Theory Of Surface Tension Research Articles

On the Theory and Computation of Surface Tension: The Elimination of Parasitic Currents through Energy Conservation in the Second-Gradient Method

Errors in the computation of fluid flows with surface tension are examined. These errors cause large parasitic flows when the capillary number is large and have often been attributed to truncation error in underresolved interfacial regions. A study using the second-gradient method reveals that when truncation error is eliminated in the computation of energy exchanges between surface and kinetic energies so that energy is strictly conserved, the parasitic currents are reduced to round-off. The results are based on general thermodynamic arguments and can be used to guide improvements in other methods, such as the continuum-surface-force (CSF) method, which is commonly used with the volume-of-fluid (VOF) method.

Einstein's theory of surface tension

Einstein's theory of surface tension

Einstein's theory of surface tension

Einstein's theory of surface tension

A Bubble Theorem

Proof of a simple mathematical inequality using elements of surface tension theory and ideal gas law to the formation and coalescence of bubbles.

A Bubble Theorem

Proof of a simple mathematical inequality using elements of surface tension theory and ideal gas law to the formation and coalescence of bubbles.

Surface tension in terms of direct correlation function c(r,r′): Proposed modelling of c(r,r′) through liquid–vapor interface

A simple model is proposed for the moments of the inhomogeneous direct correlation function entering the theory of surface tension and liquid–vapor interfacial profile for a classical fluid.

Measurements of the Surface Elasticity in Medium Frequency Range Using the Oscillating Bubble Method

Various experimental techniques are available for the investigation of dynamic surface tension, and a generally accepted theoretical model of these dynamics has been established. However, reliable rheological parameters of a fluid surface are very scarce. Therefore, comparisons of rheological parameters resulting from slow and faster processes or from theoretical calculations are required. In particular, a comprehensive experimental verification of the complex surface elasticity modulus which characterizes the dynamic behavior of a fluid surface in an appropriate manner is desirable.For this reason a new version of oscillating bubble method was developed which allows exact measurements of the complex elasticity modulus in the frequency range 3–500 Hz. With this method the assumptions of the theory of dynamic surface tension can be verified for medium frequencies. The new experimental results, in particular the experimental determination of the Gibbs elasticity, reveal that these assumptions are only approximately valid for faster processes. However, with a slight modification of the established model the experimental results can be explained.These experiments were carried out with solutions of tridecyl dimethyl phosphine oxide, fatty acids,n-alkanols, and triton X-100 at different surfactant concentrations.

Ginzburg criterion for the crossover behavior of model fluids

The Ginzburg criterion, which is based on the three coefficients in the Landau–Ginzburg (LG) expansion of the Helmholtz free energy density of a nonuniform system, is believed to give a reasonable estimate for the temperature scale on which crossover occurs. To compute the contribution of the square-gradient term in the LG expansion, we extend the van der Waals theory of surface tension and, in contrast to our earlier treatment, account for the dependence of the pair distribution function on the spatially varying density. Via this approach we calculate and compare the Ginzburg temperatures of ionic, dipolar, and simple model fluids, namely the restricted primitive model (RPM), the Onsager model, and the square-well fluid (the second and third virial coefficients, for which we also present exact results). To compute the properties of the RPM, we employ the Fisher–Levin theory and its recent extension for Debye-shielded dipole–dipole interactions and a state-dependent dielectric constant that was developed by us. In contrast to the results of our earlier work and in accordance with the calculations of Fisher and Lee, we now find that the RPM has no exceptionally small region in which mean-field theory fails.

A Two-Body Formulation for Solder Joint Shape Prediction

Solder shape prediction is essential for accurate fatigue life determination and joint design optimization. In the present paper, a new solution approach using the surface tension theory is developed to simultaneously predict standoff height, wetted surface area, contact angles, and solder shape by including energy effects between a molten solder body and an arbitrarily shaped solid body. Existing models for solder shape prediction do not appear to determine all characteristics including joint standoff height, wetted surface area, and contact angles simultaneously. A general two-body axisymmetric finite element code is developed and coupled with a constrained optimizer to solve four illustrative examples. These examples include the shape of a sessile droplet on a fixed pad, a flip-chip joint, a sessile droplet on a free surface, and a typical ceramic ball grid array solder joint. In all four examples, the results predicted by the present approach compare favorably with available experimental and numerical results.

Analysis of Adsorption Isotherms: Lattice Theory Predictions, Classification of Isotherms for Gas–Solid Equilibria, and Similarities in Gas and Liquid Adsorption Behavior

Adsorption at fluid–solid interfaces is considered in the framework of a lattice with boundaries. Using ideas proposed by S. Ono and S. Kondo (in“Molecular Theory of Surface Tension in Liquids” (S. Flügge, Ed.), Encyclopedia of Physics, Vol. 10, p. 134. Springer-Verlag, Berlin, 1960), a lattice model is derived, both rigorously and phenomenologically, and applied to macro-, meso-, and microporous adsorbents by imposing different boundary conditions. It is shown that this lattice theory can predict the entire spectrum of behavior observed when gases, liquids, or supercritical fluids adsorb on solid surfaces. In particular, it is able to predict steps in the isotherms, scaling behavior near saturation conditions, supercritical behavior, and adsorption hysteresis. It is shown that there is a profound analogy in the adsorption behavior of a one-component gas to that of a binary liquid mixture. This analysis leads to a new classification of physisorption isotherms for fluid/solid equilibria.

One-fluid theory of liquid-vapor surface tension in simple multi-component mixtures

We propose a theory of surface tension in multi-component mixtures of simple fluids far from critical conditions based on the one-fluid van der Waals approximation and statistical mechanical perturbation approach. The theory contains no adjustable parameters. By comparison with molecular dynamics simulation of binary Lennard-Jones mixtures it is shown that the theory provides a quantitative (within a tolerance of ∼15%–20%) description of surface tension.

On the Ginzburg temperature of ionic and dipolar fluids

Critical fluctuations in fluids are investigated within the framework of the generalized van der Waals theory. The square-gradient term—added to the Landau expansion of the Helmholtz free energy density—is obtained following a procedure similar to that originally proposed by van der Waals in the theory of surface tension, however replacing the Heaviside step function originally used by an approximative pair distribution function. Representative for ionic fluids we choose the restricted primitive model (RPM) and treat it within the Debye–Hückel theory, thus neglecting effects of ion pairing. The other approximative extreme—complete ion pairing resulting in a fluid of hard dipolar dumbbells—is mimicked by a fluid composed of dipolar hard spheres (DHS). For this case we use the Onsager reaction field and the second pressure virial coefficient. We calculate the amplitudes of the correlation length and the Ginzburg temperatures, and find (in reduced quantities) ξ0*=3.50 and ΔTGi*=0.0087 for the ionic system, and ξ0*=0.82 and ΔTGi*=1.63 for the dipolar fluid. For calibration we compute the same quantities for simple neutral fluids and obtain ξ0*=0.50 and ΔTGi*=2.89 for a Sutherland fluid (hard core term plus attractive r−6-potential) and ξ0*=0.43 and ΔTGi*=8.50 for a square-well fluid. The result of a smaller Ginzburg temperature for the ionic fluid than for nonionic fluids in a treatment that neglects ion pairing is clearly at variance with the results of other groups. The correlation length in the low-density limit obtained from our approach has the same functional dependencies as the Lee–Fisher expression, but differs by a numerical factor of 5.7.

Thermodynamics of the curvature effect on ice surface tension and nucleation theory

The Gibbs model of surface tension, together with the approach by Defay and Prigogine [Surface Tension and Adsorption (Longmans, New York/London, 1966)] and Dufour and Defay12 [Thermodynamic of Clouds (Academic, New York/London, 1963)] for calculation of the Gibbs adsorption Γ, is used to estimate the curvature effect on ice surface tension. For the case of the ice-water interfaces a new model for calculation of the Gibbs adsorption Γi/w is proposed. Ice surface tension decreases with the curvature which in turn reduces the equilibrium radius of an ice embryo thus making the nucleation event more probable. The comparison of the calculated transitional region between ice and supercooled water with that obtained from experiments and theoretical calculations showed that the proposed new model gives the correct sign and correct order of magnitude for Γi/w.

Microscale theory of surface tension.

A phenomenological model of capillarity, which accounts for the structure of the liquid-vapor layer, is successively derived assuming that the entropy of a particle depends on the internal energy, the density, and the gradient of the density. Employing classical thermodynamic principles in combination with the balance of mass, momentum, and energy, a rheological expression for capillary stresses is obtained in terms of the free energy of a liquid-vapor system. It is demonstrated that this model admits potential solutions, provided that external forces are potential and the effects of viscous dissipation of energy and heat conductivity are neglected. Moreover, it is shown that a variational principle can be formulated for potential flows, which generalizes Luke's variational principle for free-boundary inviscid flow. This model can be applied to flows involving a topological change of the capillary interface, such as those associated with the spontaneous growth, coalescence and breakup of vaporous bubbles in a liquid.

On the Theory of Dynamic Surface Tension of Ionic Surfactant Solutions, I: Diffusion-Convective Adsorption

A quantitative theory for dynamic surface tension has been developed for the diffusion-controlled and diffusion-convective-controlled models without any “quasi-equilibrium” hypotheses. A rigorous model considers both the diffusion and the migration of surfactant and electrolytes in the electrical field that develops as the charged surfactant adsorbs at a planar interface. The surface concentration of surfactant is calculated by using the coupled Poisson and Nernst–Planck equations. An unknown electrical potential in the Nernst–Planck equation is found from the integral form of the charge-balance equation. It is shown that the proposed system of equations describes over a wide range of time the non-steady-state process in the bulk for both ionic surfactants and electrolytes. The relaxation equation ofF(γ(t)) = log[(γ0− γ(t))/(γ(t) − γe)] =nlog(t/trel) is suitable to describe the dynamic surface tension process over a wide range of times for diffusion-controlled and diffusion-convective-controlled adsorption. The simple formula is derived to calculate the parameters ofn,trel(the relaxation time), andDeff(the effective diffusion coefficients) from the experimental data represented in the form of the relaxation functionF(t) versus log(t). The analytical expressions are obtained to describe the dynamic surface tension for short and long times by using the diffusion-controlled and diffusion-convective-controlled adsorption obeying linear, rectangular, and arbitrary adsorption isotherms.

Viscous flow at infinite Marangoni number.

We formulate the theory of surface tension driven Stokes flow set up by an active scalar with zero diffusivity. The 3D hydrodynamic problem can be reduced to a 2D nonlinear evolution equation involving only free surface quantities. For a semi-infinite layer it can be rigorously demonstrated that the solutions to this equation blow up in finite time and develop singular forms. The new type of nonlinearity plays a universal role in the description of interfacial turbulence.

Calculation of surface tension of salt solutions: effective polarizability of solvated ions

The Onsager–Samaras version of Wagner's theory of surface tension of solutions of 1:1 salts has been modified to include ion-induced dipole terms in the image potential. Effective solvated ionic polarizabilities are used, which are large and negative owing to local dielectric saturation around the ions. The modified theory agrees with the order of magnitude of the variation among various alkali halides (e.g., Δσ for 0.1 molar aqueous KI is about 70% of the corresponding value for KCl). Polarizabilities were estimated from Hasted's dielectric decrements. Keywords: surface tension, salt solutions, symmetric electrolytes, effective polarizability, theory.

The principle of corresponding states for polymer liquid surface tension

A corresponding states principle is demonstrated for polymer and oligomer liquids. Scaling the measured surface tension with the thermodynamic properties obtained from pressure, volume, and temperature ( PVT) data, one obtains a universal curve. We have developed a discrete interface cell model (DICM), which is a modified form of the Prigogine—Saraga cell model theory of surface tension which had previously been applied to polymer liquids. Using PVT data as input, the model accurately describes the surface tension data. With no adjustable parameters, the DICM theory provides a simple predictive tool for the conversion of the PVT properties of any material into surface tensions, with high accuracy at all temperatures and molecular weights. A strict adherence to the corresponding states principle is observed within each oligomer-to-polymer homologous series. This allows the model to predict the surface tension to within 1% accuracy for such a homologous series. The existence of a strict corresponding states principle implies that for the polymers studied here, the dominant contribution to the surface tension comes from the cohesive and entropic properties of the bulk liquids and is only weakly dependent on the molecular conformations and end groups. A discussion of the experimental difficulties in obtaining accurate PVT data and also extrapolating the thermodynamic properties to atmospheric pressure is given.

The perturbation approach to the statistical theory of surface tension: new analytical results

A new analytical expression for the surface tension of a planar liquid-vapour interface is obtained. It is based on the Fowler approximation and the statistical-mechanical perturbation approach in the theory of liquids. The expression obtained is explicit in the equilibrium liquid density. Theoretical predictions are shown to be consistent with the results of Monte Carlo and molecular-dynamics simulations of the Lennard-Jones fluid.

Free Surface Profile and Surface Tension in a Polymer Melt: A Monte Carlo Study

Monte Carlo (MC) lattice simulation studies have been performed for a compressible polymer melt over a wide range of cohesion energies or densities. The free surface of the melt was examined with respect to its concentration profile, surface thickness D, and surface tension u. For various reduced intersegmental energies, e (e < 0), we found that u is proportional to and that D varies linearly with (e, - e)-1/2, where e, is a critical value of the reduced intersegmental energy; both relations are analogous to those for a polymer-polymer interface. The surface thickness for a representative system was calculated to be in the range D = 1.5-4.0 nm. Relative to the values found with the equation-of-state theory for surface tension of Sanchez and Poser and with the functional integral approach of Hong and Noolandi, we observed somewhat lower densities and thicker interfaces for given cohesion energies. The observed surface profiles are also symmetric rather than asymmetric as predicted by these theories. Our equation for the dimensionless parameter urd versus Td provides realistic values for the surface tension and its temperature coefficient using parameters obtained from data for the bulk polymers. The microscopic analysis of the interface from the MC simulation results revealed a surface enrichment by chain ends, a layering of coil centers beneath the surface, a continuous variation in intersegmental contacts, and a strong deformation of coils in the interface. We also found a slight variation of coil dimensions in the bulk for different temperatures and cohesion energies.