# Contact angle of water on a model heterogeneous surface. A density
functional approach

K. Dąbrowska${}^{\textsf{\footnotesize{\mbox{?}}}}$, O. Pizio  $|$
${}^{\textsf{\footnotesize{\mbox{?}}}}$, S. Sokołowski  $|$
${}^{\textsf{\footnotesize{\mbox{?}}}}$

(Received May 11, 2022, in final form July 13, 2022)

###### Abstract

We use a density functional approach to calculate the contact angle of the
water model on a heterogeneous, graphite-like surface. The surface
heterogeneity results from the pre-adsorption of a layer of spherical species.
The pre-adsorbed molecules can also be a mixture of molecules of different
sizes. The presence of pre-adsorbed layer causes geometrical and energetical
heterogeneity of the surfaces. Two cases are considered. The pre-adsorbed
molecules can either behave like hard-sphere obstacles, or they can also
attract the molecules of water. In the first case, an increase of the amount
of pre-adsorbed species leads to an increase of the wetting temperature, but
this increase does not depend linearly on the amount of obstacles. In the case
of obstacles exerting attractive forces on water molecules, the curves
describing the dependence between the amount of pre-adsorbed species and the
contact angle can exhibit a maximum. In addition, we have also studied how the
pre-adsorbed species influence the local densities of gaseous and liquid
phases in contact with a modified solid surface.

Key words: density functional theory, contact angle, heterogeneous surface,
water model

###### Abstract

За допомогою методу функціоналу густини розраховано кут змочування у моделі
води на гетерогенній графітоподібній поверхні. Неоднорідність поверхні
створена шаром преадсорбованих сферичних частинок, або ж сумішшю молекул
різних розмірів. Наявність преадсорбованого шару призводить до виникнення
геометричної та енергетичної гетерогенності поверхонь. Розглянуто два випадки.
Преадсорбовані молекули можуть або поводити себе як перешкоди у вигляді
твердих сфер, або можуть також притягувати до себе молекули води. У першому
випадку збільшення кількості преадсорбованих складників призводить до
зростання температури змочування, яке, однак, не залежить лінійно від
кількості перешкод. У тому випадку, коли перешкоди притягують молекули води,
криві, що описують залежність кута змочування від кількості преадсорбованих
частинок, можуть мати максимум. Крім того, було досліджено, яким чином
преадсорбовані складники суміші впливають на локальні густини газоподібних і
рідких фаз при їх контакті з модифікованою твердою поверхнею.

Ключов слова: метод функціоналу густини, кут змочування, неоднорідна поверхня,
модель води

##  1 Introduction

The problem of the influence of surface heterogeneity of solid surfaces on
thermodynamic properties of fluid-solid systems has been the subject of
intensive research for a long time [1, 2, 3, 4, 5, 6, 7, 8]. Usual models of
heterogeneous surfaces have been based on distinguishing several kinds of
surface active sites. The sites differ by their interaction energy with fluid
molecules. The function describing the distribution of sites with the energy
is termed “the energy distribution function” and is commonly treated as a
characteristic property of a given heterogeneous solid [4].

Different models describing the systems with heterogeneous surfaces have been
considered. They can be classified according to the way in which the surface
active sites are defined and distributed over a surface. According to the so-
called patchwise model, the sites of the same kind are grouped into patches.
The patches are geometrically flat, energetically homogeneous and independent
of each other. Consequently, thermodynamic properties of a fluid in contact
with a patchy heterogeneous solid (e.g., adsorption isotherms) are computed as
weighted averages of the relevant properties for homogeneous surfaces with
statistical weights resulting from the energy distribution function [2, 4].
However, for several solid surfaces, such as silica gels, aluminum oxides, or
active carbons, the patchwise heterogeneous model is unrealistic [6] and
considering such systems, the models of random heterogeneous surfaces were
introduced [9, 10].

One of the possible statistical-thermodynamic methods of describing the random
heterogeneous systems comes from the theory of quenched-annealed systems [11,
12, 13]. Although the latter approach is mainly applied in the studies of
fluids confined by microporous solids, it can be equally well used for
modelling a rough solid surface, or capillaries filled with random matrices
[14, 15, 16]. The initial flat surface is covered with a layer of randomly
distributed particles, which remain frozen at fixed positions when the fluid
enters the system.

The models intermediate between patchwise and random surfaces are patterned
surface models. The sites of particular kinds occupy small regions of simple
geometrical shape (e.g., stripes, squares, or triangles). These geometrical
constructs are ordered into crystalline-like lattices. In contrast to
patchwise models, the processes occurring over particular structures are not
independent of each other. The thermodynamic properties of fluids in contact
with patterned surfaces were studied in several publications, see e.g. [17,
18, 19, 20, 21].

The heterogeneity of solid surfaces plays an important role in the adsorption
of fluids on solids and in chromatographic processes [22, 23]. Indeed, the
presence and specific distribution of adsorption sites on the surfaces can
decide about the effectiveness of the separation of fluid mixtures.
Furthermore, the surface heterogeneity can strongly influence the surface
phase transitions. The studies of the latter problems mainly concern patterned
and random surfaces [24, 25, 26, 27].

Wetting is an important phenomenon that is common in nature [28]. It is also
an essential issue for the design and development of novel substances with
desirable surface properties that can be applied in many areas of engineering,
chemistry, and biology. One of the most significant ingredients that appear in
several processes is water. In particular, the wetting behavior of water on
graphite-like substrates was the subject of our recent detailed study [29].

Experimental method for investigating the wetting is usually based on the
measurement of the static contact angle, $\theta$. A crossover from non-
wetting to wetting state takes place if the contact angle changes from a
nonzero value to zero [30]. The temperature at which this transition occurs is
called the wetting temperature, $T_{w}$. Other experiments rely on the
measurements of adsorption isotherms for gas densities up to the saturated
vapor density. In the case of first-order wetting transitions, characteristic
changes in the course of adsorption isotherms with temperature take place.
Namely, at temperatures below $T_{w}$, upon the bulk density approaching the
liquid-vapor coexistence, the thickness of the adsorbed film remains finite
and small, whereas at higher temperatures, it diverges to infinity [31, 32,
33]. Thus, the study of changes of adsorption isotherms allows the
determination of $T_{w}$. Note that the adsorption method for the first time
permitted to capture the wetting temperature for ${{}^{4}}{\rm He}$ on cesium
[34].

Of course, the value of the static contact angle, $\theta$, depends on the
surface heterogeneity. However, there is no general approach that would
describe $\theta$ in terms of parameters characterizing the surface
heterogeneity. Although several attempts to elucidate the effects of surface
heterogeneity on the values of $\theta$ were undertaken, but they concerned
specific surface models. For patchy heterogeneous surfaces, Cassie and Baxter
[35] proposed an expression relating the contact angle to the energy
distribution function. Another attempt was proposed by Wenzel [36] for
macroscopically rough surfaces. Both Cassie-Baxter and Wenzel approaches were
next applied and verified for numerous systems, cf. reference [37] and the
references quoted therein. The Cassie-Baxter and Wenzel equations can be used
if the patches are big enough, i.e., if their size is much larger than the
range of capillary forces. We would like to stress that the Cassie-Baxter
equation for the contact angle is familiar [38] to the so-called integral
equation for the adsorption isotherm, [4]. Recent studies, however,
concentrated on the description of patterned surfaces composed of alternately
arranged hydrophilic and hydrophobic regions [39, 40, 41, 42, 43, 44, 45].

One of the methods [46, 47] for modifying the surface properties of solids
relies on pre-adsorption of some selected species on bare surfaces, e.g.,
chain molecules (oligomers and polymers). This method has found particularly
important applications in developing novel chromatographic column packing with
the required properties [48, 49]. The grafted chains have also a significant
impact on the surface phase transitions, such as wetting, layering, and
capillary condensation in confined fluids [50, 51, 52]. As our recent study
indicated, the grafted chains are capable not only of quantitatively but also
qualititavely influencing the topology of the surface phase diagrams [53].

Theoretical methods for the description of surface phase transitions are
mainly based on the density functional approaches [54]. Density functional
theories can be also employed to the systems with surfaces modified by
chemically bonded spherical molecules. Therefore, in this work, we propose a
density functional method to study the changes of static contact angle with
the changes of the adsorbing surface amount and size of pre-adsorbed spherical
molecules.

The fluid-fluid interaction is selected to mimic the interaction between a
pair of water molecules. Water belongs to the class of associating fluids and
the formation of intermolecular hydrogen (associative) bonds should be taken
into account in the model. However, the application of contemporary force
fields for water with site-site electrostatic forces [55] would be
computationally prohibitive within the density functional approaches.
Therefore, we use the potential resulting from the statistical association
fluid theory (SAFT), [56, 57] with the parameters proposed by Clark et al.
[58]. This model is not only one of the most accurate in terms of evaluating
the liquid-vapor phase diagram, but it also correctly predicts the temperature
dependence of the surface tension [58, 59].

Water molecules are in contact with a modified surface of graphite. Similarly
to our previous work [29], the Lennard-Jones 10-4-3 function [60] is used to
describe the interaction of a water particle with a bare graphite surface. The
surface of graphite is covered with a layer of pre-adsorbed spherical
molecules. The pre-adsorbed layer can be either one- or many-component and we
consider the cases of purely repulsive and repulsive and attractive
interactions with water molecules. The values of the equilibrium contact angle
are then calculated from the density functional theory. The theory used for us
is a modification of the approach for grafted chain molecules [15, 16]. The
pre-adsorbed layer creates both geometrical and energetic heterogeneity of an
adsorbing surface. However, we should also mention here that there exist
alternative density functional approaches proposed by Aslymov and co-workers
[61, 62] and by Zhou [63], based on the development of an appropriate
expression for the “effective” fluid-solid that takes into accounts the
surface roughness. This effective potential is then used as an external
potential field in the classical density functional treatment. Our approach,
however, leads to modifications of the expressions for the system free energy.

The paper is arranged as follows. In the next section we briefly outline the
details of the model and the interaction potentials. Then, we describe the
basic points of the density functional theory and the method for the
evaluation of the values of the contact angle. Section 3 presents the results
obtained for the pre-adsorbed layer formed by one-component layer of hard
spheres (obstacles), built of a binary mixture of obstacles, and, finally, we
consider the case of pre-adsorbed particles interacting via repulsive and
attractive forces. The last section concludes the obtained theoretical data.

##  2 Model and theory

###  2.1 Interaction potentials

According to the model developed by Chapman, Gubbins, and Jackson [64, 65],
each water molecule possesses four associative sites $\Gamma=\\{A,B,C,D\\}$,
located at the vertices of tetrahedron inscribed into a spherical core. The
interaction energy between the molecules $i=1,2$ depends on the center-to-
center distance, $r_{12}=|\mathbf{r}_{12}|$ , and on orientations of both
molecules, ${\bm{\omega}}_{i}$,

$u(12)=u_{ff}(r_{12})+\sum_{\alpha\in\Gamma}\sum_{\beta\in\Gamma}u_{\alpha\beta}(\mathbf{r}_{\alpha\beta}).$
(2.1)

The site-site vectors, $\mathbf{r}_{\alpha\beta}$, are
$\mathbf{r}_{\alpha\beta}=\mathbf{r}_{12}+\mathbf{d}_{\alpha}(\bm{\omega_{1})}-\mathbf{d}_{\beta}(\bm{\omega}2)$,
where $\mathbf{d}_{\iota}(\bm{\omega}i)$ is the vector connecting the site
$\iota$ on molecule $i$ with its center (see also figure 1 of reference [64]).
Only the site-site association AC, BC, AD, and BD is allowed, and all
association energies are equal. The associative interaction between the sites
is

$u_{\alpha\beta}(\mathbf{r}_{\alpha\beta})=\left\\{\begin{array}[]{ll}-\varepsilon_{{\rm
as}},&0<|\mathbf{r}_{\alpha\beta}|\leqslant r_{c},\\\
0,&|\mathbf{r}_{\alpha\beta}|>r_{c},\end{array}\right.$ (2.2)

where $\varepsilon_{{\rm as}}$ is the depth and $r_{c}$ is the cut-off range
of the associative interaction.

The non-associative part of the pair potential, $u_{ff}(r)$, is described by a
square-well potential

$u_{ff}(r)=u_{{\rm hs},ff}(r)+u_{{\rm att},ff}(r),$ (2.3)

where $u_{{\rm hs},ff}(r)$ and $u_{{\rm att},ff}(r)$ are, respectively, the
hard-sphere (hs) and attractive (att) parts of the potential,

$u_{{\rm hs},ff}(r)=\left\\{\begin{array}[]{ll}\infty,&r<\sigma,\\\
0,&r\geqslant\sigma,\end{array}\right.$ (2.4)

and

$u_{{\rm att},ff}(r)=\left\\{\begin{array}[]{ll}0,&r<\sigma,\\\
\varepsilon,&\sigma\leqslant r<\lambda_{ff}\sigma,\\\
0,&r\geqslant\lambda_{ff}\sigma.\end{array}\right.$ (2.5)

In the above, $\sigma$, $\varepsilon$ and $\lambda_{ff}$ are the diameter, the
depth and the range of the non-associative water-water potential,
respectively.

The parameters for the W1 model of Clark et al. [58] are collected in table 1.

Table 1:  The parameters of the W1 water-water model potential from reference [58].  Model |  $\sigma$ (nm) |  $\varepsilon/k$ ($K$) | $\lambda$ |  $r_{c}$ (nm) | $\varepsilon_{{\rm as}}/k$ ($K$) | $|{\mathbf{d}}_{\iota}|/\sigma$
---|---|---|---|---|---|---
W1 | 0.303420 | 250.000 | 1.78890 | 0.210822 | 1400.00 | 0.25

Water molecules are in contact with a modified surface. The initial, non-
modified surface is assumed to be graphite-like. The adsorbing potential,
$v(z)$, exerted on a fluid particle is then given by the potential developed
by Steele [60].

$v(z)=\varepsilon_{fw}\left[\frac{2}{5}\left(\frac{\sigma_{fw}}{z}\right)^{10}-\left(\frac{\sigma_{fw}}{z}\right)^{4}\right.\left.-\frac{\sigma_{fw}^{4}}{3\Delta(z+0.61\Delta)^{3}}\right],$
(2.6)

where $\varepsilon_{fw}$ and $\sigma_{fw}$ are the energy and the size
parameters, respectively. The interlayer spacing of the graphite planes equals
$\Delta=0.335$ nm. The values of $\sigma_{fw}$ and $\varepsilon_{fw}$ follow
from Lorentz-Berthelot mixing rules. Thus, [29], $\sigma_{fw}/\sigma=1.06028$
and $\varepsilon_{fw}/\varepsilon=8.311$. Our calculations were also carried
out for $\varepsilon_{fw}/\varepsilon=7$ and 9.5. The potential of Steele was
widely used with success in the theory of adsorption of fluids on graphite
[29, 66]. A comprehensive discussion of this potential was provided by Zhao
and Johnson [67].

The inhomogeneity of the surface results from its chemical modification.
According to our model, the modification means the “sticking” (tethering) of
spherical particles of diameters $\sigma_{i}$ at the distance $\sigma_{i}/2$
from the surface. For the surface at $z=0$, the potential that leads to the
tethering of molecules reads

$\exp[-v_{i}^{(1)}(z)/kT]=\delta(z-\sigma_{i}/2),$ (2.7)

where $\delta$ denotes the Dirac function.

The amount of tethered species $i$ is controlled by the parameter $R_{ci}$
that gives the total surface density of pre-adsorbed species. (In the case of
one-component pre-adsorbed layer, we drop the subscript $i$ and use the symbol
$R_{c}$.)

Similarly to the case of non-associative water-water interaction, the
interaction of water molecules with pre-adsorbed species, $u_{fi}(r)=u_{{\rm
hs},fi}(r)+u_{{\rm att},fi}(r)$, is described by the square well potential

$u_{{\rm hs},fi}(r)=\left\\{\begin{array}[]{ll}\infty,&r<\sigma_{fi},\\\
0,&r\geqslant\sigma_{fi},\end{array}\right.$ (2.8)

and

$u_{{\rm att},fi}(r)=\left\\{\begin{array}[]{ll}0,&r<\sigma_{fi},\\\
\varepsilon_{fi},&\sigma_{fi}\leqslant r<\lambda_{fi}\sigma_{fi},\\\
0,&r\geqslant\lambda_{fi}\sigma_{fi}.\end{array}\right.$ (2.9)

The interactions between pre-adsorbed species, however, are assumed to be of
hard-sphere type with the cross size parameters equal to
$(\sigma_{i}+\sigma_{j})/2$.

###  2.2 Density functional theory

The system is studied using a version of the density functional theory (DF),
described already in detail in references [15, 16]. To avoid unnecessary
repetitions, we recall only basic equations.

The symbols $\rho(\mathbf{r})$ and $\rho_{i}(\mathbf{r})$ denote the local
density of water-like molecules and of tethered particles of the kind $i$. As
usual, we start with defining the excess surface free energy (the grand
canonical potential) as a functional of the local densities

${\Omega}=F[\rho(\mathbf{r}),\\{\rho_{i}(\mathbf{r})\\}]+\sum_{\\{i\\}}\int\mathrm{d}\mathbf{r}\rho_{i}v_{i}(z)+\int\mathrm{d}\mathbf{r}\rho(\mathbf{r})[v(z)-\mu],$
(2.10)

where $\mu$ is the chemical potential of water and
$F[\rho(\mathbf{r}),\\{\rho_{i}(\mathbf{r})\\}]$. At fixed values of
temperature and the amount of pre-adsorbed species of all kinds, $\Omega$ is a
function of $\mu$, or, alternatively, the bulk density of water, $\rho_{b}$,
$\Omega=\Omega(\rho_{b})$.

The free energy functional is considered as the sum of an ideal term, $F_{{\rm
id}}[\rho(\mathbf{r}),\\{\rho_{i}(\mathbf{r})\\}]$, the hard-sphere functional
$F_{{\rm hs}}[\rho(\mathbf{r}),\\{\rho_{i}(\mathbf{r})\\}]$, the functional
arising from attractive interparticle forces, $F_{{\rm
att}}[\rho(\mathbf{r}),\\{\rho_{i}(\mathbf{r})\\}]$ and the part due to the
formation of associative bonds, $F_{{\rm
as}}[\rho(\mathbf{r}),\\{\rho_{i}(\mathbf{r})\\}]$.

$\displaystyle F[\rho(\mathbf{r}),\\{\rho_{i}(\mathbf{r})\\}]$
$\displaystyle=F_{{\rm id}}[\rho(\mathbf{r})]F_{{\rm
hs}}[\rho(\mathbf{r}),\\{\rho_{i}(\mathbf{r})\\}]$ $\displaystyle+F_{{\rm
att}}[\rho(\mathbf{r}),\\{\rho_{i}(\mathbf{r})\\}]+F_{{\rm
as}}[\rho(\mathbf{r}),\\{\rho_{i}(\mathbf{r})\\}].$ (2.11)

The ideal term is known exactly

$F_{{\rm
id}}/kT=\int\mathrm{d}\mathbf{r}\rho(\mathbf{r})[\ln\rho(\mathbf{r})-1].$
(2.12)

The hard-sphere functional is evaluated according to the theory originally
developed by Rosenfeld and modified in [68]. It requires the knowledge of four
scalar [$n^{(I)}(\mathbf{r})$, $I=0,1,2,3$] and two vector
[$n^{(VI)}(\mathbf{r})$, $I=1,2$] averaged densities

$n^{(L)}(\mathbf{r})=\int\mathrm{d}\mathbf{r}^{\prime}\rho(\mathbf{r}^{\prime})w^{(L)}(|\mathbf{r}-\mathbf{r}^{\prime}|)+\sum_{\\{i\\}}\int\mathrm{d}\mathbf{r}^{\prime}\rho_{i}({\bm{r}^{\prime}})w^{(L)}_{i}(|\mathbf{r}-\mathbf{r}^{\prime}|),$
(2.13)

where $L=0,1,2,3,V1,V2$ and $w^{(L)}(|\mathbf{r}-\mathbf{r}^{\prime}|)$ are
the weight functions, see equations (11)–(14) of reference [68]. However, the
explicit equations for the hard-sphere free energy is given by equation (11)
of [15].

The attractive forces contribution results from the mean-field approximation.
According to the introduced model, the pre-adsorbed molecules do not interact
with each other by attractive forces. Thus,

$\displaystyle F_{{\rm att}}[\rho(\mathbf{r}),\\{\rho_{i}(\mathbf{r})\\}]$
$\displaystyle=\frac{1}{2}\int\int\mathrm{d}\mathbf{r}\mathrm{d}\mathbf{r}^{\prime}\rho(\mathbf{r}^{\prime})\rho(\mathbf{r})u_{{\rm
att},ff}(|\mathbf{r}^{\prime}-\mathbf{r}|)$
$\displaystyle+\sum_{\\{i\\}}\int\int\mathrm{d}\mathbf{r}\mathrm{d}\mathbf{r}^{\prime}\rho(\mathbf{r}^{\prime})\rho_{i}(\mathbf{r})u_{{\rm
att},fi}(|\mathbf{r}^{\prime}-\mathbf{r}|).$ (2.14)

Finally, the contribution arising from the formation of associative bonds
between water molecules follows from the application of the first-order
thermodynamic theory of Wertheim (TPT1). The TPT1 contribution to the free
energy is expressed in terms of the fraction of molecules not bonded at a
given site $\chi(\mathbf{r})$, which represents the statistical mechanical
analogue of the mass action law. The expression used by us for $F_{{\rm
as}}[\rho(\mathbf{r}),\\{\rho_{i}(\mathbf{r})\\}]$ reads [52]

$F_{{\rm
as}}[\rho]=4\int\mathrm{d}\mathbf{r}\,n_{0}(\mathbf{r})\zeta(\mathbf{r})\left\\{\ln\chi(\mathbf{r})-\frac{1}{2}\left[\chi(\mathbf{r})-1\right]\right\\}.$
(2.15)

All the details and the definition of the functions $\zeta(\mathbf{r})$ and
$\chi(\mathbf{r})$ are given in previous works [52, 69]. We omit here their
explicit expressions to avoid unnecessary repetitions.

The density profile $\rho(\mathbf{r})$ results from the Euler-Lagrange
equation,

$\delta\Omega/\delta\rho(\mathbf{r})=0.$ (2.16)

The density profiles of pre-adsorbed species, however, are fixed by the
external potential, equation (2.7), and the assumption about constancy of pre-
adsorbed particles of the kind $i$. This means that

$\int\mathrm{d}z\rho_{i}(z)=R_{ci},$ (2.17)

i.e., $\rho_{i}(z)=R_{ci}\delta(z-\sigma_{i}/2)$.

If the external field depends on the distance perpendicular to the solid
surface only, as in equation (2.6), then the local density is one-dimensional
as well, i.e. $\rho(\mathbf{r})\equiv\rho(z)$. The Euler-Lagrange equation for
the density profile can be solved using different numerical procedures, see
e.g., [70] and references therein. In our calculations, we used the classical
Picard iteration method.

###  2.3 Contact angle

The criterion for wetting the surface by a liquid is usually derived from the
classical Young’s equation that expresses the force balance at a three-phase
contact between a liquid drop ($l$), a solid surface ($s$) and a gas phase
($g$), in terms of the contact angle, $\theta$ [30, 31] as

$\gamma_{g}-\gamma_{l}=\gamma\cos\theta,$ (2.18)

where $\gamma_{\kappa}=\Delta\Omega_{\kappa}/A$ is the interfacial tension at
gas-solid ($\kappa=g$) and at liquid-solid ($\kappa=l$) interface, $A$ is the
interfacial surface area,
$\Delta\Omega{{}_{g}}=\Omega(\rho_{bg})-\Omega_{b}(\rho_{bg})$ and
$\Delta\Omega{{}_{l}}=\Omega(\rho_{bl})-\Omega_{b}(\rho_{bl})$ are the excess
grand thermodynamic potentials for the gas-solid and for the liquid-solid
interfaces calculated for the bulk densities $\rho_{bg}$ and $\rho_{bl}$
corresponding to the densities of coexisting gaseous and liquid bulk phases.
Next, $\Omega_{b}$ is the bulk grand thermodynamic potential at the bulk
liquid-vapor coexistence and $\gamma$ is the gas-liquid interfacial tension
(the surface tension).

The Young equation is an approximation to reality [71] as it neglects the
effect of the line tension at a three-phase contact. The wetting temperature,
$T_{w}$, is determined as the highest temperature at which the ratio
$(\gamma_{g}-\gamma_{l})/\gamma$ becomes zero for the first time. At
temperatures $T\geqslant T_{w}$, the liquid drop completely spreads across the
surface.

The applications of equation (2.18) requires the knowledge of the values of
the surface tension, $\gamma$. In order to evaluate them, one needs to
calculate the changes of the local density across the interface between semi-
infinite slab of a liquid and a semi-infinite slab of a gas. This is done by
removing the solid wall and setting the boundary conditions to
$\rho(z=-\infty)=\rho_{l}$ and $\rho(z=\infty)=\rho_{g}$, using the procedure
described in detail in reference [72].

Figure 1:  Bulk liquid-vapor phase diagram in the temperature-density plane
for the model W1 of water.

##  3 Results and discussion

All variables below are expressed in reduced units. The reduced temperature,
reduced distance, reduced diameter and a reduced amount of pre-adsorbed
particles, as well as a reduced local and bulk density are
$T^{*}=kT/\varepsilon$, $z^{*}=z/\sigma$, $\sigma_{i}^{*}=\sigma_{i}/\sigma$,
$R_{c}^{*}=R_{c}\sigma^{2}$ and $\rho^{*}(z)=\rho(z)\sigma^{3}$, and
$\rho_{b}^{*}=\rho_{b}\sigma^{3}$, respectively.

Evaluation of the contact angles requires the knowledge of two bulk densities
at the gas-liquid coexistence. The calculation of the bulk phase diagram was
self-consistently carried out using the bulk version of the density functional
theory [29], and the obtained phase diagram is given in figure 1. Similarly,
the values of the liquid-vapor surface tension were obtained using the density
functional method and all the details of these calculations are outlined in
[29].

We have already noted that similarly to our previous works [29, 53], the
calculations were carried out for the W1 model of Clark et al. [58]. The
parameters of this model were selected so as to reproduce the experimental
bulk liquid-gas phase diagram. Therefore, the agreement between theoretical
and experimental bulk dew and bubble densities is quite good, in general.
However, the data fitting of Clark was at temperatures lower than the bulk
critical temperature. Thus, some deviations can appear between bulk
theoretical and experimental data within the temperatures range in the
vicinity of the critical temperature. When assessing the results presented
below, we should remember that a lower accuracy of the bulk system description
for temperatures close to the critical temperature may have an impact on the
predictions of the applied theory. Note that the bulk critical temperature
resulting from our approach [29, 73], $T_{c}^{*}\approx 2.715$, is in a
reasonable agreement with experiment.

Figure 2: (Colour online) The dependencies of the static contact angle
$\theta$ on temperature and the amount of pre-adsorbed particles of the
diameter $\sigma_{1}=0.8$ (part a), $\sigma_{1}=1$ (part b), $\sigma_{1}=1.2$
(part c) and $\sigma_{1}=1.4$ (part d). The calculations are for
$\varepsilon_{fw}^{*}=8.311$.

The first series of calculations were carried out for the surface covered by a
one-component layer of hard-sphere obstacles, i.e., the interaction of a pre-
adsorbed molecule with water molecules was of hard-sphere type. Four sizes of
pre-adsorbed particles were studied: $\sigma_{1}^{*}=0.8$, 1, 1.2, and 1.4.

The pre-adsorbed molecules block the access to the surface of the adsorbate
molecules. As a consequence, the effective (i.e., averaged over the entire
surface) surface-water interactions become weaker. Since the wetting of a
solid surface results from a balance of solid-fluid and fluid-fluid
interactions, one can expect an increase of the wetting temperature with the
surface density $R_{c}^{*}$ of pre-adsorbed species.

In figure 2 we show three dimensional plots illustrating the dependence of the
contact angle on temperature and on $R_{c}^{*}$. The calculations are for
$\varepsilon_{fw}^{*}=8.311$.

For $\sigma_{1}^{*}=1.4$ (figure 2 d), the wetting occurs only for surface
densities $R_{c}^{*}\lesssim 0.25$ (or, for $R_{c}\sigma_{1}^{2}\lesssim
0.49$). The close-packed hexagonal surface coverage corresponds to the surface
density $R_{cp}^{*}\sigma_{1}^{2}=2/\sqrt{3}$. Therefore, the covering of
approximately 40% of the surface by hard obstacles of the diameter
$\sigma_{1}^{*}=1.4$ suffices to prohibit the wetting at all temperatures
$T^{*}<T_{c}^{*}$. In the case of the particles of diameter $\sigma_{1}^{*}=1$
(figure 2 b), the covering of the surface that inhibits the wetting is higher
and equals approximately $R_{c}\sigma_{1}^{2}\approx 0.82$, which corresponds
to 71% of the close-packed coverage. Of course, the higher efficiency of
larger hard particles in retarding the wetting is connected with stronger
lowering of the effective water-surface interactions by obstacles of larger
size.

At low temperatures, the plots of $\theta(T)$ exhibit a plateau. This type of
behavior is more evident for larger obstacles and at higher values of
$R_{c}^{*}$. In other words, with an increasing $R_{c}^{*}$, the surface
becomes more hydrophobic for a wider range of temperatures. We checked that at
$T^{*}=1.5$, the contact angle for the surface covered with the close-packed
hexagonal layer of particles of $\sigma^{*}=1$ is $\theta\approx 110^{\circ}$,
while for $\sigma^{*}_{1}=1.4$ it equals $\approx 130^{\circ}$.

Figure 3: (Colour online) Parts a and b. Local density profiles at different
temperatures for bulk density marginally lower than bulk dew density (part a)
and marginally higher than the bubble density (part b). The temperatures are
given in part b. Part c. Adsorption isotherm from gaseous phase at $T^{*}=2.2$
(red lines), 2.4 (blue lines), and 2.55 (black lines). Vertical dashed lines
mark the bulk dew densities. Part d. Local densities for coexisting thin
(black) and thick (red) adsorbed films at $T^{*}=2.55$ and for the bulk
density $\rho^{*}_{b}=0.1220$ (the bulk dew density equals $\approx 0.1256$).
All calculations are for $\sigma^{*}=1$, $R_{c}^{*}=0.28$ and for
$\varepsilon_{fw}^{*}=8.311$.

Calculations of the contact angles from equation (2.18) require the knowledge
of the local densities of the systems for bulk densities equal to the
densities of coexisting liquid and gaseous phases. Figures 3 a and 3 b show
the density profiles of water at bulk densities marginally lower (part a) and
marginally higher (part b) than the densities at the bulk coexistence. The
calculations are for $\sigma_{1}^{*}=1$ and $R_{c}^{*}=0.28$. Up to the
temperature $T_{c}^{*}=2.44$ only a thin film formation is observed for
adsorption from the gaseous phase. At $T_{c}^{*}=2.5$, however, a thick film
is formed and its thickness diverges as the bulk density approaches
$\rho_{bg}^{*}$. The crossover between thin and thick film behavior is at
$T^{*}_{w}=2.445$. At the same temperature, the value of the contact angle
becomes zero. This indicates a consistency of the values of the wetting
temperature obtained from the Young equation and from the adsorption study.

Figure 4: (Colour online) The dependencies of the static contact angle
$\theta$ on temperature and the amount of pre-adsorbed particles of the
diameter $\sigma_{1}^{*}=1$ and for $\varepsilon_{fw}^{*}=7$ (part a) and
$\varepsilon_{fw}^{*}=9.5$ (part b).

In figures 3 c and 3 d we plot the adsorption isotherms at $T^{*}=2.2$, 2.4
and $T^{*}=2.55$ (part c), as well as the density profiles (part d) at
$T^{*}=2.55$. Vertical dashed lines in part c indicate the bulk densities of
gas coexisting with a liquid. At two lower temperatures, the isotherm remains
small up to the bulk liquid-vapor coexistence. At $T^{*}=2.55$, however, the
prewetting jump on the adsorption isotherm appears at $\rho_{b}^{*}=0.1220$.
The local densities in part d show the profiles just before and after the
prewetting jump. According to our estimation, for the system under study, the
prewetting jump ends at the critical prewetting temperature,
$T_{cp}^{*}\approx 2.61$. Although we did not conduct detailed studies of the
prewetting phase transition, the obtained results indicate that for all
systems under study, if the prewetting exists, it is a first-order phase
transition [31, 33, 29].

In the case of non-modified surfaces, the wetting temperature depends on the
depth of the water-surface potential [equation (2.6)], $\varepsilon_{fw}^{*}$.
If $\varepsilon_{fw}^{*}$ increases, the wetting temperature decreases. A
similar behavior was found for surfaces modified with hard obstacles, which is
illustrated in figure 4. The calculations were carried out for
$\sigma_{1}^{*}=1$. Part a shows the dependence of $\theta$ on temperature and
$R_{c}^{*}$ for $\varepsilon_{fw}^{*}=7$, and part b — for
$\varepsilon_{fw}^{*}=9.5$. Note that according to the results of our previous
work for non-modified surfaces [29], the wetting temperature was
$T_{w}^{c}=2.29$ for $\varepsilon_{fw}^{*}=7$ and $T_{w}^{c}=1.76$ for
$\varepsilon_{fw}^{*}=9.5$.

Figure 5: (Colour online) The dependence of $T_{w}^{*}$ (part a) and
$T_{90}^{*}$ (part b) on the amount of pre-adsorbed particles for different
values of $\sigma_{1}^{*}=1$ and $\varepsilon_{fw}^{*}$, given in the figure.
The dotted line in part a denotes the bulk critical temperature.The meaning of
arrows is explained in the text. We stress that the scale of the abscissa in
part a is $R_{c}\sigma_{1}^{2}$, while in part b we use the reduced values of
$R_{c}^{*}$.

The presence of hard obstacles leads to the effect of a further lowering of
the effective adsorbing potential and thus to an increase of the wetting
temperature. For a weakly adsorbing surface, $\varepsilon_{fw}^{*}=7$, and for
high surface density of pre-adsorbed particles, $R_{c}^{*}\gtrapprox 0.7$, the
wetting transition disappears for all temperatures $T^{*}<T_{c}^{*}$. By
contrast, for $\varepsilon_{fw}^{*}=9.5$, the wetting transition was observed
for all investigated values of $R_{c}^{*}$.

Summary of our calculations for the surfaces covered with hard-sphere
obstacles is given in figure 5. Part a presents the dependence of the wetting
temperature on $R_{c}^{*}$ for different sizes of obstacles and for three
values of $\varepsilon_{fw}^{*}$. In the experimental studies of wetting, the
temperature at which the contact angle becomes equal to $90^{\circ}$ is
important since experimental works usually classify the surfaces with the
contact angle $\theta>90^{\circ}$ as hydrophobic and the surfaces with
$\theta<90^{\circ}$ — as hydrophilic. The temperature abbreviated as
$T_{90}^{*}$, separates these two regimes. We emphasize that in the case of
capillaries with hydrophobic walls, the meniscus of fluid inside the pores is
concave, while it is convex for hydrophilic walls. The relationship of
$T_{90}^{*}$ on the surface coverage and the size of obstacles is presented in
figure 5 b.

Two groups of the curves can be distinguished in figure 5 a. The first group
is at a constant value of $\varepsilon_{fw}^{*}=8.311$ and illustrates the
effect of the size of obstacles on the wetting temperature. All three these
curves originate at ($R_{c}^{*}=0,T_{c}^{*}=2.02$), i.e., at the wetting
temperature for a non-modified surface. The second group of the curves is for
the fixed value of $\sigma_{1}^{*}=1$ and for three values of
$\varepsilon_{fw}^{*}=7$, 8.311 and 9.5. Of course, in this case, the wetting
temperature for different non-modified surfaces is different.

All the curves in figure 5 exhibit non-linear behavior. Thus their shape
contradicts the predictions by Cassie and Baxter [35] theory. In the case of
four curves (marked by the arrows in figure 5 a), there exist the values of
the surface coverage $R_{w}\sigma_{1}^{2}$ at which the wetting temperature
becomes equal to the bulk critical temperature. For the coverages
$R_{c}\sigma_{1}^{2}>R_{w}\sigma_{1}^{2}$, the wetting transition is
suppressed for all temperatures up to the bulk critical temperature. The value
of $R_{w}\sigma_{1}^{2}$ is lower for higher $\sigma_{1}^{*}$ and for lower
$\varepsilon_{gs}^{*}$. For $\sigma_{1}^{*}=0.8$ and
$\varepsilon_{fw}^{*}=8.311$, as well as for $\sigma_{1}^{*}=1$ and
$\varepsilon_{fw}^{*}=9.5$, the wetting transition occurs for all surface
coverages $R_{c}^{*}$.

Figure 6: (Colour online) The dependencies of $T_{w}^{*}$ on the composition
$x=R_{c1}^{*}/(R_{c1}^{*}+R_{c2}^{*})$ of a binary pre-adsorbed phase of hard-
particles of the diameter $\sigma_{1}^{*}=0.8$ and $\sigma_{2}^{*}=1.4$. The
calculations are for fixed values of $R_{c}^{*}=R_{c1}^{*}+R_{c2}^{*}$ given
in the figure.

We also performed the calculation of the contact angles for two-component pre-
adsorbed layers consisting of hard-sphere obstacles of the diameters
$\sigma_{1}^{*}=0.8$ and $\sigma_{2}^{*}=1.4$. Figure 6 shows the dependence
of the wetting temperature on the composition of the obstacles,
$x=R_{c1}^{*}/(R_{c1}^{*}+R_{c2}^{*})$. The displayed results were obtained at
three selected constant values of the total two-dimensional density
$R_{c}^{*}=R_{c1}^{*}+R_{c2}^{*}$. Of course, for $x=0$ the and for $x=1$, the
wetting temperatures are equal to those of one-component layers consisting of
spheres of the diameter $\sigma_{1}^{*}=1.4$ and 0.8, respectively. Again,
contrary to the Cassie and Baxter [35] theory, the curves in figure 6 are non-
linear.

The second series of calculations were for a layer of pre-adsorbed molecules
interacting with water molecules via attractive forces. The interactions
between pre-adsorbed particles were still of hard-sphere type. We assumed the
diameter of pre-adsorbed species to be the same as the diameter of water
species, $\sigma_{1}^{*}=1$. The energy parameter of the potential of equation
(2.9) was treated as a free parameter,
$\varepsilon^{*}_{f1}=\varepsilon_{f1}/\varepsilon$.

Figure 7: (Colour online) The dependence of $\theta$ on $T^{*}$ and
$R_{c}^{*}$ for the system with attractive interactions between water and pre-
adsorbed species. Calculations are for $\sigma_{1}^{*}=1$ and
$\varepsilon_{f1}^{*}=0.4$.

The presence of species attracting the water molecules leads to two opposite
effects. The first one results from blocking the access of water molecules to
the surface. Consequently, the effective water-surface interactions are lower,
and the wetting temperature would increase. However, the pre-adsorbed
molecules also exert attractive forces on water molecules. If the second
effect prevails, the wetting temperature would decrease. Competition between
these two factors can lead to a dependence of the wetting temperature on
$R_{c}^{*}$ that could exhibit an extreme.

Figure 7 shows a three-dimensional plot of the dependence of $\theta$ on
temperature and $R_{c}^{*}$. The presented results are for
$\varepsilon_{f1}^{*}=0.4$. At very low values of $R_{c}^{*}$, a small
decrease of the wetting temperature compared to the bare substrate occurs. It
means that for small $R_{c}^{*}$, the increase of effective attraction between
fluid molecules and the modified solid is more important than the effect due
to the blocking of the surface. At still higher values of $R_{c}^{*}$ the
wetting temperature starts to increase and attains its maximum value for
$R_{c}^{*}\approx 0.51$. Within this region of surface coverages, the blocking
effect plays a dominant role. A further increase of $R_{c}^{*}$ leads to the
lowering of the wetting temperature.

Figure 8: (Colour online) The dependence of the wetting temperatures on
$R_{c}^{*}$ for the system with different values of $\varepsilon_{f1}^{*}$
that are given in the figure. The red solid line denotes the results for pre-
adsorbed hard-spheres. Calculations are for $\sigma_{1}^{*}=1$.

A comparison of the changes in the wetting temperature with the two-
dimensional the density of pre-adsorbed species and with the energy parameter
$\varepsilon_{f1}^{*}$ is shown in figure 8. We also included here the curve
for the pre-adsorbed layer of hard obstacles as a reference. As expected, for
a low value of $\varepsilon_{f1}^{*}=0.05$, the evaluated curve
$T_{w}^{*}(R_{c}^{*})$ is close to that for hard obstacles. However, we
observe an undulated course of this function. The changes in the values of the
contact angle result from the changes in the free energies for gaseous and
liquid phases contacting with a solid. According to perturbative treatment,
the shape of the function $T_{c}^{*}(R^{*}_{c})$ is to a great extent
determined by a delicate balance between the hard-sphere contribution to the
free energy and the mean-field term due to water-pre-adsorbed molecule
attraction. If the latter contribution prevails, the wetting temperature
decreases, but when the hard-sphere contribution becomes dominant, the wetting
temperature increases.

For $\varepsilon_{f1}^{*}=0.7$, the wetting temperature decreases and for all
values of $R_{c}^{*}$ and for $R_{c}^{*}>0.4$ this decrease is quite fast. We
did not perform the calculations at temperatures lower than $T_{c}^{*}<1.5$,
because the application of the considered version of the density functional at
the temperatures lower than the triple point temperature may be questioned.

Finally, in figure 9 we display the effect of the attractive forces on the
structure of gaseous and liquid water on modified surfaces. Left-hand panels
are at bulk densities marginally lower than the bulk dew density, while right-
hand panels are at the bulk liquid density marginally higher than the bulk
bubble density. The presence of sites attracting water molecules on the
graphite surface greately influences the structure of both gaseous and liquid
adsorbed phases. In the case of adsorption from the gaseous phase, the
attraction between pre-adsorbed species and water molecules leads to the
formation of a “knee” on the density profile (part a). At higher values of
$R_{c}^{*}$, this knee transforms into the second maximum of $\rho^{*}(z)$. At
the highest coverages of the surface with pre-adsorbed molecules, the local
density maximum at the value of $z$ corresponding to a minimum of the
potential $v(z)$ vanishes and the pre-adsorbed layer starts to play the role
of the source of an external potential.

Figure 9: (Colour online) Part a. The dependence of local densities of
coexisting gaseous (left-hand panel) and liquid (right-hand panel) phases. The
calculations were carried out at constant $R_{c}^{*}=0.4$ and for different
values of $\varepsilon_{f1}^{*}$ given in the figure. Part b. The same as in
part a, but for constant $\varepsilon_{f1}^{*}=0.4$ and for different values
of $R_{c}^{*}$, given in the figure. In all cases $\sigma_{1}^{*}=1$ and
$T^{*}=2$.

##  4 Summary

We have proposed an approach based on the density functional theory to
describe the changes in the contact angle with the surface heterogeneity.
According to the proposed model, the surface heterogeneity results from the
formation of a layer of pre-adsorbed molecules on the original, bare solid
surface. Depending on the model of interactions between the pre-adsorbed
molecules, on their size, and on their amount (expressed in terms of the
surface density, or surface coverage), different changes in the contact angle
and the wetting temperature were observed. In the case of hard-sphere fluid-
pre-adsorbed layer interactions, the presence of pre-adsorbed hard-sphere
molecules leads to an increase of the values of the contact angle and the
wetting temperature. These effects were more pronounced for larger pre-
adsorbed particles.

In the case of pre-adsorbed molecules attracting fluid molecules, the changes
in the wetting temperature with the amount of pre-adsorbed species are more
complex. For some selected values of the parameter characterizing the
attractive interaction and for the selected amount of pre-adsorbed molecules,
the wetting temperature can exhibit a maximum. Since the value of the wetting
temperature depends on the surface excess free energies of gaseous and liquid
phases in contact with the modified solid, the presence of a maximum is the
result of an interplay between particular terms in the perturbational free
energy expansion, basically on the competitions between the hard-sphere and
the contribution due to attractive water-pre-adsorbed particles interaction.

The theory considered in this work indicates that the value of the wetting
temperature does not depend linearly on the surface density of different kinds
of adsorbing sites, as predicted by the classical Cassie approach [37]. Our
treatment can also be considered as an alternative to the treatment by
Aslyamov et al. [61, 62]. The latter approach is based on the development of
an effective, one-dimensional external potential that, in turn, is next used
in the classical density functional one-dimensional density functional
expressions for evaluating the excess free energy. Instead of evaluating the
effective free energy, we propose an appropriate modification of the free
energy contributions. Our idea has its origin in an approximate treatment of
the quenched-annealed systems that was previously used to study adsorption on
heterogeneous surfaces [24, 25, 26, 27]. Unfortunately, there exist no
experimental or simulation data that would be useful for verifying our
theoretical predictions.

Basically, our calculations were carried out for one-component pre-adsorbed
phase. However, the theory was proposed for the case of a multicomponent pre-
adsorbed layer. It can be also extended to the case of a polydisperse mixture
by proceeding along the lines described in [74, 75, 76]. Next, the theory can
be also extended by assuming a multilayer structure of pre-adsorbed molecules.
The treatment of such a system would be based on the approaches used for
quenched-annealed systems [11]. All these problems are under study in our
laboratories.

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Кут змочування води на модельній гетерогенній поверхні. Метод функціоналу
густини

K. Домбровська${}^{\textsf{\footnotesize{\mbox{?}}}}$, O.
Пізіо${}^{\textsf{\footnotesize{\mbox{?}}}}$, С.
Соколовський${}^{\textsf{\footnotesize{\mbox{?}}}}$

${}^{\textsf{\footnotesize 1}}$ Факультет теоретичної хімії, Університет ім.
Марії Склодовської-Кюрі, Люблін 20-031, Польща,
email<EMAIL_ADDRESS>${}^{\textsf{\footnotesize 2}}$ Інститут
хімії, Національний автономний університет Мехіко, Circuito Exterior, 04510,
Мехіко, Мексика email<EMAIL_ADDRESS>