying regime for the
hard-wall substrate. Thus, the linear term was included by hand into
the expansion (32) Stewart and Evans (2005a).

Finally, the binding potential within the SKA for the spherical wall yields

(33) |

where terms have been neglected.

### iv.2 SIA for the spherical wall

As an alternative to SKA, Napiórkowski and Dietrich Napiórkowski and Dietrich (1986) proposed a modified version of the effective Hamiltonian, in which the liquid-gas interface was approximated in a less crude way by a continuous monotonic function, the SIA. Applied for the second-order wetting transition on a planar wall, SIA merely confirmed that SKA provides a reliable prediction for such a system. Formulated now for the spherical case, the density profile of the fluid takes the form:

(34) | |||

Thus, a non-zero width of the liquid-vapour interface, , is introduced as an additional parameter. The density profile in this region is not specified, but the following constraints are imposed:

(35) |

with an additional assumption of a monotonic behaviour of the function . An illustrative example of is given in Fig. 5. The corresponding excess grand potential takes the form

(36) |

taking as the Gibbs dividing surface (so that is a measure of the number of particles adsorbed at the wall). The binding potential (see also Appendix C.3) is obtained from

(37) |

where – see Appendix B.1 for the explicit form of the last expression.

The wall-liquid surface tension remains unchanged compared to that obtained from SKA, Eq. (III.2). However, the liquid-gas surface tension now reads (see Appendix C.1)

(38) |

where is the pressure at saturation.

From now on, we neglect the curvature dependence of and , as they would introduce higher-order corrections not affecting the asymptotic results at our level of approximation. This is also in line with previous studies which show that the Tolman length only depends on the density profile in the planar limit Fisher and Wortis (1984). Then (38) can be written as

(39) |

where is the Tolman length of the liquid-gas surface tension, as given by (Appendix C.2):

(40) |

The Tolman length is independent of the choice of the dividing surface. We also note that an immediate consequence of Eq. (40) is that within SKA the Tolman length vanishes.

The equilibrium film thickness then follows from setting the derivative of (36) w.r.t. equal to zero:

(41) | |||

The last two terms of (41) are of the form

(42) |

with and . Since is monotonic, i.e. does not change sign, the mean value theorem can be employed such that

(43) | ||||

for some , where we made use of . Substituting (43) into (41) and setting the resulting expression equal to zero, we obtain:

(44) |

So far, there is no approximation within SIA. Equation (44) can be simplified by appropriately estimating the values of the auxiliary parameters and . To this end, we employ a simple linear approximation to the density profile at the liquid-gas interface, taking in (43). Furthermore, we expand in powers of

(45) | ||||

(46) | ||||

where we assumed the distinguished limits . Inserting (45) and (46) into (43) yields for :

(47) |

From (44), we obtain to leading order,

(48) | ||||

(49) | ||||

Finally, substituting (48) and (49) into (44) we have to leading order:

(50) |

and hence, to leading order the equilibrium wetting film thickness is:

(51) |

We note that this asymptotic analysis can be extended beyond (51), by including terms , and . The latter occurs due to the “soft” treatment of the liquid-vapor interface and is thus not present in SKA.

In Fig. 6 we compare two adsorption isotherms () corresponding to wetting on a planar and a spherical wall (). The two curves are mutually horizontally shifted by a practically constant value, in accordance with Eq. (50). This implies that the curve for the spherical wall crosses the saturation line at a finite value of , and eventually converges to the saturation line as from the right, thus the finite curvature prevents complete wetting. The horizontal shift corresponds to the Laplace pressure contribution, , as verified by comparison with the numerical DFT, Fig. 7. All these conclusions are in line with SKA. However, the difference between SKA and SIA consists in a different treatment of , compare (68) and (81). This is quite obvious, since the softness of the interface influences the free energy required to increase the film thickness. We will discuss this point in more detail in the following section.

### iv.3 Comparison of SKA and SIA

We now examine the repercussions of the way the liquid-gas interface is treated on the prediction of wetting behaviour on a spherical surface. As already mentioned in Sec. IV.2, the linear correction in the curvature to the planar liquid-gas surface tension, ignored within SKA, is properly captured by SIA. Furthermore, the presence of the Laplace pressure suggests that the liquid-gas surface tension plays a strong part in the determination of the equilibrium film thickness. This contrasts to the case of a planar geometry, where the term associated with the liquid-gas surface tension has no impact on the equilibrium configuration.

To investigate this point in detail, we will first compare the
approximations of as obtained by the two approaches.
For this purpose, we start with SIA for a given parameterization of
the liquid-gas interface. As shown in Table 1,
we employ linear, cubic and hyperbolic tangent auxiliary functions,
where the latter violates condition (35)
negligibly. The particular parameters are determined by minimization
of a given function with respect to the corresponding parameters. In
Table 1 we display the planar liquid-gas
surface tension associated with a particular parameterization and
the Tolman length resulting from Eq. (40) for
the temperature . In all three cases the
surface tension is close to the one obtained from the numerical
solution of DFT and also the predictions of the Tolman length are in
a reasonable agreement with the most recent simulation
results Sampayo *et al.* (2010); Block *et al.* (2010); van Giessen and Blokhuis (2009), with thermodynamic results Bartell (2001) as well as
with results from the van der Waals square gradient theory Blokhuis and Kuipers (2006).

Auxiliary function | argument | ||
---|---|---|---|

It is reasonable to assume that from the set of considered auxiliary functions, the -approximation is the most realistic one, although the numerical results as given in Table 1 suggest that it is mainly the finite width of the liquid-gas interface, rather than the approximation of the density profile at this region, that matters. To illustrate this, we show in Fig. 8 the dependence of the surface tension on the steepness parameter , determining the shape of the function. Note that the limit corresponds to the surface tension as predicted by SKA, , for . Such a value contrasts with the result of SIA, which corresponds to the minimum of the function, and yields , in much better agreement with the numerical solution of DFT, .

Asymptotic analysis of the film thickness in Eq. (50), reveals that the film thickness for large but finite remains finite even at saturation with in line with earlier studies, e.g. Refs. Bieker and Dietrich (1998); Stewart and Evans (2005a). From Eq. (50) one also recognizes a strong dependence of on the planar liquid-gas surface tension. In Fig. 9 we present the SIA and SKA predictions of the dependence on as a function of the wall radius. The comparison with the numerical DFT results reveals that for large SIA is clearly superior, reflecting a more realistic estimation of the liquid-gas surface tension. For small values of (and ) we observe a deviation between DFT and the SIA results. This indicates a limit of validity of our first-order analysis and the assumption of large film thicknesses.

The occurrence of the undersatura