Surface energy Laplace equation

This effect assumes importance only at very small radii, but it has some applications in the treatment of nucleation theory where the excess surface energy of small clusters is involved (see Section IX-2). An intrinsic difficulty with equations such as 111-20 is that the treatment, if not modelistic and hence partly empirical, assumes a continuous medium, yet the effect does not become important until curvature comparable to molecular dimensions is reached. Fisher and Israelachvili [24] measured the force due to the Laplace pressure for a pendular ring of liquid between crossed mica cylinders and concluded that for several organic liquids the effective surface tension remained unchanged... 

Fermi-level DOS 115 Jellium model 92—97 failures 97 schematic 94 surface energy 96 surface potential 93 work function 96 Johnson noise 252 Kohn-Sham equations 113 Kronig-Penney model 99 Laplace transforms 261, 262, 377 and feedback circuits 262 definition 261 short table 377 Lateral resolution... 

Consider a one dimensional oil droplet pressed against a solid surface and surrounded by an aqueous solution of surfactant (Fig. 8). The shape of the oil-water interface far away from the sohd surface is governed by the Laplace equation [18] and close to the sohd surface is augmented by an additional term relating to the film energy given by the structural disjoining pressure [12] ... 

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 LaPlace Equation. The concept of surface energy allows us to describe a number of naturally occurring phenomena involving liquids and solids. One such situation that plays an important role in the processing and application of both liquids and solids is the pressure difference that arises due to a curved surface, such as a bubble or spherical particle. For the most part, we have ignored pressure effects, but for the isolated surfaces under consideration here, we must take pressure into account. 

This is the simplest model of an electrocatalyst system where the single energy dissipation is caused by the ohmic drop of the electrolyte, with no influence of the charge transfer in the electrochemical reaction. Thus, fast electrochemical reactions occur at current densities that are far from the limiting current density. The partial differential equation governing the potential distribution in the solution can be derived from the Laplace Equation 13.5. This equation also governs the conduction of heat in solids, steady-state diffusion, and electrostatic fields. The electric potential immediately adjacent to the electrocatalyst is modeled as a constant potential surface, and the current density is proportional to its gradient ... 

Coalescence. This is caused by rupture of the film between two emulsion drops or two foam bubbles. The driving force is the decrease in free energy resulting when the total surface area is decreased, as occurs after film rupture. The Laplace equation (Section 10.5.1) plays a key role. 

The pore size distribution can be obtained from capillary pressure measurements or mercury porosimetry. The capillary pressure is related to the specific free energies of the interface between fluids and between the fluid and the capillary wall. At mechanical equilibrium, the surface free energy between the fluids is a minimum. The equilibrium condition is expressed by the Laplace equation ... 

There is another way of looking at the same effect, since a surface energy per unit area is equivalent to a line force per unit length, or surface tension, hence there is an excess pressure in a spherical droplet given by the Laplace equation ... 

Gas adsorption can also be used in systems with mesopores to measure pore size distributions. In this range of pore sizes, the surface energy of the pore walls causes a condensation of the gas (usually N2 in practice) at pressures where it would remain in the gas state if not confined an interface forms with surface tension y and the reduced vapour pressure on the convex side of the meniscus, as expected from the Laplace equation, explains the condensation at equilibrium. Equating the... 

This effect may be understood in terms of the concept of critical nucleus size for growth of crystals. Consider a small particle of material of diameter D immersed in another medium as shown in Fig. 10.23(a). If the particle is a droplet of liquid sitting in another liquid, such that an interfacial energy y exists between the two liquids, then the particle has an excess surface energy which causes a pressure excess 4y/D inside the particle according to Laplace s equation. 

In most cases of interest, the surface excess mass E is small, so that the acceleration and body force terms may be neglected. Then Equation 1.40 simplifies to two conditions. One of them, V y = 0, requires that interfacial tension be uniform. The other is the Young-Laplace equation (Equation 1.22), which was obtained previously from thermodynamics for situations where body force and acceleration terms were unimportant. That the same equation (Equation 1.22) results from independent thermodynamic and mechanical derivations implies that interfacial tension must have the same value whether it is defined as in Equation 1.9 from energy considerations or as in Equation 1.39 from force considraations. Simply put, the force and energy definitions of interfacial tension are eqnivalrait, a conclusion emphasized in the work of Buff (1956). 

Now let us consider two immersed particles of radii R i, R i-The free energy is then defined by the same Equation [76] with Ip = 0 at the two particle surfaces and ip = ipo far away from both particles. The free energy depends on the distance D between the particle centers W=W D). The force Fs = -dWl 3D of effective interaction between the particles can be found using the Laplace Equation [77]. The problem is analogous to the interaction of two conducting charged spheres. The result for D Rj is... 

Having defined what we mean by the surface of a liquid and its curvature, we now consider the derivation of Laplace s law for a spherical drop of liquid with radius R. In this case the radii of curvature at every point on the surface are simply equal to R. The usual derivation of the Laplace equation follows from a consideration of the net change in free energy of the liquid droplet resulting from a change in the radius R. Consider the Helmholtz free energy of a spherical drop of liquid consisting of only one type of molecule... 

An alternative derivation of Young s equation follows the same route as the derivation of the Laplace equation using a notional change of the location of the dividing surface. Consider the surface free energy of the system depicted in Fig. 4. Around the line of three-phase contact a cylinder is drawn with length L and radius R, and implicitly we assume that R and L approach... 

It is emphasized that the proper surface quantity to be used in Equations (6.11) and (6.12) is a, not y. It is only for the special case of fluid systems, where a = y, that the surface energy may be used in the Laplace equation and... 

Techniques that use the Laplace equation to measure surface energy... 

At all channel wall surfaces, the no-slip boundary condition is applied to the velocity field (the Navier-Stokes equation), the fixed zeta-potential boundary condition is imposed on the EDL potential field (the Poisson-Boltzmann equation), and the insulation boundary condition is assigned to the applied electric field (the Laplace equation), and the no-mass penetration condition is specified for the solute mass concentration field (the mass transport equation). In addition, the third-kind boundary condition (i. e., the natural convection heat transfer with the surrounding air) is applied to the temperature field at all the outside surfaces of the fabricated channels to simultaneously solve the energy equation for the buffer solution together with the conjugated heat conduction equation for the channel wall. 

Finally, let us briefly address some of the principal methods that are used to measure surface tension (free surface energy) in liquids. Nearly all of these methods rely on the Laplace equation ... 

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