Energy of Physical Adsorption
The red parts represent the positive charges, mainly given by the nuclei of the atoms. The dashed blue lines are the negative electron positions without vdW forces, and the solid blue lines are with vdW forces. a) London dispersion force: vdW interaction between non-polarized particles (London, 1937; Leite et al., 2012).
LJ Expression of vdW Potential
The blue parts represent the negative charges, mostly given by the electrons of the atoms. Suppose the electron cloud in the atom behaves like a harmonic oscillator around the nucleus, and the frequency is𝜔. The initial Hamiltonian without the vdW force is given by: .
DFT Algorithms for vdW Potential
- DFT-D and DFT-D2
- DFT-D3 and Tkachenco-Scheffler (TS)
- Many-Body Dispersion (MBD)
According to the results in fig. 4.1, the energy change is small along the carbon surfaces. According to the results in fig. 4.3, the energy change is large vertically in relation to the carbon surfaces.
Thermodynamic Analysis of Physisorption
Clausius-Clapeyron Equation
It derives the enthalpy of phase change with the following equation: whereℎ is the enthalpy and 𝑣 is the volume. For the phase transition between the gas and a condensed phase, the adsorbed phase, for example, the volume is considered very small compared to the gas phase.
Henry’s Law
At a phase transition, the difference between the Gibbs free energy of the two phases is zero. 2.6), the Clapeyron equation presented in Eq. Assuming that the volume is negligible for the adsorbed phase, its internal energy is equal to its enthalpy.
Langmuir Model
Compared to Eqn (2.13), with this derivation,𝐾. Here, 𝐾L is proportional to𝑇−5/2if𝜖 is temperature independent. During the adsorption, the chemical potential 𝜇 is equal to the gas phase chemical potential shown in Eq.
Surface Dynamics
- Variants of Langmuir Model
However, it is not impossible to add dynamics and preserve the form of the Langmuir model shown in Eq. These results are exactly the same as the Cartesian coordinate system results, such as the quadratic potential 𝜀. A fundamental part of the Langmuir model is the configurational entropy in Eq. 2.32), and assumes that the particles and adsorbed sites are identical.
Compared to the total area 𝐴, 𝑁 𝐷𝑥𝐷𝑦 is the total area of the sites, which is smaller. The difference is that the calculation of the partition function for individual particles Eq. must be following.
Multi-Site Adsorption
The difference is that the calculation of the partition function for individual particles Eq. must be following. Another equivalent method to obtain the isosteric adsorption energy is to calculate the partition function from Eq. 2.47), for multiple sites, the partition function is:. 2.56). For the multi-site Langmuir model, it is assumed that the total number of sites is 𝑁 and the fractions of each type of site are 𝛼.
Slit-Pore Model and NLDFT Method
For example, for the potential of 8.4 Å shown in Fig. 4.6), the flat end of the potential contributes significantly to the partition function. For each potential shown in Fig. 4.4), the first 400 eigenenergies are calculated and used to calculate the partition function. The corresponding internal energies of the partition functions given by Eq. c) Heat capacities given by the derivatives of the results in (b).
Also according to fig. (4.7), the biggest changes occur for the energetically less favorable layer spacings. The VdW potential can be calculated with the previously used constants and the results are shown in Fig. B.1), the bulk vdW potential significantly underestimated the magnitude of the vdW potential.
Experimental Observation of the Temperature Dependence of
Experimental Setup
Two types of porous carbon were used as the adsorbent: MSC-30 from Kansai Coke & Chemicals Company Ltd. The TriStar II 3020 system from Micromeritics was used to collect argon adsorption isotherms for the measurement of the pore size and surface area distribution and was used to collect krypton adsorption isotherms for the adsorption energy analysis. The methanol in the refrigerator passes through the condensing coil in the bath bottle and supplies heat to the methanol in the bath.
The pore size distributions were given by the NLDFT analysis in the Micromeritics system with the “Carbon-Ar-87 K, 2D-NLDFT. A methanol bath kept the temperature stable during the data collection process and the temperature fluctuations were within 0.5 K.
Pore-Size Measurement and Isotherms
Data Analysis
- Henry’s Law Analysis
- Langmuir Model Analysis
Although the prefactor forms given by Fig (3.9) show a good match with the results in Fig (3.8), the match with other forms of prefactors is also practiced, and the results are shown in Fig. The corresponding internal energies of the partition functions given by Eq. c) The heat capacities given from the derivatives of the results in (b). change dominates, and the partition function decreases. The results and corresponding energies and heat capacities are shown in Fig. Compared to the results in Fig. 4.8) shown, the partition functions and the internal energies are almost exactly the same in Fig.
The distances between the layers on the same side of krypton are 𝐷. Similar to the previous calculation, the vdW potential along the dashed line is shown in Figure 4.4), the shapes of the potentials are almost similar. For the case of 1 layer, the vdW potential is 𝜀. The calculation of 2 layers and 100 layers is also shown.
Computational Calculation of Adsorbed Phase Internal Energies . 33
Slit-Pore Model
- Potentials of the Pores
- Thermodynamic Analysis of Internal Energy of the Pores
- Classical Versus Quantum
- Two Layers Versus Four Layers
A 1 K temperature step is used for𝛿𝑇 and𝛿𝑞𝑧. The corresponding heat capacities are given from the derivatives of the internal energies, and the results are shown in Fig. 4.8a shown), for small layer spacings the partition functions are small because the potentials increase rapidly from below, as the 6.4 Å potential shown in Fig. For example, the 2nd energy state belongs to the wave function inside the left potential well and has an energy of 5.52 meV. The 3rd energy state belongs to the wave function inside the right potential well and has an energy of 5.61 meV. The 4th energy state belongs to the wave function inside the left potential well and has an energy of 9.11 meV. The energy difference between the 2nd and 3rd energy states is much lower than that of jumping from the 2nd to the 4th state. For a further comparison between the classical and quantum theories, the calculations were performed with a larger temperature range from 10 K to 410 K. The results are shown in Fig.
Therefore, some lines similar to the dashed line in Fig. 4.3) are chosen to calculate the 𝜀 value. The choice of lines is shown in Fig. 4.12), and the corresponding vdW potentials are shown in Fig. The vdW potentials in Figs. 4.15b) are close to the sum of the potentials of two double layers with layer distances of 𝐷 and 𝐷+ 2𝐷. Take as an example the layer distance𝐷 =6.2 Å, 𝐷+2𝐷. The 12.9 Å layer distance potential is weak in the middle part between the two inner layers, where the 6.2 Å layer distance has the strongest potential.
Thermal Vibration
- Simulation with TDEP Method
- Monte Carlo Calculation
- Impact of vdW Potentials on Thermal Vibration
This is due to the change in the location of the potential minima away from the center between carbon atoms. Using the Einstein solid model and referring to the "ZO" mode frequency in Fig. 4.17), the carbon atoms can initially be considered as a harmonic oscillator with ℏ𝜔 = 100 meV. The ground state probability function is a Gaussian function with a height of 9.56 Å−1 and a standard deviation of 0.022 Å. Asℏ𝜔is large energy compared to room temperature 𝑘. B𝑇, the thermal averaged probability functions have no significant deviation from the ground state probability function. The third chapter of this work investigates the temperature dependence of krypton adsorption energy and the heat capacity of the adsorbed phase.
Using the zero-coverage Henry's constants, the rate of change of adsorption energy per atom is 4.35𝑘. By incorporating higher coverage results into the Langmuir model, the rate of change of adsorption energy per atom is 2.10𝑘. The DFT-based computational study with the slit-pore model showed the vdW potential of different-sized pores. The distribution of pore sizes plays a decisive role in the temperature dependence of the overall adsorption energy.
Discussion on the Temperature Dependence of Adsorption Energy 64
Other Possible Effects
The gas-gas interaction can be significant if the initial gas adsorption takes place in a process similar to the nucleation, where the other gas atoms also attract the newly adsorbed gas atoms. Then the gas-gas vdW potential can enhance the adsorption by the surface and provide more energy states for Boltzmann scattering. This number is much smaller than the krypton adsorption energies in the slit pores, and the formation of the 2D crystal is almost impossible.
Compared to the results in Fig. 5.3), these small pores are extremely favorable for the gas. temperature, the gas molecules in these small pores are transferred to larger pores. The solid lines are the surface area distribution shown in fig. b) Isosteric adsorption energies with the new modified surface area distribution are shown with the dashed lines.
Future Outlook
The changes in these internal degrees of freedom can lead to extra shifts in the temperature dependence of the adsorption energies. A thermodynamic analysis of the 2D ideal gas and the slit pore models highlighted the role of vdW potentials in the adsorption energy and isotherm fitting methods, including Henry's law and the Langmuir model. Meanwhile, at the edge of the slit pore, adsorbate molecules also feel the 𝑥 − 𝑦 direction vdW force.
In thermodynamics, the work done by the vdW edge force does not introduce new terms in the vdW potential. advertising was ignored in most previous discussions. The ideal chemical potential of a gas is: gas is the Gibbs energy of the gas phase. gas, It is a function of 𝑃and 𝑇. Meanwhile, the chemical potential of the adsorbed phase is:
Comparison to Infinite Layers
Therefore, it can be concluded that for the multilayer configurations having two layers on one side is already a good approximation.
Comparison to Continuum Adsorbent
A consistent and accurate ab initio density functional dispersion correction (DFT-D) parameterization for 94 H-Pu elements. Significant temperature dependence of isosteric heats of gas adsorption in zeolites demonstrated by experiments and molecular simulations. Failure of density functional dispersion correction in metallic systems and its possible solution using modified many-body dispersion correction”.
Perspective: Progress and challenges in the treatment of van der Waals dispersion forces in density functional theory”. Measurement and prediction of the heat of adsorption and equilibrium concentration of CO2 on zeolite 13X”.
