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 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 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 . 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: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.
Simplified 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.
History
As early as 1886, Robert von Helmholtz (son of the German physicist Hermann von Helmholtz) had observed that finely dispersed liquids have a higher vapor pressure.Robert von Helmholtz (1886) "Untersuchungen über Dämpfe und Nebel, besonders über solche von Lösungen" (Investigations of vapors and mists, especially of such things from solutions), Annalen der Physik, 263 (4) : 508–543. From page 525: "Eine zufällig von mir gemachte Beobachtung dürfte vielleicht eine experimentelle Bestätigung dieser Resultate enthalten: Wenn nämlich auf einer Glasscheibe ein feiner Beschlag gebildet ist, über den dickere Tropfen zerstreut sind, so bildet sich bald um die letzteren herum eine Scheibe, welche vom feineren Beschlag befreit ist, ein Beweis, dass die kleinen in die grossen Tropfen überdestillirt sind." (An observation that I made by chance perhaps might contain an experimental confirmation of this result: namely, if a fine mist forms on a pane of glass, over which large drops are scattered, then around the latter, a disk soon forms, which is free of fine mist — evidence that the small [droplets] are "distilled" into the big ones.) By 1906, the German physical chemist Friedrich Wilhelm Küster (1861–1917) had predicted that since the vapor pressure of a finely pulverized volatile solid is greater than the vapor pressure of the bulk solid, then the melting point of the fine powder should be lower than that of the bulk solid.Friedrich Wilhelm Küster (1906) Lehrbuch der allgemeinen physikalischen und theoretischen Chemie ... [Textbook of general physical and theoretical chemistry ... ] (Heidelberg, Germany: Carl Winter, 1906), v.1, p. 189. The relevant passage is reprinted on page 189 of volume 1 of the 1913 edition: § 127. Schmelzen feinster Pulver. (Melting of the finest powder). From page 189: "Folglich ist die Schmelztemperatur des Pulvers, t1°, niedriger als die der Kristalle, t°. Der Unterschied ist jedoch so gering, daß er noch nicht zur Beobachtung gelangt ist (vgl. weiter unter §. 131)." (Consequently, the melting temperature of the powder, t1°, is lower than that of the [bulk] crystal, t°. However, the difference is so small that it still hasn't been observed (compare §. 131 below).) Investigators such as the Russian physical chemists Pavel Nikolaevich Pavlov (or Pawlow (in German), 1872–1953) and Peter Petrovich von Weymarn (1879–1935), among others, searched for and eventually observed such melting point depression.As early as 1906, the Austrian mineralogist Cornelio August Doelter (1850-1930) had attempted to determine the melting points of various minerals via a microscope and had observed that finely powdered silicates melted over a range of as much as 100°C. See pp. 618-619 of: Doelter. C (17 August 1906) "Bestimmung der Schmelzpunkte vermittelst der optischen Methode" (Determination of melting points by means of an optical method), Zeitschrift für Elektrochemie und angewandte physikalische Chemie, 12 (33) : 617-621. From p. 618: " … wir erkennen, dass zwischen Beginn der Schmelzung und diesem Punkt bei manchen Silikaten ein erheblicher Temperaturunterschied — bis 100° — liegen kann, … " ( … we discern that between the beginning of melting and this point [i.e., at which molten droplets join together] there can lie, in the case of some silicates, a considerable difference in temperature — up to 100°C … ) By 1932, Czech investigator Paul Kubelka (1900–1956) had observed that the melting point of iodine in activated charcoal is depressed as much as 100 °C.See: Kubelka, Paul (July 1932) "Über den Schmelzpunkt in sehr engen Capillaren" (On the melting point in very narrow capillaries), Zeitschrift für Elektrochemie und angewandte physikalische Chemie (Journal for Electrochemistry and Applied Physical Chemistry), 38 (8a) : 611–614. Available on-line in English translation at: National Research Council Canada . From page 614: "Tests which will be reported in detail by the author elsewhere enable us to prove ... that iodine in activated charcoal is still liquid at room temperature, i.e., approximately 100° below the melting point." Investigators recognized that the melting point depression occurred when the change in surface energy was significant compared to the latent heat of the phase transition, which condition obtained in the case of very small particles.See, for example: Павлов П. Н. (Pavlov, P.N.) (1908) "О зависимости температуры плавления от поверхностной энергии твердого тела" (On the dependence of the melting point on the surface energy of a solid body), Журнал Русского Физико-химического Общества (Journal of the Russian Physical-Chemical Society), (in Russian) 40 : 1022–1067. Pawlow, P. N. (1909) "Über die Abhängigkeit des Schmelzpunktes von der Oberflächenenergie eines festen Korpers" (On the dependency of the melting point on the surface energy of a solid body), Zeitschrift für physikalische Chemie, 65 : 1–35; 545–548. Pawlow, P. N. (1910) "Über den Dampfdruck der Körner einer festen Substanz" (On the vapor pressure of granules of a solid substance), Zeitschrift für physikalische Chemie, 68 : 316–322. Pawlow, P. N. (1910) "Über die Schmeltztemperatur der Körner des Salols" (On the melting temperature of granules of salol), Zeitschrift für physikalische Chemie, 74 : 562–566. Pawlow, P. N. (1910) "Ueber den Einfluss der Oberfläche einer festen Phase auf die latente Wärme und die Temperatur des Schmelzens" (On the effect of the surface of a solid phase on the latent heat and the temperature of melting), Zeitschrift für Chemie und Industrie der Kolloide, 7 : 37–39. P. P. von Weimarn (October 1910) "Der Einfluß des Dispersitätsgrades eines festen Kristalles auf seine Schmelztemperatur" [Influence of the degree of dispersion of a solid crystal on its melting temperature], Zeitschrift für Chemie und Industrie der Kolloide, 7 (4) : 205–208. H. Leitmeier (June 10, 1913) "Zur Kenntnis der Schmelzpunkte von Silikaten. Der Einfluß der Korngröße auf den Schmelzpunkt. Bestimmung des Schmelzpunktes einiger Silikate durch längeres Erhitzen" ((Contribution) to (our) knowledge of the melting point of silicates. The influence of particle size on the melting point. Determination of the melting point of some silicates by prolonged heating) Zeitschrift für anorganische und allgemeine Chemie, 81 (1) : 209–232.
Neither Josiah Willard Gibbs nor William Thomson (Lord Kelvin) derived the Gibbs–Thomson equation.Jeong-Myeong Ha, Crystallization and Thermotropic Properties of Organic Solids in Nanoscopic Reactors, Ph.D. thesis: University of Minnesota, 2006, pages 26–28. Also, although many sources claim that British physicist J. J. Thomson derived the Gibbs–Thomson equation in 1888, he did not.Sir Joseph John Thomson derived Kelvin's equation (page 163) and the depression of the melting point of ice by pressure (page 258), but he did not derive the Gibbs–Thomson equation. However, on pages 251–252, Thomson considered the effects of temperature and surface tension on the solubility of salts in spherical droplets, and he obtained an equation for that phenomenon which has a form similar to that of the Gibbs–Thomson equation. See: Thomson, J.J., Applications of dynamics to physics and chemistry (London, England: Macmillan and Co., 1888). Early in the 20th century, investigators derived precursors of the Gibbs–Thomson equation.See: Pawlow, P. N. (1909) "Über die Abhängigkeit des Schmelzpunktes von der Oberflächenenergie eines festen Korpers" (On the dependency of the melting point on the surface energy of a solid body), Zeitschrift für physikalische Chemie, 65 : 545–548.