Gibbs–Thomson equation
Encyclopedia
The Gibbs–Thomson effect in Physics
refers to the observation that small crystals of a liquid melt at a lower temperature than the bulk liquid.
The melting point depression is inversely proportional to the pore size, as given by the Gibbs–Thomson 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 Themoporometry 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 (c.f. 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.
The geometry of the crystal–liquid interface determines the value of the constant in the Gibbs-Thomson equation – the conventional "4" only applies to a spherical interface in a cylindrical pore.
in its own liquid, the Gibbs-Thomson Equation for the structural melting point depression can be written:
Where: TmB=Bulk Melting temperature
The "4" in the above equation comes from the spherical geometry of the solid-liquid interface.
Note:
is used for the pore size rather than 
for a number of reasons :
. 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 
in cylindrical pores.
Thus for a spherical interface between a non-wetting crystal and its own liquid, in an infinite cylindrical pore of diameter
, the structural melting point depression
is given by:
where the Gibbs-Thomson Coefficient
assumes different values for different liquids and different interfacial geometries (spherical/cylindrical/planar).
In more detail:,
Where :
is a geometric constant dependent on the interfacial shape,
has deservedly given his name to the first part of this equation, as it a particular case of his generalised Gibbs Equations.
It is somewhat less clear where the "Thomson" part came from.
The easy assumption is that it was William Thomson, who later as Lord Kelvin
gave his name to the Kelvin equation
. However, it was probably experiments by his brother James Thomson
(also known for his innovations in the analysis of regelation
– the effect of pressure on the freezing point of water) that originated the subject. There are recorded letters between Gibbs and the two Thomson brothers on these subjects. However the original version of the Gibbs-Thomson equation appears to occur in J. J. Thomson
's book on "generalised dynamics", on page 258 (but the notation is very different).
Thus all three Thomsons appear to share the honours.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
refers to the observation that small crystals of a liquid melt at a lower temperature than the bulk liquid.
The melting point depression 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 equationKelvin equation
The Kelvin equation describes the change in vapour pressure due to a curved liquid/vapor interface with radius r . The Kelvin equation is used for determination of pore size distribution of a porous medium using adsorption porosimetry...
. They are both particular cases of the Gibbs Equations of Josiah Willard Gibbs
Josiah Willard Gibbs
Josiah Willard Gibbs was an American theoretical physicist, chemist, and mathematician. He devised much of the theoretical foundation for chemical thermodynamics as well as physical chemistry. As a mathematician, he invented vector analysis . Yale University awarded Gibbs the first American Ph.D...
: 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 Themoporometry 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 (c.f. Mercury Intrusion
Porosimetry
Porosimetry is an analytical technique used to determine various quantifiable aspects of a material's porous nature, such as pore diameter, total pore volume, surface area, and bulk and absolute densities....
), 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.
The geometry of the crystal–liquid interface determines the value of the constant in the Gibbs-Thomson equation – the conventional "4" only applies to a spherical interface in a cylindrical pore.
Gibbs–Thomson equation for particles
For an isolated spherical solid particle of diameter in its own liquid, the Gibbs-Thomson Equation for the structural melting point depression can be written: 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
The "4" in the above equation comes from the spherical geometry of the solid-liquid interface.
Note:
is used for the pore size rather than for a number of reasons :
-
- It is consistent with the original published notation.
- The equation can be used with planar geometry (with a change of constant).
- For consistency with the related Strange–Rahman–Smith equationStrange–Rahman–Smith equationThe Strange–Rahman–Smith equation is used in the cryoporometry method of measuring porosity.NMR cryoporometry is a recent technique for measuring total porosity and pore size distributions...
where the symbol
is used for the differential operatorDifferential operatorIn mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
.
Gibbs–Thomson equation 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. 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 in cylindrical pores.Thus for a spherical interface between a non-wetting crystal and its own liquid, in an infinite cylindrical pore of diameter
, the structural melting point depressionis given by:
Simplified Gibbs-Thomson equation
The Gibbs-Thomson equation may be written in a compact form:where the Gibbs-Thomson Coefficient
assumes different values for different liquids and different interfacial geometries (spherical/cylindrical/planar).In more detail:,
Where :
is a geometric constant dependent on the interfacial shape,
-
-
is a constant involving parameters specific to the crystalline solid of solid–liquid system, and -
is an interfacial energy term.
-
Origins of the Gibbs–Thomson equation
Josiah Willard GibbsJosiah Willard Gibbs
Josiah Willard Gibbs was an American theoretical physicist, chemist, and mathematician. He devised much of the theoretical foundation for chemical thermodynamics as well as physical chemistry. As a mathematician, he invented vector analysis . Yale University awarded Gibbs the first American Ph.D...
has deservedly given his name to the first part of this equation, as it a particular case of his generalised Gibbs Equations.
It is somewhat less clear where the "Thomson" part came from.
The easy assumption is that it was William Thomson, who later as Lord Kelvin
William Thomson, 1st Baron Kelvin
William Thomson, 1st Baron Kelvin OM, GCVO, PC, PRS, PRSE, was a mathematical physicist and engineer. At the University of Glasgow he did important work in the mathematical analysis of electricity and formulation of the first and second laws of thermodynamics, and did much to unify the emerging...
gave his name to the Kelvin equation
Kelvin equation
The Kelvin equation describes the change in vapour pressure due to a curved liquid/vapor interface with radius r . The Kelvin equation is used for determination of pore size distribution of a porous medium using adsorption porosimetry...
. However, it was probably experiments by his brother James Thomson
James Thomson (engineer)
right|300px|James Thomson was an engineer and physicist whose reputation is substantial though it is overshadowed by that of his younger brother William Thomson .-Biography:Born in Belfast, he grew up mostly in Glasgow...
(also known for his innovations in the analysis of regelation
Regelation
Regelation is the phenomenon of melting under pressure and freezing again when the pressure is reduced. Many textbooks and reference books as well as websites often claim that regelation can be demonstrated by looping a fine wire around a block of ice , with a heavy weight attached to it...
– the effect of pressure on the freezing point of water) that originated the subject. There are recorded letters between Gibbs and the two Thomson brothers on these subjects. However the original version of the Gibbs-Thomson equation appears to occur in J. J. Thomson
J. J. Thomson
Sir Joseph John "J. J." Thomson, OM, FRS was a British physicist and Nobel laureate. He is credited for the discovery of the electron and of isotopes, and the invention of the mass spectrometer...
's book on "generalised dynamics", on page 258 (but the notation is very different).
Thus all three Thomsons appear to share the honours.