Solid-liquid interface, Gibbs

Table VI. Gibbs Energies of Solid-Liquid Interfaces, ergs/sq. cm., 25°C.

The cosine cannot exceed one. Then we might ask What happens, if 7s — 7sl — 7l > 0 or 7S - 7Si is higher than 7// Does this not violate Young s equation No, it does not because in thermodynamic equilibrium 7s — Ysl — 1l can never become positive. This is easy to see. If we could create a situation with 7s > 7SL + 7l, then the Gibbs free energy of the system could decrease by forming a continuous liquid film on the solid surface. Vapor would condense onto the solid until such a film is formed and the free solid surface would be replaced to a solid-liquid interface plus a liquid surface. 

On a liquid-gas interface, the partial pressure of the adsorbed gas is substituted in Equation 1.59. On the solid-gas and solid-liquid interfaces, only the excess surface concentration can be measured directly, and not the surface tension. The Gibbs adsorption isotherm is suitable for the calculation of the change of surface tension. 

In geological surfaces, the solid-gas and solid-liquid interfaces are important, so the correct thermodynamic adsorption equation (Gibbs isotherm) cannot be used. Instead, other adsorption equations are applied, some of them containing thermodynamic approaches, and others being empirical or semiempirical. One of the most widespread isotherms is the Langmuir equation, which was derived for the adsorption of gas molecules on planar surfaces (Langmuir 1918). It has four basic assumptions for adsorption (Fowler 1935) ... 

With a liquid-vapor interface, Gibbs [36] has developed a thermodynamic treatment of the variation of surface tension with composition. This derivation comes from the book Physical Chemistry of Surfaces by Adamson [2, p. 340]. This derivation sets the stage for adsorption at the solid—liquid interface, which will be discussed next. 

For many purposes it is conducive to start analyses with thermodynamic considerations. In this way, it is often possible to find laws of general validity and to determine the boundaries between which models can be developed. For the study of (relaxed) double layers the Gibbs adsorption equation is the starting point. Although the interfacial tension of a solid-liquid interface cannot be measured, this equation remains useful because it helps to distinguish measurable and Inoperable variables, and because it can be used to correlate surface concentrations of different species (Including the surface ions), some of which may not be analytically accessible. 

The Gibbs adsorption equation (equations 2.19, 2.19a) at the solid-liquid interface can be written as... 

Any flexible polymer adsorbing to a solid/liquid interface must (1) be transported from the solution towards the surface, (2) establish a sufficient number of attachment points, and (3) change its conformation to achieve the state of lowest Gibbs energy, by accommodating to the geometrical properties of the substrate [1]. 

The Gibbs equation applies also to the solid/liquid interface. Direct measurements of the surface tension of a solid are possible only in very special circumstances. On the other hand, as outlined in Appendix III, adsorption at a solid surface is usually more... 

If we deform the liquid surface slightly so that the area of the solid-liquid interface increases by dA i, then the Gibbs energy change is... 

Thus, if (7 is positive, (dG/dA i) is negative, and the Gibbs energy will decrease as the solid-liquid interface enlarges the liquid will spread spontaneously. If = 0, the configuration is stable (in equilibrium) with respect to variations in the area of the solid-liquid interface. If is negative, the liquid will contract and decrease spontaneously. Combining Eqs. (18.21) and (18.22) we get... 

The adsorption of polymers at the liquid/liquid interface is somewhat different from that at the solid/liquid interface as the polymer can penetrate both phases, x determines the adsorption behavior of polymers at liquid/liquid interfaces. The presence of the polymer at the interface between the two immiscible liquids lowers the surface tension. Determination of the adsorption isotherm (see Section IX.B) is more straightforward compared to particulate dispersions as surface tension measurements, interpreted using the Gibbs equation, can be used to give accurate adsorbed amounts. 

The free enthalpy of adsorption of solid/liquid interfaces can be calculated with the Gibbs equation [48-50] and knowledge of the isotherm = /(xj) ... 

It follows the Gibbs equation for solid-liquid interfaces ... 

Except for the term I Afi, this expression was in fact already employed by Gibbs [1]. Gibbs, however, tacitly assumed that yi = j2, which implies that C = 0, an assumption that presumably holds for a vast majority of interfaces in the liquid state (as opposed to the gel state) but not necessarily for interfaces of a more complicated structure, such as biological membranes or, for the part, rubber membranes and many solid-liquid interfaces. In the above expression, y denotes the ordinary (thermodynamic) surface tension and is defined as the mean value of the principal components yi and j2 of the surface tension tensor... 

The adsorption isotherms in Table 4.2 can be applied to both fluid and solid interfaces. The surface tension isotherms in Table 4.2, which relate o and Fj, are usually applied to fluid interfaces, although they could also be used also for solid-liquid interfaces if o is identified with the Gibbs [4] superficial tension. (The latter is defined as the force per unit length which opposes every increase of the wet area without any deformation of the solid.)... 

The preferential sorption-capillary flow model starts from the consideration of the solid-liquid interface. For example, aqueous sodium chloride solution is in contact with a solid surface. Sodium chloride solution represents the reverse osmosis system where the separation of solute (sodium chloride) flrom solvent (water) occurs. This system also represents one of the most important applications of reverse osmosis, i.e., seawater desalination. A concentration gradient should inevitably appear at the solution-solid interface, as shown in Figure 6.1. The Gibbs adsorption isotherm... 

Fig. 1. Gibbs free energy curves with respect to composition near a solid—liquid interface (schematically). C, and Cj points correspond to an equilibrium state at the interface, and chemical potentials of components in the phases (tangents to the curves at these points) are equal. Deviation of the actual liquid composition (C,) at the interface temperature T, ftom the equilibrium concentration C, generates a tendency to restore the equilibrium (C — Ci, C2) or a driving force (AG,) for nucleation.

Surfactants lower the surface tension of the liquid, yiy, and they also adsorb at the solid/liquid interface, lowering yg. The adsorption of surfactants at the liquid/air interface can be easily described by the Gibbs adsorption equation [5],... 

Recently, Angioletti-Uberti et al. [24, 25] have used a novel approach based on MTD to calculate Ysi at the melting temperature Tm- Since the chemical potentials lx P,T)(P denotes the pressure) of the solid and liquid bulk phases coincide at Tm, the Gibbs free energy G (P, 7] ) of a system containing a solid-liquid interface can be written as ... 

This section will deal with the above interfacial aspects starting with the equilibrium aspects of surfactant adsorption at the air/water and oil/water interfaces. Due to the equilibrium aspects of adsorption (rate of adsorption is equal to the rate of desorption) one can apply the second law of thermodynamics as analyzed by Gibbs (see below). This is followed by a section on dynamic aspects of surfactant adsorption, particularly the concept of dynamic surface tension and the techniques that can be applied in its measurement. The adsorption of surfactants both on hydrophobic surfaces (which represent the case of most agrochemical solids) as well as on hydrophilic surfaces (such as oxides) will be analyzed using the Langmuir adsorption isotherms. The structure of surfactant layers on solid surfaces will be described. The subject of polymeric surfactant adsorption will be dealt with separately due to its complex nature, namely irreversibility of adsorption and conformation of the polymer at the solid/liquid interface. 

Of importance in the case of mineral fibers is the adsorption of substances from a solution in which they are submerged. Willard Gibbs published an isotherm for the adsorption of dissolved substances onto a solid substrate. It was shown that a substance which lowered the surface tension of a liquid at the solid-liquid interface would usually be more concentrated on the surface of a submerged solid than in the body of the solution. These are usually referred to as surface-active agents. On the other hand, substances that raise the surface tension of a liquid in which they are dissolved will be negatively adsorbed at a surface. The concentration of these salts will be greater in the body of the solution than at an interface. In company reports, I have called these salts body active salts. Many electrolytes will raise the surface tension of water. 

But, when the curvature of the interface becomes very strong the Gibbs-Thomson effect close to the crucible wall has to be taken into account [55]. The consequence is that the phase boundary must not follow completely the melting isotherm and an extended undercooled region exists in front of the solid/liquid interface that also increases the probability for defect formation as shown in Fig. 5.17. The extension of the undercooled region can be expressed by, for example, increasing the temperature gradient in front of the solid/liquid interface. 

---