Thickness equations describe the effective depth of the adsorbed layer as a function of pressure. The most noteworthy of which are those of Harkins and Jura, Halsey, and Broekhoff-de Boer.
For each point designated, the following parameters are used in thickness curve calculations:
For the calculation, we use the refernce implementation from the report-models-python on github from Micromeritics. This tool not only provides example the reference caclulations, but aslo provides exampel data to work with, and utilities to make working with the data better.
In [1]:
%matplotlib inline
import numpy as np
from micromeritics import thickness as t, plots
Prel = np.array([0.000000448802,0.000000936935,0.00000148652,0.00000209607,0.00000276863,0.00000351456,
0.00000435161,0.00000526938,0.00000630655,0.00000748296,0.00000884365,0.0000104439,0.0000123147,
0.0000146023,0.0000174301,0.0000209452,0.0000255346,0.0000314851,0.000039581,0.0000510105,
0.0000673633,0.0000921119,0.000132109,0.000203374,0.000346635,0.000670347,0.00109569,0.00127345,
0.00140321,0.00155907,0.00185297,0.00253435,0.00347369,0.00463502,0.00627777,0.00823441,0.0108343,
0.0308038,0.066713,0.0813908,0.100712,0.120399,0.14077,0.160733,0.180374,0.20067,0.250146,0.301205,
0.351224,0.400769,0.450922,0.50095,0.551038,0.601054,0.65121,0.701325,0.751307,0.801299,0.820996,
0.850996,0.875747,0.900984,0.925356,0.949191])
$\displaystyle{t_\mathrm{i} = \left(\frac{C_\mathrm{1}}{C_\mathrm{2}-\log{(P_\mathrm{rel,\ i})}}\right)^{C_\mathrm{3}}}$
where typically $C_\mathrm{1} = 60.6500$, $C_\mathrm{2} = 0.03071$, and $C_\mathrm{3} = 0.3968$
M. Kruk, M. Jaroniec, and A. Sayari. "Application of Large Pore MCM-41 Molecular Sieves To Improve Pore Size Analysis Using Nitrogen Adsorption Measurements." Langmuir, 1997, 13 (23), pp 6267–6273
M. Kruk, M. Jaroniec, and A. Sayari. "Adsorption Study of Surface and Structural Properties of MCM-41 Materials of Different Pore Sizes." J. Phys. Chem. B, 1997, 101 (4), pp 583–589
Michal Kruk, Mietek Jaroniec, and Abdelhamid Sayari. "Relations between Pore Structure Parameters and Their Implications for Characterization of MCM-41 Using Gas Adsorption and X-ray Diffraction." Chem. Mater., 1999, 11 (2), pp 492–500
In [2]:
plots.plotThickness( Prel, t.KrukJaroniecSayari()(Prel), 'Kruk-Jaroniec-Sayari' )
The Halsey equation assumes the adsorbed liquid monolayer has the same density and packing as the normal liquid. The values 3.54 and 5.00 are empirical and are user adjustable.
$\displaystyle{t_\mathrm{i} = C_\mathrm{1}\left(\frac{C_\mathrm{2}}{\ln{(P_\mathrm{rel,\ i})}}\right)^{C_\mathrm{3}}}$
where typically $C_\mathrm{1} = 3.540$, $C_\mathrm{2} = -5.0001$, and $C_\mathrm{3} = 0.333$
Halsey, G.D., J. Chem. Phys., 16, 931 (1948).
In [3]:
plots.plotThickness( Prel, t.Halsey()(Prel), 'Halsey')
The work of Harkins and Jura has shown that a plot of $\log{(P_\mathrm{rel})} = B - A/V_\mathrm{a}^\mathrm{2}$ returns a linear region where the film is condensed (where B is the intercept and A is the slope of the linear region). The relative pressure can be related to the statistical thickness $(t_\mathrm{i})$ of the adsorbed film with the following association:
$\displaystyle{t_\mathrm{i} = \left(\frac{13.99}{0.034-\log{(P_\mathrm{rel,\ i})}}\right)^{\frac{1}{2}}}$
where the empirical values 13.99 and 0.034 are substituted for the slope A and intercept B, respectively. Note that these values are user adjustable which gives the equation:
$\displaystyle{t_\mathrm{i} = \left(\frac{C_\mathrm{1}}{C_\mathrm{2}-\log{(P_\mathrm{rel,\ i})}}\right)^{C_\mathrm{3}}}$
Harkins, W.D. and Jura, G., J. Am. Chem. Soc., 66, 1366 (1944).
In [4]:
plots.plotThickness( Prel, t.HarkinsJura()(Prel), 'Harkins and Jura')
$\displaystyle{\log{(P_\mathrm{rel,\ i})} = \frac{C_\mathrm{1}}{t_\mathrm{i}^{2}}+C_\mathrm{2}e^{C_\mathrm{3}t_\mathrm{i}}}$
where typically $C_\mathrm{1} = -16.1100$, $C_\mathrm{2} = 0.1682$, and $C_\mathrm{3} = -0.1137$
Broekhoff, J.C.P. and de Boer, J.H., "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).
In [5]:
plots.plotThickness( Prel, t.BroekhoffDeBoer()(Prel), 'Broekhoff-de Boer')
In [6]:
plots.plotThickness( Prel, t.CarbonBlackSTSA()(Prel), 'Carbon Black STSA')