Laplace equation derivation

Several graphical curve-fitting techniques have been developed (see Padday [53] for details) that can be used in conjunction with the numerical integration of the Laplace equation by Bashforth and Adams (and by subsequent workers) to determine d and to obtain y v. Smolders [54,55] used a number of coordinate points of the profile of the drop for curve fitting. If the surface tension of the liquid is known and if 0 > 90, a perturbation solution of the Laplace equation derived by Ehrlich [56] can be used to determine the contact angle, provided the drop is not far from spherical. Input data are the maximum radius of the drop and the radius at the plane of contact of the drop with the solid surface. The accuracy of this calculation does not depend critically on the accuracy of the interfacial tension. 

Curvature effect. For a curved interface, the gas and liquid phase pressures are related by the well-known Young-Laplace equation derived in Chapter 2,... 

Harmonic Functions Both the real and the imaginary )arts of any analytic function/= u + iij satisfy Laplaces equation d /dx + d /dy = 0. A function which possesses continuous second partial derivatives and satisfies Laplace s equation is called a harmonic function. 

A variety of methodologies have been implemented for the reaction field. The basic equation for the dielectric continuum model is the Poisson-Laplace equation, by which the electrostatic field in a cavity with an arbitrary shape and size is calculated, although some methods do not satisfy the equation. Because the solute s electronic strucmre and the reaction field depend on each other, a nonlinear equation (modified Schrddinger equation) has to be solved in an iterative manner. In practice this is achieved by modifying the electronic Hamiltonian or Fock operator, which is defined through the shape and size of the cavity and the description of the solute s electronic distribution. If one takes a dipole moment approximation for the solute s electronic distribution and a spherical cavity (Onsager s reaction field), the interaction can be derived rather easily and an analytical expression of theFock operator is obtained. However, such an expression is not feasible for an arbitrary electronic distribution in an arbitrary cavity fitted to the molecular shape. In this case the Fock operator is very complicated and has to be prepared by a numerical procedure. 

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. 

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 ... 

Similar results can also be derived by using the Laplace equation (Equation 2.21) (1/radius = 1/R) ... 

Young and Laplace (1805) derived meniscus curvature equation. 

Since it is relatively easy to transfer molecules from bulk liquid to the surface (e.g. shake or break up a droplet of water), the work done in this process can be measured and hence we can obtain the value of the surface energy of the liquid. This is, however, obviously not the case for solids (see later section). The diverse methods for measuring surface and interfacial energies of liquids generally depend on measuring either the pressure difference across a curved interface or the equilibrium (reversible) force required to extend the area of a surface, as above. The former method uses a fundamental equation for the pressure generated across any curved interface, namely the Laplace equation, which is derived in the following section. 

The Young Equation. The principle of balancing forces used in the derivation of the Laplace equation can also be used to derive another important equation in surface thermodynamics, the Young equation. Consider a liquid droplet in equilibrium... 

Can you prove why this is so ) When x, y, and z are thermodynamic quantities, such as free energy, volume, temperature, or enthalpy, the relationship between the partial differentials of M and N as described above are called Maxwell relations. Use Maxwell relations to derive the Laplace equation for a... 

The presence of surface tension has an important implication for the pressures across a curved interface and, as a consequence, for phase equilibria involving curved interphase boundaries. The equation that relates the pressure difference across an interface to the radii of curvature, known as the Laplace equation, is derived in Section 6.4, and the implications for phase equilibria are considered for some specific cases in Section 6.5. 

This expression is known as the Laplace equation and was derived in 1805. 

The Laplace equation applied specifically to spherical surfaces can be derived in a variety of ways. Example 6.1 considers an alternative derivation that points out the thermodynamic character of the result quite clearly. 

EXAMPLE 6.1 Laplace Equation for Spherical Surfaces A Thermodynamic Derivation. The Maxwell relations play an important role in thermodynamics. By including the term dA in the usual differential form for cfG, show that (dVfdA)pJ = (dyidp)AJ. Evaluate (dVidA)pJ assuming a spherical surface and, from this, derive the Laplace equation for this geometry. 

Laplace equation A thermodynamic derivation Determining surface tension from the Kelvin equation Heat of immersion from surface tension and contact angle Surface tension and the height of a meniscus at a wall Interfacial tensions from the Girifalco-Good-Fowkes equation... 

At this point we mention a simple, alternative way of deriving the Laplace equation. In equilibrium we have dF/dA = 0. It leads to... 

The most common technique for the derivation of fundamental solutions is to use integral transforms, such as, Fourier, Laplace or Hankel transforms [29, 39]. For simple operators, such as the Laplacian, direct integration and the use of the properties of the Dirac delta are typically used to construct the fundamental solution. For the case of a two-dimensional Laplace equation we can use a two-dimensional Fourier transform, F, to get the fundamental solution as follows,... 

Single phase microemulsions are treated in the next section. Two general thermodynamic equations are derived from the condition that the free energy of the system should be a minimum with respect to both the radius r of the globules as well as the volume fraction of the dispersed phase. The first equation can be employed to calculate the radius while the second, a generalized Laplace equation, can be used to explain the instability of the spherical shape of the globules. The two and three phase systems are examined in Sections III and IV of the paper. 

PROBLEM 6.21.1. Using Laplace transforms, from Fick s second equation, derive Eq. (6.21.2). 

Note that for 9 > 90°, ze is positive i.e., it corresponds to a depression for 9 < 90°, ze is negative and corresponds to a capillary rise. Equation (1.55) can also be derived by a mechanical approach, considering the hydrostatic pressure APh = pgz and the capillary pressure APC. Applying the Laplace equation (1.20) to the capillary configuration with R] = R2 = -r/cos0 (see Figure 1.37), APcis ... 

Figure A.l. Displacement of a liquid surface allowing derivation of the Laplace equation.

We give below a simple method to derive an approximate solution to the hnear-ized Poisson-Boltzmann equation (1.9) for the potential distribution i/ (r) around a nearly spherical spheroidal particle immersed in an electrolyte solution [12]. This method is based on Maxwell s method [13] to derive an approximate solution to the Laplace equation for the potential distribution around a nearly spherical particle. 

The task is to derive an expression for Ap, because from that efo) and K[(o] are immediately obtained, using [4.8.4 and 5). In practice d is computed and [4.8.61 used. The static dipole moment has been given in [3.13.4-6]. The equation from which is obtained is the Laplace equation. 

Oil Entrapment Mechanisms. Enhanced oil recovery processes depend in large part on the elimination or reduction of capillary forces. Capillary forces are the strongest that occur under typical reservoir conditions, and are most responsible for oil entrapment. Viscous forces, which act to displace oil, are composed of the applied pressure gradient, gravity, density differences between phases, and viscosity ratio. In a permeable medium, capillary forces result when the pores constrain the oil-water interface to a high degree of curvature. From the Laplace equation, the capillary pressure in a capillary tube can be derived ... 

The potential energy of the dipole is pBi = —p2R. The part depending on (e — eo) represents the dipole-dipole interactions while the (k<7d)2 term concerns the dipole-ion interactions. For k 0, Eq. (10) reduces to the reaction-field expression of dielectric theory [54] derived from Laplace equation. Interestingly, even for e = eo a reaction field results because of the polarization of the ionic cloud. 

The relation between pore size r(nm) and the extent of temperature depression AT (Fig. 4.8) is obtained from the equations derived by Brun from Gibbs-Duhem and Laplace equations. For cylindrical pores with water inside (0 < AT < 0 and f = 0.8 nm), it leads to... 

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