Surface phase diagrams for wetting on heterogenous substrates

Wetting behavior of a colloidal particle trapped at a composite liquid-vapor interface of a binary liquid mixture

A partially miscible binary liquid mixture, composed of A and B particles, is considered theoretically under conditions for which a stable A-rich liquid phase is in thermal equilibrium with the vapor phase. The B-rich liquid is metastable. The liquids and the thermodynamic conditions are chosen such that the interface between the A-rich liquid and the vapor contains an intervening wetting film of the B-rich phase. In order to obtain information about the large-scale fluid structure around a colloidal particle, which is trapped at such a composite liquid-vapor interface, three related and linked wetting phenomena at planar liquid-vapor, wall-liquid, and wall-vapor interfaces are studied analytically, using classical density functional theory in conjunction with the sharp-kink approximation for the number density profiles of the A and B particles. If in accordance with the so-called mixing rule the strength of the A-B interaction is given by the geometric mean of the strengths of the A-A and the B-B interactions, and similarly the ratio between the wall-A and the wall-B interaction, the scenario, in which the colloid is enclosed by a film of the B-rich liquid, can be excluded. Up to six distinct wetting scenarios are possible, if the above mixing rules for the fluid-wall and for the fluid-fluid interactions are relaxed. The way the space of system parameters is divided into domains corresponding to the six scenarios, and which of the domains actually appear, depends on the signs of the deviations from the mixing rule prescriptions. Relevant domains, corresponding, e.g., to the scenario in which the colloid is enclosed by a film of the B-rich liquid, emerge, if the ratio between the strengths of the wall-A and the wall-B interactions is reduced as compared to the mixing rule prescription, or if the strength of the A-B interaction is increased to values above the one from the mixing rule prescription. The range, within which the contact angle may vary inside the various domains, is also studied.

Symmetry-breaking morphological transitions at chemically nanopatterned walls

We study the structure and morphological changes of fluids that are in contact with solid composites formed by alternating and microscopically wide stripes of two different materials. One type of the stripes interacts with the fluid via long-ranged Lennard-Jones-like potential and tends to be completely wet, while the other type is purely repulsive and thus tends to be completely dry. We consider closed systems with a fixed number of particles that allows for stabilization of fluid configurations breaking the lateral symmetry of the wall potential. These include liquid morphologies corresponding to a sessile drop that is formed by a sequence of bridging transitions that connect neighboring wet regions adsorbed at the attractive stripes. We study the character of the transitions depending on the wall composition, stripes width, and system size. Using a (classical) nonlocal density functional theory (DFT), we show that the transitions between different liquid morphologies are typically weakly first-order but become rounded if the wavelength of the system is lower than a certain critical value L_{c}. We also argue that in the thermodynamic limit, i.e., for macroscopically large systems, the wall becomes wet via an infinite sequence of first-order bridging transitions that are, however, getting rapidly weaker and weaker and eventually become indistinguishable from a continuous process as the size of the bridging drop increases. Finally, we construct the global phase diagram and study the density dependence of the contact angle of the bridging drops using DFT density profiles and a simple macroscopic theory.

Microscopic aspects of wetting using classical density functional theory

Wetting is a rather efficient mechanism for nucleation of a phase (typically liquid) on the interface between two other phases (typically solid and gas). In many experimentally accessible cases of wetting, the interplay between the substrate structure, and the fluid-fluid and fluid-substrate intermolecular interactions brings about an entire 'zoo' of possible fluid configurations, such as liquid films with a thickness of a few nanometers, liquid nanodrops and liquid bridges. These fluid configurations are often associated with phase transitions occurring at the solid-gas interface and at lengths of just several molecular diameters away from the substrate. In this special issue article, we demonstrate how a fully microscopic classical density-functional framework can be applied to the efficient, rational and systematic exploration of the rich phase space of wetting phenomena. We consider a number of model prototype systems such as wetting on a planar wall, a chemically patterned wall and a wedge. Through density-functional computations we demonstrate that for these simply structured substrates the behaviour of the solid-gas interface is already highly complex and non-trivial.

Classical density functional study of wetting transitions on nanopatterned surfaces

Even simple fluids on simple substrates can exhibit very rich surface phase behaviour. To illustrate this, we consider fluid adsorption on a planar wall chemically patterned with a deep stripe of a different material. In this system, two phase transitions compete: unbending and pre-wetting. Using microscopic density-functional theory, we show that, for thin stripes, the lines of these two phase transitions may merge, leading to a new two-dimensional-like wetting transition occurring along the walls. The influence of intermolecular forces and interfacial fluctuations on this phase transition and at complete pre-wetting are considered in detail.

The order of condensation in capillary grooves

We consider capillary condensation in a deep groove of width L. The transition occurs at a pressure p(co)(L) described, for large widths, by the Kelvin equation p(sat) - p(co)(L) = 2σ cosθ/L, where θ is the contact angle at the side walls and σ is the surface tension. The order of the transition is determined by the contact angle of the capped end θcap; it is continuous if the liquid completely wets the cap, and first-order otherwise. When the transition is first-order, corner menisci at the bottom of the capillary lead to a pronounced metastability, determined by a complementary Kelvin equation Δp(L) = 2σ sinθcap/L. On approaching the wetting temperature of the capillary cap, the corner menisci merge and a single meniscus unbinds from the bottom of the groove. Finite-size scaling shifts, crossover behaviour and critical singularities are determined at mean-field level and beyond. Numerical and experimental results showing the continuous nature of condensation for θcap = 0 and the influence of corner menisci on adsorption isotherms are presented.

Fluid bridges confined between chemically nanopatterned solid substrates

We discuss equilibrium properties of classical fluids confined to nanoscopic volumes by solid substrates. The substrates themselves are endowed with wettable chemical patterns of variable symmetry. We develop a thermodynamic description suitable for these highly anisotropic systems. Based upon a combination of Monte Carlo simulations in the grand canonical ensemble and lattice density functional theory at mean-field level we analyze the structure and phase behaviour of the confined fluid. Under suitable thermodynamic conditions the fluid may condense partially in regions controlled by the wettable nanopatterns. The resulting fluid bridges are established as thermodynamic phases and exhibit unique rheological features.

Adsorption of polymers on a brush: Tuning the order of the wetting phase transition

We develop a computational methodology for the direct measurement of a wetting transition and its order via the effective interface potential. The method also allows to estimate contact angles in the nonwet state and to study adsorption isotherms. The proposed methodology is employed in order to study the wetting behavior of polymers on top of a brush consisting of identical polymers. In the absence of long-range forces, the system shows a sequence of nonwet, wet, and nonwet states as the brush density is increased. Including attractive long-range interactions we can make the polymer liquid wet the bush at all grafting densities, and both first- and second-order wetting transitions are observed. The latter case is limited to a small interval of grafting densities where the melt wets the brush in the absence of long-range interactions. Second-order wetting transitions are preceded by a first-order surface transition from a thin to a thick adsorbed layer. The interval of second-order wetting transitions is limited at low grafting densities by a surface critical end point and at high grafting densities by a tricritical wetting point. Our study highlights the rich wetting behavior that results when competing adsorbent-substrate interactions of different scales are tuned over a broad range.

Wetting of rings on a nanopatterned surface: a lattice model study

We perform mean-field density functional theory calculations on a lattice model to study the wetting of a solid substrate decorated with a ring pattern of nanoscale dimensions. We have found three different liquid morphologies on the substrate: a ring morphology where the liquid covers the pattern, a bulge morphology where a droplet is forming on one side of the ring, and a morphology where the liquid forms a cap spanning the nonwetting disk inside the pattern. We investigate the relative stability of these morphologies as a function of the ring size, wall-fluid interaction, and temperature. The results found are in very good agreement with experiments and calculations performed on similar systems at a micrometer length scale. The bulge morphology has also been observed in Monte Carlo simulations of the lattice model. Our results show that (i) morphologies of wetting patterns previously observed on a much larger (microm) scale can also form on a nm length scale, (ii) whether or not this happens depends crucially on the size of the wettable pattern, and (iii) the wettable ring may only be partially wet by the bulge morphology of the fluid. This morphology is a result of a spontaneously broken symmetry in the system.

The validity of Cassie's law: A simple exercise using a simplified model

The contact angle of a macroscopic droplet on a heterogeneous but flat substrate is studied using the interface displacement model which can lead to the augmented Young-Laplace equation. Droplets under the condition of constant volume as well as constant vapor pressure are considered. By assuming a cylindrical liquid-vapor surface (meniscus) and minimizing the total free energy of the interface displacement model, we derive an equation which is similar but different from the well-known Cassie's law. Our modified Cassie's law is essentially the same as the formula obtained previously by Marmur [J. Colloid Interface Sci. 168 (1994) 40]. A few consequences from this modified Cassie's law are briefly described in the following sections of this paper. Several sets of recent experimental results seem to support the validity of our modified Cassie's law.

Extended wedge covariance for wetting and filling transitions

Fluid adsorption on nonplanar and heterogeneous substrates is studied using a simple interfacial model. For systems with short-ranged forces, we find that, by tuning the local strength of the substrate potential, it is possible to find the exact equilibrium interfacial profile as a functional of the wall shape psi x. The tuning of the local substrate potential takes the form of a gauge condition theta x=+/-psi x, where theta x can be interpreted as a local effective contact angle. For wedgelike geometries with asymptotic tilt angle alpha, the midpoint interfacial height and roughness satisfy the same covariance relations previously found for simple linear wedges. For troughlike geometries satisfying the gauge condition, covariance is also found for the two-point correlation function. Predictions for more microscopic Landau and Ising models are also discussed.

Ellipsometric study of adsorption on nanopatterned block copolymer substrates

We report ellipsometrically obtained adsorption isotherms for a carefully chosen test liquid on block copolymer films of Kraton G1650, compared with adsorption isotherms on homogeneous films of the constituent polymers. Standard atomic force microscopy images imply the outer surface of Kraton G1650 is chemically patterned on the nanoscale, but this could instead be a reflection of structure buried beneath a 10 nm layer of the lower energy component. Our test liquid was chosen on the basis that it did not dissolve in either component and in addition that it was nonwetting on the lower energy polymer while forming thick adsorbed films on pure substrates of the higher energy component. Our ellipsometry data for Kraton G1650 rule out the presence of segregation by the lower energy constituent to the outer surface, implying a mixed surface consistent with Cassie's law. We discuss implications of our findings and related work for the outer surface structures of block copolymer films.

On the validity of the Cassie equation via a mean-field free-energy lattice Boltzmann approach

We studied wetting phenomena on heterogeneous surfaces by a mean-field free-energy lattice Boltzmann method recently proposed [Phys. Rev. E 69 (2004) 32,602]. Our results suggest that the Cassie equation in macroscopic contact angle measurements is in general not valid. It was found that the Cassie equation is valid only when the patch size is on the same order of the liquid-vapor interfacial thickness. We also demonstrated that contact angle manifests itself from local surface properties near the contact point and does not result from the specific solid-liquid interactions across the contact area.

Aspects of prewetting at nonplanar surfaces

We employ Monte Carlo simulations in the grand canonical ensemble (GCEMC) to investigate the impact of nonplanarity of a solid substrate on the locus of the prewetting phase transition. The substrate is modelled as a periodic sequence of furrows of depth D and periodicity sx in the x direction; the furrows are infinitely long in the y direction. Our results indicate that a necessary prerequisite for a prewetting transition is the formation of a(n approximately) planar interface between molecularly thin films and an adjacent (bulk) gas. Thus, in general the prewetting transition is shifted to larger chemical potentials because the formation of a planar film-gas interface is more difficult next to a nonplanar compared with a planar solid surface. However, this shift turns out to be nonmonotonic depending on D on account of subtle packing effects manifested in the deviation of the local density Deltarho(x,Deltaz;D) at the nonplanar solid surface from that at a planar substrate. If D becomes sufficiently large prewetting as a discontinuous phase transition is suppressed because inside the furrow a highly ordered film forms that prevents a planar film-gas interface from forming.

Dewetting of conducting polymer inkjet droplets on patterned surfaces

The manufacture of high-performance electronic devices with micrometre or even submicrometre dimensions by solution processing and direct printing, requires the ability to control accurately the flow and spread of functional liquid inks on surfaces. This can be achieved with the help of surface-energy patterns causing inks to be repelled and dewetted from pre-defined regions of the substrate. To exploit this principle for the fabrication of submicrometre device structures, a detailed understanding of the factors causing ink droplets to dewet on patterned surfaces is required. Here, we use hydrophobic surface-energy barriers of different geometries to study the influence of solution viscosity, ink volume, and contact angle on the process of dewetting of inkjet-printed droplets of a water-based conducting polymer. We demonstrate polymer field-effect transistor devices with channel length of 500 nm fabricated by surface-energy-assisted inkjet printing.

Wetting of nanopatterned surfaces: The hexagonal disk surface

Metropolis Monte Carlo simulations are used to investigate the wetting of chemically nanopatterned surfaces, for the case of hexagonal disk patterns where liquid wishes to wet high-energy circular patches but not wet the background surface. We calculate the density profiles of saturated liquid adsorbed on a variety of such substrates, spanning the nanoscale to atomic scale patterns. In addition, statistical mechanical sum rules are used to obtain interfacial order parameters and interfacial free energies. We observe that Cassie's law is typically obeyed, together with an associated breakdown of the mechanical interpretation of Young's equation, for pattern wavelengths greater than 15 molecular diameters. Here, the adsorbed fluid exists as an array of hemi-drops. At about half this wavelength, the breakdown of Cassie's law lies within realistic energy scales and is associated with the unbending of the outer surface of adsorbed films. For atomic scale patterns, the usual interpretation of Young's equation is restored for films thicker than one monolayer. At high chemical contrast, when the monolayer in contact with high-energy regions would prefer to be crystalline, we observe a variety of exotic interfacial phenomena that may have technological significance.