Models Based on Classical Theories
Both surface energy distribution and pore size distribution may be evaluated using classical approaches to model kernel functions for use with equation (1) of the DFT Theory. The Calculations document can be found on the Micromeritics web page (www.Micromeritics.com). Be aware that the deconvolution method only provides a fitting mechanism; it does not overcome any inherent shortcomings in the underlying theory.
Surface Energy
The use of classical theories to extract adsorptive potential distribution is mostly of historical interest. At a minimum, the equation must contain a parameter dependent on adsorption energy and another dependent on monolayer capacity or surface area. This is sufficient to permit the calculation of the set of model isotherms that is used to create a library model. The Langmuir equation has been used in the past, as have the Hill-de Boer equation and the Fowler-Guggenheim equation. All of these suffer from the fact that they only describe monolayer adsorption, whereas the data may include contributions from multilayer formation.
Pore Size
It is well established that the pore space of a mesoporous solid fills with condensed adsorbate at pressures somewhat below the prevailing saturated vapor pressure of the adsorptive. When combined with a correlating function that relates pore size with a critical condensation pressure, this knowledge can be used to characterize the mesopore size distribution of the adsorbent. The correlating function most commonly used is the Kelvin equation. Refinements make allowance for the reduction of the physical pore size by the thickness of the adsorbed film existing at the critical condensation pressure. Still further refinements adjust the film thickness for the curvature of the pore wall.
The commonly used practical methods of extracting mesopore distribution from isotherm data using Kelvin-based theories, such as the BJH method, were for the most part developed decades ago and were designed for hand computation using relatively few experimental points. In general, these methods visualize the incremental decomposition of an experimental isotherm, starting at the highest relative pressure or pore size. At each step, the quantity of adsorptive involved is divided between pore emptying and film thinning processes and exactly is accounted for. This computational algorithm frequently leads to inconsistencies when carried to small mesopore sizes. If the thickness curve used is too steep, it finally will predict a larger increment of adsorptive for a given pressure increment than is actually observed; since a negative pore volume is non-physical, the algorithm must stop. Conversely, if the thickness curve used underestimates film thinning, accumulated error results in the calculation of an overly large volume of (possibly nonexistent) small pores.
The use of equation (1) represents an improvement over the traditional algorithm. Kernel functions corresponding to various classical Kelvin-based methods have been calculated for differing geometries and included in the list of models.
Models Included
Kelvin Equation with Halsey Thickness Curve
N2 - Halsey Thickness Curve
Geometry: |
Slit |
Substrate: |
Average |
Category: |
Porosity |
Method: |
Nitrogen 77 K |
The kernel function is calculated using the Halsey equation with standard parameters:
The nitrogen properties used in the Kelvin equation are:
Surface tension = |
8.88 dynes cm-1 |
Molar density = |
0.02887 g cm-3 |
N2 - Halsey Thickness Curve
Geometry: |
Cylinder |
Substrate: |
Average |
Category: |
Porosity |
Method: |
Nitrogen 77 K |
The calculation is the same as above except that cylindrical geometry is assumed.
Reference: |
G. Halsey, J. Chem. Phys 16, 931 (1948). |
Kelvin Equation with Harkins and Jura Thickness Curve
N2 - Harkins and Jura Thickness Curve
Geometry: |
Slit |
Substrate: |
Average |
Category: |
Porosity |
Method: |
Nitrogen 77 K |
The kernel function is calculated using the Harkins and Jura equation with standard parameters:
The nitrogen properties used in the Kelvin equation are:
Surface tension = |
8.88 dynes cm-1 |
Molar density = |
0.02887 g cm-3 |
Geometry: |
Cylinder |
Substrate: |
Average |
Category: |
Porosity |
Method: |
Nitrogen 77 K |
The calculation is the same as above except that cylindrical geometry is assumed.
References: |
W. D. Harkins and G. Jura, J.A.C.S. 66, 1366 (1944). J. H. DeBoer et al., J. Colloid and Interface Sci. 21, 405 (1966). |
Kelvin Equation with Broekhoff-de Boer Thickness Curve
N2 - Broekhoff-de Boer Model
Geometry: |
Cylinder |
Substrate: |
Average |
Category: |
Porosity |
Method: |
Nitrogen 77 K |
The kernel function is calculated using the Broekhoff-de Boer equation with standard parameters:
The nitrogen properties used in the Kelvin equation are:
Surface tension = |
8.88 dynes cm-1 |
Molar density = |
0.02887g cm-3 |
N2 - Broekhoff-de Boer Model
Geometry: |
Cylinder |
Substrate: |
Average |
Category: |
Porosity |
Method: |
Nitrogen 77 K |
The calculation is similar to the above except that cylindrical geometry is assumed, and the film thickness depends on pore size (see reference).
References: |
Specifically, equations 20 and 21 in: J.C.P. Broekhoff and J.H. de Boer, “The Surface Area in Intermediate Pores,” Proceedings of the International Symposium on Surface Area Determination, D.H. Everett, R.H. Ottwill, eds., U.K. (1969). |
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