Interfacial Hamiltonian Model Research Articles

Critical effects and scaling at meniscus osculation transitions.

We propose a simple scaling theory describing critical effects at rounded meniscus osculation transitions which occur when the Laplace radius of a condensed macroscopic drop of liquid coincides with the local radius of curvature R_{w} in a confining parabolic geometry. We argue that the exponent β_{osc} characterizing the scale of the interfacial height ℓ_{0}∝R_{w}^{β_{osc}} at osculation, for large R_{w}, falls into two regimes representing fluctuation-dominated and mean-field-like behavior, respectively. These two regimes are separated by an upper critical dimension, which is determined here explicitly and depends on the range of the intermolecular forces. In the fluctuation-dominated regime, representing the universality class of systems with short-range forces, the exponent is related to the value of the interfacial wandering exponent ζ by β_{osc}=3ζ/(4-ζ). In contrast, in the mean-field regime, which was not previously identified and which occurs for systems with longer-range forces (and higher dimensions), the exponent β_{osc} takes the same value as the exponent β_{s}^{co} for complete wetting, which is determined directly by the intermolecular forces. The prediction β_{osc}=3/7 in d=2 for systems with short-range forces (corresponding to ζ=1/2) is confirmed using an interfacial Hamiltonian model which determines the exact scaling form for the decay of the interfacial height probability distribution function. A numerical study in d=3, based on a microscopic model density-functional theory, determines that β_{osc}≈β_{s}^{co}≈0.326 close to the predicted value of 1/3 appropriate to the mean-field regime for dispersion forces.

Height of a liquid drop on a wetting stripe.

Adsorption of liquid on a planar wall decorated by a hydrophilic stripe of width L is considered. Under the condition that the wall is only partially wet (or dry) while the stripe tends to be wet completely, a liquid drop is formed above the stripe. The maximum height ℓ_{m}(δμ) of the drop depends on the stripe width L and the chemical potential departure from saturation δμ where it adopts the value ℓ_{0}=ℓ_{m}(0). Assuming a long-range potential of van der Waals type exerted by the stripe, the interfacial Hamiltonian model is used to show that ℓ_{0} is approached linearly with δμ with a slope which scales as L^{2} over the region satisfying L≲ξ_{∥}, where ξ_{∥} is the parallel correlation function pertinent to the stripe. This suggests that near the saturation there exists a universal curve ℓ_{m}(δμ) to which the adsorption isotherms corresponding to different values of L all collapse when appropriately rescaled. Although the series expansion based on the interfacial Hamiltonian model can be formed by considering higher order terms, a more appropriate approximation in the form of a rational function based on scaling arguments is proposed. The approximation is based on exact asymptotic results, namely, that ℓ_{m}∼δμ^{-1/3} for L→∞ and that ℓ_{m} obeys the correct δμ→0 behavior in line with the results of the interfacial Hamiltonian model. All the predictions are verified by the comparison with a microscopic density functional theory (DFT) and, in particular, the rational function approximation-even in its simplest form-is shown to be in a very reasonable agreement with DFT for a broad range of both δμ and L.

Curvature corrections to the nonlocal interfacial model for short-ranged forces.

In this paper we revisit the derivation of a nonlocal interfacial Hamiltonian model for systems with short-ranged intermolecular forces. Starting from a microscopic Landau-Ginzburg-Wilson Hamiltonian with a double-parabola potential, we reformulate the derivation of the interfacial model using a rigorous boundary integral approach. This is done for three scenarios: a single fluid phase in contact with a nonplanar substrate (i.e., wall); a free interface separating coexisting fluid phases (say, liquid and gas); and finally a liquid-gas interface in contact with a nonplanar confining wall, as is applicable to wetting phenomena. For the first two cases our approaches identifies the correct form of the curvature corrections to the free energy and, for the case of a free interface, it allows us to recast these as an interfacial self-interaction as conjectured previously in the literature. When the interface is in contact with a substrate our approach similarly identifies curvature corrections to the nonlocal binding potential, describing the interaction of the interface and wall, for which we propose a generalized and improved diagrammatic formulation.

Crossover scaling of apparent first-order wetting in two-dimensional systems with short-ranged forces.

Recent analyses of wetting in the semi-infinite two-dimensional Ising model, extended to include both a surface coupling enhancement and a surface field, have shown that the wetting transition may be effectively first-order and that surprisingly the surface susceptibility develops a divergence described by an anomalous exponent with value γ_{11}^{eff}=3/2. We reproduce these results using an interfacial Hamiltonian model making a connection with previous studies of two-dimensional wetting, and we show that they follow from the simple crossover scaling of the singular contribution to the surface free-energy, which describes the change from apparent first-order to continuous (critical) wetting due to interfacial tunneling. The crossover scaling functions are calculated explicitly within both the strong-fluctuation and intermediate-fluctuation regimes, and they determine uniquely and more generally the value of γ_{11}^{eff}, which is nonuniversal for the latter regime. The location and the rounding of a line of pseudo-prewetting transitions occurring above the wetting temperature and off bulk coexistence, together with the crossover scaling of the parallel correlation length, are also discussed in detail.

Geometrical Aspects of Drying in a Capped Capillary: a DFT Study

The order of the liquid–solid phase transition of a fluid confined by a capillary of two parallel walls can be changed from first to second order if the capillary is capped at one end by a third wall. This change of order of the phase transition was predicted using an effective interfacial Hamiltonian model and was recently confirmed by a microscopic density functional theory study. In this study some geometrical aspects of the continous transition in a capped capillary are considered within the framework of density functional theory.

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.

The critical wetting saga: how to draw the correct conclusion

A long-standing problem in condensed matter physics concerns the nature ofthe critical wetting phase transition in the Ising model or, more generally, in3D systems with short-ranged forces. This is of fundamental interest because3D corresponds to the upper critical dimension of the transition and it is notclear a priori whether the behaviour of the system will be mean-field-like orfluctuation-dominated. Renormalization group studies of the standard coarse-grainedeffective interfacial Hamiltonian model famously predict strong non-universalcritical exponents which depend on the value of the so-called wetting parameterω. However, these predictions are at odds with extensive Monte Carlo simulations of wettingin the Ising model, due to Binder, Landau and coworkers, which appear to bemore mean-field-like. Further amendments to the interfacial Hamiltonian, whichincluded the presence of a position-dependent stiffness, worsened the problemby paradoxically predicting fluctuation-induced first-order wetting behaviour.Here we show from re-analysis of a microscopic Landau–Ginzburg–Wilson model of 3Dshort-ranged wetting that correlation functions are characterized by two diverging parallellength scales, not one, as previously thought. This has a simple diagrammatic explanationusing a non-local interfacial Hamiltonian and yields a thermodynamically consistent theoryof wetting in keeping with exact sum rules. The non-local model crucially containslong-ranged two-body interfacial interactions, characterized by the new length scale, whichwere missing in earlier treatments. For critical wetting the second length cuts off thespectrum of interfacial fluctuations determining the repulsion from the wall. We show howthis corrects previous renormalization group predictions for fluctuation effects, basedon local interfacial Hamiltonians. In particular, lowering the cut-off leads to asubstantial reduction in the effective value of the wetting parameter controllingthe non-universality and also prevents the transition being driven first-order.Quantitative comparison with the Ising model simulation studies is also made.

Derivation of a non-local interfacial Hamiltonian for short-ranged wetting: I. Double-parabolaapproximation

We derive a non-local effective interfacial Hamiltonian model for short-ranged wettingphenomena using a Green’s function method. The Hamiltonian is characterized by abinding potential functional and is accurate to exponentially small order in the radii ofcurvature of the interface and the bounding wall. The functional has an elegantdiagrammatic representation in terms of planar graphs which represent different classes oftube-like fluctuations connecting the interface and wall. For the particular cases of planar,spherical and cylindrical interfacial (and wall) configurations, the binding potentialfunctional can be evaluated exactly. More generally, the non-local functional naturallyexplains the origin of the effective position-dependent stiffness coefficient in thesmall-gradient limit.

Interfacial structure at a two-dimensional wedge filling transition: Exact results and a renormalization group study.

Interfacial structure and correlation functions near a two-dimensional wedge filling transition are studied using effective interfacial Hamiltonian models. An exact solution for short range binding potentials and results for Kratzer binding potentials show that sufficiently close to the filling transition a new length scale emerges and controls the decay of the interfacial profile relative to the substrate and the correlations between interfacial positions above different positions. This new length scale is much larger than the intrinsic interfacial correlation length, and it is related geometrically to the average value of the interfacial position above the wedge midpoint. The interfacial behavior is consistent with a breather mode fluctuation picture, which is shown to emerge from an exact decimation functional renormalization group scheme that keeps the geometry invariant.

Three-dimensional wedge filling in ordered and disordered systems

We investigate interfacial structural and fluctuation effects occurring at continuous fillingtransitions in 3D wedge geometries. We show that fluctuation-induced wedgecovariance relations that have been reported recently for 2D filling and wetting havemean-field or classical analogues that apply to higher-dimensional systems. Classicalwedge covariance emerges from analysis of filling in shallow wedges based on asimple interfacial Hamiltonian model and is supported by detailed numericalinvestigations of filling within a more microscopic Landau-like density functionaltheory. Evidence is presented that classical wedge covariance is also obeyed forfilling in more acute wedges in the asymptotic critical regime. For sufficientlyshort-ranged forces mean-field predictions for the filling critical exponents andcovariance are destroyed by pseudo-one-dimensional interfacial fluctuations. Weargue that in this filling fluctuation regime the critical exponents describing thedivergence of length scales are related to values of the interfacial wandering exponentζ(d) defined for planar interfaces in (bulk) two-dimensional(d = 2) andthree-dimensional (d = 3) systems. For the interfacial height , with θ the contactangle and α the wedgetilt angle, we find βw = ζ(2)/2(1−ζ(3)). For pure systems (thermal disorder) we recover the known resultβw = 1/4 predicted by interfacial Hamiltonian studies whilst for random-bond disorder we predictthe universal critical exponent even in the presence of dispersion forces. We revisit the transfer matrix theory ofthree-dimensional filling based on an effective interfacial Hamiltonian model and discussthe interplay between breather, tilt and torsional interfacial fluctuations. We show that thecoupling of the modes allows the problem to be mapped onto a quantum mechanicalproblem as conjectured by previous authors. The form of the interfacial height probabilitydistribution function predicted by the transfer matrix approach is shown to be consistentwith scaling and thermodynamic requirements for distances close to and far from the wedgebottom respectively.

Wedgecovariance for two-dimensional filling and wetting

A comprehensive theory of interfacial fluctuation effectsoccurring at two-dimensional wedge (corner) filling transitions in pure(thermal disorder) and impure (random bond disorder) systems ispresented. Scaling theory and the explicit results of transfermatrix and replica trick studies of interfacial Hamiltonianmodels reveal that, for almost all examples of intermolecularforces, the critical behaviour at filling isfluctuation dominated, characterized by universal criticalexponents and scaling functions that depend only on thewandering exponent ζ. Within this filling-fluctuation (FFL) regime, the critical behaviour of themidpoint interfacial height, probability distribution function,local compressibility and wedge free energy are identical tocorresponding quantities predicted for the strong-fluctuation (SFL) regime for critical wetting transitions atplanar walls. In particular the wedge free energy is related tothe SFL regime point tension, which is calculated for systemswith random bond disorder using the replica trick. Theconnection with the SFL regime for all these quantities can beexpressed precisely in terms of special wedge covariancerelations, which complement standard scaling theory and restrictthe allowed values of the critical exponents for both FFLfilling and SFL critical wetting. The predictions for the valuesof the exponents in the SFL regime recover earlier results basedon random walk arguments. The covariance of the wedge freeenergy leads to a new, general relation for the SFL regime pointtension, which derives the conjectured Indekeu-Robledo criticalexponent relation and also explains the origin of thelogarithmic singularity for pure systems known from exact Isingstudies due to Abraham and co-workers. Wedge covariance is alsoused to predict the numerical values of critical exponents andposition dependence of universal one-point functions for puresystems.

A disorder point for filling transitions in 1+1 dimensions

We study a continuum interfacial Hamiltonian model of fluid adsorption in a (1+1)-dimensional wedge geometry, which is known to exhibit a filling transition when the contact angle θπ = α, with α the wedge angle. We extend the transfer matrix analysis of the model to calculate the interfacial height probability distribution function P(l;x), for arbitrary positions x along the wedge. The asymptotics of this function reveal a fluctuation-induced disorder point (non-thermodynamic singularity) that occurs prior to filling when θπ = 2α, where there is a change of length scales determining the decay of P(l;x).

Two-dimensional filling in ordered and disordered systems

We consider filling or wedge-wetting transitions occurring in a (1 + 1)-dimensional wedge geometry with both thermal and random-bonddisorder using effective interfacial Hamiltonian models. Forordered systems the problem may be solved using transfer-matrixmethods for quite arbitrary choices of interfacial binding potentialand gives a complete classification of the possible criticalbehaviours. For random bonds the transition is studied forshort-ranged forces using the replica trick and the wedge-wettingcritical exponents are identified. Our results establish aremarkable relation between the mid-point height probabilitydistribution PF(l0) at filling transitions and the appropriatelydefined height distribution function Pπ(l;θπ) atplanar wetting transitions. We observe that provided the wettingspecific heat component αs = 0, then, in the scaling limit,PF(l0) = Pπ(l;θπ-α)where θπ is the contact angle and α is the tilt angleof the wedge. This relation completely determines the allowed values ofthe filling critical exponents in the fluctuation-dominatedregimes. Conjectures regarding interfacial fluctuation effects infinite-size two-dimensional systems are also made.

Wetting on non-planar and heterogeneous substrates

We report results of wetting on non-planar and heterogeneous surfaces calculated from an effective interfacial Hamiltonian model. The lack of translational invariance along the substrate induces a series of structural changes of the interface such as unbending and a number of non-thermodynamic singularities and can modify the location of the wetting transition. We show that the order of the wetting transition in the planar homogeneous system strongly affects the behaviour of the non-planar and heterogeneous surfaces.

Universality for 2D Wedge Wetting

We study 2D wedge wetting using a continuum interfacial Hamiltonian model which is solved by transfer-matrix methods. For arbitrary binding potentials, we are able to exactly calculate the wedge free-energy and interface height distribution function and, thus, can completely classify all types of critical behaviour. We show that critical filling is characterized by strongly universal fluctuation dominated critical exponents, whilst complete filling is determined by the geometry rather than fluctuation effects. Related phenomena for interface depinning from defect lines in the bulk are also considered.

Coupled Hamiltonians and three-dimensional short-range wetting transitions

We address three problems faced by effective interfacial Hamiltonian models of wetting based on a single collective coordinate ℓ( y ) representing the position of the unbinding fluid interface. Problems (P1) and (P2) refer to the predictions of non-universality at the upper critical dimension d=3 at critical and complete wetting, respectively, which are not borne out by Ising model simulation studies. (P3) relates to mean-field correlation function structure in the underlying continuum Landau model. Building on earlier work by Parry and Boulter we investigate the hypothesis that these concerns arise due to the coupling of order parameter fluctuations near the unbinding interface and wall. For quite general choices of collective coordinates X i( y ) we show that arbitrary two-field models H[ X 1, X 2] can recover the required anomalous structure of mean-field correlation functions (P3). To go beyond mean-field theory we introduce a set H of Hamiltonians based on proper collective coordinates s( y ) near the wall which have both interfacial and spin-like components. We argue that an optimum model H[s,ℓ]∈ H , in which the degree of coupling is controlled by an angle like variable δ ∗ , best describes the non-universality of the Ising model and investigate its critical behaviour. For critical wetting the appropriate Ginzburg criterion shows that the true asymptotic critical regime for the local susceptibility χ 1 is dramatically reduced consistent with observations of mean-field behaviour in simulations (P1). For complete wetting the model yields a precise expression for the temperature dependence of the renormalised critical amplitude θ in good agreement with simulations (P2). We highlight the importance of a new wetting parameter which describes the physics that emerges due to the coupling effects.

Universal and non-universal short-distance expansions in the strong, weak and intermediate fluctuation regimes of critical wetting

The author studies the scaling properties of the density profile rho (z) of a fluid near a wall undergoing a critical wetting transition in the strong-fluctuation and in the weak-fluctuation scaling regimes. A critical exponent theta describing the short-distance algebraic decay of the density profile (for distances z normal to the wall much less than the perpendicular correlation length zeta perpendicular to ) is calculated explicitly in dimension d=2 using an effective interfacial Hamiltonian model. The exponent theta is found to be universal in the strong-fluctuation regime ( theta =1) and in the weak-fluctuation regime ( theta =3). In contrast to the other standard critical exponents theta is non-universal at the weak-fluctuation/mean-field borderline. The author discusses the origin of the universality of theta in the different fluctuation regimes and derives expressions for theta , valid for general d, in terms of known critical exponents.

Short-distance expansion for the local susceptibility and pair correlation function at continuous wetting transitions

A scaling ansatz is proposed for the density profile rho (z) (with z the distance normal from the substrate) for a fluid undergoing a continuous wetting transition at a wall. The ansatz predicts that, in the fluctuation dominated regime, the local susceptibility ( delta rho (z)/ delta mu )T, with mu the chemical potential, has a simple power law position dependence for short distances z<< zeta perpendicular to , the perpendicular correlation length. The power law is determined by the wetting critical exponents, which is different for complete and critical wetting. The scaling predictions are confirmed in dimension d=2 by explicit analysis for an interfacial Hamiltonian model of the wetting transition. For critical wetting a scaling ansatz for the form of the transverse moments of the two-point correlation function G is postulated which now yields the z1, z2 dependence of G in terms of critical exponents. Calculations again confirm the scaling theory for d=2. The analysis highlights the different qualitative and quantitative features of the response functions for critical and complete wetting for fluids with short-ranged forces.