t-plot#

Module doc#

This module contains the t-plot calculation.

pygaps.characterisation.t_plots.t_plot(isotherm: PointIsotherm | ModelIsotherm, thickness_model: str | t.Callable[[float], float] = 'Harkins/Jura', branch: str = 'ads', t_limits: tuple[float, float] = None, verbose: bool = False)[source]#

Calculate surface area and pore volume using a t-plot.

The thickness_model parameter is a string which names the thickness equation which should be used. Alternatively, a user can implement their own thickness model, either as an Isotherm or a function which describes the adsorbed layer. In that case, instead of a string, pass the Isotherm object or the callable function as the thickness_model parameter. The t_limits specifies the upper and lower limits of the thickness section which should be taken for analysis.

Parameters:

  • isotherm (PointIsotherm, ModelIsotherm) -- The isotherm of which to calculate the t-plot parameters.

  • thickness_model (str, PointIsotherm, ModelIsotherm, callable, optional) -- Name of the thickness model to use. Defaults to the Harkins and Jura thickness curve.

  • branch ({'ads', 'des'}, optional) -- Branch of the isotherm to use. It defaults to adsorption.

  • t_limits (tuple[float, float], optional) -- Thickness range in which to perform the calculation.

  • verbose (bool, optional) -- Prints extra information and plots graphs of the calculation.

Returns:

dict -- A dictionary containing the t-plot curve, as well as a list of dictionaries with calculated parameters for each straight section. The basis of these results will be derived from the basis of the isotherm (per mass or per volume of adsorbent):

  • thickness curve (list) : Calculated thickness curve

  • results (list of dicts):

    • section (array) : the points of the plot chosen for the line

    • area (float) : calculated surface area, from the section parameters

    • adsorbed_volume (float) : the amount adsorbed in the pores as calculated per section

    • slope (float) : slope of the straight trendline fixed through the region

    • intercept (float) : intercept of the straight trendline through the region

    • corr_coef (float) : correlation coefficient of the linear region

Raises:

Notes

Description

The t-plot method [1] attempts to relate the adsorption on a material with an ideal curve which describes the thickness of the adsorbed layer on a surface. A plot is constructed, where the isotherm loading data is plotted versus thickness values obtained through the model.

It stands to reason that, in the case that the experimental adsorption curve follows the model, a straight line will be obtained with its intercept through the origin. However, since in most cases there are differences between adsorption in the pores and ideal surface adsorption, the t-plot will deviate and form features which can be analysed to describe the material characteristics.

  • a sharp vertical deviation will indicate condensation in a type of pore

  • a gradual slope will indicate adsorption on the wall of a particular pore

The slope of the linear section can be used to calculate the area where the adsorption is taking place. If it is of a linear region at the start of the curve, it will represent the total surface area of the material. If at the end of the curve, it will instead represent external surface area of the material. The formula to calculate the area is:

\[A = \frac{s M_m}{\rho_{l}}\]

where \(\rho_{l}\) is the liquid density of the adsorbate at experimental conditions

If the region selected is after a vertical deviation, the intercept of the line will no longer pass through the origin. This intercept be used to calculate the pore volume through the following equation:

\[V_{ads} = \frac{i M_m}{\rho_{l}}\]

Limitations

Since the t-plot method is observing the differences between the isotherm and an ideal adsorption curve, care must be taken to ensure that the model actually describes the thickness of a layer of adsorbate on the surface of the adsorbent. This is more difficult than it appears as no universal thickness curve exists. When selecting a thickness model, make sure that it is applicable to both the material and the adsorbate. Interactions at loadings that occur on the t-plot lower than the monolayer thickness do not have any physical meaning.

A second notable caveat is that the geometry of the pore will change the packing of the molecules, therefore the total thickness. Therefore, at smaller pores (<10 x the size of the guest molecule)

References

See also

pygaps.characterisation.t_plots.t_plot_raw

low level method

pygaps.characterisation.t_plots.t_plot_raw(loading: list[float], pressure: list[float], thickness_model: Callable[[float], float], liquid_density: float, adsorbate_molar_mass: float, t_limits: tuple[float, float] = None)[source]#

Calculate surface area and pore volume using a t-plot.

This is a 'bare-bones' function to calculate t-plot parameters which is designed as a low-level alternative to the main function. Designed for advanced use, its parameters have to be manually specified.

Parameters:

  • loading (list[float]) -- Amount adsorbed at the surface, mmol/material.

  • pressure (list[float]) -- Relative pressure corresponding to the loading.

  • thickness_model (callable[[float], float]) -- Function which which returns the thickness of the adsorbed layer at a pressure p.

  • liquid_density (float) -- Density of the adsorbate in the adsorbed state, in g/cm3.

  • adsorbate_molar_mass (float) -- Molar mass of the adsorbate, in g/mol.

  • t_limits (tuple[float, float], optional) -- Thickness range in which to perform the calculation.

Returns:

  • results (list) -- A list of dictionaries with the following components:

    • section (array) : the indices of points chosen for the line fit

    • area (float) : calculated surface area, from the section parameters

    • adsorbed_volume (float) : the amount adsorbed in the pores as calculated per section

    • slope (float) : slope of the straight trendline fixed through the region

    • intercept (float) : intercept of the straight trendline through the region

    • corr_coef (float) : correlation coefficient of the linear region

  • thickness_curve (array) -- The generated thickness curve at each point using the thickness model.

Available thickness models#

See Available thickness models.