Rapid Communication In ﬂ uence of the Disjoining Pressure on the Equilibrium Interfacial Pro ﬁ le in Transition Zone Between a Thin Film and a Capillary Meniscus

Thebehaviourofliquidlayersonsolidsubstratesdependsonanumberoffactors,themostimportantofwhichis the action of surface forces in the vicinity of the three phase contact line. The equilibrium interfacial (gas/liquid) pro ﬁ leinthetransition zone between thethin ﬂ at ﬁ lmandthe sphericalpart of a meniscus isdeterminedbythe combined action of the disjoining/conjoining and capillary pressures. The disjoining/conjoining pressure is considered to include the electrostatic, van der Waals and structural components. The Poisson – Boltzmann equation is also solved with various boundary conditions to calculate the electrostatic component of the disjoining/conjoining pressure. Wetting conditions are considered and the interfacial pro ﬁ le is determined for various parameters governing the surface interactions, as well as the ratio between the disjoining/conjoining and capillary pressures.


Introduction
DCP 1 is a manifestation of the surface forces acting on thin liquid layers and this concept was introduced, and successfully investigated, in the pioneering works of Derjaguin [1,2]. The well-known DLVO (Derjaguin-Landau-Verwey-Overbeek) theory of colloidal stability is based on DCP acting between colloidal particles/droplets [1]. DCP also acts in the vicinity of the three-phase contact line in the case of wetting/spreading [3], and for historical reasons, the action of DCP under these conditions has received less attention. However, there are notable examples where the effect of surface forces on wetting and spreading phenomena of liquids on solid substrates are considered (see [4][5][6][7] and references therein).
Flat wetting films on solid substrates exist because the DCP inside the liquid film is balanced by CP 2 in the neighbouring meniscus or droplet. Within the spherical part of the meniscus or droplet, the separation between the liquid-vapour and solid-liquid interfaces is high and the DCP is negligible. Hence, the shape of the meniscus or droplet is determined by the action of CP only [3]. Therefore, a transition zone must exist between the bulk meniscus or droplet and the flat film wherein DCP and CP act simultaneously [3,5,7]. Since measurements of equilibrium/hysteresis contact angles and surface curvature of bulk liquids are carried out outside the transition zone, its size and profile are of interest. The latter provide information on the DCP-isotherm for thin liquid films on a solid substrate. The transition profile of the meniscus was calculated in [3] for the case of complete wetting. The primary aim of this article is to determine the shape of the transition zone for various types of the DCP-isotherm. The exact numerical solutions for PW conditions and for complicated forms of the DCP-isotherm were obtained for the first time.

Model Assumptions
The transition zone II (Fig. 1) under equilibrium conditions is located between a two dimensional capillary meniscus (I) and a flat wetting film (II). A rectangular coordinate system, (x 1 , x 2 ) is used, in which x 1 and x 2 are the lateral and normal coordinates, respectively. The width of the capillary, 2H, is assumed to be much larger than the thickness of the equilibrium flat film, h e . In the case of CW 3 (see Fig. 1,a), the continuation of the spherical meniscus (broken line) of radius r e does not intersect either the solid walls of the capillary or the thin liquid film of thickness h e . The case of PW 4 is shown in Fig. 1,b; the continuation of the spherical meniscus intersects the boundary at the contact angle θ e .
Under equilibrium conditions, there is no flow, and zero ion fluxes. The surface tension is assumed to be constant, which is valid in the absence of surfactants (and thermal gradients) [8].

Expression for the DCP Isotherm
The full DCP is a sum of electrostatic, van der Waals and structural components: The van der Waals component [1] is given by Here A = −A H , A H is the Hamaker constant; and h is a thickness of the liquid film. The influence of the structural component, Π S , is discussed later (see Effect of the structural component section). An example of the graphic dependences of the DCP components will be given in Fig. 5,a.
To derive an expression for the electrostatic component, Π E , the Poisson-Boltzmann equation in the small-slope approximation, h′ 2 ≪ 1, where the prime denotes differentiation with respect to x 1 , is used: where φ = ΦF/(RT) is a dimensionless potential in which the dimensional potential, Φ, is scaled on F/RT, wherein F is the Faraday constant, R is the gas constant, and T denotes the temperature, respectively; c 0 is the molar electrolyte concentration; ε and ε 0 are the dielectric constants of water and vacuum, respectively. In the case of equilibrium, the momentum equations in the x i directions (i = 1,2) are expressed by the equations of electrohydrodynamics [9,10]: where q = Fc 0 (exp(−φ)−exp(φ)) is a volume charge density; and p is the pressure in the liquid.
Eqs. (3) and (4) are used in the normal stress balance at gas-liquid interface in the transition zone [3]: Taking into account in the last equation that ∂ , the following expression for the electrostatic component in the case of the smallslope profiles is obtained: The following boundary conditions are used here: Π(h → ∞) = 0 and φ(h → ∞) = 0, which correspond to the decay conditions for the DCP and the electric potential at long distances. The detailed description of the derivation of Eq. (5) and an analysis of the results for CW conditions are given in [11].
Eq. (5) coincides with Derjaguin's expression [1] for flat films. However, there is a substantial distinction of the expression (5) from that deduced in [1]: Eq. (5) is valid for non-flat thin liquid films in the case of small-slope approximation.
Two types of boundary conditions for Eq. (3) are used below to find the distribution of φ(x 2 ) and ∂φ/∂x 2 across the liquid film: (1) constant surface electrical potentials on both liquid-solid and liquid-vapour interfaces: φ s , φ h = const; (2) constant surface charge densities σ s , σ h = const.
The situations when φ s ≠ φ h or σ s ≠ σ h are rather common and will be studied below.
The equilibrium interfacial profile under the action of the surface forces is described by the augmented Young-Laplace equation [1,3]: Here P e is the excess pressure equal to the capillary pressure for the spherical meniscus, P e = γ/(H − h*).
Eq. (6) representing the normal stress balance, is solved for the case γ = const. The calculation results are discussed in the Equilibrium interfacial profile in the transition zone section. For the systems with the surface tension gradients, the significant tangential stress along 4 PWpartial wetting.
the interface can arise, and the tangential stress balance must also be taken into account. Let us first consider the case in which both φ s and φ h are constant but φ s ≠ φ h for which the results are presented in Fig. 2. The distribution of potentials is asymmetric (Fig. 2,b), and since both φ s and φ h are constant, the surface charges σ h,s are not, as demonstrated by the varying values of the potential slopes at the boundaries in Fig. 2,b. The isotherm Π E has a maximum in this case (Fig. 2,c): at high separations, repulsion prevails, whereas at small distances attraction is observed. Since the electric potentials on the boundaries are constant, the DCP can change only due to the variation of the Maxwell part (corresponding to the last term) in Eq. (5).

Results for the DCP Calculation
For low h values (region 1 in Fig. 2), the gradient ∂φ/∂x 2 tends to infinity because for the case φ h,s = constant, the difference Δφ = φ h − φ s is also constant at x 2 → 0 ( Fig. 2,a, case 1). According to Eq. (5), this results in Π → −∞ at low separations h between the surfaces. In region 2, of the intermediate values of h, the gradient ∂φ/∂x 2 decreases and at a certain h it becomes equal to zero. Under these conditions, the Maxwell part of the DCP is absent and the DCP takes its maximum value Π max , which is equal to the excess osmotic pressure in the film. Finally, at large distances, h, the gradient ∂φ/∂x 2 becomes negative, reaches its limiting value, and does not depend anymore on the increase of h (Fig. 2, region 3). In this region, the surfaces are non-interacting, the Maxwell part for them becomes constant and equals to the excess osmotic pressure, Π max .
In view of the above, the maximum of the DCP isotherm is related to the sign change of the gradient ∂φ/∂x 2 , which is proportional to the surface electric charge σ: one of the surfaces changes its charge sign in response to the variation in h. A physical explanation for this behaviour could be the phenomenon of a surface charge reversal or overcharging [12,13]. However, the DCP changes its sign not at the point of overcharging, but at the point where the Maxwell part of the DCP in Eq. (5) becomes equal to the osmotic part.
A similar analysis can be made for the case σ h,s = constant; the minimum on the DCP isotherm is explained here by the variation of the osmotic contribution of the DCP with the Maxwell part remaining constant.

Equations for the Equilibrium Interfacial Profile
The equilibrium interfacial profile is found from the following equation obtained by integration of Eq. (6) using the boundary condition h′(h → H) = − ∞: Here γ/r e = P e in which r e = H − h ⁎ is the radius of the equilibrium meniscus (see Fig. 1 for definition of h ⁎ ), and the upper limit of integration was replaced by infinity because the DCP vanishes at distances compared with a capillary width scale.
The solution of Eq. (7) must satisfy the boundary conditions in the flat film region: h(∞) = h e ; h′(∞) = 0. Substitution of these conditions into Eq. (7) gives the following relation: If the contact angle θ e is determined by the intersection of the spherical meniscus with the flat film, the equilibrium contact angle from Eq. (8) is: The latter expression was firstly obtained by Derjaguin and Frumkin from the Gibbs adsorption equation [14,15]. The dependence Π(h) in the above equations can be calculated using Eqs.
As was mentioned above, the structural component of Π is not used in Eq. (10). After the values h ⁎ and h e are found, the interface profile in the transition zone is calculated using Eq. (7).

Results of the Equilibrium Interfacial Profile Calculation
The electrostatic component of the DCP was calculated for the case of constant surface charge density σ s,h = const, with σ s = − 150 mC and σ h = 150 mC at the boundaries. The van der Waals component of the DCP was also included in calculations. The effect of the structural component is considered in the Effect of the structural component section.
As shown in the inset to Fig. 3,a, there are three points of intersection of the DCP with P e line: the points h e and h β correspond to the stable and metastable equilibrium states. The CW conditions are observed; this follows from the form of the DCP isotherm: the integral in Eq. (9) is positive. There is a small-slope plateau in the convex part corresponding to the metastable state h β (Fig. 3,a). Fig. 3,b shows that DCP (curve 1) has a maximum value exceeding the excess pressure P e (line 4) in the vicinity of point x 1 ≈ 2.3 × 10 −5 m, i.e. in the interval between h u and h β . Since the total pressure (i.e. the sum of DCP and the local CP) must be constant, the magnitude γK becomes negative in this interval (curve 2). As a result, the profile curvature K changes its sign and the convex part appears in the interface profile (Fig. 3,a). It is obvious that the necessary condition for the presence of the convex part is the triple intersection of the DCP with P e , i.e. the multiplicity of the equilibrium states [1].
In PW case the condition cos θ e b 1 is valid, hence according to Eq. (9) the condition ∫ ∞ he Πdhb0 is satisfied and the DCP-isotherm includes a negative (attractive) part. The interface profile including the convex part for the case of PW conditions is presented in Fig. 4, for which θ e = 0.9°. There are three intersection points in the inset, hence the convex part is   Fig. 4,b) has a maximum where Π N P e (line 4). As a result, K b 0 here (curve 2) which corresponds to the convex part in Fig. 4,a; the convex is most pronounced at low values of the contact angle for the case of PW conditions.

Effect of the Structural Component
In this section, the structural component of the DCP is included in calculation in addition to the electrostatic and van der Waals components. The structural component Π S of the DCP was calculated using the following dependence [16]: where K 1 , K 2 and λ 1 , λ 2 are the parameters related to the magnitude and the characteristic length of the structural forces. The indexes 1 and 2 correspond to the short-range and long-range structural interactions, respectively. The structural component can arise due to the changes in the orientation structure of polar liquid near the surface [1,16]. A long-range structural attraction causes an additional secondary minimum in the DCP isotherms (Fig. 5,a, inset). This attraction (the negative values of the DCP shown by curve 1 in Fig. 5,b) is the reason that the values of the profile curvature (curve 2, Fig. 5,b) exceed the curvature of the spherical meniscus (line 4, Fig. 5,b). As a result, the interface profile is curved more sharply than the spherical meniscus is, and the convex part of the interface is located as shown in Fig. 5,a. This behaviour takes place for both CW and PW conditions. It should be noted that the convex part is observed in the case of the equilibrium states multiplicity only as it was mentioned above.

Conclusion
Calculation of the equilibrium interfacial profile inside the transition zone between a spherical meniscus and a thin liquid film has been carried out; this calculation has also yielded the equilibrium contact angle, the disjoining/conjoining pressure (DCP) isotherms, and the distribution of electric potential and charge in the liquid phase. The precise numerical integration of the augmented Young-Laplace equation was performed for the first time for various types of the DCP isotherm. The obtained results indicate that the form of the isotherm has a significant effect on the shape of the equilibrium interfacial profile in the transition zone: the convex part in the interface profile depends on the equilibrium states between the DCP and excess pressure. For complete wetting conditions, the convex part of the interface is usually observed in front of the spherical meniscus, whereas for partial wetting, it is located behind it. Inclusion of the van der Waals component in the DCP leads to an increase in the latter, and, as a result, a growth of the flat film thickness and a decrease of the meniscus radius. The structural component leads to the interface profile shape with a convex part in front of the meniscus for both complete and partial wetting conditions.