The BJH method is used to create pore volume and surface area distributions based on a complete desorption or adsorption isotherm. The isotherm plots the amount of gas adsorbed onto a material at varying relative pressures. The distribution displays the amount of volume or area that is found within pores of a certian range of radii, ranges calculated from the ranges of pressure found on the isotherm.
The original method developed by Barrett, Joyner, and Halenda for calculating pore size distributions involved analyzing a desorption (or reversed adsorption) isotherm in order to allocate the desorbing gas into two groups: vaporizing cores (or inner capillaries) which make up a considerable portion of the pores' total volume as a cylindrical tube which is concentric with the pore, followed by the gradual thinning of multimolecular films exposed in pores whose cores have been empied during prior desorption intervals. This is the basis for MicroActive's implimented BJH methods: The Faas Correction developed by George S. Faass, the Standard Correction, and the Kruk-Jaroniec-Sayari correction which adds a slight modification to the Kelvin equation.
All three methods apply the Kelvin equation which calculates the core radii of an inner capillary that would vaporize at a given relative pressure. The Kelvin radius can be calculated for a relative pressure point as follows:
$\displaystyle{ \log \frac{P}{P_\mathrm{0}} = \frac{2\gamma V_\mathrm{m}}{r_\mathrm{c}RT} }$
where
$P/P_\mathrm{0}$ = relative pressure
$\gamma$ = surface tension
$V_\mathrm{m}$ = liquid molar volume
$r_\mathrm{c}$ = Kelvin radius
$R$ = universal gas constant
$T$ = temperature
Because the surface tension, liquid molar volume, gas constant, and temperature remain constant during the calculation these terms are combined into a single term. The Adsorbate Property Factor (APF) is calculated as follows:
$\displaystyle{ A = -\frac{2 \gamma V_\mathrm{m}}{RT} \ \overset{\circ}{A} }$
Substituting the APF into the Kelvin equation and solving for the Kelvin radius gives:
$\displaystyle{ r_\mathrm{c} = \frac{-A}{\log \left(\frac{P}{P_\mathrm{0}}\right)} \ \overset{\circ}{A} }$
Once a pore has been emptied of its capillary condensate during a particular decrement of relative pressure, further decrements will result in the gradual thinning of its multimolecular film until the relative pressue reaches zero and all the adsorbate has desorbed. This can be visualized as the pore shedding off an annulus shaped tube of adsorbate during each desorption interval. The thickness of the tube can be calculated by a thickness equation given the starting and ending pressure points.
A thickess equation calculates the thickness of an adsorbed layer as a function of relative pressure. For example, if the Harkins and Jura thickness equation were used the multilayer thickness could be found from the following equation:
$\displaystyle{ Tw = \sqrt{\frac{13.99}{0.034-\log_{10}{(P/P_\mathrm{0})}}}\ \overset{\circ}{A} }$
See here for more details.
The standard method in MicroAcrtive for BJH calculations is more closely based off of the original method developed in 1951 without the thinning correction introduced by the Faas correction. Let
$i$ refer to the current desorption interval,
$j$ refer to all previous intervals in which new pores were found, and
$k$ refer to the total number of intervals in which new pores are found.
If no new pores are found during the interval only a thinning of the adsorbed layer of previously opened pores occurs and a point is not added to the distribution.
It is assumed that the starting relative pressure point is near one and therefor nearly all pores will be filled. It is then understood that for the first pressure decrement nearly all gas desorbed will be from emptying cores and we calculate the Kelvin radius for the upper bound of the first desorption interval.
$\displaystyle{ Rc_\mathrm{0} = \frac{-A}{Pr_\mathrm{0}} \ \overset{\circ}{A} }$
For each interval the volume of liquid adsorbate that will desorb off of walls previously exposed is calculated. This is done by finding the annulus cross sectional area of the region of gas to desorb and multiplying it by the length of the pore, for each interval. The volume $Vw_\mathrm{i}$ is calculated as follows:
$\displaystyle{ Vw_\mathrm{i} = \sum ^k _{j=0} \pi \left[ (Rp_{avg_\mathrm{j}} - Tw_\mathrm{i} + \Delta Tw_\mathrm{i})^2 - (Rp_{avg_\mathrm{j}} - Tw_\mathrm{i})^2 \right] \cdot Lp_\mathrm{j} \cdot 10^{-16} \ \mathrm{cm}^3 }$
where $\Delta Tw$ is the change in film thickness observed during the interval, $Tw_\mathrm{i}$ is the thickness found using the lower relative pressure bound (i.e. the larger of the two pressures), and $10^{-16}$ is the conversion for square centimeters per square angstrom.
When new pores are found in an intevral they are collectively seen as a single pore with a total length $Lp$ and an average radius $Rp_{avg}$ representing the actual pore radius. The quantity $Lp$ is calculated at the end of each interval and is used in the calculation of the volume of gas required for wall layer thinning, $Vw$, for all subsequent intervals.
The quantity $V_\mathrm{w}$ is then compared to the incremental liquid volume desorbed during the interval, $\Delta Vl_\mathrm{i} = Vl_\mathrm{i} - Vl_\mathrm{i+1}$. The terms $Vl_\mathrm{i}$ are equivalent to the quantity desorbed times the density conversion factor.
If $V_\mathrm{w}$ is less than the incremental volume desorbed during the current interval then the remaining volume is attributed to the opening of new pores. The capillary volume of pores opened during this interval is found as follows:
$\displaystyle{ Vc_\mathrm{i} = \Delta Vl_\mathrm{i} - Vw_\mathrm{i} \ \mathrm{cm}^3 }$
The Kelvin radius at the end of the interval, $Rc$, is calculated using the Kelvin equation and the average Kelvin radius for the interval is found as the weighted average of the starting and ending Kelvin radii for the interval.
$\displaystyle{ Rc_{avg_\mathrm{i}} = \frac{Rc_\mathrm{1} Rc_\mathrm{2} (Rc_\mathrm{1} + Rc_\mathrm{2})} {{Rc_\mathrm{1}}^2 + {Rc_\mathrm{2}}^2} \ \overset{\circ}{A} }$
The average pore radius for the interval is calculated by adding the average Kelvin radius and the corresponding film thickness at that point which is approximated using an inverse Kelvin equation.
$\displaystyle{ Pr = \exp \left( \frac{-A}{Rc_{avg_\mathrm{i}}} \right) }$
$\displaystyle{ Rp_{avg_\mathrm{i}} = Rc_{avg_\mathrm{i}} + \sqrt{\frac{13.99}{0.034-\log_{10}{Pr}}} \ \overset{\circ}{A} }$
The pore length of all pores exposed during the current interval is found using the geometric relationship between the capillary volume and the core radius.
$\displaystyle{ Lp_\mathrm{i} = \frac{Vc_\mathrm{i}}{\pi (Rc_{avg_\mathrm{i}})^2 \cdot 10^{-16}} \ \mathrm{cm} }$
The total number of intervals in which new pores are found, $k$, is increased by one.
If $V_\mathrm{w}$ is greater than or equal to the volume desorbed during the current interval then desorption from walls only is occuring. This interval will be added onto the next interval such that the upper bound of the next interval will be the upper bound of the current interval plus one but the lower bound will remain the same. E.g. if no pores are found from (Prel_15 to Prel_16) then the next interval will include (Prel_15 to Prel_17). This is also done with the quantity desorbed data.
The Kruk-Jaroniec-Sayari method follows the same procedure as the Standard method but with a modified Kelvin equation (and inverse Kelvin equation).
$\displaystyle{ Rc = \frac{-A}{P/P_\mathrm{0}} + 3.0 \ \overset{\circ}{A} }$
$\displaystyle{ Pr = \exp \left(\frac{-A}{Rc - 3.0} \right) }$
The Faas method was delveloped by George Steven Faass in 1981 in order to create a more accurate distribution. The Faas method includes a correction that adjusts the change in multilayer thickness during intervals in which no cores are emptied. A distribution calculation using the Faas correction is explained below. For this example let
$i$ refer to the current desorption interval,
$j$ refer to all previous intervals in which new pores were found, and
$k$ refer to the total number of intervals in which new pores are found.
If no new pores are found during a desorption interval only a thinning of the adsorbed layer of previously opened pores occurs and a point is not added to the distribution.
It is assumed that the starting relative pressure point is near one and therefor nearly all pores will be filled. It is then understood that for the first pressure decrement nearly all gas desorbed will be from emptying cores and we calculate the Kelvin radius for the upper bound of the first desorption step.
$\displaystyle{ Rc_\mathrm{0} = \frac{-A}{\log(Pr_\mathrm{0})} \ \overset{\circ}{A} }$
For each interval the volume of liquid adsorbate that will desorb off of walls previously exposed is calculated. This is done by finding the annulus cross sectional area of the region of gas to desorb and multiplying it by the length of the pore, $Lp$, for each previous interval $j$. The volume $Vw_\mathrm{i}$ is calculated as follows:
$\displaystyle{ Vw_\mathrm{i} = \sum ^k _{j=0} \pi \left[(Rc_{avg_\mathrm{j}} + \Delta Tw_\mathrm{i})^2 - {Rc_{avg_\mathrm{j}}}^2 \right] \cdot Lp_\mathrm{j} \cdot 10^{-16} \ \mathrm{cm}^3 }$
where $\Delta Tw$ is the change in film thickness observed during the interval and $10^{-16}$ is the conversion for square centimeters per square angstrom.
When new pores are found in an intevral they are collectively seen as a single pore with a total length $Lp$ and an average radius $Rc_{avg}$ which originaly represents the radius of the core but is increased each interval by the change in film thickness to represent the distance from the pore center to the exposed wall of the multilayer film. The quantity $Lp$ is calculated at the end of each interval and is used in the calculation of the volume of gas required for wall layer thinning, $Vw$, for all subsequent intervals.
The quantity $V_\mathrm{w}$ is then compared to the incremental liquid volume desorbed during the interval, $\Delta Vl_\mathrm{i} = Vl_\mathrm{i} - Vl_\mathrm{i+1}$. The terms $Vl_\mathrm{i}$ are equivalent to the quantity desorbed times the density conversion factor.
If $V_\mathrm{w}$ is less than the incremental volume desorbed during the current interval then the remaining volume is attributed to the opening of new pores. The capillary volume of pores opened during this interval is found as follows:
$\displaystyle{ Vc_\mathrm{i} = \Delta Vl_\mathrm{i} - Vw_\mathrm{i} \ \mathrm{cm}^3 }$
The pore radius at the end of the interval, $Rc$, is calculated using the Kelvin equation and the average pore radius for the interval is found as the weighted average of the starting and ending Kelvin radii for the interval.
$\displaystyle{ Rc_{avg_\mathrm{i}} = \frac{Rc_\mathrm{1} Rc_\mathrm{2} (Rc_\mathrm{1} + Rc_\mathrm{2})} {{Rc_\mathrm{1}}^2 + {Rc_\mathrm{2}}^2} \ \overset{\circ}{A} }$
The immediate thinning of films from pores exposed during the current interval (excluding pores emptied during prior intervals) is approximated using an inverse Kelvin equation to calculate the relative pressure at the average kelvin radius for the interval. This relative pressure is then used to find the film thickness at that point and the ending film thickness for the interval is subtracted to find the change in film thickness for pores exposed during the current interval. This quantity is then added to the interval's average Kelvin radius.
$\displaystyle{ Pr = \exp\left(\frac{-A}{Rc_{avg_\mathrm{i}}}\right) }$
$\displaystyle{ Rc_{avg_\mathrm{i}} \mathrel{+}= \sqrt{\frac{13.99}{0.034-\log_{10}{Pr}}} \ \overset{\circ}{A} }$
The pore length of all pores exposed during the current interval is found using the geometric relationship between the capillary volume and the pore radius.
$\displaystyle{ Lp_\mathrm{i} = \frac{Vc_\mathrm{i}}{\pi (Rc_{avg_\mathrm{i}})^2 \cdot 10^{-16}} \ \mathrm{cm} }$
It is important to note that the average pore radius does not represent the radius of the pore but instead represents the radius from the pore center to the film wall. This is done by increasing the average pore radius by $\Delta Tw$ during each subsequent interval. The total number of intervals in which new pores are found, $k$, is increased by one.
If $V_\mathrm{w}$ is greater than or equal to the volume desorbed during the current interval then desorption from walls only is occuring. In many instances $V_\mathrm{w}$ will be greater than $\Delta Vl$ and a thinning correction is done to calculate a new change in thickness for the interval, $\Delta Tw$, which wont overcompensate for the actual volume desorbed during the interval. This is done as follows:
The total surface area of films currently exposed is calculated
$\displaystyle{ Aw_\mathrm{i} = \sum ^k _{j=0} 2 \pi \cdot Rc_{\mathrm{avg_j}} \cdot Lp_\mathrm{j} \cdot 10^{-8} \ \mathrm{cm}^2 }$
where $10^{-8}$ converts angstroms to centimeters. The new change in film thickness is calculated as follows:
$\displaystyle{ \Delta Tw_\mathrm{i} = \frac{\Delta Vl_\mathrm{i}}{Aw_\mathrm{i}} \cdot 10^{8} \ \overset{\circ}{A} }$
At the end of each interval the average Kelvin radii and pore radii are adjusted to accurately represent the current distance from the pore center to the multilayer film. This is done by adding the change in film thickness observed during the inverval to both radii for all prior intervals. This is done to allow for the calculation of the quantity of gas that will desorb off of multilayers during the next interval.
$\displaystyle{ Rc_{avg_\mathrm{j}} \mathrel{+}= \Delta Tw }$
$\displaystyle{ Rc_\mathrm{j} \mathrel{+}= \Delta Tw }$
for all $j$
After the Faas, Standard, or Kruk-Jaroniec-Sayari method have been used the distribution is calculated using the pore length, $Lp$, and average pore radii, $Rp_{avg}$, data as follows:
$\displaystyle{ V_\mathrm{p} = \pi \cdot {Rp_{avg}}^2 \cdot Lp \cdot 10^{-16} \ \mathrm{cm}^3 }$
$\displaystyle{ A_\mathrm{p} = 2 \pi \cdot Rp_{avg} \cdot Lp \cdot 10^{-12} \ \mathrm{m}^2 }$
Note: For the Faas method the average pore radii is the average core radii, $Rc_{avg}$, which has been adjusted each interval to represent the total pore radii.
After the incremental quantities have been calculated a cumulative distribution can be made using pore radii within specified limits e.g. 2.0 to 2000.0 $\overset{\circ}{A}$.
In [1]:
%matplotlib inline
import bjh_ui
bjh_ui.bjh_display()