Abstract
The technical term, ‘capillarity’, applies broadly to a large class of interfacial phenomena that deal with a range of thermodynamic influences associated with internal and external surfaces. Geometric aspects of surfaces and interfaces also connote associated metric quantities, including volumes, areas, lengths, and curvatures, all of which determine the relative importance of capillary phenomena in determining the behavior of a thermodynamic system.
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Notes
- 1.
Surface and interfacial energies, expressed on a per atom basis, are often not more than about 10−3 times the bulk free energy per atom developed by supercooling a melt ca. 1 K, or slightly supersaturating a crystalline phase. Although seemingly inconsequential, surface energies remain important in many crystal growth and solidification processes.
- 2.
The scientific literature on capillary phenomena also employs interface curvatures termed ‘mean curvature’ and ‘total curvature’, but defined instead as the total trace of the curvature tensor. This difference from the definition given in the text—viz., a factor of 2—if not explicitly noted or specified, can be a source of confusion.
- 3.
See Appendix A on the definition of the Helmholz free energy or potential, \(F(T,V, N_B)\), which is the Legendre transform of the internal energy, \(U(S,V, N_B)\), where temperature replaces the entropy as the independent variable. This choice assumes that the system is maintained both at a constant temperature and constant volume by its surroundings.
- 4.
When an energetically isotropic, planar interface is displaced normal to itself, additional freezing or melting occurs. The energy changes associated with this displacement include only the swept volume contributions, as the interfacial area, and its energy, remain constant. In the case of a curved interface, however, both the area and volume change with displacement, in accord with the kinematics required by the mean curvature, \({H=dA/dV}\), so additional energy changes occur.
- 5.
Standard capillarity equations, such as the Gibbs–Thomson relationship, have been shown to be accurate provided \({\mathcal{H}<2\times10^5}\;{\textrm{cm}}^{-1}\).
- 6.
Kulkarni and DeHoff provide a thorough discussion of both pressure changes and chemical potential shifts at curved interfaces in their important paper on this subject [15]. Those authors point out inconsistencies from classical capillarity developed for liquid- or solid-gas interfaces that have been promulgated through papers and reviews on capillarity-induced shifts between condensed phases such as solid–liquid systems.
- 7.
The exploration of curvature on vapor pressure and chemical potential to be demonstrated here uses isothermal steps applied to a pure condensed phase and its vapor. Previously, the Gibbs-Thomson equation was derived for two condensed phases (solid plus liquid) using isobaric freezing or melting processes to determine the curvature-induced shift in the melting temperature. The conditions imposed on the intensive variables and the nature of the phases are both critically important to these derivations and their correct thermodynamic interpretation [15].
- 8.
The conceptual devices (frictionless pump and valves) for accomplishing this isothermal reversible compression and transfer are not indicated in Fig. 8.13.
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Glicksman, M.E. (2011). Crystal-Melt Interfaces. In: Principles of Solidification. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7344-3_8
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DOI: https://doi.org/10.1007/978-1-4419-7344-3_8
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