Curvature Kelvin equation

Here, r is positive and there is thus an increased vapor pressure. In the case of water, P/ is about 1.001 if r is 10" cm, 1.011 if r is 10" cm, and 1.114 if r is 10 cm or 100 A. The effect has been verified experimentally for several liquids [20], down to radii of the order of 0.1 m, and indirect measurements have verified the Kelvin equation for R values down to about 30 A [19]. The phenomenon provides a ready explanation for the ability of vapors to supersaturate. The formation of a new liquid phase begins with small clusters that may grow or aggregate into droplets. In the absence of dust or other foreign surfaces, there will be an activation energy for the formation of these small clusters corresponding to the increased free energy due to the curvature of the surface (see Section IX-2). 

Bikerman [179] has argued that the Kelvin equation should not apply to crystals, that is, in terms of increased vapor pressure or solubility of small crystals. The reasoning is that perfect crystals of whatever size will consist of plane facets whose radius of curvature is therefore infinite. On a molecular scale, it is argued that local condensation-evaporation equilibrium on a crystal plane should not be affected by the extent of the plane, that is, the crystal size, since molecular forces are short range. This conclusion is contrary to that in Section VII-2C. Discuss the situation. The derivation of the Kelvin equation in Ref. 180 is helpful. 

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. 

At the junction of the adsorbed film and the liquid meniscus the chemical potential of the adsorbate must be the resultant of the joint action of the wall and the curvature of the meniscus. As Derjaguin pointed out, the conventional treatment involves the tacit assumption that the curvature falls jumpwise from 2/r to zero at the junction, whereas the change must actually be a continuous one. Derjaguin put forward a corrected Kelvin equation to take this state of affairs into account but it contains a term which is difficult to evaluate numerically, and has aroused little practical interest. 

At the middle of the capillary where the eflect of the walls on chemical potential is negligible, the radius of curvature will be equal to r as calculated by the Kelvin equation (3.20) but it will become progressively larger as the wall is approached. 

In our derivation of the Kelvin equation, the radius is measured in the liquid. For a gas bubble in a liquid, the same equilibrium is involved, but the bubble radius is measured on the opposite side of the surface. As a consequence, a minus sign enters Equation (40) when it is applied to bubbles. Since y is on the order of millijoules while RT is on the order of joules, Equation (40) predicts that In (p/p0) is very small. It is important to realize, however, that y/R is also divided by the radius of the spherical particle and therefore becomes more important as Rs decreases. For water at 20°C, the Kelvin equation predicts values of p/p0 equal to 1.0011, 1.0184, 1.1139, and 2.94 for drops of radius 10 6, 10"7, 10 8, andl0 9m, respectively. For bubbles of the same sizes in liquid water, p/p0 equals 0.9989, 0.9893, 0.8976, and 0.339. These calculations show that the effect of surface curvature, while relatively unimportant even for particles in the micrometer range, becomes appreciable for very small particles. 

The Kelvin effect is not limited to spheres. For example, the neck of liquid between two supports is described by a paraboloid of revolution. In this case the radius of curvature of the concave surface is outside the liquid and therefore introduces a minus sign into Equation (40). In addition, the factor of 2 is not required to describe the Kelvin effect for this geometry. Example 6.2 illustrates a test of the Kelvin equation based on this kind of liquid neck. 

EXAMPLE 6.2 Use of the Kelvin Equation for Determining Surface Tension. Figure 6.5 shows a plot of experimental data that demonstrates the validity of the Kelvin effect. Necks of liquid cyclohexane were formed between mica surfaces at 20°C, and the radius of curvature was measured by interferometry. Vapor pressures were measured for surfaces with different curvature. Use these data to evaluate 7 for cyclohexane. Comment on the significance of the fact that the linearity of Figure 6.5 extends all the way to a p/p0 value of 0.77. 

FIG. 6.5 Verification of the Kelvin equation for cyclohexane necks of different curvature between mica surfaces (discussed in Example 6.2). (Redrawn from L. R. Fisher and J. N. Israelach-vili, Nature, 277, 548 (1979).)... 

In this chapter we get to know the second essential equation of surface science — the Kelvin5 equation. Like the Young-Laplace equation it is based on thermodynamic principles and does not refer to a special material or special conditions. The subject of the Kelvin equation is the vapor pressure of a liquid. Tables of vapor pressures for various liquids and different temperatures can be found in common textbooks or handbooks of physical chemistry. These vapor pressures are reported for vapors which are in thermodynamic equilibrium with liquids having planar surfaces. When the liquid surface is curved, the vapor pressure changes. The vapor pressure of a drop is higher than that of a flat, planar surface. In a bubble the vapor pressure is reduced. The Kelvin equation tells us how the vapor pressure depends on the curvature of the liquid. 

An important application of the Kelvin equation is the description of capillary condensation. This is the condensation of vapor into capillaries or fine pores even for vapor pressures below Pequilibrium vapor pressure of the liquid with a planar surface. Lord Kelvin was the one who realized that the vapor pressure of a liquid depends on the curvature of its surface. In his words this explains why moisture is retained by vegetable substances, such as cotton cloth or oatmeal, or wheat-flour biscuits, at temperatures far above the dew point of the surrounding atmosphere [17]. 

Attention has to be paid as to which radius is inserted into the Kelvin equation. Generally, there is no rotational symmetric geometry. Then 2/rc has to be substituted by / R + 1/R >. In a fissure or crack, one radius of curvature is infinitely large. Instead of 2/rc there should be 1/r in the equation, with r being the bending radius vertical to the fissure direction. 

The vapor pressure of a liquid depends on the curvature of its surface. For drops it is increased compared to the vapor pressure of a planar surface under the same conditions. For bubbles it is reduced. Quantitatively this is described by the Kelvin equation. 

This expression, known as the Kelvin equation, has been verified experimentally. It can also be applied to a concave capillary meniscus in this case the curvature is negative and a vapour pressure lowering is predicted (see page 125). 

A. u = Tc( + ) M 111P. Laplace-Kelvin equation. Difference in fluid pressure A.11 across two-fluid interface. Related to surface tension Tc and the curvature radii r and r2... 

The vapor pressure over a pure liquid droplet at equilibrium ps depends on its radius of curvature. The Kelvin equation gives this relationship as ... 

All of these methods assume the following [4] (i) Kelvin equation is applicable over the complete mesopore range (ii) the meniscus curvature is controlled by the pore size and shape, and 0 (contact angle) is 0 (iii) the pores are rigid and of well-defined shape (iv) the distribution is confined to the mesopore range (v) the filling (or emptying) of each pore does not depend on its location, and (vi) the adsorption on pore walls follows the same mechanism as on the open surface. 

Although Kelvin equation is based on well-defined thermodynamic principles, its applicability to the mesopore size range, especially the lower ones, is questionable due to the uncertainties between the meniscus curvature and the pore size and shape. More extensive discussion about Kelvin-based meso-PSD methods and their applicability can be found in the literature [4,5,7],... 

Here ps is the saturated vapor pressure at temperature T, y the surface tension, Vm the molar volume of the liquid, and the curvature radius r is conventionally taken as negative for concave interfaces. Kelvin equation for a non-ideal multicomponent mixture was derived by Shapiro and Stenby (1997). 

Langmuir s equation indicates that a droplet will not evaporate when S = pip, a 1.0. But according to the Kelvin equation, curvature effects will cause small droplets to evaporate, even when S exceeds 1.0. How can this apparent contradiction be resolved One way is to replace the p, - pt term in Eq. 15.14 with an equivalent term from Kelvin s equation which takes curvature into account. Recalling Kelvin s equation... 

A saddle-shaped meniscus (or pendular ring) is developed in the first stage of condensation. Application of Equation (7.10) now requires the designation of two radii of curvature of opposite sign, one being concave and the other convex. The Kelvin equation therefore takes the form... 

We turn now to the question of validity of the Kelvin equation. Although the thermodynamic basis of the Kelvin equation is well established (Defay and Prigogine 1966), its reliability for pore size analysis is questionable. In this context, there are three related questions (1) What is the exact relation between the meniscus curvature and the pore size and shape (2) Is the Kelvin equation applicable in the range of narrow mesopores (say >vp < 5 nm) (3) Does the surface tension vary with pore width The answers to these questions are still elusive, but recent theoretical work has improved our understanding of mesopore filling and the nature of the condensate. 

It was recognized many years ago (Foster, 1932) that the Kelvin equation is likely to break down as the meniscus curvature approaches a limiting value. Molecular simulation studies (Jessop et ai, 1991) have indicated that the Kelvin equation fails to account for the effects of the fluid-wall interactions and the associated inhomogeneity of the pore fluid. These and other studies (Lastoskie et ai., 1993) reveal that the Kelvin equation probably underestimates the pore size and that its reliability may not extend below a pore size of 7.5 nm. 

Many early attempts were made to correct the Kelvin equation (see Brunauer, 1945). As already indicated, when the Kelvin equation is applied to capillary condensation it is normally assumed that the reduction in chemical potential is entirely dependent on the curvature of the meniscus. This assumption implies a sharp discontinuity between the state of the adsorbed layer and the condensate. However, as Detjaguin first pointed out (1957), the transition is more likely to be a gradual one. This problem was also discussed by Everett and Haynes (1973). 

According to the classical interpretation, the phenomenon is explained by application of the Kelvin equation this gives the relationship between the relative pressure and the meniscus curvature of liquid condensed in a pore. When the Kelvin equation is used to evaluate the pore size, it is tacitly assumed that there is a direct relationship between the meniscus curvature and the dimensions of the mesopores. In principle, it is also necessary to define the pore shape. 

The surface stress of some solids in a liquid might be determined by measuring solubility changes of small particles [97,98]. As small liquid drops have an increased vapor pressure in gas, small crystals show a higher solubility than larger ones. The reason is that, due to the curvature of the particles surface, the Laplace pressure increases the chemical potential of the molecules inside the particle. This is described by the Kelvin equation, which can be written (Ref. 3, p. 380)... 

Vapor Transport The svirface curvatures in Figure 16.5(a) introduces an excess pressure according to the Kelvin equation... 

This result is the Kelvin equation. In this approximation, we have further assumed that the particle is essentially a flat particle (with a vapor pressure of Pq) compared to the radius of curvature of the neck. We can calculate the rate at which the neck increases by equating the rate of material transfer to the surface of the lens between the spheres with the increase in its volume. The rate of condensation, m, is proportional to the difference in equilibrium vapor pressure, ZiP, as given by... 

The gas-liquid permporometry combines the controlled stepwise blocking of membrane pores by capillary condensation of a vapor, present as a component of a gas mixture, with the simultaneous measurement of the free diffusive transport of the gas through the open pores of the membrane. The condensable gas can be any vapor provided it has a reasonable vapor pressure and does not react with the membrane. Methanol, ethanol, cyclohexane and carbon tetrachloride have been used as the condensable gas for inorganic membranes. The noncondensable gas can be any gas that is inert relative to the membrane. Helium and oxygen have been used. It has been established that the vapor pressure of a liquid depends on the radius of curvature of its surface. When a liquid is contained in a capillary tube, this dependence is described by the Kelvin equation, Eq. (4-4). This equation which governs the gas-liquid equilibrium of a capillary condensate applies here with the usual assumption of a=0 ... 

It is well established that the pore space of a mesoporous solid fills with condensed adsorbate at pressures somewhat below the prevailing saturated vapor pressure of the adsorptive. When combined with a eorrelating function that relates pore size with a critical condensation pressure, this knowledge can be used to characterize the mesopore size distribution of an adsorbent from its adsorption isotherm. The correlating function most commonly used is the Kelvin equation [1], Refinements make allowance for the reduction of the physical pore size by the thickness of the adsorbed film existing at the critical condensation pressure [1-2]. Still further refinements adjust the film thickness for the curvature of the pore wall [3]. 

Nitrogen sorption isotherms at 77 K were calculated by means of the simulated 3D networks. Besides the Kelvin equation, necessary for determining the critical radius of curvature Rc, at which condensation and evaporation would occur, it is also necessary to consider specific menisci interactions and network effects that can influence the sorption phenomena [5, 7]. The existence of an adsorbed layer is indeed of great importance on the outcome of a sorption process, but for simplicity it will not be considered in this treatment. 

Using the conventional Lifshitz-Slyozov arguments [74,75] based on the Kelvin equation, one might expect [73,76] that the 3D crystallite growth, occurring via 2D diffusion, should follow Eq. (18) with n = 4. This value, however, is much lower than observed in experiments. The appreciable difference between the theory and experiment is actually not surprising, because the applicability of the Lifshitz-Slyozov model to nm crystallites is far from obvious (e.g., the curvature of such crystallites is an ill-defined quantity). 

---