Disjoining Forces Research Articles

Effect of equilibrium contact angle on water equilibrium film thickness for the carbon dioxide–brine–mineral system based on surface force theory

The thickness of the thin wetting film depends on disjoining pressure forces, and it evolves with pH evolution due to brine acidification at the physical and chemical conditions of geological carbon dioxide storage becoming thinner in response to dewetting. In the literature, molecular dynamic simulation (MDS) studies have been employed to understand the effect of pressure/capillary pressure on the thin wetting film evolution. In this paper, a theoretical approach based on the Frumkin–Derjaguin Equation (FDE), models of electric double layer repulsion, and van der Waals forces have been used for the calculation of the wetting film thickness. The approach excluded hydration forces contribution to disjoining pressure forces due partly to its poorly understood nature, and partly to the high salinity conditions encountered in geological carbon storage. Due to its promising global storage capacity compared to other lithologies, the carbon dioxide–brine–silica systems was chosen to simulate sandstone saline aquifers. The validation of the model benefited much from literature resources on data and a universal model of carbon dioxide–brine interfacial tension. Calculated results confirm pH-induced dewetting and they follow trends controlled by pH and pressure as found in the literature. The novelty of the paper can be seen from the fact that it has demonstrated a theoretical supplement to MDS studies in addition to justifying the fundamental utility and versatility of the FDE. Moreover, the paper links for the first time, a transcendental equation to the thin wetting film theory encountered in the carbon dioxide–solid–brine system found in geological carbon storage.

Structure-property relations of β-lactoglobulin/κ-carrageenan mixtures in aqueous foam

There is a great interest in mixed protein/polysaccharide layers at air-water interfaces due to their ability to stabilize aqueous foams. Properties of β-lactoglobulin (BLG)/κ-carrageenan mixtures at different κ-carrageenan concentrations were studied using foam analysis and molecular-level characterizations at the air-water interface. Because air-water interfaces are a major element in aqueous foams, they were investigated with a combination of interface-sensitive methods such as sum-frequency generation (SFG) as well as surface tensiometry. Using SFG spectroscopy we provide new information on the changes in the interfacial charging state and composition when the BLG/κ-carrageenan mixing ratio changes. κ-carrageenan is only weakly surface active as it does not lower the surface tension to a large extend. However, BLG/κ-carrageenan mixtures showed a synergistic effect because the formation of complexes at the air-water interface and in the bulk solution can favorably influence foam stability. BLG (7.5 µM) was mixed with different κ-carrageenan concentrations from 0 to 1 mM while the bulk pH was adjusted to pH values which are below (4.5), slightly above (5.6) and well above (7.0) the isoelectric point of BLG. At pH 4.5, the positive ζ-potential of BLG was drastically reduced to negative values and large aggregates in the bulk formed that showed a reduced surface activity and caused also a reduced net charging state at the air-water interface. The interfacial and bulk properties at pH 4.5, resulted in a poor stability of foams that were likely only stabilized by the weak electrostatic disjoining forces in the foam lamella. At pH 5.6 and 7, cooperative effects of BLG and κ-carrageenan dominate and, in particular, at pH 5.6 give rise to an increase in BLG surface coverage when the κ-carrageenan concentration increases. This resulted in a substantially improved foam stability.

Disjoining-pressure-induced acceleration of mass shedding in solid-state dewetting.

Surface-diffusion mediated solid-state dewetting has been observed and studied in a number of different systems during the past two decades. This process can be accompanied by the pinching of the film at a finite distance from the retracting triple line. The repetition of this pinching is often referred to as periodic mass shedding. We show that the disjoining pressure of the film can accelerate mass shedding by orders of magnitude in ultrathin films with nanometric thickness. In the presence of power-law disjoining pressures induced by van der Waals forces, the mass shedding time exhibits an approximate power-law dependence on film thickness t_{ms}∼h[over ¯]^{ν}, with ν≈6. Exponentially decaying disjoining forces also give rise to a strong acceleration of mass shedding. However, due to the finite range of the exponential potential, the mass shedding time does not exhibit a simple power-law dependence on the thickness, and is controlled by a cutoff thickness. In addition, two-dimensional simulations indicate that, within the range of thicknesses that we have studied and for isotropic dynamics, the transversal instability of a straight front does not lead to fingering, and mass shedding is the dominant instability of the dewetting front. Finally, we also show that no significant difference is observed in the dewetting dynamics between simulations based on a model with a wetting potential integrated over the film surface area, or over the projected substrate area.

Micromechanical approach to swelling behavior of capillary‐porous media with coupled physics

SummaryA detailed multiscale analysis is presented of the swelling phenomenon in unsaturated clay‐rich materials in the linear regime through homogenization. Herein, the structural complexity of the material is formulated as a three‐scale, triple porosity medium within which microstructural information is transmitted across the various scales, leading ultimately to an enriched stress‐deformation relation at the macroscopic scale. As a side note, such derived relationship leads to a tensorial stress partitioning that is reminiscent of a Terzaghi‐like effective stress measure. Otherwise, a major result that stands out from previous works is the explicit expression of swelling stress and capillary stress in terms of micromechanical interactions at the very fine scale down to the clay platelet level, along with capillary stress emerging due to interactions between fluid phases at the different scales, including surface tension, pore size, and morphology. More importantly, the swelling stress is correlated with the disjoining forces due to electrochemical effects of charged ions on clay minerals and van der Waals forces at the nanoscale. The resulting analytical expressions also elucidate the role of the various physics in the deformational behavior of clayey material. Finally, the capability of the proposed formulation in capturing salient behaviors of unsaturated expansive clays is illustrated through some numerical examples.

Controls on microstructural features during solidification of colloidal suspensions

We present a mathematical model of the directional freezing of colloidal suspensions. Key ingredients of the model are the disjoining forces between the colloidal particles and the solidified suspending fluid, flow of the suspending fluid towards the solidification front through an accumulating layer of particles, and flow through microscopic films of unfrozen liquid separating particles from the freezing front. Our model predicts three different modes of solidification leading to different microstructures: dendritic formations; laddered structures of ice spears and lenses; a frozen fringe, from which transverse ice lenses can form. It explains why different researchers have reported the existence of ice lensing with and without the pre-existence of frozen fringes. Our quantitative predictions are encapsulated within a universal, dimensionless phase diagram showing which microstructure is to be expected under which operating conditions, and we show that these predictions are consistent with previous experimental studies as well as new experiments that we present here.

Adsorption-Induced Deformation of a Nanoporous Material: Influence of the Fluid–Adsorbent Interaction and Surface Freezing on the Pore-Load Modulus Measurement

Liquid adsorption in nanoporous materials induces their deformation due to strong capillary forces. The linear relationship between the liquid pressure and the solid strain (pore-load modulus) provides an experimental technique to determine the mechanical properties of nanosized solids. Puzzling experimental results have often been reported, leading to a severe reconsideration of the mechanical properties of the thin walls, the introduction of surface stresses, and the suggestion of a mutual influence of fluid adsorption and matrix deformation. This work presents a molecular simulation examination of the fundamentals of the pore-load measurement technique. The pore-load protocol is reproduced as in experiments by measuring the solid deformation in presence of the liquid ("numerical experiment"), and the result is compared to the expected mechanical response of the solid. Focusing on a single nanoplatelet mimicking silicon stiffness, we show that the pore-load protocol is valid as long as the liquid in the pores remains liquid. However, when an ordered layer can form at the solid surface, it significantly affects the pore-load measurement. It is shown that this may happen above the freezing point even for moderately strong fluid-solid interactions. This observation could help for the interpretation of experimental data, in particular in porous silicon, where the expected presence of atomically smooth surfaces could favor the formation of highly ordered fluid layers.

Application of a Multi-Scale form of Terzaghi’s Effective Stress Principle for Unsaturated Expansive Clays to Simulate Hydro-Mechanical Behavior During Hydration

Our recently developed multi-scale form of Terzaghi’s effective stress principle for unsaturated swelling clays that was rigorously derived by periodic homogenization starting from micro- and nano-mechanical analyses is applied to numerically simulate one-dimensional swelling pressure tests of compacted bentonites during hydration. The total macroscopic stress captures the coupling between disjoining forces at the nanoscopic scale of clay platelets and capillary effects at the microscopic scale of clay aggregates over the entire water content range. The numerical results allow to draw conclusions on the water transfer mechanism between inter- and intra-aggregate pores during hydration and consequently on the evolution of the external swelling pressure resulting from the competition between capillary and disjoining forces. In addition, such application highlights the abilities and the limits of the electrical double-layer theory to compute the disjoining pressure in the nano-pores. For large platelet distances, in the range of osmotic swelling, the nature of the disjoining pressure is electro-chemical and can be computed from Poisson-Boltzmann theory. Conversely, at small distances, in the crystalline swelling, a solvation component has to be added to account for the molecular nature of the solvent. As a first improvement of the nano-scale description the solvent is treated as a hard sphere fluid using Density Functional Theory.

Influence of short-range forces on melting along grain boundaries

We investigate a model which couples diffusional melting and nanoscale structural forces via a combined nano-mesoscale description. Specifically, we obtain analytic and numerical solutions for melting processes at grain boundaries influenced by structural disjoining forces in the experimentally relevant regime of small deviations from the melting temperature. Though spatially limited to the close vicinity of the tip of the propagating melt finger, the influence of the disjoining forces is remarkable and leads to a strong modification of the penetration velocity. The problem is represented in terms of a sharp interface model to capture the wide range of relevant length scales, predicting the growth velocity and the length scale describing the pattern, depending on temperature, grain boundary energy, strength and length scale of the exponential decay of the disjoining potential. Close to equilibrium the short-range effects near the triple junctions can be expressed through a contact angle renormalisation in a mesoscale formulation. For higher driving forces strong deviations are found, leading to a significantly higher melting velocity than predicted from a purely mesoscopic description.

A Modified Effective Stress Principle for Unsaturated Swelling Clays Derived from Microstructure

A three‐scale model is proposed to describe electro‐chemo‐mechanical couplings in unsaturated swelling clays characterized by two porosity levels and three separate length scales. The nanoscale portrait consists of charged clay particles separated by a nanoporous network saturated by a binary monovalent aqueous electrolyte solution. Local ion distribution and electric potential are governed by the Poisson–Boltzmann problem. At the microscale, the system is represented by swollen clay clusters separated from each other by a network of micropores filled with a mixture of bulk water and air. Under mechanical equilibrium characterized by the competition between disjoining forces of electrochemical nature and capillary attraction effects, we derived a novel form of the effective stress principle in the asymptotic limit of scale separation. Such form includes a new macroscopic equivalent pore pressure weighted by a two‐scale effective Bishop coefficient that incorporates the effects of the adsorbed water at the secondary nanopore level in addition to the contributions of the water–air interfaces at the micropore level. Within the thermodynamic context for constructing macroscopic constitutive laws based on stress–strain variables, the three‐scale model leads to a set of three work‐conjugated state variables. In addition to the contact stress between particles and the new effective Bishop‐type component, the novel form includes salinity as an additional stress‐conjugated variable and a three‐scale version of the electrochemical swelling stress. The potential of the multiscale approach in capturing the complex features of unsaturated expansive clays is illustrated by numerical reconstruction of the effective coefficients in a simplified isotropic microstructure.

The drag on a flattened bubble moving across a plane substrate

AbstractThe equilibrium shape of an axisymmetric liquid drop or gas bubble in an immiscible supporting liquid held under gravity against a horizontal plane rigid surface is derived. A thin film of supporting liquid remains between the drop/bubble surface and the rigid substrate at equilibrium, of thickness ${h}_{0} $ determined by the balance of the disjoining pressure $\Pi (h)$ between the drop and the substrate and the internal Laplace pressure of the drop/bubble. The interface is macroscopically flat around the drop axis out to a radius $a$ but matches smoothly into the outer shape of radius $A$ through a boundary layer region of width $a{ H}_{0}^{1/ 2} $ where ${H}_{0} = 2{h}_{0} A/ {a}^{2} $ is a small parameter. The outer drop shape is determined by a balance of buoyancy forces and local Laplace pressure and is roughly spherical if $A/ \lambda \ll 1$, where $\lambda = \sqrt{\gamma / \mrm{\Delta} \rho g} $ is the capillary length in the interface with a logarithmic correction due to the action of the disjoining pressure across the flattened region. With the shape determined, we calculate the drag force on this flattened bubble to lowest order in the velocity as it moves across the rigid substrate using a lubrication approximation valid to terms of $O( \mathop{ ({h}_{0} / a)}\nolimits ^{2} )$ as an integral over the flattened bubble surface of the hydrodynamic pressure. The lubrication theory of itself is not sufficient to determine the drag due to the divergence of that integral if the outer flow field properties are neglected. By using the known exact result for the drag force on an undistorted bubble, the drag on the flattened bubble can be computed as an integral over the lubrication region alone. We derive the drag as a series expansion in the small parameter ${H}_{0} $ by means of a fairly intricate boundary layer analysis. The logarithmic divergence of the translational drag with film thickness ${h}_{0} $ for the undistorted bubble is replaced by the stronger divergence ${ h}_{0}^{\ensuremath{-} 1/ 2} $ to leading order for the flattened bubble case. We present explicit numerical results for the first few terms in the expansion for the case of an exponentially repulsive disjoining pressure, and analytic expressions for these terms in the limit of very short-range disjoining pressure forces. The results of this calculation are compared with recent work (Hodges, Jensen & Rallison, J. Fluid Mech., vol. 512, 2004, p. 95) where disjoining pressure is neglected and hydrodynamic pressure is balanced against buoyancy and Laplace forces. The limits of validity of this linear drag theory are also presented.

Role of solid surface structure on evaporative phase change from a completely wetting corner meniscus

The transport processes that occur at small length scales are greatly influenced by interfacial and intermolecular forces. Surface roughness at the nanoscale generates additional intermolecular interactions that arise due to the increased surface area. In this work, we have experimentally studied how the magnitude as well as the shape of surface roughness influences the microscale transport processes that occur in the contact line region of a liquid corner meniscus. The surface roughness contribution to the interaction potential was calculated and a direct relationship between the wetting properties of the liquid and the underlying surface properties was obtained. Since the underlying roughness alters the surface potential, the shape of the meniscus and in turn, the resulting capillary and disjoining pressure forces also changed. Atomic force microscopy was utilized to obtain a detailed characterization of the shape of the prepared surfaces. Surface morphology features were obtained from a height-height correlation function. These features were related to the wetting and transport properties of the meniscus at the contact line. Finally, the modified capillary and disjoining pressure forces on the structured surfaces were observed to influence the evaporative heat transfer from the corner meniscus.

Ice growth in a spherical cavity of a porous medium

AbstractWe consider an idealized problem of a sphere of ice growing symmetrically in a spherical cavity within a porous rock in order to identify and quantify different physical mechanisms that can result in fracturing the rock. We show that if the permeability of the rock is very small then high pressures can develop in the cavity as the water inside it expands on freezing. However, given typical permeabilities of most rocks, the pressure is relieved by flow out of the cavity through the rock pores. When ice fills the cavity, there remains a microscopic film of water separating the ice from the rock, owing to disjoining forces, and these forces can stress the rock and have the potential to fracture it. The elastic pressure in the rock depresses the freezing temperature, which can limit the potential for fracturing. This simple example reveals the important interactions between disjoining forces, elasticity and fluid flow in determining the pressure exerted during freezing of water-saturated cavities in rocks.

Surface Forces and Drainage Kinetics of Protein-Stabilized Aqueous Films

Surface or disjoining forces between protein layers adsorbed at the air/water interface were measured in single, isolated films. Two proteins, β-casein and bovine serum albumin (BSA), were investigated under varying conditions of pH, ionic strength, and degree of aging at the interface. Force−distance curves were determined with a modified thin-film balance and interferometer, using both equilibrium and dynamic methods. Dynamic surface tension and ellipsometry data for β-casein and BSA adsorption at the air/water interface are also reported. Charged β-casein and BSA molecules do not strongly adsorb at the air/water interface. Accordingly, stable films were observed when electrostatic interactions were screened or the proteins were near their isoelectric points. For β-casein, the force−distance curves indicated a transition from an outer to an inner branch at a distance equivalent to the diameter of β-casein. This depended on the interfacial structuring of adsorbed β-casein multilayers at the air/water int...

Simulation of Droplet Motion on Low-Energy and Heterogeneous Surfaces

A method of calculation is introduced that allows the simulation of the time-dependent three-dimensional motion of liquid droplets on solid substrates for systems exhibiting finite equilibrium contact angles. The contact angle is a prescribed function of position on the substrate. An evolution equation is presented, using the lubrication approximation, that includes viscous, capillary, disjoining, and gravitational forces. Motion to and from dry substrate regions is made possible by use of a thin energetically stable wetting layer. Axisymmetric spreading on a uniform substrate is calculated, and it is found, in agreement with reported experiments, that spreading rates are independent of the contact angle until the drop has almost stabilized. We simulate motion on a heterogeneous substrate composed of two different materials having widely different contact angles. Motion proceeds in an almost discontinuous fashion as the initial droplet breaks up into smaller pieces through the action of the wetting forces. Various forms of the disjoining energy functional are employed; the particular choice is found to have only a limited quantitative effect of the drop dynamics. Experimental observations confirm the basic features of the simulation, although a time-scale correction needs to be applied.

Thermal Behavior of a Nonfreezing Water Interlayer

The temperature dependence of the thickness of a nonfreezing water interlayer confined between an ice core and a smooth surface of quartz capillary is studied here theoretically. By using the principle of conservation of energy a first order differential equation is found. The equation relates the temperature and the film thickness via the film pressure (disjoining pressure). Molecular and structural components of this pressure are examined. Nonequilibrium and equilibrium solutions are compared with the respective experimental data. It is shown that disjoining pressure forces cause the lowering of the temperature in systems having confined phases.

A general theoretical approach to the behaviour of foreign particles at advancing solid-liquid interfaces

In the present paper expressions for the viscous drag and disjoining forces upon a foreign spherical body at a solid-liquid interface are developed under general conditions, with clearly denned boundary conditions, and for various situations (planar, concave or convex interface, a smooth particle, a particle with coating or absorption).

Theory of flotation of small and medium-size particles

The paper describes a theory of flotation of small and medium-size particles less than 50μ in radius) when their precipitation on a bubble surface depends more on surface forces than on inertia forces, and deformation of the bubble due to collisions with the particles may be neglected. The approach of the mineral particle to the bubble surface is regarded as taking place in three stages corresponding to movement of the particles through zones 1, 2 and 3. Zone 3 is a liquid wetting layer of such thickness that a positive or negative disjoining pressure arises in this intervening layer between the particle and the bubble. By zone 2 is meant the diffusional boundary layer of the bubble. In zone 1, which comprises the entire liquid outside zone 2, there are no surface forces. Precipitation of the particles is calculated by considering the forces acting in zones 1, 2 and 3. The particles move through zone 1 under the action of gravity and inertia. Analysis of the movement of the particles under the action of these forces gives the critical particle size, below which contact with the bubble surface is impossible, if the surface forces acting in zones 2 and 3 be neglected. The forces acting in zone 2 are ‘diffusio-phoretic’ forces due to the concentration gradient in the diffusional boundary layer. The concentration and electric field intensity distribution in zone 2 is calculated, taking into account ion diffusion to the deformed bubble surface. An examination is made of the ‘equilibrium’ surface forces acting in zone 3 independent of whether the bubble is at rest or in motion. These forces, which determine the behaviour of the thin wetting intervening layer between the bubble and the mineral particle and the height of the force barrier against its rupture, may be represented as results of the disjoining pressure forces acting on various parts of the film. The main components of the disjoining pressure are van der Waals forces, forces of an iono-electrostatic nature and forces related to structural changes in the boundary layers. A quantitative examination of the first two kinds of forces makes it possible (by neglecting the forces of the third kind) to obtain the condition of disappearance of the force barrier, i.e. of unhindered rupture of the wetting film and formation of a wetting perimeter. When this condition is fulfilled the kinetics of flotation recovery depends only on stages 1 and 2. Calculation of the forces acting in zone 2 and of their influence on the velocity of precipitation of the particles is given separately for small particles, the size of which does not exceed the thickness of the diffusional boundary layer, and for relatively large (‘medium’) particles, whose size is greater than the thickness of the diffusional boundary layer. The possibility and rate of precipitation of small particles are determined by diffusio-phoretic relationships in the concentration and electric fields of the diffusional boundary layer. A formula is derived for the resultant velocity of precipitation of small particles on a bubble surface under the action of gravity and diffusio-phoretic forces (in the absence of a force barrier in zone 3), and this serves as a basis for calculating the effectiveness of precipitation and the critical particle size below which contact is impossible at certain values of the cationic and anionic components of the ζ-potential (even for hydrophobic particles). The paper then deals with the specific forces of a diffusio-electric nature, which arise when the particle acquires or surpasses the ‘average’ size of zone 2, in the process of desorption of the flotation reagent from its surface, its diffusion to the bubble surface and adsorption on it. It is shown that these forces favour thinning of the liquid layer between the bubble and particle surfaces, the viscous resistance of which in their absence cannot be overcome in a number of cases during the ‘contact’ time. ‘Medium’-sized particles, and under certain conditions small particles as well, pass through stage 2 ‘unhindered’, and then flotation efficiency depends on the forces acting in zone 3. In particular, there may be cases where the main controlling factor is the ζ-potential.

Cohesive properties of thin films of liquids adhering to a solid surface

The process of forming and rupturing a thin liquid film at a solid surface is described thermodynamically for both high and low energy solid surfaces. In part 1 the build-up of thin films on high-energy surfaces from the first monolayer is considered and reviewed. Components of the surface free energy of formation of the thin film (disjoining pressure) are defined. For curved surfaces the disjoining forces should be combined with the Laplace capillary pressure to give a correct form of the Kelvin equation. It is suggested from the early work of Bangham and Deryaguin that thin liquid layers have anomalous physical properties. These studies are discussed in relation to the thickness of the liquid films.