Thomson-Freundlich equation

Changes in crystal dispersion due to surface effects are typically based on the Gibbs-Thomson equation (sometimes called the Ostwald-Freundlich equation), which describes the effects of particle size on equilibrium conditions (Hartel 2001). For solution systems, the equation is given as... 

As early as 1878, Gibbs concluded that the breakdown or growth of a crystal was not a continuous transformation, as the gas-liquid transition was considered to be. Thomson derived what has come to be known as the Gibbs-Thomson equation, relating the vapor pressure of liquid droplets to the size of the droplets. Ostwald extended the concept to the problem of solubility, but made a numerical error later corrected by Freundlich. Similar to the Gibbs-Thomson equation, the Ostwald-Freundlich equation was expressed by... 

Solubility is also affected by particle size, small crystals (<1 pm say) exhibiting a greater solubility than large ones. This relationship is quantified in the Gibbs-Thomson, Ostwald-Freundlich equation (see Mullin, 2001)... 

For a given solid material, a progressive reduction of particle size corresponds to increases in the surface/volume ratio and the escaping tendency of the molecules until the nature of the surface dominates the properties of the material. Two related thermodynamic consequences of this effect are an increase of solubility in any solvent and an increase of vapor pressure as the size of the particle is reduced. For a spherical particle of radius r, thermodynamic arguments lead to the Thomson-Freundlich equation [19] ... 

Thomson-Freundlich equation relates the solubility of a particle to its curvature radius (e.g., Swalin, 1962). Using this equation, Kirkaldy and Young (1987) show how periodic precipitation may result from the capillary resistance of the matrix to grow small precipitates. Formally, the conditions read... 

The difference in thermodynamic equilibrium between crystals of different size is defined by the Gibbs-Thomson or Ostwald-Freundlich equation (Mullin 1993). The original Kelvin equation was written to describe droplets of liquid in a vapor in terms of vapor pressures around droplets of different size. However, in crystallization studies, the vapor pressure may be related, through the solution activity, to the equilibrium concentration in solution for crystals of different size, faking the assumption that concentration can be used directly fpr vapor pressure gives... 

Confusingly, but for understandable reasons, equation 3.58 is referred to in the literature by a variety of names such as the Gibbs-Thomson, Gibbs-Kelvin and Ostwald-Freundlich relation. For consistency, however, the designation Gibbs-Thomson will be used throughout this work. 

Equation (6.76) is known as the Thomson-Freundlich equation, which can also be written as... 

Closely similar expressions for particle (grain) growth in dispersed systems of widely-separated particles by solution of the smaller particles in the matrix and precipitation on the larger particles have been derived by Greenwood ( °), Wagner ( ), Lifshitz and Slezov ( ), and Li and Oriani ( ). All these authors assumed the particles to be spherical, so that the variation of solubility with particle size was given by the Thomson-Freundlich equation,... 

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