Liquid meniscus, thermodynamic

As with all thermodynamic relations, the Kelvin equation may be arrived at along several paths. Since the occurrence of capillary condensation is intimately, bound up with the curvature of a liquid meniscus, it is helpful to start out from the Young-Laplace equation, the relationship between the pressures on opposite sides of a liquid-vapour interface. 

Before we start a discussion on the thermodynamics of a contact between particles, it is worthwhile to briefly address the phenomena taking place at the three-phase contact line, and in particular, wetting and capillary forces acting within a liquid meniscus. We will also briefly summarize the principal methods of surface tension measurement. 

O Figure 4.14 illustrates the situation before and after wetting at thermodynamic equilibrium. For simplicity, consider a solid with at least one smooth and flat part of its surface. In the initial state (superscript 1), the solid has a phase boundary with the common vapor only. Hence, there are two D-faces with the areas Ajy, in the initial state. Then, the solid is lowered until the flat part of its surface is just isothermally wetted at constant pressure by the flat liquid meniscus. This is the final state (superscript 2) with three D-faces (areas A y, A y, A l) and one triple line. (Any immersion of the solid into the liquid would simply create new area of the D-faces. Such motion only complicates the thermodynamic free energy balance without giving any additional insight.)... 

The topic of capillarity concerns interfaces that are sufficiently mobile to assume an equilibrium shape. The most common examples are meniscuses, thin films, and drops formed by liquids in air or in another liquid. Since it deals with equilibrium configurations, capillarity occupies a place in the general framework of thermodynamics in the context of the macroscopic and statistical behavior of interfaces rather than the details of their molectdar structure. In this chapter we describe the measurement of surface tension and present some fundamental results. In Chapter III we discuss the thermodynamics of liquid surfaces. 

A capillary system is said to be in a steady-state equilibrium position when the capillary forces are equal to the hydrostatic pressure force (Levich 1962). The heating of the capillary walls leads to a disturbance of the equilibrium and to a displacement of the meniscus, causing the liquid-vapor interface location to change as compared to an unheated wall. This process causes pressure differences due to capillarity and the hydrostatic pressures exiting the flow, which in turn causes the meniscus to return to the initial position. In order to realize the above-mentioned process in a continuous manner it is necessary to carry out continual heat transfer from the capillary walls to the liquid. In this case the position of the interface surface is invariable and the fluid flow is stationary. From the thermodynamical point of view the process in a heated capillary is similar to a process in a heat engine, which transforms heat into mechanical energy. 

Most models to calculate the pore size distributions of mesoporous solids, are based on the Kelvin equation, based on Thomson s23 (later Lord Kelvin) thermodynamical statement that the equilibrium vapour pressure (p), over a concave meniscus of liquid, must be less than the saturation vapour pressure (p0) at the same temperature . This implies that a vapour will be able to condense to a liquid in the pore of a solid, even when the relative pressure is less than unity. This process is commonly called the capillary condensation. 

Since the capillary condensate in a particular mesopore is in thermodynamic equilibrium with the vapour, its chemical potential, p°, must be equal to that of the gas (under the given conditions of T and p). As we have seen, the difference between p° and p1 (the chemical potential of the free liquid) is normally assumed to be entirely due to the Laplace pressure drop, Ap, across the meniscus. However, in the vicinity of the pore wall a contribution from the adsorption potential, 0(z), should be taken into account. Thus, if the chemical potential is to be maintained constant throughout the adsorbed phase, the capillary condensation contribution must be reduced. 

Another pore filling model based upon capillary equilibrium in cylindrical pores has recently been proposed in which the condition of thermodynamic equilibrium is modified to include the effects of surface layering and adsorbate-adsorbent interactions [135-137]. Assuming that the vapor-liquid interface is represented by a cylindrical meniscus during adsorption and by a hemispherical meniscus during desorption, and invoking the Defay-Prigogene expression for a curvature-dependent surface tension [21], the equilibrium condition for capillary coexistence in a cylindrical pore is obtained as... 

One can observe that a meniscus attached to a horizontal plate spontaneously falls off at a certain critical height of the plate. On the other hand, if the plate is immersed into a liquid bath, the liquid spontaneously spreads and wets the entire plate at a critical depth. In this section, we first discuss the unstable phenomenon of a two-dimensional meniscus under a horizontal plate from a thermodynamic viewpoint based on Eq. (10) above. Then, in order to verify... 

It can see from the above-mentioned discussion that capillary cohesion is closely related to the curved liquid surface. The pressme boimdary causes capillary cohesion — the critical vapor pressure relates to the cmvatme radius of liquid surface. Kelvin equation has been derived from thermodynamics, where the curvature radius (rjs) of the meniscus of hemispherical (concave) liquid and the equilibrium vapor pressure (p) has the following relationships ... 

In thermodynamic equilibrium, a small vertical displacement of the meniscus together with the liquid column below it, 8s, implies that the weight of the liquid column performs an amount of work per unit length of the column that is equal to an increase of the free energy of the film. It is then possible to find the thermodynamic thickness of the wetting film on the channel wall from the known isotherm of the disjoining pressure and the film tension ... 

We write the conditions of quasi-equilibrium of the meniscus in region 1, in the boundaries of which fluxes can be neglected and where the excess pressure can be considered to be constant at all points and equal to P = P h- AP = const. We assume that Equation 3.311 still describes the quasi-equihbrium of profile of the liquid, h(x), in region 1 in the absence of full thermodynamic equilibrium in the whole system. That is, we adopt ... 

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