The Gibbs–Thomson effect, in common physics usage, refers to variations in vapor pressure or chemical potential across a curved surface or interface. The existence of a positive interfacial energy will increase the energy required to form small particles with high curvature, and these particles will exhibit an increased vapor pressure. See Ostwald–Freundlich equation. More specifically, the Gibbs–Thomson effect refers to the observation that small crystals that are in equilibrium with their liquid, melt at a lower temperature than large crystals. In cases of confined geometry, such as liquids contained within porous media, this leads to a depression in the freezing point / melting point that is inversely proportional to the pore size, as given by the Gibbs–Thomson equation.
Introduction The technique is closely related to using gas adsorption to measure pore sizes, but uses the Gibbs–Thomson equation rather than the Kelvin equation. They are both particular cases of the Gibbs Equations of Josiah Willard Gibbs: the Kelvin equation is the constant temperature case, and the Gibbs–Thomson equation is the constant pressure case. This behaviour is closely related to the capillary effect and both are due to the change in bulk free energy caused by the curvature of an interfacial surface under tension. The original equation only applies to isolated particles, but with the addition of surface interaction terms (usually expressed in terms of the contact wetting angle) can be modified to apply to liquids and their crystals in porous media. As such it has given rise to various related techniques for measuring pore size distributions. (See Thermoporometry and cryoporometry.) The Gibbs–Thomson effect lowers both melting and freezing point, and also raises boiling point. However, simple cooling of an all-liquid sample usually leads to a state of non-equilibrium super cooling and only eventual non-equilibrium freezing. To obtain a measurement of the equilibrium freezing event, it is necessary to first cool enough to freeze a sample with excess liquid outside the pores, then warm the sample until the liquid in the pores is all melted, but the bulk material is still frozen. Then, on re-cooling the equilibrium freezing event can be measured, as the external ice will then grow into the pores. This is in effect an "ice intrusion" measurement (cf. mercury intrusion), and as such in part may provide information on pore throat properties. The melting event can be expected to provide more accurate information on the pore body.
For particles For an isolated spherical solid particle of diameter d {\displaystyle d} in its own liquid, the Gibbs–Thomson equation for the structural melting point depression can be written:
Δ T m = T m B − T m ( d ) = T m B 4 σ s l H f ρ s d {\displaystyle \Delta \,T_{m}=T_{mB}-T_{m}(d)=T_{mB}{\frac {4\sigma _{sl}}{H_{f}\rho _{s}d}}}
where:
TmB = bulk melting temperature σsl = solid–liquid interface energy (per unit area) Hf = bulk enthalpy of fusion (per gram of material) ρs = density of solid d = nanoparticle size
For liquids in pores Very similar equations may be applied to the growth and melting of crystals in the confined geometry of porous systems. However the geometry term for the crystal-liquid interface may be different, and there may be additional surface energy terms to consider, which can be written as a wetting angle term cos ϕ {\displaystyle \cos \phi \,} . The angle is usually considered to be near 180°. In cylindrical pores there is some evidence that the freezing interface may be spherical, while the melting interface may be cylindrical, based on preliminary measurements for the measured ratio for Δ T f / Δ T m {\displaystyle \Delta \,T_{f}/\Delta \,T_{m}} in cylindrical pores. Thus for a spherical interface between a non-wetting crystal and its own liquid, in an infinite cylindrical pore of diameter x {\displaystyle x} , the structural melting point depression is given by:
Δ T m ( x ) = T m B − T m ( x ) = − T m B 4 σ s l cos ϕ H f ρ s x {\displaystyle \Delta \,T_{m}(x)=T_{mB}-T_{m}(x)=-T_{mB}{\frac {4\sigma \,_{sl}\cos \phi \,}{H_{f}\rho \,_{s}x}}}
Simplified equation The Gibbs–Thomson equation may be written in a compact form:
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