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Carbon Dioxide Adsorption on Akageneite, Sphere and Particle Packing, and Ordered Amorphous and Microporous Molecular Sieve Silica
Published in Rolando M.A. Roque-Malherbe, Adsorption and Diffusion in Nanoporous Materials, 2018
Therefore, as the grand potential is defined as Ω=U−TS−μn, Ωa=Ua−TSa−μana−Φ. Thereafter [50], () dΩa=dUa−TdSa−SadT−dμana−μadn−dΦ
Nucleation on a sphere: the roles of curvature, confinement and ensemble
Published in Molecular Physics, 2018
Jack O. Law, Alex G. Wong, Halim Kusumaatmaja, Mark A. Miller
In order to develop a model that explains these features, we start with the result from classical nucleation theory (CNT). In the grand canonical ensemble, the thermodynamic potential (free energy) is the grand potential , where F is the Helmholtz free energy. The change in grand potential for the creation of a nucleus is where γ is the line tension, is the bulk free energy of the nucleating phase relative to the parent phase per unit area, P is the perimeter of the nucleus and is the area of the nucleus as a function of its perimeter. Figure 3 includes an attempted fit of Equation (4) to the simulated results, assuming the usual planar relation . It can be seen that Equation (4) not only fails to capture the increase in the free energy at large cluster sizes, it also cannot fit the shape of the initial barrier.
Phase behaviour of a simple fluid confined in a periodic porous material
Published in Molecular Physics, 2021
Daniel Stopper, Gerd E. Schröder-Turk, Klaus Mecke, Roland Roth
The phase coexistence between a liquid and a gas phase in a porous material can be found by equating the grand potential (per unit cell) at the same temperature and chemical potential or reservoir packing fraction . As an example we show in Figure 4 the behaviour of per unit cell of the gas and the liquid phase for a square-well fluid, confined by a gyroid minimal surface in nodal approximation with unit cell length . Since the confining walls are hard, the contact angle of a square-well fluid at a planar wall is 180 degrees, so that one can find capillary evaporation, i.e. capillary condensation of the gas phase, in the pore. For a given temperature T and unit cell length L we start with a stable bulk liquid at a reservoir packing fraction of and reduce its value, or equivalently the chemical potential, and thereby approaching the bulk coexistence – see Figure 1. The DFT is minimised, resulting in an inhomogeneous density distribution and a corresponding grand potential . In Figure 4, the full line corresponds to the liquid branch of the grand potential. At some sufficiently low value of the liquid phase cannot be stabilised in the pore anymore and the grand potential jumps onto its gas branch (dotted line). This happens when the spinodal of the confined fluid is reached. The gas branch of (dotted line) can be mapped out by starting with a sufficiently low value of , or corresponding chemical potential, and increasing its value in small steps until the upper spinodal is reached. At some point between the two spinodals one finds an intersection of the two branches of the grand potential. This intersection point, , corresponds to the phase coexistence between liquid and gas. The thermodynamic stable phase for a given value of is that with the lowest value of the grand potential. For the stable phase is the inhomogeneous gas and the liquid that can be found is meta stable. For , one finds the opposite situation with an inhomogeneous stable liquid and a meta stable gas.