A Robust Framework for Generating Adsorption Isotherms to Screen Materials for Carbon Capture

To rank the performance of materials for a given carbon capture process, we rely on pure component isotherms from which we predict the mixture isotherms. For screening a large number of materials, we also increasingly rely on isotherms predicted from molecular simulations. In particular, for such screening studies, it is important that the procedures to generate the data are accurate, reliable, and robust. In this work, we develop an efficient and automated workflow for a meticulous sampling of pure component isotherms. The workflow was tested on a set of metal–organic frameworks (MOFs) and proved to be reliable given different guest molecules. We show that the coupling of our workflow with the Clausius–Clapeyron relation saves CPU time, yet enables us to accurately predict pure component isotherms at the temperatures of interest, starting from a reference isotherm at a given temperature. We also show that one can accurately predict the CO2 and N2 mixture isotherms using ideal adsorbed solution theory (IAST). In particular, we show that IAST is a more reliable numerical tool to predict binary adsorption uptakes for a range of pressures, temperatures, and compositions, as it does not rely on the fitting of experimental data, which typically needs to be done with analytical models such as dual-site Langmuir (DSL). This makes IAST a more suitable and general technique to bridge the gap between adsorption (raw) data and process modeling. To demonstrate this point, we show that the ranking of materials, for a standard three-step temperature swing adsorption (TSA) process, can be significantly different depending on the thermodynamic method used to predict binary adsorption data. We show that, for the design of processes that capture CO2 from low concentration (0.4%) streams, the commonly used methodology to predict mixture isotherms incorrectly assigns up to 33% of the materials as top-performing.

The Clausius-Clapeyron relation is used in this study to predict pure component isotherms at different temperatures given an isotherm at a reference temperature. Figure S1 helps understand how the relation is applied in practice. The workflow developed to compute pure component isotherms ensures that the reference CO 2 and N 2 isotherms are thermodynamically consistent, i.e., the uptake is a monotonically increasing function of pressure: Where the numerator and denominator are both positive.
Consequently, the extrapolated isotherms, using the Clausius-Clapeyron relation, must also be thermodynamically consistent, as they are derived from the reference isotherms.
Given that Clausius-Clapeyron shifts the pressure values, keeping the uptake constant, it follows from equation S1 that, for extrapolated isotherms to be monotonically increasing, S-3 the relation described in equation S2 must hold: Substituting equation 13 into equation S2 and rearranging terms gives the following relation: In the grand canonical ensemble, the isosteric heat of adsorption is calculated from fluctuations in the energy of the system and the number of particles. S1 Therefore, at higher density, i.e., higher isotherm pressure, very long simulations are required to accurately calculate the isosteric heat of adsorption. Therefore, the isosteric heat of adsorption data is computed with high uncertainties.
The noisy nature of the isosteric heat data makes it almost impossible to meet the criterion described in equation S3. A trivial solution would be to use the average value of the isosteric heat of adsorption. However, reducing the data to a single value fails to include the effect of density on the heat of adsorption. In an attempt to reduce the noise in the data, we implemented fitting (linear regression and cubic spline approximation) and smoothing techniques (cumulative and exponential moving averages). They all failed to produce thermodynamically consistent isotherms when extrapolating to different temperatures.
Therefore, we implemented an exponential filtering technique to reduce the noise in the data and meet the criterion. The technique can be summarized in the following relation: S-4 Where α is the specific damping factor that is unique to each structure, an optimum value of α exists for each temperature. The lower the value of α, the higher the range of temperature that can be covered by the Clausius-Clapeyron relation. Therefore, α was computed in such a way to include the temperatures of interest up to 125°C.
In this study, the Clausius-Clapeyron relation is used to predict the pure component isotherms for CO 2 and N 2 at 50, 75, 100, and 125°C for each of the 50 sub-selected structures given a reference temperature of 25°C. Then, the extrapolated isotherms were used to compute the binary adsorption data for a range of CO 2 partial pressures (0.0004 to 0.95) using IAST. These calculations were compared to the values obtained given the pure component isotherms generated using the workflow at the same temperatures. Figure S2 and Figure S3 show the error profiles for CO 2 and N 2 respectively. The percent errors were obtained using the following relation: Where i can be either CO 2 or N 2 S-5 In figures S2 and S3, the percent error distribution of the predicted binary CO 2 and N 2 uptakes, respectively, for all temperature differences display a peak between 1% and 3%.
This shows that using Clausius-Clapeyron to extrapolate to different temperatures is accurate enough for most of the binary CO 2 and N 2 predictions. However, as the temperature difference increases, the error distribution flattens, and higher errors are attained. For example, in the extreme case of 125°C, very few predictions are calculated with 25% error.
One way to overcome this issue is to use another reference isotherm at a relatively higher temperature, 75°C for example, to minimize the temperature difference when extrapolating S-7 at high temperatures.

Dual-Site Langmuir Fitting
In the following section, we discuss the shortcomings of DSL fitting affecting mainly the fits of pure component CO 2 isotherms, given that CO 2 is the strongly adsorbed component.
Also, we show that using IAST to predict binary adsorption data is a more reliable tool than extended DSL for the full set of structures.
For instance, a different set of model parameters can be obtained by using different initial guesses. Figure S4 shows, for four structures, that in the high-pressure region, where the majority of data points lie, the different sets of parameters give similar fits. However, in the vicinity of the Henry regime, different initial guesses give very different predictions. It is also evident from figure S4 that for a given set of initial guesses corresponding to the solid lines, DSL fits describe better the pure component data. Those "educated guesses" follow some simple rules: (1) the saturation loadings, q sat1 and q sat2 in equation 16, are estimated using the highest uptake value for CO 2 and (2) Figure S4: DSL fits for four different structures at 25, 50 and 75°C, corresponding to blue, red, and green, respectively. Data points, dotted lines, and solid lines correspond to pure component data, DSL fits using random guesses, and DSL fits using "educated guesses," respectively.
In addition, fitting can be done either by putting more weight on data points at low pressure to force the correct Henry coefficient or by considering all data points to have the same weight. Figure S5) shows that forcing the correct Henry coefficient affects the accuracy of the fits at high pressure. It also shows that the accuracy of the fits at low pressure is S-9 compromised when we consider all data points to have the same weight. Finally, we predicted binary adsorption data for CO 2 and N 2 using IAST and extended DSL with and without forcing the correct Henry coefficient, given the structures for which the fitting went well (400 out of 500). Those predictions were compared to binary GCMC adsorption data at the adsorption conditions of temperature and CO 2 partial pressure for S-12 three different case studies: coal-fired power plant (CF-PP), natural gas-fired power plant (NG-PP), and confined spaces (CS). Refer to table S1 for the gas composition and adsorption temperature. Figure S8 shows that for the three cases and both guest molecules, IAST predictions are more reliable and accurate than using extended DSL.  Figure S8: CO 2 (left) and N 2 (right) binary adsorption using IAST (blue) and extended DSL with and without forcing the correct Henry coefficient (green and red respectively) for coal-fired power plant (a), natural gas-fired power plant (b), and confined spaces (c).

S-14
The TSA model implemented in this study is built in python around some basic assumptions: • The inlet stream is a dry binary mixture of CO 2 and N 2 and is considered as an ideal gas. It also goes without saying that both components are adsorbable.
• Equilibrium between the gas and adsorbed phases is reached instantaneously because mass transfer resistance is negligible.
• Heating and cooling are assumed to be carried out with heating fluids flowing along the column with a homogeneous temperature distribution. Also, the heat transfer resistance is negligible.
• The radial gradients, thermal dispersion, and axial mixing are negligible.
• Pressure drop across the bed is negligible.
In this work, we consider capturing CO 2 from three different sources at ambient pressure  Table S1 summarizes the process parameters.
Using the TSA process, one can rank materials based on different key performance indicators (KP Is) such as purity, working capacity (W C), and thermal energy (E thermal ) requirements. Those KPIs are defined as: Here N CO 2 ,product and N product are the amount of CO 2 and the total amount leaving the column in step (ii) (refer to figure 7), respectively and m s is the mass of the solid. S-15 Here, ∆H sens and ∆H ads are the process total sensible heat and heat of adsorption (kJ mol −1 ) requirements per TSA cycle. ∆H j sens and ∆H j ads are the sensible heat and heat of adsorption at step j. T is the temperature of the column, c p denotes the specific heat capacity, N i denotes the amount of species i, ∆H i refers to the heat of adsorption of species i, and ∆N i is the difference in the amount of species i between step j and step (j − 1). The subscripts col, ads, and s refer to the column, the adsorbed phase, and the solid, respectively.

Materials Ranking
In figure 8, we showed the materials ranking for the three case studies based on the purity of the CO 2 rich product stream. In figures S9 and S10, we show the materials ranking using two different KPIs, working capacity (mol kg −1 ) and specific heat requirements (MJ mol −1 ) respectively.  Figure S10: Specific heat requirements (MJ mol −1 ) of the output stream for a coal-fired power plant (CF-PP), natural gas-fired power plant (NG-PP), and confined spaces (CS). In these figures we compare IAST with DSL and DSL Henry , see also the caption to figure 8