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Phase equilibrium Laplace equation

The purpose of this chapter is to introduce the effect of surfaces and interfaces on the thermodynamics of materials. While interface is a general term used for solid-solid, solid-liquid, liquid-liquid, solid-gas and liquid-gas boundaries, surface is the term normally used for the two latter types of phase boundary. The thermodynamic theory of interfaces between isotropic phases were first formulated by Gibbs [1], The treatment of such systems is based on the definition of an isotropic surface tension, cr, which is an excess surface stress per unit surface area. The Gibbs surface model for fluid surfaces is presented in Section 6.1 along with the derivation of the equilibrium conditions for curved interfaces, the Laplace equation. [Pg.158]

Equation (6.27) is the Laplace equation, or Young-Laplace equation, which defines the equilibrium condition for the pressure difference over a curved surface. In Section 6.2 we will examine the consequences of surface or interface curvature for some important heterogeneous phase equilibria. [Pg.164]

The Laplace equation (eq. 6.27) was derived for the interface between two isotropic phases. A corresponding Laplace equation for a solid-liquid or solid-gas interface can also be derived [3], Here the pressure difference over the interface is given in terms of the factor that determines the equilibrium shape of the crystal ... [Pg.167]

The fact that the curvature of the surface affects a heterogeneous phase equilibrium can be seen by analyzing the number of degrees of freedom of a system. If two phases a and are separated by a planar interface, the conditions for equilibrium do not involve the interface and the Gibbs phase rule as described in Chapter 4 applies. On the other hand, if the two coexisting phases a and / are separated by a curved interface, the pressures of the two phases are no longer equal and the Laplace equation (6.27) (eq. 6.35 for solids), expressed in terms of the two principal curvatures of the interface, defines the equilibrium conditions for pressure ... [Pg.175]

The Young-Laplace equation gives the equilibrium pressure difference (mechanical equilibrium) at the menisci between liquid water in membrane pores and vapor in the adjacent phase ... [Pg.372]

The controlled drop tensiometer is a simple and very flexible method for measuring interfacial tension (IFI) in equilibrium as well as in various dynamic conditions. In this technique (Fig. 1), the capillary pressure, p of a drop, which is formed at the tip of a capillary and immersed into another immiscible phase (liquid or gas), is measured by a sensitive pressure transducer. The capillary pressure is related to the IFT and drop radius, R, through the Young-Laplace equation [2,3] ... [Pg.2]

It is well known (Defay and Prigogine, 1951) that a spherical interface of radius of curvature r and surface tension y can maintain mechanical equilibrium between two fluids at different pressures p" and p. The phase on the concave side of the interface experiences a pressure p" which is greater than that on the convex side. The mechanical equilibrium condition is given by the Laplace equation ... [Pg.192]

Thus, substituting distortion vector by u = Uq + grad (p, the respective Laplace equation for the distortion potential tp will be Atp(r) = 0. Consider now a grain of phase A, which has several border interfaces with other A grains as well as with B-matrix and the internal volume of this grain will be a positive direction (Fig.l). The picture on Fig.l is taken from the real structure of W-Cu infiltrated composite with a graded distribution of diamonds [4]. The equilibrium requirements stipulate that distortion potential on positive (inside) and negative (outside) sides of the interfaces must be coupled as follows, so far the material is not a subject of an application of external forces ... [Pg.23]

This simple form of the Young-Laplace equation shows that if the radius of the sphere increases, AP decreases, and when sph—> °°, AP —> 0, so that when the curvature vanishes and transforms into a flat Euclidean plane, there will be no pressure difference, and the two phases will be in hydrostatic equilibrium as stated above. [Pg.128]

Wenzel s relation has been confirmed in terms of the first two laws of thermodynamics. Huh and Mason, in 1976, used a perturbation method for solving the Young-Laplace equation while applying Wenzel s equation to the surface texture. Their results can be reduced to Wenzel s equation for random roughness of small amplitude. They assume that hysteresis was caused by nonisotropic equilibrium positions of the three-phase contact line, and its movement was predicted to occur in jumps. On the other hand, in 1966, Timmons and Zisman attributed hysteresis to microporosity of solids, because they found that hysteresis was dependent on the size of the liquid molecules or associated cluster of molecules (like water behaves as an associated cluster of six molecules). [Pg.326]

The Young-Laplace equation (equation 5) describes the force balance in terms of capillary pressure for two fluid phases in contact with each other and a surface. If one of the phases is present as a thin film, the equilibrium relationship that accounts for the thin film is the augmented Young-Laplace equation [6-S],... [Pg.164]

Thus, the charge density defines the curvature of the potential profile in an electrically neutral medium, the right-hand side of Equation 3.5 is 0 (Laplace equation), and consequently, the potential profile should be a straight line (i.e., a constant electric field). The presence of net charge at some region in space is required to find a curved potential profile this is, unless special circumstances arise, the case when interfaces are present. On the other hand, in a bulk phase such au aqueous solution, only in conditions far from equilibrium, a charge imbalauce cau be found. [Pg.29]

Extend the derivation of Prob. 8.1, concerning a liquid droplet of radius r suspended in a gas, to the case in which the liquid and gas are both mixtures. Show that the equilibrium conditions are T = T, pf = fi (for each species i that can equilibrate between the two phases), and p = p -l-2y/r, where y is the surface tension. (As in Prob. 8.1, the last relation is the Laplace equation.)... [Pg.280]

For an ordinary fluid two-phase system, minimization of the free energy at constant temperature flrst of all results in the well-known chemical potential condition for diffusive equilibrium with respect to the soluble components present. Furthermore, the mathematical extremum condition for the free energy contains the pressure difference AP across the interface as a parameter. Solving for AP, this condition takes the form of a generalized Laplace equation. Whether or not this equation signifies a stable equilibrium is, however, often a rather complex issue where the detailed system properties may enter in a crucial manner. [Pg.558]

An entirely different situation may arise when the interfadal tension y becomes small enough to exhibit an appreciable curvature dependence that, approximately at least, is accounted for by the Helfrich expression, Eq. (52). In the Winsor I case, for instance, where we have small oil droplets dressed by surfactant dispersed in water and where an excess oil phase is present (Fig. 5), the (spherical) droplet free energy function 4 tP y passes through a minimum for P = Peq. At this particular radius, the Laplace equation written in the form of Eq. (76) yields AP = 0, and thus the condition of equal chemical potential in the droplet and the excess phase is satisfied. Around the minimum (where the curvature of the free energy function amounts to 8717 ), equilibrium fluctuations in size and shape occur, which have important entropic implications [40]. [Pg.583]

The capillary rise method was the earliest technique by which surface tension was measured and, indeed, was the technique by which the force itself was recognized. If a narrow tube of radius r is partially inserted into a liquid, the liquid rises up inside the tube to some equilibrium position as shown in Fig. 22. This occurs because the attractive interaction of the wetting liquid (aqueous solution) with the solid surface is stronger than that of the gas phase. Gravity opposes the rise, and the equilibrium height H corresponds to the minimum free energy of the system. The treatment is based on the Laplace equation that gives the pressure difference across a curved interface due to the surface or interfacial tension of the liquid [62]. Let us assume that we have a spherical bubble Of gas in a liquid... [Pg.85]

The last equation is a highly generalized form of the famous Laplace equation. Its particular well-known form for the case of the capillary equilibrium will be discussed in Section HI. Equation (16) shows the effect of the excess phase on the equdibrium condition, which would otherwise be w = 0. [Pg.380]

The example discussed suggests another more realistic formulation of the equilibrium problem. Under the conditions of the previous problem, instead of P, and Pg, we specify the pressure in one phase, say, Pg. The system of Eqs. (8) and (9) is completed by the Laplace equation in the form of Eqs. (23). In the second equation, (23), the dependence /(/ ) is supposed to be known from the geometry of the porous space. The surface tension and the wetting angle are defined as known functions of the thermodynamic conditions (e.g., the surface is assumed to be wet by the condensate and the surface tension is calculated by the parachor method). The volumetric hquid... [Pg.387]

The interfaces between two bulk phases considered so far have been planar, and the pressures in the two bulk phases under equilibrium have been equal for gas-liquid and liquid-liquid systems. When the interface has a curvature, mechanical equilibrium requires different values of the pressures in the two phases. The general relation (the Young-Laplace equation) governing the pressure difference between bulk phases 1 and 2 is as follows (Adamson, 1967 Guggenheim, 1967) ... [Pg.136]

We provide here a derivation of the Kelvin equation based on the Young-Laplace equation and phase equilibrium principles. [Pg.91]

The equation of Young and Laplace describes one of the fundamental laws in interface science If an interface between two fluids is curved, there is a pressure difference across it provided the system is in equilibrium. The Young-Laplace equation relates the pressure difference between the two phases AP and the curvature of the interface. In the absence of gravitation, or if the objects are so small that gravitation is negligible, the Young-Laplace equation is... [Pg.128]

Let us start with the action of Young-Laplace law (Equation 9.6), which determines the equilibrium configuration of the fluids (liquid and liquid-like phases) and the driving force of mass transfer that cause the spontaneous formation of equilibrium configurations. [Pg.267]

In addition to the enhanced diffusivity effect, another issue needs to be taken into account when considering stationary-phase mass transfer in CEC with porous particles. The velocity difference between the pore and interstitial space may be small in CEC. Under such conditions the rate of mass transfer between the interstitial and pore space cannot be very important for the total separation efficiency, as the driving mechanism for peak broadening, i.e., the difference in mobile-phase velocity within and outside the particles, is absent. This effect on the plate height contribution II, s has been termed the equilibrium effect [35], How to account for this effect in the plate height equation is still open to debate. Using a modified mass balance equation and Laplace transformation, we first arrived at the following expression for Hc,s, which accounts for both the effective diffusivity and the equilibrium effect [18] ... [Pg.199]

An approximate treatment of the phenomenon of the capillary rise can be easily made in terms of Laplace s equation. If the liquid wets the wall of the capillary, the liquid surface is forced to lie parallel to the wall, and the liquid surface has to be concave in shape. The pressure in the liquid below the surface is less than that in the gas phase above the liquid surface. If the capillary is circular in cross section and not too wide in radius, the meniscus will be approximately hemispherical, as is illustrated in Figure 6.5. Such a case is described well by Eq. (6.11). If h denotes the height of the meniscus above the flat liquid surface, then at equilibrium, AP must also be equal to the hydrostatic pressure of the liquid column inside the capillary. Thus AP = Apgh, where Ap is the difference in density between the liquid and gas phases and g is the acceleration due to gravity. Equation (6.11) then becomes... [Pg.290]

The Laplace-Kelvin equation predicts that an isolated gas bubble should redissolve if its size is below a critical threshold, and conversely it should grow if its radius rj, exceeds the critical value given by Eq. (43) [50], where surface tension, Pyne is the local pressure over the bubble (it may be simply the hydrostatic pressure, but in other circumstances it may be controlled by medium elasticity [61]), and Pj, is the pressure inside the bubble, equal to the sum of the partial vapor pressures due to inert gases and volatile components if physical equilibrium and gas-phase ideality hold. So, it is possible that no bubbling occurs if the pressure is high enough or the content of volatiles is too low, but this is often not the case. [Pg.78]

The classical analysis of Young and Laplace of static wetting problems rests on the characterization of each interface by a macroscopic surface tension. At the intersection of three bulk phases, the three phase contact line is at rest only if the capillary forces represented by these surface tensions balance. When the three phases are a solid substrate S, a wetting liquid L and a vapor V, the mechanical equilibrium condition parallel to the solid gives the Young-Dupr6 equation for the contact angle Oq... [Pg.221]


See other pages where Phase equilibrium Laplace equation is mentioned: [Pg.365]    [Pg.229]    [Pg.72]    [Pg.106]    [Pg.595]    [Pg.596]    [Pg.10]    [Pg.61]    [Pg.255]    [Pg.574]    [Pg.239]    [Pg.133]    [Pg.290]    [Pg.160]    [Pg.467]    [Pg.461]    [Pg.116]   
See also in sourсe #XX -- [ Pg.227 , Pg.228 ]




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