Continuum model of adsorption

In order to get rid of undesired lattice effects like the almost cuboid form of the most compact adsorbed conformations in the subphases of AC2 in the phase diagram of a polymer on a simple-cubic lattice (see Fig, 13.2), we now investigate the stmcture of conformational phases of a semiflexible off-lattice polymer near an attractive substrate [304,307,308]. 

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Classical thermodynamic models of adsorption based upon the Kelvin equation [21] and its modihed forms These models are constructed from a balance of mechanical forces at the interface between the liquid and the vapor phases in a pore filled with condensate and, again, presume a specihc pore shape. Tlie Kelvin-derived analysis methods generate model isotherms from a continuum-level interpretation of the adsorbate surface tension, rather than from the atomistic-level calculations of molecular interaction energies that are predominantly utihzed in the other categories. 

A continuum description becomes increasingly inaccurate as distances from the interface become comparable to the size of a solute molecule. Another crude concept used in simple continuum models of interfaces for calculating adsorption free energy and electronic spectra involves the use of an effective interfacial dielectric constant. For example, the reduced orientational freedom of interfacial water molecules and their reduced density result in a smaller effective dielectric constant than in the bulk. This is consistent with assigning the water liquid/vapor a polarity value similar to that of CCI4. Finally, we mention that a molecular theory of a local dielectric constant, which reproduces interfacial electric fields, can be developed with the aid of molecular dynamics simulation as described by Shiratori and Morita. ... 

In these hybrid simulations, coupling happened through the boundary condition. In particular, the fluid phase provided the concentration to the KMC method to update the adsorption transition probability, and the KMC model computed spatially averaged adsorption and desorption rates, which were supplied to the boundary condition of the continuum model, as depicted in Fig. 7. The models were solved fully coupled. Note that since surface processes relax much faster than gas-phase ones, the QSS assumption is typically fulfilled for the microscopic processes one could solve for the surface evolution using the KMC method alone, i.e., in an uncoupled manner, for a combination of fluid-phase continuum model parameter values to develop a reduced model (see solution strategies on the left of Fig. 4). Note again that the QSS approach does not hold at very short (induction) times where the microscopic model evolves considerably. 

The rationale of using hybrid simulation here is that a classic diffusion-adsorption type of model, Eq. (2), can efficiently handle large distances between steps by a finite difference coarse discretization in space. As often happens in hybrid simulations, an explicit, forward discretization in time was employed. On the other hand, KMC can properly handle thermal fluctuations at the steps, i.e., provide suitable boundary conditions to the continuum model. Initial simulations were done in (1 + 1) dimensions [a pseudo-2D KMC and a ID version of Eq. (2)] and subsequently extended to (2 + 1) dimensions [a pseudo-3D KMC and a 2D version of Eq. (2)] (Schulze, 2004 Schulze et al., 2003). Again, the term pseudo is used as above to imply the SOS approximation. Speedup up to a factor of 5 was reported in comparison with KMC (Schulze, 2004), which while important, is not as dramatic, at least for the conditions studied. As pointed out by Schulze, one would expect improved speedup, as the separation between steps increases while the KMC region remains relatively fixed in size. At the same time, implementation is definitely complex because it involves swapping a microscopic KMC cell with continuum model cells as the steps move on the surface of a growing film. 

The formulation of a proper surface boundary condition is a delicate matter, as noted by DiMarzio (1965) and de Gennes (1969). Lattice models simply require that P(i, s) = 0 for layers i < 0, a form proven correct by DiMarzio (1965). In continuum models, chains intersecting the surface undergo both reflection and adsorption, the relative amount of each depending on the energy of contact at the surface. The result is a mixed boundary condition expressed by de Gennes (1969) as... 

The above techniques have been used in numerous calculations of solute free energy profiles. Wilson and Pohorille [52] and Benjamin[53] have determined the free energy profiles for small ions at the water liquid/vapor interface and compared the results to predictions of continuum electrostatic models. The transfer of small ions to the interface involves a monotonic increase in the free energy which is in qualitative agreement with the continuum model. This behavior is consistent with the increase in the surface tension of water with the increase in the concentration of a very dilute salt solution, and it represents the fact that small ions are repelled from the liquid/vapor interface. On the other hand, calculations of the free energy profile at the water liquid/vapor interface of hydrophobic molecules, such as phenol[54] and pentyl phenol[57] and even molecules such as ethanol [58], show that these molecules are attracted to the surface region and lower the surface tension of water. In addition, the adsorption free energy of solutes at liquid/liquid interfaces[59,60] and at water/metal interfaces[61-64] have been reported. 

While studying adsorption in mesopores using the molecular continuum model we have found [4,6,7] that there exist two critical diameters based on thermodynamic analysis of the adsorption, and two more when the mechanical stabihty of the meniscus is considered. These criticalities refer to the critical pore diameter below which there either exists a different mechanism of adsorption, or the adsorption is reversible. Here we provide a brief outline of these criticalities. The chemical potential of the fluid adsorbed in a cyfindrical pore of radius R can be expressed as [6,7] (r,R) = /jj (r,R) + (f>(R-r,R) = constant(/ ). After considering... 

It is then of great importance to develop simple models capable of describing the energetic topography on the basis of a few parameters and to study the effects of these parameters on several surface processes, with the hope that, in such a process, methods to obtain the relevant parameters from experimental data will be envisaged. These models can be of two kinds continuum models or lattice-gas models. The former are more suited to mobile adsorption, generally physisorption, and then more closely related to the surface energetic characterization problem, whereas the latter are more suited to locaHzed adsorption (e.g., chemisorption). 

Apart from diffusion in continuum phase, the transport of surface-adsorbed molecules and capillary condensate takes place in meso- and macroporous media. In order to model transport of adsorbable vapor at elevated pressure, it is necessary to consider the type of adsorption occurring monolayer adsorption, multilayer adsorption, or capUlary condensation [38]. Models for surface diffusion have been proposed... 

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