Liquid-Vapour Interfaces

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

The non-consen>ed variable (.t,0 is a broken symmetry variable, it is the instantaneous position of the Gibbs surface, and it is the translational synnnetry in z direction that is broken by the inlioinogeneity due to the liquid-vapour interface. In a more microscopic statistical mechanical approach 121, it is related to the number density fluctuation 3p(x,z,t) as... 

Holcomb C D, Clancy P and Zollweg J A 1993 A critical study of the simulation of the liquid-vapour interface of a Lennard-Jones fluid Mol. Phys. 78 437-59... 

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. 

Figure 2.10 Schematic diagram showing that the equilibrium vapour pressure changes with the curvature of the liquid-vapour interface.

For some applications flat heat pipe panels (HPP) have advantages over conventional cylindrical heat pipes, such as geometry adaptation, ability for localized heat dissipation and the maintenance of an entirely flat isothermal surface (Fig. 14). The liquid-vapour interface formed in capillary channels inside the heat pipe panel is capable to generate self-sustained thermally driven oscillations. Thin layer (several mm) of the sorbent between mini-fins on the outer side of the heat pipe panel ensures an advanced heat and mass transfer during the cycle adsorption/de sorption. 

The few investigators who have attempted to use the original Harkins-Junt method have encountered a number of inherent difficulties. A major problem is that it is virtually impossible to avoid some interparticle capillary condensation as p/p° —+ 1. This inevitably reduces the extent of the available liquid-vapour interface (Wade and Hackerman, 1960), Moreover, the thickness of a pre-adsorbed film as p/p° — 1 is highly dependent on the shape, size and roughness of the particles. 

The question of the constancy of the surface tension in porous media has been under consideration for many years and has been taken up again recently by Grown et al. (1997). Formerly, it was thought that for a concave liquid-vapour interface the surface tension should increase with increased curvature. The experimental findings that the hysteresis critical temperature is generally appreciably lower that the bulk critical temperature (see Section 7.5) is considered to be a strong indication that the surface tension of a capillary-condensate is reduced below the bulk value. More work on model pore structures is evidently required to settle this question. 

With a drop of liquid in contact with a solid (Figure C2-7), there are three interfaces the solid/liquid the solid/vapour and the liquid/vapour interfaces. Each of these has its own interface tension. For a drop that partially wets a solid, the horizontal components of the interface tensions must be in equilibrium. This determines the value of the contact angle 0... 

A small contact angle implies that the drop spreads over the surface if the contact angle is zero, the surface will be wetted completely (Figure C2-8). This happens when the solid/vapour tension is much larger than the solid/liquid tension the system then avoids any solid/vapour interface. If the contact angle is equal to Jt, there is no wetting. This happens when the solid/liquid tension is much higher than the solid/vapour tension. The system then minimizes the liquid/vapour interface, as with a drop of mercury on paper. 

In the case of a liquid/vapour interface it is straightforward to reduce the above procedure of approximating the inhomogeneous direct correlation functions Cayivi, F2) given by Equations (19)-(20) and (23)-(26) to the non-linear interpolation between two phases, vapour and liquid. 

The structure of the liquid/vapour interface of ambient methanol obtained in the SS-LMBW/RISM-KHM approach is depicted in Figure 5.5. Compared to the non-polar n-hexane, the decay of the inhomogeneity away from the interface is noticeably quicker for the polar methanol 10 A to the gas and 30 A to the liquid phase. The coarse-grained... 

FIGURE 5.4. Site density profiles of the planar liquid/vapour interface of n-hexane at ambient temperature, following from the SS-LMBW/RISM-KHM theory. Outer 1,6-CH3, middle 2.5-CH2 and inner 3.4-CH2 groups of the n-hexane molecule are represented by solid, dashed and dotted lines, respectively. The density profiles are normalized to the side densities in the liquid phase, pi". The lower part zooms in the interface region. 

Inhomogeneous Integral Equation Theory of a Liquid/Vapour Interface... 

FIGURE 5.9. Reduced density profiles of the planar liquid/vapour interface of the Lennard-Jones fluid, obtained from the lOZ-KH/LMBW theory at reduced temperature T = ksT/s = 0.725, 0.825, 0.925, 1.000, 1.050, 1.075. Bolder lines correspond to lower temperatures. 

FIGURE 5.10. Sections of the inhomogeneous two-particle distribution function g(s, zi, Z2> of the Lennard-Jones fluid along the liquid/vapour interface for reduced temperature T = 1.15 at the distance z = Z = Z2 = -lOcr 0 -HOct from the interface, respectively in the vapour phase (thin solid line) at the interface (bold solid line) and in the liquid phase (dashed line). Predictions of the lOZ-KHM/LMBW theory. 

FIGURE 5.11. Sections of the inhomogeneous two-paiticle direct conelation function c(s, z, zi) of the Lennard-Jones fluid along the liquid/vapour interface (part a), and its Hankel transform c(p, zt,zz) (ptut b). Line nomenclature is the same as in Figure 5.10. Results of die lOZ-KHM/LMBW theory. The inset in part a zooms in the long-range tail of c(s, Zi,Z2) with the asynqitodcs (47). 

In mesopores a multilayer film will be adsorbed at the pore wall as the saturation pressure is approached. The stability of this film is determined by the interaction with the wall, e.g. long-range Van der Waals Interaction, and by the surface tension and curvature of the liquid-vapour interface. Saam and Cole -have advanced a theory, showing how the curved film becomes unstable at a certain critical thickness t = a-r. The adsorption process is shown schematically in fig. 1.32a (1) (3). During desorption (4) -> (6) an asymmetrical state... 

Simulations like those discussed In sec. 2.2c, but now at the water-air Interface, support this trend. Using Monte Carlo (MC) or Molecular Dynamics (MD) the preferential orientation of the dipoles can be found, from which the dipolar part of x follows. When p Is the value of the dipole moment, P, (z) the number density of the dipoles, and (cos 6) the average orientation in a position z in the surface layer (z is normal to the liquid-vapour interface), x follows from... 

Standard heat pipe is shown on Figure 1. Basic phenomena and equations are related with liquid-vapour interface, heat transport between the outside and the interface ( radial heat transfer), vapour flow and liquid flow. 

R. Evans, The Nature of the Liquid-Vapour Interface and other Topics in the Statistical Mechanisms of Non-Uniform ClassiccLl Fluids. Adv. Phys. 28 (1979) 143-200. (Theoretical emphasis on simple molecules, extensive discussion of the... 

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