Implications of the phase rule for capillary systems containing surfaces and three-phase contact lines with surface and linear constraint relations

The phase rule is generalized to heterogeneous systems with moderately curved surfaces and linear or line-phase boundaries. It will be shown that the number of degrees of freedom or variancef of a capillary sysem is, in general, larger than that predicted by the classical Gibbs' phase rule. Restrictions on the value off for capillary systems is intimately connected with the imposition of mechanical constraints like the Laplace and Young equations of capillarity. As a direct consequence, it may be shown that the variance for anr-component, three-phase solid-liquid-vapour system (i.e., a liquid drop on a solid surface) is different from that of anr-component, liquid-liquid-vapour system. This conclusion has important implications for the existence or possible existence of equation-of-state type relations.     

Critical Eotvos numbers for buoyancy-induced oil drop detachment based on shape analysis

Critical values of the Eotvos number, which is half the Bond number, above which buoyancy induced drop detachment occurs, are estimated based on force balance equations available in the literature [Colloids Surf. A: Physicochem. Eng. Aspects 178 (2001) 249]. Since there are two significantly different expressions of the capillary retention force responsible for holding oil drops on a solid substrate in an aqueous phase, the critical dimensionless number is estimated with these two distinct equations. The differential equation defining the drop shape, with the constraints of the drop volume and the 'pinned' or 'receding' contact line, is numerically solved. The equilibrium drop shapes predicted are shown to match the experimentally observed variations in drop shape. From the numerical solution, it is observed that for interfacial tension (IFT) values lower than a certain limit for a given drop size, no numerically estimated drop shape can fulfil the drop volume constraint. Similarly, for the dimensionless number above a critical value, no shape can meet all the constraints. These critical Eotvos numbers are estimated, based on the above numerical approach, for initial contact angles measured in oil varying from 20 degrees to 90 degrees. It is found that the critical Eotvos numbers estimated from the numerical shape analysis are between the critical values estimated from the two force-balance equations. Near 90 degrees, the critical values estimated from the drop shape analysis matches the values from one of the force balance estimates, but merges with the critical values of the dimensionless number, estimated from the other force balance model near 10 degrees. From this analysis, it appears that a combination of the two equations for the capillary retention force is required, with one dominating when the contact angles are high, while the other applies for low values of the contact angle.     

Wetting and surface forces

In this review we discuss the fundamental role of surface forces, with a particular emphasis on the effect of the disjoining pressure, in establishing the wetting regime in the three phase systems with both plane and curved geometry. The special attention is given to the conditions of the formation of wetting/adsorption liquid films on the surface of poorly wetted substrate and the possibility of their thermodynamic equilibrium with bulk liquid. The calculations of contact angles on the basis of the isotherms of disjoining pressure and the difference in wettability of flat and highly curved surfaces are discussed. Mechanisms of wetting hysteresis, related to the action of surface forces, are considered.     

Desorption kinetics of surfactants at fluid interfaces by novel coaxial capillary pendant drop experiments

Surfactants appear in multiphase fluid systems in which the interface and the adjacent bulk phase have been removed from equilibrium. Here, a new method is described for the measurement of rate constants of desorption of surface-active materials from fluid/fluid interfaces and the extent to which adsorption is reversible: the coaxial capillary pendant drop experimental technique.

Kinetic constants are determined by desorption experiments in pendant drops in which the interface adjacent to a surfactant solution is removed from equilibrium by replacing the subphase of the drop with pure water. Further, we demonstrate that although the rate of subphase exchange is comparatively slow with respect to the desorption timescale, it is possible to resolve desorption processes which occur under local equilibrium with the adjacent bulk phase from those that are determined in part by sorption kinetics. Experiments which measure the desorption kinetic coefficient, , using a homologous series of n-alkyl (C₈, C₁₀, C₁₂, C₁₄) dimethyl phosphine oxides are presented.     

Density functional theory study on the structure and capillary phase transition of a polymer melt in a slitlike pore: effect of attraction

A density functional theory is proposed to investigate the effects of polymer monomer-monomer and monomer-wall attractions on the density profile, chain configuration, and equilibrium capillary phase transition of a freely jointed multi-Yukawa fluid confined in a slitlike pore. The excess Helmholtz energy functional is constructed by using the modified fundamental measure theory, Wertheim's first-order thermodynamic perturbation theory, and Rosenfeld's perturbative method, in which the bulk radial distribution function and direct correlation function of hard-core multi-Yukawa monomers are obtained from the first-order mean spherical approximation. Comparisons of density profiles and bond orientation correlation functions of inhomogeneous chain fluids predicted from the present theory with the simulation data show that the present theory is very accurate, superior to the previous theory. The present theory predicts that the polymer monomer-monomer attraction lowers the strength of oscillations for density profiles and bond orientation correlation functions and makes the excess adsorption more negative. It is interesting to find that the equilibrium capillary phase transition of the polymeric fluid in the hard slitlike pore occurs at a higher chemical potential than in bulk condition, but as the attraction of the pore wall is increased sufficiently, the chemical potential for equilibrium capillary phase transition becomes lower than that for bulk vapor-liquid equilibrium.     

Application of the projection operator formalism to non-hamiltonian dynamics

Reconstruction of equations of motion from incomplete or noisy data and dimension reduction are two fundamental problems in the study of dynamical systems with many degrees of freedom. For the latter, extensive efforts have been made, but with limited success, to generalize the Zwanzig-Mori projection formalism, originally developed for hamiltonian systems close to thermodynamic equilibrium, to general non-hamiltonian systems lacking detailed balance. One difficulty introduced by such systems is the lack of an invariant measure, needed to define a statistical distribution. Based on a recent discovery that a non-hamiltonian system defined by a set of stochastic differential equations can be mapped to a hamiltonian system, we develop such general projection formalism. In the resulting generalized Langevin equations, a set of generalized fluctuation-dissipation relations connect the memory kernel and the random noise terms, analogous to hamiltonian systems obeying detailed balance. Lacking of these relations restricts previous application of the generalized Langevin formalism. Result of this work may serve as the theoretical basis for further technical developments on model reconstruction with reduced degrees of freedom. We first use an analytically solvable example to illustrate the formalism and the fluctuation-dissipation relation. Our numerical test on a chemical network with end-product inhibition further demonstrates the validity of the formalism. We suggest that the formalism can find wide applications in scientific modeling. Specifically, we discuss potential applications to biological networks. In particular, the method provides a suitable framework for gaining insights into network properties such as robustness and parameter transferability.     

Time propagation of constrained coupled Gaussian wave packets

The dynamics of quantum systems can be approximated by the time propagation of Gaussian wave packets. Applying a time dependent variational principle, the time evolution of the parameters of the coupled Gaussian wave packets can be calculated from a set of ordinary differential equations. Unfortunately, the set of equations is ill behaved in most practical applications, depending on the number of propagated Gaussian wave packets, and methods for regularization are needed. We present a general method for regularization based on applying adequate nonholonomic inequality constraints to the evolution of the parameters, keeping the equations of motion well behaved. The power of the method is demonstrated for a nonintegrable system with two degrees of freedom.     

An experimental investigation of retention of liquids in corners of a square capillary

The retention of liquids in the corners of a 0.03-cm square capillary after the passage of a gas slug was studied experimentally as a function of capillary number in the range from 10(-3) to 10(-6). In gas-wetting liquid systems, for capillary number greater than 5 x 10(-4), the retention of a wetting liquid in the corners showed a strong dependence on the capillary number; i.e., the retention of the liquid decreased with decreasing capillary number. For capillary number less than 10(-4), the retention of a wetting liquid was found to be determined by the capillary forces and the rate (or viscous) effect was negligible. In gas-oil-water systems involving double displacements--gas was displacing oil which was in turn displacing water--the total retention of water and oil vs capillary number curve showed the same trend as the retention of a wetting phase in a gas-wetting liquid system. However, because of the viscous effect, the water retention showed a continuous decrease with decreasing capillary number and could be lower than the capillary equilibrium value at very low capillary numbers. As a result of this, the oil retention vs capillary number curve in the double displacement process showed a minimum; i.e., oil retention increased with decreasing capillary number in the range of very low capillary numbers.     

A generalized scale of free energy of excess adsorption of solute and absolute composition of the interfacial phase

Moles of a surfactant (gamma2(1)) absorbed per unit area of the solid-liquid interface estimated analytically from the difference of the solute molality in the bulk phase before and after adsorption have been quantitatively related to the absolute compositions deltan1 and deltan2 of the solvent and solute forming the inhomogeneous surface phase in contact with the bulk phase of homogeneous composition. By use of isopiestic experiments, negative values of gamma2(1) for the adsorption of inorganic salts onto a solid-liquid interface have been calculated in the same manner. From the linear plot of gamma2(1) versus the ratio of the bulk mole fractions of the solute and solvent, values of deltan1 and deltan2 have been evaluated under a limited range of concentrations. For the adsorption of the surfactant and the inorganic salt respectively onto the fluid interface, gamma2(1) values have been evaluated from the surface tension concentration data using the Gibbs adsorption equation. Gamma2(1) based on the arbitrary placement of the Gibbs dividing plane near the fluid interface is quantitatively related to the composition of the inhomogeneous surface phase. Also, the Gibbs equation for multicomponent solutions has been appropriately expressed in terms of a suitably derived coefficient m. Integrating the Gibbs adsorption equation for a multicomponent system, the standard free energy change, deltaG degrees, per unit of surface area as a result of the maximum adsorption gamma2(m) of the surfactant at fluid interfaces due to the change of the activity alpha2 of the surfactant in the bulk from zero to unity have been calculated. A similar procedure has been followed for the calculation of deltaG degrees for the surfactant adsorption at solid-liquid interfaces using thermodynamically derived equations. deltaG degrees values for surfactant adsorption for all such systems are found to be negative. General expressions of deltaG degrees for negative adsorption of the salt on fluid and solid-liquid interfaces respectively have also been derived on thermodynamic grounds. deltaG degrees for all such systems are positive due to the excess spontaneous hydration of the interfacial phase in the presence of inorganic salt. Negative and positive values of deltaG degree for excess surfactant and salt adsorption respectively have been discussed in light of a generalized scale of free energy of adsorption.     

Introducing phase transitions to quantum chemistry: from Trouton's rule to first principles vaporization entropies

In the present study, we employ quantum cluster equilibrium calculations on a small water cluster set in order to derive thermochemical equilibrium properties of the liquid phase as well as the liquid-vapor phase transition. The focus is set on the calculation of liquid phase entropies, from which entropies of vaporization at the normal boiling point of water are derived. Different electronic structure methods are compared and the influences of basis set size and of cooperative effects are discussed. In line with a previous study on the subject [B. Kirchner, J. Chem. Phys. 123, 204116 (2005)], we find that the neglect of cooperativity leads to large errors in the equilibrium cluster populations as well as in the obtained entropy values. In contrast, a correct treatment of the intermolecular many-body interaction yields liquid phase entropies and phase transition entropies being in very good agreement with the experimental reference, thus demonstrating that the quantum cluster equilibrium partition function intrinsically accounts for the shortcomings of the ideal gas partition function often employed in first principles entropy calculations. Comparing the calculated vaporization entropies to the value predicted by Trouton's rule, it is observed that for entropy calculations the consideration of intracluster cooperative effects is more important than the explicit treatment of the intercluster association even in a highly associated liquid such as water. The decomposition of entropy into contributions due to different degrees of freedom implies the need for the accurate treatment of particle indistinguishability and free volume of translation, whereas minor influences should be expected from the vibrational and rotational degrees of freedom and none from the electronic degrees of freedom.     

Thermodynamic equilibrium of adsorbed phases

The types of phase equilibrium behavior for adsorbed binary mixtures that can be predicted by an equation of state (EOS) based on the lattice gas theory are investigated. The equilibrium conditions were obtained by solving the isofugacity equations between adsorbed phases. It is observed that the investigated EOS can predict complex behavior for adsorbed phases such as the existence of azeotropes, and retrograde and double retrograde phase transition phenomena, that are analogous to those found in bulk phase equilibrium. Furthermore, it was possible to find systems that presented phase equilibrium between two dense adsorbed phases, analogously to liquid–liquid equilibrium for bulk phases. Experimental data would be necessary to confirm the types of adsorbed phase behavior predicted by the calculations presented.     

Considering vibrations in the thermodynamic functions of a solid adsorbate in slit-like pores

Problems that arise in refining calculations of the equilibrium thermodynamic functions of an adsorbate located near an adsorbent’s surface or inside slit-like pores when its collective vibrational motions are considered are discussed. A technique is proposed for calculating collective vibrations for a small number of vacancies in a defective heterogeneous adsorbate from changes in the number of degrees of freedom and the local frequencies in the frequency distribution function in an ideal bulk crystal.     

Vapor-liquid equilibria for benzyl alcohol with carbon dioxide, ethane, or nitrogen at elevated pressures

Vapor-liquid equilibrium (VLE) phase compositions were measured for the binary systems of benzyl alcohol with carbon dioxide, ethane, or nitrogen at temperatures from 333.15 K to 453.15 K and pressures up to 19 MPa. Henry's constants were calculated from the isothermal equilibrium data. The new VLE data were correlated by the Patel-Teja equations of state with three different types of mixing rules. In general, using the one-fluid, two-parameter van der Waals mixing rule yielded the best representation for the investigated systems. The validity of a generalized Soave model was also tested with the equilibrium data of carbon dioxide + benzyl alcohol.     

Theoretical prediction of the coordination number, local composition, and pressure-volume-temperature properties of square-well and square-shoulder fluids

By assuming a Boltzmann distribution for the molecular equilibrium between local and bulk environments, a general model is derived for the prediction of coordination numbers and local compositions of square-well and square-shoulder fluids. The model has no empirical parameter fitted from the data of square-well and square-shoulder fluids, but is valid from the low-density limit to the high-density limit. The applicable width of well or shoulder covers the commonly used range varying from 1.0 to 2.0. The model can accurately predict the coordination numbers of pure square-well and square-shoulder fluids, so the equation of state derived from it is superior to other equations of state based on the existing coordination number models. The model also accurately predicts the local compositions of mixtures in wide ranges of density and size ratio (1.0-8.0), as well as the configuration energy of lattice gases and highly nonideal lattice mixtures. It is remarkable that the model correctly predicts temperature-dependent coordination numbers and local compositions for both equal- and unequal-sized mixtures at close packing, which cannot be predicted by the existing coordination number models. Our derivation demonstrates that the energy parameters in local composition models should represent the potential difference of a molecule between the local and bulk environments, not the pair-interaction potential, and depend on the system conditions and different kinds of pair-interaction parameters. This result is very useful for the development of local composition and activity coefficient models and the mixing rules of equations of state.     

Influence of surface processes on the dilational visco-elasticity of surfactant solutions

The mechanical properties of liquid-fluid systems, like the dynamic interfacial tension and interfacial rheology are closely related to the kinetic processes involved and to the behaviour of the adsorbed molecules. Therefore, provided suitable models and experimental methods are set, investigating these properties allows qualitative and quantitative information on these processes to be drawn. This paper presents recent developments in dilational rheology of liquid-fluid adsorption layers, including experimental methods, models and experimental data concerned with surfactants undergoing transformations in the adsorption layer. Models account both for relaxation due to surfactant diffusion and to processes internal to the adsorption layer. In particular surfactant reorientation, aggregation phase transitions and interfacial chemical reactions have been considered as possible reorganisation processes. The presented approach, allows the dilational viscoelasticity to be derived as a function of the perturbation frequency and of the equilibrium and kinetic parameters of the system. The results can also be easily specified for insoluble monolayer. The principal experimental techniques are reviewed and the recent progresses in the implementation of an Oscillation Bubble/Drop method for Capillary Pressure Tensiometer are discussed in detail. Two experimental studies of surfactants characterised by re-orientation and aggregation phase transition are presented. Beside providing a wider comprehension of these mechanisms, the interpretation of the dilational visco-elasticity data, according to the developed models, allows the effective estimation of the equilibrium and kinetic parameters.     

Thermodynamic forces in highly curved fluid interfaces

The identification of the force distribution in curved interfaces as a thermodynamic force [Baus and Lovett, J. Chem. Phys. 101, 377 (1995)] can be interpreted as a relation between the force distribution and the grand canonical free energy difference between two distinct systems. Using this interpretation, molecular expressions are developed for the force distribution in cylindrical and spherical interfaces that remain valid for very highly curved interfaces.     

Dynamic behavior of interfaces: Modeling with nonequilibrium thermodynamics

In multiphase systems the transfer of mass, heat, and momentum, both along and across phase interfaces, has an important impact on the overall dynamics of the system. Familiar examples are the effects of surface diffusion on foam drainage (Marangoni effect), or the effect of surface elasticities on the deformation of vesicles or red blood cells in an arterial flow. In this paper we will review recent work on modeling transfer processes associated with interfaces in the context of nonequilibrium thermodynamics (NET). The focus will be on NET frameworks employing the Gibbs dividing surface model, in which the interface is modeled as a two-dimensional plane. This plane has excess variables associated with it, such as a surface mass density, a surface momentum density, a surface energy density, and a surface entropy density. We will review a number of NET frameworks which can be used to derive balance equations and constitutive models for the time rate of change of these excess variables, as a result of in-plane (tangential) transfer processes, and exchange with the adjoining bulk phases. These balance equations must be solved together with mass, momentum, and energy balances for the bulk phases, and a set of boundary conditions coupling the set of bulk and interface equations. This entire set of equations constitutes a comprehensive continuum model for a multiphase system, and allows us to examine the role of the interfacial dynamics on the overall dynamics of the system. With respect to the constitutive equations we will focus primarily on equations for the surface extra stress tensor.