Two-stage capillary condensation in pores with structured walls: A nonlocal density functional study

Phase behavior of fluids in undulated nanopores

The geometry of walls forming a narrow pore may qualitatively affect the phase behavior of the confined fluid. Specifically, the nature of condensation in nanopores formed of sinusoidally shaped walls (with amplitude A and period P) is governed by the wall mean separation L as follows. For L>L_{t}, where L_{t} increases with A, the pores exhibit standard capillary condensation similar to planar slits. In contrast, for L<L_{t}, the condensation occurs in two steps, such that the fluid first condenses locally via bridging transition connecting adjacent crests of the walls, before it condenses globally. For the marginal value of L=L_{t}, all the three phases (gaslike, bridge, and liquidlike) may coexist. We show that the locations of the phase transitions can be described using geometric arguments leading to modified Kelvin equations. However, for completely wet walls, to which we focus on, the phase boundaries are shifted significantly due to the presence of wetting layers. In order to take this into account, mesoscopic corrections to the macroscopic theory are proposed. The resulting predictions are shown to be in a very good agreement with a density-functional theory even for molecularly narrow pores. The limits of stability of the bridge phase, controlled by the pore geometry, is also discussed in some detail.

Breaking Cassie's Law for Condensation in a Nanopatterned Slit

We study the phase transitions of a fluid confined in a capillary slit made from two adjacent walls, each of which are a periodic composite of stripes of two different materials. For wide slits the capillary condensation occurs at a pressure which is described accurately by a combination of the Kelvin equation and the Cassie law for an averaged contact angle. However, for narrow slits the condensation occurs in two steps involving an intermediate bridging phase, with the corresponding pressures described by two new Kelvin equations. These are characterised by different contact angles due to interfacial pinning, with one larger and one smaller than the Cassie angle. We determine the triple point and predict two types of dispersion force induced Derjaguin-like corrections due to mesoscopic volume reduction and the singular free-energy contribution from nanodroplets and bubbles. We test these predictions using a fully microscopic density functional model which confirms their validity even for molecularly narrow slits. Analogous mesoscopic corrections are also predicted for two-dimensional systems arising from thermally induced interfacial wandering.

Three-Phase Fluid Coexistence in Heterogenous Slits

We study the competition between local (bridging) and global condensation of fluid in a chemically heterogeneous capillary slit made from two parallel adjacent walls each patterned with a single stripe. Using a mesoscopic modified Kelvin equation, which determines the shape of the menisci pinned at the stripe edges in the bridge phase, we determine the conditions under which the local bridging transition precedes capillary condensation as the pressure (or chemical potential) is increased. Provided the contact angle of the stripe is less than that of the outer wall we show that triple points, where evaporated, locally condensed, and globally condensed states all coexist are possible depending on the value of the aspect ratio a=L/H, where H is the stripe width and L the wall separation. In particular, for a capillary made from completely dry walls patterned with completely wet stripes the condition for the triple point occurs when the aspect ratio takes its maximum possible value 8/π. These predictions are tested using a fully microscopic classical density functional theory and shown to be remarkably accurate even for molecularly narrow slits. The qualitative differences with local and global condensation in heterogeneous cylindrical pores are also highlighted.

Conformal Sites Theory for Adsorbed Films on Energetically Heterogeneous Surfaces

We present a conformal sites theory for a solid substrate whose surface is both geometrically and energetically heterogeneous and that interacts with an adsorbed film. The theory is based on a perturbation expansion for the grand potential of a real system with a rough surface about that of a reference system with an ideal reference surface, thus mapping the real system onto a much simpler interfacial system. The expansion is in powers of the intermolecular potential parameters, and leads to mixing rules for the potential parameters of the reference system. Grand canonical Monte Carlo simulations for the adsorption of argon at 87.3 K, carbon dioxide at 273 K, and water vapor at 298 K on heterogeneous carbon surfaces are investigated to explore the limits of applicability of the theory. Simulation results indicate that the theory works well with typical asymmetry of the potential parameters in the force field. However, care should be taken when applying the theory to strongly associating fluids and in the low-pressure region where the active surface sites play an important role. The conformal sites theory can be used to predict the adsorption properties and to characterize the solid substrate by taking advantage of the corresponding states principle. Other possible applications are also discussed.

The Structural Properties and Diffusion of a Three-Dimensional Isotropic Core-Softened Model Fluid in Disordered Porous Media. Molecular Dynamics Simulation

The microscopic structure and dynamic properties of an isotropic three-dimensional core-softened model fluid in disordered matrices of Lennard-Jones particles have been studied. Molecular dynamics computer simulations in Grand Canonical ensemble were used as the methodological tools. It was shown that the microscopic structure of the fluid is characterized by anomalies similar to those found in a bulk model, but that it is affected by the fluid-matrix interactions. The dynamic properties also exhibit anomalous dependence on fluid density, but the magnitude of these anomalies is suppressed in comparison to the bulk fluid model. The anomalous behaviour of the diffusion coefficient is attributed to structural changes in the first coordination shell of a given fluid particle. It seems that the anomalies can only be suppressed at matrix densities which are higher than those studied in the present work.

Adsorption of fluids in a pore with chemical heterogeneities: the cooperative effect

In this work, we study the cooperative adsorption of fluids in a heterogeneous pore, in which the pore walls are composed of homogeneous substrates with chemical groups (CGs) decorating them. The adsorption caused by the homogeneous substrates alone and that by CGs do not add up to the overall adsorption, indicating the existence of a cooperative effect. The cooperative effect is the source of cooperative adsorption, and is characterized in this work by the ratio of the overall adsorption to the sum of adsorption by the substrate only and that by CGs. It is found that the cooperative adsorption does not depend monotonically on the substrate or the CGs. Two different origins of the cooperative adsorption play different roles depending on which one dominates the overall adsorption. Our simulations reveal that, when the homogeneous substrate dominates the overall adsorption, weakening of the attractive fluid-substrate interaction or alternatively strengthening of the fluid-CGs interaction leads to a stronger cooperative effect and enhances the cooperative adsorption. However, when CGs dominate the overall adsorption, weakening of the attractive fluid-CG interaction or strengthening the fluid-substrate interaction results in strong cooperative adsorption. In order to investigate the effects of the distribution of CGs on cooperative adsorption, a design-test method is generalized and used in this work. Simulation results show that the overall adsorption can be significantly affected by the CG distribution.

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.

Spinodal instability and pattern formation in thin liquid films confined between two plates

The instability, morphology and pattern formation engendered by the van der Waals force in a thin liquid film of thickness h confined between two closely placed solid surfaces (at distance d > h) are investigated based on nonlinear 3D simulations. The initial and the final stages of dewetting and pattern formation are found to be crucially dependent on the volumetric (thickness) ratio of air and liquid and its deviation from the location of the maximum of the spinodal parameter versus volumetric ratio curve. On a low energy surface, relatively thinner films and wider air gaps favor initial dewetting of the lower plate by the formation of holes, whereas thicker films with thinner air gaps initially evolve by the formation of columns/bridges that join the upper plate. In the later stage of evolution, the initial holes in thinner films evolve into columns/drops, while a rapid coalescence of columns in the thicker films eventually causes formation of holes. Thus, a phase inversion, either from liquid-in-air to air-in-liquid dispersion or vice versa, occurs during the final stages of evolution. A thin film confined between two high-energy solid surfaces forms columns (bridges) only when its mean thickness, h0, is greater than a critical thickness (hc) or the air gap is smaller than a critical distance. The patterns can be aligned by using a topographically patterned confining surface. Conditions on pattern periodicity, amplitude, and the volumetric ratio of air and liquid in the gap are explored for the formation of various types of ordered patterns including annular rings of columns, concentric ripples, parallel channels and a rectangular array of complex features. The results are of significance in soft lithographies such as LISA, soft stamping and capillary force lithography.

Nanoscopic Liquid Bridges between Chemically Patterned Atomistic Walls

A binary liquid mixture, containing the Lennard-Jones molecules A and B, in equilibrium with a bulk liquid reservoir near the point of phase separation, confined between atomistic chemically patterned walls, is studied by grand canonical Monte Carlo simulations. In the bulk, the B-rich phase is stable and the A-rich phase is metastable. The walls bear patches attractive to A; when the walls are close, A-rich liquid bridges condense between the patches. The normal and lateral forces on the walls are measured as a function of the wall separation and of the lateral displacement between the patches on opposite walls. When there are one or two molecular layers in the bridge and the wall lattice constant is close to that of crystalline A, the normal and lateral forces depend strongly on the registry of the wall lattices, varying in an oscillatory manner.

Adsorption of fluids in slitlike pores containing a small amount of mobile ions

We apply density functional theory to investigate changes in the phase behavior of a fluid caused by the presence of mobile ions inside the pore. The approach has been based on the fundamental measure density functional theory and on the theory of nonuniform electrolytes developed recently by O. Pizio, A. Patrykiejew, S. Sokołowski [J. Chem. Phys. 121 (2005) 11,957]. We have evaluated capillary condensation phase diagrams for pores of different widths and for different concentrations of confined ions. The calculations have demonstrated that the presence of ions leads to lowering the critical temperature and to an increase of the value of the chemical potential at the capillary condensation point.

Thermodynamic Characterization of Fluids Confined in Heterogeneous Pores by Monte Carlo Simulations in the Grand Canonical and the Isobaric−Isothermal Ensembles

Materials presenting nanoscale porosity are able to condense gases in their structure. This "capillary condensation" phenomenon has been studied for more than one century. Theoretical models help to understand experimental results but fail in explaining all experimental features. Most of the time, the difficulties in making quantitative or even qualitative predictions are due to the geometric complexity of the porous materials, such as large pore size distribution, chemical heterogeneities, or pore interconnections. Numerical calculations (lattice gas models or molecular simulations) are of considerable interest to calculate the adsorption properties of a fluid confined in a porous model with characteristic sizes up to several tens of nanometers. For instance, the grand canonical Monte Carlo method allows one to compute the average amount of fluid adsorbed in the porous model as a function of the temperature and the chemical potential of the fluid. However, the grand potential, necessary for a complete characterization of the system, is not a direct output of the algorithm. It is shown in this paper that the use of the isobaric-isothermal (NPT) ensemble allows one to circumvent this problem; that is, it is possible to get in one single Monte Carlo run the absolute grand potential for any given thermodynamic state of the fluid. A simplified thermodynamic integration scheme is then used to evaluate the grand potential over the whole isotherm branch passing through this initially given point. Since the usual NPT technique is a priori limited to homogeneous pores, it is proposed, for the first time, to generalize this procedure to a pore presenting a chemical heterogeneity along its axis. The new method gives the same results as the previous for homogeneous pores and allows new predictions for chemically heterogeneous pores. Comparison with the full integration scheme shows that the proposed direct calculation is faster since it avoids multiple Monte Carlo runs and more precise because it avoids the possible cumulative errors of the integration procedure.

Grand Potential, Helmholtz Free Energy, and Entropy Calculation in Heterogeneous Cylindrical Pores by the Grand Canonical Monte Carlo Simulation Method

The adsorption of fluids in porous media is still an open area of research, since no model is able to explain all experimental features. The difficulties rise from the complexity of the real porous materials which present surface heterogeneities, large pore size distributions, and complex networks of interconnected pores. In parallel to experimental efforts trying to produce more ordered porous materials, theoreticians try to introduce more disorder in their models, with the help of molecular simulation for instance. This grand canonical Monte Carlo simulation study concentrates on the adsorption of a simple Lennard-Jones fluid in three porous substrates, to compare the effect of purely geometric heterogeneity (spatial deformation of the external potential) as opposed to purely chemical heterogeneity (amplitude variations of the external potential). This separation is unrealistic, since geometric fluctuations of a real pore diameter along its axis generally induce variations in the amplitude of the external potential created by the pore. However it enables one to compare both effects. In this paper, a thermodynamic integration scheme is applied to a complete set of adsorption/desorption isotherms. The grand potential, free energy, and entropy are calculated, which allows one to discuss the features of the phase diagrams. It is shown that a purely geometric deformation (undulation) of the external potential does not affect the thermodynamic characteristics of the confined fluid. On the other hand, amplitude modulation of the external potential (chemical heterogeneity) strongly distorts the phase diagram. This heterogeneity is actually able to stabilize a "bridgelike" phase which corresponds to an accumulation of molecules in the most attractive region of the pore.

Torsion-induced phase transitions in fluids confined between chemically decorated substrates

In this paper we investigate the phase behavior of a "simple" fluid confined to a chemically heterogeneous slit pore of nanoscopic width s(z) by means of Monte Carlo simulations in the grand canonical ensemble. The fluid-substrate interaction is purely repulsive except for elliptic regions of semiaxes A and B attracting fluid molecules. On account of the interplay between confinement (i.e., s(z)) and chemical decoration, three fluid phases are thermodynamically permissible, namely, gaslike and liquidlike phases and a "bridge phase" where the molecules are preferentially adsorbed by the attractive elliptic patterns and span the gap between the opposite substrate surfaces. Because of their lack of cylindrical symmetry, bridge phases can be exposed to a torsional strain 0<or=theta;<or=pi/2 by rotating the upper substrate while holding the lower one in position. Depending on the thermodynamic state of the confined fluid, torsion-induced first-order phase transitions are feasible during which a bridge phase may be transformed into either a gaslike (evaporation) or a liquidlike phase (condensation). Since the chemical patterns decorating the substrates are finite in size, system properties are not translationally invariant in any spatial direction. Therefore, in order to study these phase transitions, we resorted to the thermodynamic integration scheme developed earlier to calculate the grand potential Omega in a system of low symmetry.

Shearing of nanoscopic bridges in two-component thin liquid layers between chemically patterned walls

Bridge phases associated with a phase transition between two liquid phases occur when a two-component liquid mixture is confined between chemically patterned walls. In the bulk the liquid mixture with components A, B undergoes phase separation into an A-rich phase and a B-rich phase. The walls bear stripes attractive to A. In the bridge phase A-rich and B-rich regions alternate. Grand canonical Monte Carlo studies are performed with the alignment between stripes on opposite walls varied. Misalignment of the stripes places the nanoscopic liquid bridges under shear strain. The bridges exert a Hookean restoring force on the walls for small displacements from equilibrium. As the strain increases there are deviations from Hooke's law. Eventually there is an abrupt yielding of the bridges. Molecular dynamics simulations show the bridges form or disintegrate on time scales which are fast compared to wall motion and transport of molecules into or from the confined space. Some interesting possible applications of the phenomena are discussed.

Application of density functional theory to analysis of energetic heterogeneity and pore size distribution of activated carbons

A new approach based on the nonlocal density functional theory to determine pore size distribution (PSD) of activated carbons and energetic heterogeneity of the pore wall is proposed. The energetic heterogeneity is modeled with an energy distribution function (EDF), describing the distribution of solid-fluid potential well depth (this distribution is a Dirac delta function for an energetic homogeneous surface). The approach allows simultaneous determining of the PSD (assuming slit shape) and EDF from nitrogen or argon isotherms at their respective boiling points by using a set of local isotherms calculated for a range of pore widths and solid-fluid potential well depths. It is found that the structure of the pore wall surface significantly differs from that ofgraphitized carbon black. This could be attributed to defects in the crystalline structure of the surface, active oxide centers, finite size of the pore walls (in either wall thickness or pore length), and so forth. Those factors depend on the precursor and the process of carbonization and activation and hence provide a fingerprint for each adsorbent. The approach allows very accurate correlation of the experimental adsorption isotherm and leads to PSDs that are simpler and more realistic than those obtained with the original nonlocal density functional theory.

Nanoscopic liquid bridges exposed to a torsional strain

In this paper we investigate the response to a torsional strain of a molecularly thin film of spherically symmetric molecules confined to a chemically heterogeneous slit pore by means of Monte Carlo simulations in the grand canonical ensemble. The slit pore comprises two identical plane-parallel solid substrates, the fluid-substrate interaction is purely repulsive except for elliptic regions attracting fluid molecules. Under favorable thermodynamic conditions the confined film consists of fluid bridges where the molecules are preferentially adsorbed by the attractive elliptic regions, and span the gap between the opposite substrate surfaces. By rotating the upper substrate while holding the lower one in position, bridge phases can be exposed to a torsional strain 0< or =theta< or =pi/2 and the associated torsional stress T(theta) of the (fluidic) bridge phases can be calculated from molecular expressions. The obtained stress curve T(theta)(theta) is qualitatively similar to the one characteristic of sheared confined films: as the torsion strain increases, T(theta) rises to a maximum (yield point) and then decays monotonically to zero. By changing the ellipses' aspect ratio while keeping their area constant, we also investigate the influence of the attractive elliptic patterns' shape on T(theta)(theta).

Connecting local structure to interface formation: a molecular scale van der Waals theory of nonuniform liquids

This article reviews a new and general theory of nonuniform fluids that naturally incorporates molecular scale information into the classical van der Waals theory of slowly varying interfaces. The method optimally combines two standard approximations, molecular (mean) field theory to describe interface formation and linear response (or Gaussian fluctuation) theory to describe local structure. Accurate results have been found in many different applications in nonuniform simple fluids and these ideas may have important implications for the theory of hydrophobic interactions in water.

Adsorption and Phase Transitions in Slit-like Pores with Differently Adsorbing Walls

We study the adsorption of a lattice gas in a slit-like pore with different walls. The density profiles are evaluated in the framework of the mean-field approximation. We show that changes in the potential field exerted by one wall can lead to substantial modifications of the phase behavior of the fluids. Copyright 2001 Academic Press.

Correlation of stress and structure in a simple fluid confined to a pore with furrowed walls

A Lennard-Jones (12,6) film confined between two furrowed walls was simulated by the grand canonical ensemble Monte Carlo method. The walls are constructed by gouging triangular grooves in planar substrates that are structureless on the molecular scale. The furrows are infinitely long in one transverse direction (y) and of nanoscopic width (s(x)) and depth (D). The furrows in the two walls are maintained parallel and in register. The diagonal components of the stress tensor (T(alphaalpha), alpha=x,y,z) are computed as functions of D and the separation between the substrates (s(z)) at fixed temperature, chemical potential, and s(x). The T(alphaalpha) for the film between the furrowed walls are strongly shifted from their counterparts for the film between flat (i.e., planar) walls. The shifts are rationalized in terms of the structure of the film, which becomes more ordered as the furrows deepen and the packing of film molecules becomes more restricted in the two dimensions normal to the y direction. The results demonstrate the profound impact of the coupling between molecular and nanoscopic scales on the properties of geometrically constrained fluids.

Effects of Pore-Pore Correlations on Capillary Condensation in an Ensemble of Slit-like Pores: Application of a Density Functional Theory

Using a density functional method we study how the correlations between particles adsorbed in neighboring pores, forming a network of slit-like pores, influence the capillary condensation and the structure of adsorbed Lennard-Jones fluid. The calculations indicate that if the distance between two pores is small enough, these correlations lead to pronounced changes in the density profiles, to an increase of the critical temperature, and to the modifications in the coexistence envelope. Copyright 2000 Academic Press.

Wetting films on chemically heterogeneous substrates

Based on a microscopic density functional theory, we investigate the morphology of thin liquid-like wetting films adsorbed on substrates endowed with well-defined chemical heterogeneities. As paradigmatic cases we focus on a single chemical step and on a single stripe. In view of applications in microfluidics, the accuracy of guiding liquids by chemical microchannels is discussed. Finally we give a general prescription of how to investigate theoretically the wetting properties of substrates with arbitrary chemical structures.