Thomson — Gibbs equation

For the equilibrium form, the Gibbs-Wulff-Kaischew theorem can be applied. It states that the normal distances hi of all faces from the Wulff point, including the top face and the contact face /, are proportional to the specific surface energies (cf. eq. (4.10)). Hence, the specific surface energies in eq. (4.21b) can be replaced by the respective distances, hi/A = ov, so that 

Here Fcrit = - Z Acrit,j is the volume of the critical cluster. Replacing Vent with 

This is a general form of the well-known Gibbs-Thomson (Lord Kelvin) equation applied to the case of electrochemical metal deposition. It gives the size of the critical nucleus and its equilibrium form in terms of the normal distances of the equilibrium form faces from the Wulff point, hi, as a function of the overvoltage. When this form is a regular polyhedron (cr,- = const.), the size of the nucleus can be given by the radius of the inscribed sphere, perit = h, so that 

Substituting fj in eq. (4.25c) with A,/A from eq.(4.14), having also in mind that Ai = fNcnt and AGcrit = Ncntze ri, one obtains from eq. (4.14) similar relations for the 2D case  

Under the acting overpotential, the probabilities for growth and dissolution of a critically sized cluster are equal. In other words, the nucleus of the new phase stays in equilibrium with the ambient phase at a more negative potential than the equilibrium potential of the bulk crystal. It has to be emphasized, however, that this equilibrium is metastable, and the smallest change of the cluster dimensions will result in either further growth or complete dissolution. 

4 EQUILIBRIUM AT A CURVED INTERFACE C.4.1 Gibbs-Thomson Equation 

Consider a two-phase system of fixed total volume, with constant T and p (an open system with respect to matter flow), as illustrated in Fig. C.5. Under these conditions, the function Cl = E — TS — piNi — P2N2 is the appropriate thermodynamic potential. For any small variation at equilibrium, such as an infinitesimal variation 

An open isothermal system that allows for reversible motion of a curved 

Since the total volume is constant, dVa = -dV0. For an interface of area A moving a distance dX, as illustrated in Fig. C.5, dV0 = AdX and the interfacial area change according to Section C.2.1 is dA = KdV, where dV = dV0. So the free-energy change is 

Equation C.23 is the form of the Gibbs-Thomson equation introduced in Eq. C.17. It is a conditionfor mechanical equilibrium in a two-phase system with a curved interface. The phase located on the side of the interface toward its center of curvature (e.g. the (3 phase in Fig. C.5), has the higher pressure. Note also that for a flat interface, Eq. C.23 gives Pa = P0, as expected. 

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Gibbsitic [14762-49-3] Gibbs-Kelvin equation Gibbs phase rule Gibbs s phase rule Gibbs s theorem Gibbs-Thomson equation... 

NMR cryoporometry relies on the melting point depression, i.e., the difference in the melting point of crystals with a finite size d, Tm(d), relative to the value of the bulk liquid Tm, which is given by the simplified Gibbs-Thomson equation [16] ... 

Isothermal crystallization was carried out at some range of degree of supercooling (AT = 3.3-14 K). AT was defined by AT = T - Tc, where Tj is the equilibrium melting temperature and Tc is the crystallization temperature. T s was estimated by applying the Gibbs-Thomson equation. It was confirmed that the crystals were isolated from each other by means of a polarizing optical microscope (POM). 

They were evaluated from our analysis of the primary nucleation and lateral growth rates and that of the l dependence to the melting temperature Tm using the Gibbs-Thomson equation. Insertion of the parameters given by Eq. 20 into Eq. 6 shows that the shape of a nucleus is a long thin rectangular parallelepiped with the ratio of... 

The activities of precipitate particle components depend explicitly on their size and form. The quantitative relation which describes this fact is the so-called Gibbs-Thomson equation... 

Up to this point we have dealt with the thermodynamics of planar boundaries. Let us add several relations for curved interfaces. First, we have to establish an equivalent to the Gibbs-Thomson equation which holds for curved external surfaces in a multi-component system. For incoherent (fluid-like) interfaces, this can be done by considering Figure 10-5. From the equilibrium condition at constant P and T, one has... 

The surface energy per area, 7, has the same units as a force per length and for some interfacial geometries can lead to an interfacial net force that is balanced by a difference in pressure between the two adjacent phases. If 7 is isotropic, this pressure difference is directly proportional to the interfacial curvature through the the Gibbs-Thomson equation (see Sections C.2.1 and C.4.1),... 

Another approach to estimating the size of a latent image center uses redox buffers to determine the size at the transition potential between dissolution and growth. These buffers either develop the latent image centers, bleach them, or do neither. Konstantinov and associates (203,204) used the Gibbs-Thomson equation to analyze results obtained by this method. 

They arrived at a size of the order of 100 silver atoms for a bare metal nucleus, but only 10 atoms for a nucleus embedded in gelatin. Moisar and associates (205) estimated the minimum size as four silver atoms. Ronde (206) revised their calculations and obtained 10 for an unsensitized silver bromide emulsion and six for a S-sensitized emulsion. The validity of the Gibbs-Thomson equation for such small sizes, however, is questionable. 

The spatial microstructure of the interface is strongly influenced by its surface energy, which appears in the Gibbs-Thomson equation (87) for the melting temperature of a curved interface... 

The relationship between particle size, solubility and supersaturation is expressed by the well-known Gibbs-Thomson equation [5a] ... 

As shown in Equation 10.4, the depression of the melting point of a given confined solvent is related to the geometry of the pores of the confining material. In principle, measurement of AT can give access to the pore size. Three main techniques have been developed to measure porosity in solids via the use of the Gibbs-Thomson equation thermoporosimetry, NMR cryporometry and surface force apparatus. These techniques are secondary methods since they require pre-... 

In 2006, McKenna and co-workers reviewed recent work on the use of TPM to study polymer heterogeneity [35]. In this work, uncrosslinked and crosslinked polysisoprene was studied with benzene and hexadecane as swelling solvents. The authors were able to distinguish contributions coming from the confinement as described by the Gibbs-Thomson equation and contributions from polymer-solvent interactions described by the Flory-Huggins (FH) theory [36, 37]. For the first time, it was shown that for an uncrosslinked sample an excess shift AT is... 

If we examine the competitive growth of multiple-size crystallites in solutions, we find the growth occurring along concentration gradients. The concentration gradients around the particles are explained with the Gibbs-Thomson equation ... 

Once formed, the groove continues to grow, driven by differences in curvature k at the S/L interface. These differences result in variations of the chemical potential of the solid, n, according to the Gibbs-Thomson equation ... 

Changes in crystal dispersion due to surface effects are typically based on the Gibbs-Thomson equation (sometimes called the Ostwald-Freundlich equation), which describes the effects of particle size on equilibrium conditions (Hartel 2001). For solution systems, the equation is given as... 

In equation (2), Yq has been deduced according to Gibbs-Thomson equation [7],... 

Fig. 2. Shift in the melting temperature AT as a function of 1/H for Aniline in CPG, Vycor and MCM-41. The straight line is a fit to the data for H values of 25 nm and ak)ve, and is consistent with the Gibbs-Thomson equation.

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