Wetting of curved surfaces

Abstract

As a first step towards a microscopic understanding of the effective interaction between colloidal particles suspended in a solvent, we study the wetting behavior of one-component fluids at spheres and fibers. We describe these phenomena within density functional theory which keeps track of the microscopic interaction potentials governing these systems. In particular, we properly take into account the power-law decay of both the fluid–fluid interaction potentials and the substrate potentials. The thicknesses of the wetting films as a function of temperature and chemical potential as well as the wetting phase diagrams are determined by minimizing an effective interface potential which we obtain by applying a sharp-kink approximation to the density functional. We compare our results with previous approaches to this problem.

Introduction

If colloidal particles are dissolved in a solvent consisting of a binary liquid mixture, necessarily one of these two components is preferentially adsorbed on the spherical surfaces of the colloidal particles. Near a first-order phase separation of the bulk solvent this adsorption may lead to the coating of the colloids by wetting films which snap into a bridge-like structure if two colloids come close to each other. This coagulation can become so pronounced that it results in flocculation 1, 2, 3, 4, 5. Such experiments have inspired several theoretical investigations 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 which have shed light on various features of these complex phenomena.

Although there are indications that for higher concentrations of colloidal particles their effective mutual interactions are beyond a superposition of effective pair potentials [11], for low concentrations the knowledge of the effective interaction between two isolated colloidal particles at a fixed distance immersed into the solvent is important. Moreover, the latter configuration is interesting in its own, because the bridge formation between individual pairs is not solely accessible indirectly via flocculation but can be studied also directly with atomic or surface force apparatuses [17]. Besides colloidal systems, these problems are in addition relevant for technical applications such as the contact between lubricant films on the disk surfaces in magnetic recording with the recording head [18], sintering [19], latex film formation [20], catalyst wetting efficiency in trickle-bed reactors [21], or for the bonding mechanism of emulsified adhesives in granular and fibrous substrates [22].

As a prerequisite for understanding the effective interaction between such spherical particles immersed in a solvent, one must know the structural and thermal properties of these systems in the case that the distance between two colloidal particles is macroscopically large. Near a first-order phase transition of the solvent this amounts to studying the wetting behavior of the solvent at the surface of a spherical substrate. The paradigmatic case is the formation of a liquid-like film at the curved substrate–vapor interface upon approaching the liquid–vapor coexistence curve in the bulk of a one-component fluid from the vapor side. This problem has been studied theoretically by several authors 23, 24, 25, 26, 27, 28, 29 based on various versions of phenomenological models. They confirm the general expectation (see subsection X.B in Ref. [30]) that the positive curvature of the surface of a substrate prevents the build-up of macroscopically thick wetting films because, in contrast to a planar geometry, the area of the emerging liquid–vapor interface increases with the thickness of the wetting film. This increasing cost of the free energy of the film has effectively the same consequence as if the bulk fluid is kept off liquid–vapor coexistence. Therefore, continuous wetting transitions are eliminated and first-order wetting transitions are reduced to quasi-first-order transitions between small and large, but finite, film thicknesses, smeared out due to the finite size of the substrate area.

However, beyond these general aspects, it is important to know to which extent the film thicknesses of these wetting films are limited and how these limitations depend on the radius of the colloidal particles, on the character and the form of the substrate potential and the interaction potential between the solvent molecules, and on the size of the solvent molecules. The aforementioned phenomenological models do not allow one to answer these questions because they do not keep track of the microscopic details of the system. In particular, the experience with wetting phenomena on flat substrates tells that it is essential to take into account equally the power-law decay of the substrate potential and of the interaction potential between the fluid particles 30, 31. Inter alia, the long range of the forces between the fluid particles causes the break down of a gradient expansion for treating the deviations of the emerging liquid–vapor interface from its flat configuration and leads to a nonlocal Hamiltonian [32]. For simple geometric shapes of the interfaces involved, this long-ranged character of the dispersion forces acting in and on fluids is taken into account by the Dzyaloshinskii–Lifshitz–Pitaevskii (DLP) theory [33] which has been applied to spheres and cylinders [34]. For a given system, this allows one to compute the cost in free energy $\text{Ω}{}_{\mathrm{s}}\text{(l)}$ to maintain a thick liquid film of prescribed thickness l adsorbed on the substrate in terms of the frequency dependences of the permittivities of the substrate, the bulk liquid, and the bulk vapor. Although this approach has the advantage to take into account many-body forces and retardation, it suffers also from some shortcomings. (i) The DLP theory gives access only to the leading asymptotic behavior of $\text{Ω}{}_{\mathrm{s}}\text{(l→∞)}$ and thus does not allow one to describe critical wetting transitions which result from the competition between the leading and next-to-leading order term in $\text{Ω}{}_{\mathrm{s}}\text{(l)}$ for large l 30, 31. (ii) Details of the substrate potential, which are important for first- and second-order wetting transitions 30, 31, are not captured by the DLP theory. (iii) $\text{Ω}{}_{\mathrm{s}}{}^{\mathrm{(DLP)}}\text{(l)}$ exhibits an unphysical divergence for l→0 because this approach ignores the repulsive part of both the substrate potential and the fluid–fluid interaction potential and thus the structure of the emerging substrate–liquid interface. (iv) The broadening of the emerging liquid–vapor interface upon raising the temperature cannot be accounted for. (v) The dependence of $\text{Ω}{}_{\mathrm{s}}{}^{\mathrm{(DLP)}}\text{(l)}$ on temperature and chemical pressure, in particular close to phase transitions in the bulk fluid, is not transparent. (vi) In the present context, as expounded above, the most severe drawback of the DLP approach is that the DLP results for wetting of a single sphere or cylinder [34] cannot be used as a building block for investigating later on the effective interactions between two such objects because the shape of the bridging wetting film between them is not known in advance which precludes the practical application of the DLP theory to the problem described in the beginning.

Therefore, in this paper, we compute (Section 2) and discuss (Section 3) the effective interface potential $\text{Ω}{}_{\mathrm{s}}\text{(l)}$ for the wetting of a single sphere and of a single cylinder on the basis of density functional theory [35]. This microscopic approach allows one to address the aforementioned points (i) – (vi) and overcomes the shortcomings of the phenomenological theories. The price to be paid is that the results do not account for the effects of many-body forces.

In the beginning of this introduction, we have focused on the physical interest in spheres exposed to a fluid. It turns out that there is also substantial interest in the cylindrical geometry. In view of its importance for technical processes such as the lubrication of textile fibers, optical fiber processing, and the formation of fiber-reenforced resins [36] there are many studies of the statics and dynamics of adsorption on cylinders 9, 26, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47. In addition the adsorption on cylinders is important for the vibrating-wire microbalance as a standard technique to study wetting phenomena [48]. However, one has to keep in mind that the convoluted surface of graphite fibers impedes the interpretation of these experiments on the basis of theoretical models which assume a smooth cylindrical surface [49]. (Such surface inhomogeneities may even play a role for spherical colloidal particles [50].) For the same reasons given above for the spherical geometry we apply a microscopic density functional theory also to the problem of wetting of wires in order to obtain a detailed picture of their behavior in systems with long-ranged interactions.

We conclude the introduction with two remarks. First, the present study is dealing with volatile liquids, i.e., it is important that the liquid phase and the surrounding vapor are in thermal equilibrium. Second, we do not address the problem of interfacial wetting at curved surfaces, i.e., the substrate is passive. Therefore, our results do not apply directly to such phenomena as surface melting of small particles and wires.