Lippmann Equation Research Articles

Thermodynamic aspects of capillarity and electrocapillarity of solid interfaces

In the previous paper (Gutman, JOSSEC 18:3217–3237, 2014), we have shown that the main problem in capillarity and electrocapillarity of solid surfaces is the lack of clarity in determining the surface stress and basic equations. Now, we continue the survey of efforts to solve this problem and show origins of erroneous results, accenting some important items: comparative analysis of Gibbs and Guggenheim approaches in surface thermodynamics (a geometrical dividing surface and finite-thickness surface layer, respectively), transformation of fundamental equations on per-unit-area basis to obtain Gibbs adsorption equation for finite-thickness surface layer, different attempts to derive the thermodynamic definition of “surface stress” in frames of Gibbs’ theory (including Shuttleworth’s approach), atomistic calculations of surface stress, surface stress in rational continuum mechanics, “modifications” of Gibbs–Duhem relations made for solid interface, and Maxwell relations in capillarity and electrocapillarity of solid interface. It is shown that the erroneous Shuttleworth’s approach is present in an explicit or implicit form in all efforts to introduce the surface stress in frames of Gibbsian theory (although Gibbs did not introduce surface stress). Therefore, “modernizations” or “generalizations” of the Gibbs–Duhem relation, the Gibbs adsorption equation, and the Lippmann equation to adopt them for a solid surface are unnatural and not necessary. Therefore, we recommend withdrawing the Shuttleworth equation and its consequences from circulation, including the IUPAC Recommendations.

Electrowetting force and velocity dependence on fluid surface energy

Electrowetting on dielectric is a phenomenon in which the shape and apparent contact angle of a droplet changes when an electric field is applied across the droplet interface. If the field is asymmetric with respect to the droplet, then a net force can be applied to the droplet. In this work, we have measured the electrowetting force by confining the droplet shape beneath a glass plate and measuring the force on the plate. The force was measured as a function of voltage for a range of fluids with different surface energy. Measured forces show excellent agreement with predictions based on the Young–Lippmann equation with measured contact angles. Results also show that the electrowetting force is independent of fluid surface energy below saturation but that the peak force is proportional to the surface tension. This work shows that lowering the surface energy of the fluid can induce larger contact angle change under the same voltage, but it has no beneficial impact on the actuation force in droplet-based actuators. In contrast, velocity tests with deformable droplets show higher speeds for lower surface energy fluids, even above their saturation voltage. However, when the droplet’s shape is restrained, the highest velocity is achieved with high surface energy fluids due to the larger electrowetting actuation forces applied.

Theoretical problems in solid electrocapillarity

The main problem in electrocapillarity of solid electrodes is the lack of clarity in determining the surface stress and basic equations. Within the framework of the Gibbs concept of geometrical dividing surface, the “surface stress” cannot be defined because methods of continuum mechanics can be applied to a physical surface layer (of finite thickness), but not to a mathematical surface. Gibbs never used the concept of surface stress, introducing only “surface tension” for a liquid electrode and “closely related quantity” for a solid electrode. Revisiting the derivation of the Gibbs adsorption equation, we prove its applicability to solid surfaces without the limiting requirement of constant state of strain, which was undeservedly interpreted by Eriksson as a shortcoming of the Gibbs theory caused to look for other approaches to surface stress problem. A critical analysis shows that the attempts (Shuttleworth, Eriksson, Couchman, Gokhstein, Weissmuller, etc.) to create a thermodynamic definition of the surface stress (as well as the formulation of fundamental thermodynamic equations and Maxwell relations operating with surface stresses) contain mathematical defects. It is shown that confusing interpretations of some Gibbs’ concepts encountered in the literature have led to “modifications” of the Lippmann equation based on the critical error in the Gibbs–Duhem relation due to the occurrence of an extensive variable, which is inadmissible. The famous Lippmann equation should not be modified, and it remains a unique electrocapillary relation applicable to liquid and solid electrodes.

Study of Electrocapillarity in Dielectrics Using Palm Oil

This paper presents analytical and experimental study of electrocapillarity in dielectrics under the frame work of parallel plates using palm oil.. First, Mathematical approach was used to derive an expression relating capillary ascent with the applied field. The results obtained from the experiment and that of the analytical results were compared with the expected results from the equation relating the liquid ascent and the applied field obtained from Lippmann equation, where it was observed that there is no evidence of linear relation between the height of the capillary rise of oil in the tube and the applied field.

Neither Lippmann nor Young: Enabling Electrowetting Modeling on Structured Dielectric Surfaces

Aiming to illuminate mechanisms of wetting transitions on geometrically patterned surfaces induced by the electrowetting phenomenon, we present a novel modeling approach that goes beyond the limitations of the Lippmann equation and is even relieved from the implementation of the Young contact angle boundary condition. We employ the equations of the capillary electrohydrostatics augmented by a disjoining pressure term derived from an effective interface potential accounting for solid/liquid interactions. Proper parametrization of the liquid surface profile enables efficient simulation of multiple and reconfigurable three-phase contact lines (TPL) appearing when entire droplets undergo wetting transitions on patterned surfaces. The liquid/ambient and the liquid/solid interfaces are treated in a unified context tackling the assumption that the liquid profile is wedge-shaped at any three-phase contact line. In this way, electric field singularities are bypassed, allowing for accurate electric field and liquid surface profile computation, especially in the vicinity of TPLs. We found that the invariance of the microscopic contact angle in electrowetting systems is valid only for thick dielectrics, supporting published experiments. By applying our methodology to patterned dielectrics, we computed all admissible droplet equilibrium profiles, including Cassie-Baxter, Wenzel, and mixed wetting states. Mixed wetting states are computed for the first time in electrowetting systems, and their relative stability is presented in a clear and instructive way.

Application of the Lippmann electrocapillary equation to adsorption potential shift at the air/solution interface – Analysis for undissociated n-decanoic acid

The equilibrium surface pressure (Πe) and electric surface potential (Δχe) vs. concentration (c) isotherms of n-decanoic acid (DA) in 5×10−3mol/dm3 HCl were concurrently measured at the air/solution interface – the Π with a platinum Wilhelmy plate, the Δχ with the vibrating plate method (named also the dynamic condenser). The Πe–Δχe dependence was used by us for calculating the surface charge’ density basing on the Lippmann first electrocapillary equation. It was applied under the assumption that: (i) the surface charge is solely formed by orientation of the adsorbate, and (ii) the changing DA bulk concentration does not deprive the Πe–Δχe dependence of consistency with the Lippmann equations. The charge (Qe) expressed per the adsorbate molecule (basing on the Γ–c dependence predicted by the best fit of Frumkin model to the experimental Πe–c results) increases with the adsorption coverage (Θ) within a fraction of electron per the adsorbate molecule – the reasonable result for a non ionic surfactant. The Qe vs. Θ course shows a great analogy to that of quotient of the effective dipole moment to the interface’s permittivity, μ⊥εsvs.Θ, derived from the Δχe–Γ dependence. The two dependencies: Qevs.Θ and μ⊥εsvs.Θ, show a plateau within the coverage range of 20<Θ<80% and a steep increase above it. Recalculation of the Qevs.Θ dependence for the equivalent dipole moment, under the assumption that the dipole length is comparable to length of a polar bond of the head group (ca. 1.2Å) implies the bond’s inclination to the interface of 12°. Such low inclination is reasonable in view of an angular deviation of molecular dipole moment from amphiphile molecule’ long axis.

Comment on “On the ‘simple check’ of electrocapillarity” by AY Gokhshtein

It is shown that the so-called “equation of solid-state electrocapillarity” derived by Gokhshtein from the simplest thermodynamic model, earlier used by Lippmann, is an incorrect modification that is returned to the classic Lippmann equation if to take into account the usual definition of differential capacity of the electrical double layer. Consequently, Gokhshtein’s experiments can actually confirm only the validity of the Lippmann equation (with deviations caused by physicochemical processes in the double layer, which could not be taken into account in the thermodynamic equations). The thermodynamic model is insufficient for the interpretations of these experimental results, whose understanding requires physical models of the different phenomena occurring at the electrode surface.

A generalized electrowetting equation: Its derivation and consequences

The thermodynamics of electrowetting is treated. A general equation of electrowetting is derived from the first principles. It is demonstrated that the well-known Lippmann Equation describes a particular case of electrowetting when the radial derivative of the capacitance of the double layer is constant. The apparent contact angle of electrowetting depends on the gradient of capacity of a double layer in the vicinity of the triple line. The role of the area adjacent to the triple line in constituting the equilibrium apparent contact angle of electrowetting is emphasized.

Interaction of Magnesium Ions with Pristine Single-Layer and Defected Graphene/Water Interfaces Studied by Second Harmonic Generation

This work reports thermodynamic and electrostatic parameters for fused silica/water interfaces containing cm(2)-sized graphene ranging from a single layer of pristine graphene to defected graphene. Second harmonic generation (SHG) measurements carried out at pH 7 indicate that the surface charge density of the fused silica/water interface containing the defected graphene (-0.009(3) to -0.010(3) C/m(2)) is between that of defect-free single layer graphene (-0.0049(8) C/m(2)) and bare fused silica (-0.013(6) C/m(2)). The interfacial free energy of the fused silica/water interface calculated from the Lippmann equation is reduced by a factor of 7 in the presence of single-layer pristine graphene, while defected graphene reduces it only by a factor of at most 2. Subsequent SHG adsorption isotherm studies probing the Mg(2+) adsorption at the fused silica/water interface result in fully reversible metal ion interactions and observed binding constants, Kads, of 4(1) - 5(1) × 10(3) M(-1) for pristine graphene and 3(1) - 4(1) × 10(3) M(-1) for defected graphene, corresponding to adsorption free energies, ΔGads, referenced to the 55.5 molarity of water, of -30(1) to -31.1(7) kJ/mol for both interfaces, comparable to Mg(2+) adsorption at the bare fused silica/water interface. Maximum Mg(2+) ion densities are obtained from Gouy-Chapman model fits to the Langmuir adsorption isotherms and found to range from 1.1(5) - 1.5(4) × 10(12) ions adsorbed per cm(2) for pristine graphene and 2(1) - 3.1(5) × 10(12) ions adsorbed per cm(2) for defected graphene, slightly smaller than those of for Mg(2+) adsorption at the bare fused silica/water interface ((2-4) × 10(12) ions adsorbed per cm(2)), assuming the magnesium ions are bound as divalent species. We conclude that the presence of defects in the graphene sheet, which we estimate here to be around 1.3 × 10(11) cm(2), imparts only subtle changes in the thermodynamic and electrostatic parameters quantified here.

Thermodynamics of the Polymer-Concrete Interface

Property of polymer-concrete in the interface has been dealt with from the view of thermodynamics. Gibbs function or free enthalpy was applied to characterize the binding ability of the interface. AC impedance spectroscopy was used to measure the electrical capacitance of the interface between the polymer particle and mortar. Lippmann equation was used to estimate the thermodynamic functions for the estimation of the binding ability. A typical example was given.

Calculation of the surface potential and surface charge density by measurement of the three-phase contact angle

The silica/silicon wafer is widely used in the semiconductor industry in the manufacture of electronic devices, so it is essential to understand its physical chemistry and determine the surface potential at the silica wafer/water interface. However, it is difficult to measure the surface potential of a silica/silicon wafer directly due to its high electric resistance. In the present study, the three-phase contact angle (TPCA) on silica is measured as a function of the pH. The surface potential and surface charge density at the silica/water surface are calculated by a model based on the Young–Lippmann equation in conjunction with the Gouy–Chapman model for the electric double layer. In measurements of the TPCA on silica, two distinct regions were identified with a boundary at pH 9.5—showing a dominance of the surface ionization of silanol groups below pH 9.5 and a dominance of the dissolution of silica into the aqueous solution above pH 9.5. Since the surface chemistry changes above pH 9.5, the model is applied to solutions below pH 9.5 (ionization dominant) for the calculation of the surface potential and surface charge density at the silica/aqueous interface. In order to evaluate the model, a galvanic mica cell was made of a mica sheet and the surface potential was measured directly at the mica/water interface. The model results are also validated by experimental data from the literature, as well as the results obtained by the potentiometric titration method and the electro-kinetic measurements.

Active surface tension driven micropump using droplet/meniscus pressure gradient

An active micropump with a simple layout and no moving parts is designed and fabricated which has on demand flow on/off capability. The micropump is based on droplet/meniscus pressure gradient generated by electrowetting on dielectric (EWOD). By altering the contact angle between liquid and solid using an electric field a pressure gradient was induced and a small droplet was pumped into the channel via a uniform flow rate. A surface tension based propellant method was introduced as a low power consumption actuation method in microfluidic devices. The liquid contact angle on the EWOD substrate was measured vs. electric potential and was used to obtain the capacitance of the substrate by fitting Young–Lippmann's equation. The capacitance of the EWOD substrate was also calculated to be 10±0.6μF/m2 by measuring the dielectric layer thickness which showed excellent agreement with the former method. EWOD setup parameters such as capacitance, saturation contact angle, hysteresis contact angle and onset voltage were discussed. A coupled theoretical–experimental model was developed to predict how much voltage is needed to start the micropump for different droplet sizes. The modeling results revealed that for droplets with a radius smaller than 0.4mm the droplet will start going into the channel even when no voltage is applied. For any larger droplet, a certain voltage is needed to start the pump. It was also shown that decreasing the size of the input droplet and increasing the voltage will result in an increase in the pump flow. A model for describing the shrinkage of the micropump input droplet was developed, based on direct observations, which was in agreement with the forced wetting described in literature. This model was compared to the other models used to describe passively pumped droplets and evaporating microdrops.

Dynamics of Droplet Motion under Electrowetting Actuation

The static shape of droplets under electrowetting actuation is well understood. The steady-state shape of the droplet is obtained on the basis of the balance of surface tension and electrowetting forces, and the change in the apparent contact angle is well characterized by the Young-Lippmann equation. However, the transient droplet shape behavior when a voltage is suddenly applied across a droplet has received less attention. Additional dynamic frictional forces are at play during this transient process. We present a model to predict this transient behavior of the droplet shape under electrowetting actuation. The droplet shape is modeled using the volume of fluid method. The electrowetting and dynamic frictional forces are included as an effective dynamic contact angle through a force balance at the contact line. The model is used to predict the transient behavior of water droplets on smooth hydrophobic surfaces under electrowetting actuation. The predictions of the transient behavior of droplet shape and contact radius are in excellent agreement with our experimental measurements. The internal fluid motion is explained, and the droplet motion is shown to initiate from the contact line. An approximate mathematical model is also developed to understand the physics of the droplet motion and to describe the overall droplet motion and the contact line velocities.

Current state and problems in surface tension of solids, logical and experimental analysis of basic equations

Of the four basic thermodynamic equations for the surface tension of solids: the generalized and classical Lippmann equations, and the Shuttleworth and Gokhshtein equations, only the Gokhshtein equation has been confirmed experimentally. The generalized Lippmann equation is generally considered to be more universal, since three other equations could be derived from it. This fact has been widely accepted, but recently it was reevaluated in two opposite ways. In the first approach, the experimental verification of the Gokhshtein equation should support the correctness of the generalized Lippmann and Shuttleworth equations. In the second approach, the incompatibility of the Shuttleworth equation with Hermann’s mathematical structure of thermodynamics raises doubts upon all its corollaries, including the generalized Lippmann and Gokhshtein equations. Both these logical approaches are shown to be erroneous, since the Gokhshtein equation cannot be correctly derived from any of the above-mentioned equations, and the opposite is also true: neither the generalized Lippmann nor Shuttleworth equations could be derived from the Gokhshtein equation. The high sensitivity of the contact electric resistance method to negative adsorbate charge allows detail investigation of the kinetics of the charge transfer during anion adsorption. For the fist time, this permits a tentative approach to quantify the potential dependence of the partial charge transfer during halide ions adsorption on IB metals.

Electrowetting of Aqueous Solutions of Ionic Liquid in Solid−Liquid−Liquid Systems

The influence of water on the electrowetting performance of the ionic liquid (IL) [BMIM][BF4] in contact with an ambient phase of hexadecane was studied. Electrowetting experiments on a fluoropolymer surface with IL concentrations between 0 and 99% enabled investigation of various electrowetting parameters, including the maximum range of accessible contact angles, the saturation contact angle (and voltage), and spreading dynamics. Experiments carried out using dc and ac potentials were successfully described with the Young−Lippmann equation prior to contact angle saturation. The saturation contact angles differed for ac and dc electrowetting, indicating distinct saturation mechanisms. The dilution of [BMIM][BF4] by water, within the range of concentrations studied, had no direct impact on the electrowetting behavior, aside from indirect effects due to altered viscosity and interfacial tension.

Concept of reversible cleavage in surface tension of solids

“Wholly plastic (as in cleavage)” surface area changing is the widely accepted concept formally allowing the application of Gibbs thermodynamics to surface tension of solids considered as a reversible process of partly plastic and partly elastic deformation as recommended by the IUPAC. However, this concept contains some internal contradictions that are serious enough to raise doubts about its correctness and, consequently in validity of the thermodynamic analysis of surface tension of solid electrodes based on this concept. This analysis leads to the generalized Lippmann equation and eventually approves using the classical Lippman equation for solids. The latter is shown to be erroneous since it is based on the misleading concept of “reversible cleavage” and contradicts all available in situ experimental data.

General thermodynamic equations for the surface tension of liquids and solids

The old-but-new Frumkin derivation of the first and second Gokhshtein equations follows from one thermodynamic relationship general to the surface tension of both solid and liquid electrodes: ∂ γ / ∂ E = - ∂ Q / ∂ A The classical Lippman equation is also derived from this general relationship as a particular case of the second Gokhshtein equation. We have considered the hierarchy of these equations and discussed the straightforward application of the classical Lippmann equation for solids, accounting for the elasticity of the surface structured layers of liquids.

Hierarchy of equations for the surface tension of solids

Of the four main equations in thermodynamics for the surface tension of condensed matter, i.e. the generalized and classical Lippmann equations and the Shuttleworth and Gokhshtein equations, only the classical Lippmann and Gokhshtein equations have been confirmed experimentally. The generalized Lippmann (Couchman–Davidson) equation is considered to be more universal, since three other equations could be derived from it. Although this fact has been widely accepted, it was recently reevaluated in two opposite ways. In the first approach, the experimental verification of the Gokhshtein equation should support the correctness of the generalized Lippmann and Shuttleworth equations. In the second approach, the incompatibility of the Shuttleworth equation with Hermann's mathematical structure of thermodynamics throws doubts upon all its corollaries, including the generalized Lippmann and Gokhshtein equations. However, both of these approaches are here shown to be erroneous, since the Gokhshtein equation cannot be correctly derived from any of the above-mentioned equations, and the opposite is also true: neither the generalized Lippmann nor Shuttleworth equations could be derived from the Gokhshtein equation.

Vague concept of “reversible cleavage” in the theory of the surface tension of solids

Numerous derivations of the well-known Shuttleworth equation have been based on the unclear concept of “reversible cleavage” leading to the decisive step in any derivation - equalization of the surface free energy and surface stress. This is the key concept in contemporary surface thermodynamics of solids. But “cleavage” is not a surface process and, in this field, it cannot be a reversible operation. Besides, the “reversible cleavage” has no formal definition in the domain of the surface tension of solids that is an abnormal for any exact science. Consequently, this concept and all its corollaries including the Shuttleworth and generalized Lippmann equations have to be recognized as incorrect.

Electrowetting of Ionic Liquids: Contact Angle Saturation and Irreversibility

In this work, electrowetting of several room temperature ionic liquids having in common the cation or the anion, [EMIM][EtSO4], [EMPy][EtSO4], [EMIM][NTf2], [BMIM][BF4], [BMIM][TFA], and [OMIM][BF4], on poly(tetrafluoroethylene) (PTFE) was investigated. The typical behavior of aqueous salt solutions was found: symmetric parabolic curves of contact angle versus (positive and negative) applied voltage, at low voltages, and contact angle saturation after a threshold voltage. Furthermore, the contact angle did not recover its initial value when the voltage decreased, and this irreversibility was found even at low voltages. The dependence of the contact angle with the applied voltage is generally described by the Young−Lippmann equation before contact angle saturation. In contrast, the Langevin function as well as a modified form of the Young−Lippmann equation were found to fit the experimental electrowetting curves in the whole range of voltages. A correlation between the electrowetting behavior and the surfa...