Gibbs-Thomson relationship

When this force is equal to the force to grow grains, growth by boundary migration will cease. If the limiting grain diameter is D then by the Gibbs-Thomson relationship... 

The basic Gibbs-Thomson relationship (section 3.7) for a non-electrolyte may be written... 

The driving force for ripening is the difference in solubihty between small and large particles, as given by the size solubihty (Gibbs Thomson) relationship (equation 3.58) which for the present purpose may be written as... 

This equation describes both exact Gibbs-Thomson relationship and random deviation in the X value at a certain T value because of nonuniformity of the adsorbent studied. Additionally, the formation of a solid (ice)—liquid— pore wall sandwich structure can lead to deviation from the Gibbs-Thomson equation (Petrov and Fur6 2009). In the latter review, they considered the effects of the formation of the mentioned sandwich on the dependence of the melting temperature on the pore size expressed via the surface area and the volume characteristics of the sandwich model. Some results discussed in this book could be interpreted in the term of this sandwich model because of broad PSDs, which cause a significant difference in the values of the force field near the pore walls and in the center of broad mesopores of macropores. 

Crystal Growth in Small Molecular Systems the Gibbs-Thomson relationship ... 

Many conditions of this general type, relating surface curvature to energy densities, are commonly identified as a Gibbs-Thomson relationship. 

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)... 

The relationship between particle size, solubility and supersaturation is expressed by the well-known Gibbs-Thomson equation [5a] ... 

The melting point of nitrobenzene in the pore is always depressed. The linear relationship between the shift in the pore melting temperature and the inverse pore diameter is consistent with the Gibbs-Thomson equation for larger pore sizes. The deviations from linearity, and hence from the Gibbs-Thomson equation are appreciable at pore widths as small as 4.0 nm. The quantitative estimates of the rotational relaxation times in the fluid and crystal phases of confined nitrobenzene support the existence of a contact layer with dynamic and structural properties different than the inner layers. The Landau free... 

Measurements of whisker crystallization rates by VLS have revealed that the linear growth rate is proportional to the whisker diameter, with larger fibers growing faster than smaller ones. The observed diameter-dependence has been attributed to differences in seed saturation concentration resulting from the curved interface of the seeds. The Gibbs-Thomson equation provides the relationship between the sphere diameter, d, and the chemical potential difference between the nutrient phase (e.g., molecular Si in the vapor) and the liquid alloy droplet Ap ... 

There is a linear relationship between the crystallization temperatures and the inverse lamella thicknesses, which is quite in accordance with Gibbs-Thomson equation. There is also a linear relationship between the melting temperatures and the inverse lamella thicknesses. Crossover of these two linear curves is considered to be the triple point of mesophase transition. Recently, the crossover was reproduced in the molecular simulations of lattice polymers, and the interpretation was updated to an uplimit of instant thickening at the lateral growth front of lamellar crystals (Jiang et al. 2016). 

This relationship is known as the Gibbs-Thomson equation. Thus, the chemical potential in the droplet is greater than that in the bulk liquid and the difference increases as the size of the droplet decreases. 

A similar relationship may be derived for each component in a multi-component liquid using the transfer shown schematically in Figure 6.4 in which component i is transferred. In this case the free-energy change for the transfer is Vi A P and application of the Laplace equation yields a Gibbs-Thomson equation for each component in the liquid ... 

The driving force for sintering (a reduction in excess surface free energy) is translated into a driving force that acts at the atomic level (thus resulting in atomic diffusion) by means of differences in curvature that inherently occur in different parts of the three-dimensional compact. These differences in curvature create chemical potential and vacancy concentration differences, and thus control the direction of matter transport. The relationship that links surface energy, curvature and concentration differences is the Gibbs-Thomson equation ... 

At the end of this Chapter it is worth commenting upon the correlation between the atomistic thermodynamic treatment of the n A p ) relationship (Figure 1.29) and the continuity, classical approach leading to the Gibbs-Thomson equation (1.62, 1.75, 1.136), (Figure 1.16). 

The relationship between particle size and solubility, originally derived for vapour pressures in liquid-vapour systems by Thomson (who became Lord Kelvin in 1892) in 1871, utilized later by Gibbs, and applied to solid liquid systems by Ostwald (1900) and Freundlich (1926) may be expressed in the form... 

DSC is another tool giving access to the lamellar thickness distribution. Based on the form of the DSC melting curve, the average distribution of the lamellae thickness may be determined. This procedure is based on the assumption that the melting temperature is related to the thickness L of a crystalline lamella, by the Thomson-Gibbs relationship [8] ... 

In addition large crystallites have a higher melting temperature than smaU ones. The relationship between the thickness, /, of a crystal and its melting temperature, T, is known as the Thomson-Gibbs equation ... 

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