Microscopic determination of correlations in the fluid interfacial region in the presence of liquid-gas asymmetry.

In a recent article, we showed how the properties of the density-density correlation function and its integral, the local structure factor, in the fluid interfacial region, in systems with short-ranged forces, can be understood microscopically by considering the resonances of the local structure factor [A. O. Parry and C. Rascón, Nat. Phys. 15, 287 (2019)NPAHAX1745-247310.1038/s41567-018-0361-z]. Here, we illustrate, using mean-field square-gradient theory and the more microscopic Sullivan density functional model, and how this approach generalizes when there is liquid-gas asymmetry, i.e., when the bulk correlation lengths of the coexisting liquid and gas phases are different. In particular, we are able to express the correlation function exactly as a simple average of contributions arising from two effective Ising-symmetric systems referred to as the symmetric gas and symmetric liquid. When combined with our earlier results, this generates analytical approximations for the correlation function and the local structure factor, which are near indistinguishable from the numerical solution to the Ornstein-Zernike equations over the whole range of wave vectors. Our results highlight how asymmetry affects the correlation function structure and describes the crossover from a long-ranged Goldstone mode to short-ranged properties determined by the local density as the wave vector increases.

Correlation-function structure in square-gradient models of the liquid-gas interface: Exact results and reliable approximations.

In a recent article, we described how the microscopic structure of density-density correlations in the fluid interfacial region, for systems with short-ranged forces, can be understood by considering the resonances of the local structure factor occurring at specific parallel wave vectors q [Nat. Phys. 15, 287 (2019)1745-247310.1038/s41567-018-0361-z]. Here we investigate this further by comparing approximations for the local structure factor and pair correlation function against three new examples of analytically solvable models within square-gradient theory. Our analysis further demonstrates that these approximations describe the pair correlation function and structure factor across the whole spectrum of wave vectors, encapsulating the crossover from the Goldstone mode divergence (at small q) to bulklike behavior (at larger q). As shown, these approximations are exact for some square-gradient model potentials and never more than a few percent inaccurate for the others. Additionally, we show that they describe very accurately the correlation function structure for a model describing an interface near a tricritical point. In this case, there are no analytical solutions for the correlation functions, but the approximations are nearly indistinguishable from the numerical solutions of the Ornstein-Zernike equation.

First-order wedge wetting revisited.

We consider a fluid adsorbed in a wedge made from walls that exhibit a first-order wetting transition and revisit the argument as to why and how the pre-filling and pre-wetting coexistence lines merge when the opening angle is increased approaching the planar geometry. We clarify the nature of the possible surface phase diagrams, pointing out the connection with complete pre-wetting, and show that the merging of the coexistence lines lead to new interfacial transitions. These occur along the side walls and are associated with the unbinding of the thin-thick interface, rather than the liquid-gas interface (meniscus), from the wedge apex. When fluctuation effects, together with the influence of dispersion forces are included, these transitions display strong non-universal critical singularities that depend on the opening angle itself. Similar phenomena are also shown to occur for adsorption near an apex tip.

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.

Complete prewetting.

We study continuous interfacial transitions, analagous to two-dimensional complete wetting, associated with the first-order prewetting line, which can occur on steps, patterned walls, grooves and wedges, and which are sensitive to both the range of the intermolecular forces and interfacial fluctuation effects. These transitions compete with wetting, filling and condensation producing very rich phase diagrams even for relatively simple prototypical geometries. Using microscopic classical density functional theory to model systems with realistic Lennard-Jones fluid-fluid and fluid-substrate intermolecular potentials, we compute mean-field fluid density profiles, adsorption isotherms and phase diagrams for a variety of confining geometries.

The local structure factor near an interface; beyond extended capillary-wave models.

We investigate the local structure factor S (z;q) at a free liquid-gas interface in systems with short-ranged intermolecular forces and determine the corrections to the leading-order, capillary-wave-like, Goldstone mode divergence of S (z;q) known to occur for parallel (i.e. measured along the interface) wavevectors [Formula: see text]. We show from explicit solution of the inhomogeneous Ornstein-Zernike equation that for distances z far from the interface, where the profile decays exponentially, S (z;q) splits unambiguously into bulk and interfacial contributions. On each side of the interface, the interfacial contributions can be characterised by distinct liquid and gas wavevector dependent surface tensions, [Formula: see text] and [Formula: see text], which are determined solely by the bulk two-body and three-body direct correlation functions. At high temperatures, the wavevector dependence simplifies and is determined almost entirely by the appropriate bulk structure factor, leading to positive rigidity coefficients. Our predictions are confirmed by explicit calculation of S (z;q) within square-gradient theory and the Sullivan model. The results for the latter predict a striking temperature dependence for [Formula: see text] and [Formula: see text], and have implications for fluctuation effects. Our results account quantitatively for the findings of a recent very extensive simulation study by Höfling and Dietrich of the total structure factor in the interfacial region, in a system with a cut-off Lennard-Jones potential, in sharp contrast to extended capillary-wave models which failed completely to describe the simulation results.

Liquid-gas asymmetry and the wave-vector-dependent surface tension.

Attempts to extend the capillary-wave theory of fluid interfacial fluctuations to microscopic wavelengths, by introducing an effective wave-vector (q)-dependent surface tension σeff(q), have encountered difficulties. There is no consensus as to even the shape of σeff(q). By analyzing a simple density functional model of the liquid-gas interface, we identify different schemes for separating microscopic observables into background and interfacial contributions. In order for the backgrounds of the density-density correlation function and local structure factor to have a consistent and physically meaningful interpretation in terms of weighted bulk gas and liquid contributions, the background of the total structure factor must be characterized by a microscopic q-dependent length ζ(q) not identified previously. The necessity of including the q dependence of ζ(q) is illustrated explicitly in our model and has wider implications; i.e., in typical experimental and simulation studies, an indeterminacy in ζ(q) will always be present, reminiscent of the cutoff used in capillary-wave theory. This leads inevitably to a large uncertainty in the q dependence of σeff(q).

Capillary contact angle in a completely wet groove.

We consider the phase equilibria of a fluid confined in a deep capillary groove of width L with identical side walls and a bottom made of a different material. All walls are completely wet by the liquid. Using density functional theory and interfacial models, we show that the meniscus separating liquid and gas phases at two phase capillary coexistence meets the bottom capped end of the groove at a capillary contact angle θ(cap)(L) which depends on the difference between the Hamaker constants. If the bottom wall has a weaker wall-fluid attraction than the side walls, then θ(cap) > 0 even though all the isolated walls are themselves completely wet. This alters the capillary condensation transition which is now first order; this would be continuous in a capped capillary made wholly of either type of material. We show that the capillary contact angle θ(cap)(L) vanishes in two limits, corresponding to different capillary wetting transitions. These occur as the width (i) becomes macroscopically large, and (ii) is reduced to a microscopic value determined by the difference in Hamaker constants. This second wetting transition is characterized by large scale fluctuations and essential critical singularities arising from marginal interfacial interactions.

Pair correlation functions and the wavevector-dependent surface tension in a simple density functional treatment of the liquid-vapour interface.

We study the density-density correlation function G(r, r') in the interfacial region of a fluid (or Ising-like magnet) with short-ranged interactions using square gradient density functional theory. Adopting a simple double parabola approximation for the bulk free-energy density, we first show that the parallel Fourier transform G(z, z'; q) and local structure factor S(z; q) separate into bulk and excess contributions. We attempt to account for both contributions by deriving an interfacial Hamiltonian, characterised by a wavevector dependent surface tension σ(q), and then reconstructing density correlations from correlations in the interface position. We show that the standard crossing criterion identification of the interface, as a surface of fixed density (or magnetization), does not explain the separation of G(z, z'; q) and the form of the excess contribution. We propose an alternative definition of the interface position based on the properties of correlations between points that 'float' with the surface and show that this describes the full q and z dependence of the excess contributions to both G and S. However, neither the 'crossing-criterion' nor the new 'floating interface' definition of σ(q) are quantities directly measurable from the total structure factor S(tot)(q) which contains additional q dependence arising from the non-local relation between fluctuations in the interfacial position and local density. Since it is the total structure factor that is measured experimentally or in simulations, our results have repercussions for earlier attempts to extract and interpret σ(q).

Renormalization group calculations for wetting transitions of infinite order and continuously varying order: local interface Hamiltonian approach.

We study the effect of thermal fluctuations on the wetting phase transitions of infinite order and of continuously varying order, recently discovered within a mean-field density-functional model for three-phase equilibria in systems with short-range forces and a two-component order parameter. Using linear functional renormalization group calculations within a local interface Hamiltonian approach, we show that the infinite-order transitions are robust. The exponential singularity (implying 2-α(s)=∞) of the surface free energy excess at infinite-order wetting as well as the precise algebraic divergence (with ß(s)=-1) of the wetting layer thickness are not modified as long as ω<2, with ω the dimensionless wetting parameter that measures the strength of thermal fluctuations. The interface width diverges algebraically and universally (with ν([perpendicular])=1/2). In contrast, the nonuniversal critical wetting transitions of finite but continuously varying order are modified when thermal fluctuations are taken into account, in line with predictions from earlier calculations on similar models displaying weak, intermediate, and strong fluctuation regimes.

Capillary emptying and short-range wetting.

We consider a liquid trapped in a narrow horizontal capillary, under the influence of gravity. As the slit is widened, the meniscus, separating the capillary liquid from gas, deforms and develops a long tongue extending along the bottom wall. As a critical slit width is approached, the length of the tongue diverges continuously, leading to the emptying of the capillary. We show that the critical singularities characterizing emptying are the same as those at short-ranged wetting transitions, but at a scale set by the capillary length rather than the bulk correlation length. These meso- or macroscopic versions of both complete and critical wetting are observable in the laboratory and are studied here using a colloid-polymer mixture.

The order of filling transitions in acute wedges.

Using a square-gradient density functional model we test the prediction that the filling transition for a fluid in a wedge geometry changes from continuous to first-order as the wedge becomes more acute. Our numerical findings confirm such a change of order, but the value of the tilt angle at which it occurs, α* ≈ 45°, is considerably smaller than the original theoretical prediction. We critically reassess this work, which was based on allowing for the self-interaction of the fluid interface, and argue that the interfacial curvature and effective wavevector dependent surface tension can further lower the predicted value of α*, in keeping with our numerical findings. Interfacial fluctuation effects, occurring beyond mean-field level, are also discussed using effective Hamiltonian theory and are shown to substantially increase the value of α*.

An interpretation of covariance relations for wetting and wedge filling transitions.

Recent studies have shown that there exist precise connections (or covariance relations) between adsorption properties for substrates with different shapes. This occurs, for example, when a fluid is adsorbed in a linear wedge. In this case, the influence of the geometry is to shift effectively the contact angle from theta to theta-alpha, where alpha is the tilt angle. Despite the fact that these relations are obeyed both at mean-field level and also exactly in two dimensions (when fluctuation effects dominate), their fundamental origin has been unclear. Here, we show that they can be traced to a symmetry present in interfacial Hamiltonian models, and further relate this to surface thermodynamics and the nonlocal nature of interfacial interactions in systems with short-ranged forces.

Derivation of a non-local interfacial model for 3D wetting in an external field.

We extend recent studies of 3D short-ranged wetting transitions by deriving an interfacial Hamiltonian in the presence of an arbitrary external field. The binding potential functional, describing the interaction of the interface and the substrate, can still be written in a diagrammatic form, but now includes new classes of diagrams due to the coupling to the external potential, which are determined exactly. Applications to systems with long-ranged (algebraically decaying) and short-ranged (exponentially decaying) external potentials are considered at length. We show how the familiar 'sharp-kink' approximation to the binding potential emerges, and determine the corrections to this arising from interactions between bulk-like fluctuations and the external field. A connection is made with earlier local effective interfacial Hamiltonian approaches. It is shown that, for the case of an exponentially decaying potential, non-local effects have a particularly strong influence on the approach to the critical regime at second-order wetting transitions, even when they appear to be sub-dominant. This is confirmed by Monte Carlo simulation studies of a discretized version of a non-local interfacial model.

3D short-range wetting and nonlocality.

Analysis of a microscopic Landau-Ginzburg-Wilson model of 3D short-ranged wetting shows that correlation functions are characterized by two length scales, not one, as previously thought. This has a simple diagrammatic explanation using a nonlocal interfacial Hamiltonian and yields a thermodynamically consistent theory of wetting in keeping with exact sum rules. For critical wetting the second length serves to lower the cutoff in the spectrum of interfacial fluctuations determining the repulsion from the wall. We show how this corrects previous renormalization group predictions for fluctuation effects, based on local interfacial Hamiltonians. In particular, lowering the cutoff leads to a substantial reduction in the effective value of the wetting parameter and prevents the transition being driven first order. Quantitative comparison with Ising model simulation studies due to Binder, Landau, and co-workers is also made.

Intrusion of fluids into nanogrooves: how geometry determines the shape of the gas-liquid interface.

We study the shape of gas-liquid interfaces forming inside rectangular nanogrooves (i.e., slit-pores capped on one end). On account of purely repulsive fluid-substrate interactions the confining walls are dry (i.e., wet by vapor) and a liquid-vapor interface intrudes into the nanogrooves to a distance determined by the pressure (i.e., chemical potential). By means of Monte Carlo simulations in the grand-canonical ensemble (GCEMC) we obtain the density rho(z) along the midline (x = 0) of the nanogroove for various geometries (i.e., depths D and widths L) of the nanogroove. We analyze the density profiles with the aid of an analytic expression which we obtain through a transfer-matrix treatment of a one-dimensional effective interface Hamiltonian. Besides geometrical parameters such as D and L , the resulting analytic expression depends on temperature T , densities of coexisting gas and liquid phases in the bulk rho g,l(x) and the interfacial tension gamma. The latter three quantities are determined in independent molecular dynamics simulations of planar gas-liquid interfaces. Our results indicate that the analytic formula provides an excellent representation of rho(z) as long as L is sufficiently small. At larger L the meniscus of the intruding liquid flattens. Under these conditions the transfer-matrix analysis is no longer adequate and the agreement between GCEMC data and the analytic treatment is less satisfactory.

Condensation in a capped capillary is a continuous critical phenomenon.

We show that condensation in a capped capillary slit is a continuous interfacial critical phenomenon, related intimately to several other surface phase transitions. In three dimensions, the adsorption and desorption branches correspond to the unbinding of the meniscus from the cap and opening, respectively, and are equivalent to 2D-like complete-wetting transitions. For dispersion forces, the singularities on the two branches are distinct, owing to the different interplay of geometry and intermolecular forces. In two dimensions we establish precise connection, or covariance, with 2D critical-wetting and wedge-filling transitions: i.e., we establish that certain interfacial properties in very different geometries are identical. Our predictions of universal scaling and covariance in finite capillaries are supported by extensive Ising model simulation studies in two and three dimensions.

Tests of nonuniversality and finite-size scaling for two-dimensional wetting with long-ranged forces.

Effective Hamiltonian models predict nonuniversal critical singularities for two-dimensional wetting transitions with marginal long-ranged forces. We test these predictions by studying interfacial delocalization transitions in an infinitely long Ising strip, of width L (lattice spacings), with external fields that are long ranged and have opposite signs at each surface. Finite-size scaling suggests that the shift of the delocalization temperature T(c)(L) below the (semi-infinite) wetting temperature T(w) scales as L(-1/beta(s)) with beta(s) the adsorption critical exponent. Density-matrix renormalization-group methods allow us to study the behavior of T(c)(L) for L up to several hundred lattice spacings. For short-ranged forces the method recovers the universal value of beta(s)=1 known from the exact solution. While marginal long-ranged forces strongly influence the finite-size scaling of T(c)(L) , the extrapolated asymptotic value for the exponent beta(s) does not appear to confirm the predicted nonuniversality, but instead approaches the same universal value representative of systems with short-ranged forces.

Derivation of a non-local interfacial Hamiltonian for short-ranged wetting: II. General diagrammatic structure.

In our first paper, we showed how a non-local effective Hamiltonian for short-ranged wetting may be derived from an underlying Landau-Ginzburg-Wilson model. Here, we combine the Green's function method with standard perturbation theory to determine the general diagrammatic form of the binding potential functional beyond the double-parabola approximation for the Landau-Ginzburg-Wilson bulk potential. The main influence of cubic and quartic interactions is simply to alter the coefficients of the double parabola-like zigzag diagrams and also to introduce curvature and tube-interaction corrections (also represented diagrammatically), which are of minor importance. Non-locality generates effective long-ranged many-body interfacial interactions due to the reflection of tube-like fluctuations from the wall. Alternative wall boundary conditions (with a surface field and enhancement) and the diagrammatic description of tricritical wetting are also discussed.

Point tension in adsorption at a chemically inhomogeneous substrate in two dimensions.

We study adsorption of liquid at a one-dimensional substrate composed of a single chemical inhomogeneity of width 2L placed on an otherwise homogeneous, planar, solid surface. The excess point free energy eta(L,T) associated with the adsorbed layer's inhomogeneity induced by the substrate's chemical structure is calculated within exact continuum transfer-matrix approach. It is shown that the way eta(L,T) varies with L depends sensitively on the temperature regime. It exhibits logarithmic divergence as a function of L in the limit L-->infinity for temperatures such that the chemical inhomogeneity is completely wetted by the liquid. In the opposite case eta(L,T) converges for large L to 2eta0, where eta0 is the corresponding point tension, and the dominant L-dependent correction to 2eta0 decays exponentially. The interaction between the liquid layer inhomogeneities at -L and L for the two temperature regimes is discussed and compared to earlier mean-field theory predictions.