The gibbs-thomson effect in dilute binary systems

Abstract

The solubility of a precipitate, which is commonly referred to as the composition of the matrix in equilibrium with the precipitate in the presence of a curved interface, has been well documented. For a pure precipitate ␤, which consists of component B, in an ␣ matrix that is a dilute solution of B in A, the Gibbs-Thomson effect is well known as [1,2] xwhere x ␣ B (r) and x ␣ B (ϱ) are the equilibrium atomic fractions of B in ␣ at a curved interface of a spherical precipitate (␤ ) of radius r and a planar interface, respectively. The term ␣␤ is the specific interfacial free energy, which is assumed to be isotropic and independent of precipitate size (r), and V ␤ m is the molar volume of ␤. When ␤ is not pure B, e.g., Fig. 3-The temperature dependence of the specific strength for the its composition is given by x ␤ B , a number of different exprestested alloyssions have been proposed or employed. These include [3][4][5][6][7][8][9][10][11] x

Phase Evolution and Crystallization Mechanism of Glass Ceramic Solid-State Electrolyte from In Situ Synchrotron X-ray Diffraction

“…The incorporation of Al will also increase the amount of Li into the structure to charge balance the composition, furthermore, this behavior is promoted by the bulk nucleation mechanism in the LAGP system (schematically shown in Figure b). An increase of solubility due to small grain size was also reported by Perez and Qian et al , Figure c shows the final surface morphology of the LAGP samples after the analysis in in situ SR-XRD and heat treatment. In the beginning of the nucleation stage, the LiGe 2 (PO 4 ) 3 nuclei are evenly dispersed in the bulk sample, the interfacial tension drives the Al ion into the crystal, forming Li 1+ x Al x Ge 2– x (PO 4 ) 3 crystals, which also increases the Li + for charge balancing of the structure.…”

“…However, Figure c shows that EDS analysis detected higher concentration of Al content in the crystallized region than in the glass region, which suggests that Al incorporation into the crystals can be achieved by heat treatment at lower temperatures, especially in the nucleation stage of the crystallization process. This observation can be explained by the Gibbs–Thompson effect, a theory commonly used to describe the change of solubility during phase precipitation in alloy systems. − The Gibbs–Thompson effect can be expressed mathematically in eqs and normalΔ P = 2 γ r normalΔ G = 2 σ α β V normalm r where Δ P is the interfacial tension exerted on the precipitating phase in a matrix, r is the effective curvature of precipitating phase, assuming a spherical shape, γ is the interfacial tension, σ αβ is the interfacial stress, and Δ G is the increase of Gibbs free energy. Equation indicates that the compressive stress (Δ P ) is large on the lattice when r is small, and Δ P decreases as the nucleation size increases, which explains the change of Al solubility in the LiGe 2 (PO 4 ) 3 crystal as shown in eq X normalA normall normalL normalG normalP ( r ) = X normalA normall ∞ normalL normalG normalP .25em exp ( 2 γ V m r R T ) = X normalA normall ∞ normalL normalG normalP .25em exp …”

Alloy development for the enhanced stability of Ω precipitates in Al-Cu-Mg-Ag alloys

“…It is this ''free'' magnesium, in conjunction with an ample supply of heterogeneous nucleation sites, that led to the precipitation of fine S-phase particles in the double-aged condition with coarse V plates. Enhanced solubility of solute, particularly magnesium, to the Gibbs-Thompson effect [54] may also be a contributing factor behind the lack of further S-phase precipitation in the near-peak-aged microstructures. Presumably, the double-aging at 200°C led to recovery of the matrix dislocations, which limited the number of resulting S-phase precipitation events.…”

Theoretical and Experimental Approach to the Production of ZnO Nanoparticles by Controlled Precipitation Method in Methanol

“…Although the routes to produce zinc oxide at a nanometric scale are well known, controlling the crystalline structure, size, and dispersion in solvents, such as water or alcohols, is still challenging. The effect of the size of the solid phase on the solubility of a precipitate can be evaluated through the Gibbs‐Thompson relation [37] . The deviation caused by the curvature of a surface modifies the stability of a solid particle in comparison to the bulk phase.…”