1.4 Properties of Physisorption Materials
1.4.1 Physisorption Thermodynamics
Derivation of the surface thermodynamics for physisorption processes also begin with equivalence of the free energies of gaseous and bound hydrogen, given in Equation 1.3a and Equation 1.3b. How- ever, unlike absorption materials the hydrogen molecule does not lose its identity, and no chemical reaction occurs. Instead, the hydrogen molecule exists in both solid and gaseous states with differing potentials, with the solid-state potential modified by interaction with the substrate. This process
parallels phase transition models. Taking this approach, Equation 1.3a and 1.3b may be used di- rectly to arrive at pressure and temperature dependence of the system. For phase equilibrium of a single species, Equation 1.3b can be rewritten as
Vgas∂P−Sgas∂T = Vsolid∂P−Ssolid∂T
Where gas and solid subscripts describe hydrogen within those phases. Also, using the definition G
= H - TS in Equation 1.3a, the entropy can be related to the enthalpy
Hgas−TSgas= Hsolid−TSsolid
Combining these two equations yields the Clapeyron equation:
∂P
∂T
= Sgas−Ssolid
Vgas−Vsolid
=∆Hdes
T∆V (1.4)
where ∆Hdes= Hgas−Hsolid is the desorption enthalpy, analogous to the heat of vaporization in liquid-vapor equilibrium, and ∆V = Vgas−Vsolid and P is the saturation pressure corresponding to equilibrium. The Clapeyron equation exactly describes the thermodynamic system. However, it is difficult to work with, and the specific volume of hydrogen at the surface of the substrate is difficult to define. As with absorption materials and the van’t Hoff equation, assumptions are made to simplify the use of the Clapeyron equation to a more usable form.
While the volume of the adsorbed hydrogen cannot be directly assessed, the specific volume of hydrogen on the solid surface is assumed similar to the specific volume of solid hydrogen, and small relative to the specific volume of the gas. This assumption holds for pressures of less than 100 bar and temperatures greater than 77K.21At higher pressures and lower temperatures, hydrogen gas density becomes an appreciable percentage of the solid-state density, and validation of this assumption is required. Applying this assumption to the Clapeyron equation gives
∂P
∂T
= ∆Hdes
TVgas
To eliminate the volume of the gas and reduce the equation to a function of temperature and pressure only, and equation of state for hydrogen can be used, and typically the ideal gas equation is used.
Applying the ideal gas equation yields the Clausius-Clapeyron equation, which is typically used in the literature to evaluate the enthalpy of physisorption storage materials54, 55
∂ln(P)
∂T
= ∆Hdes RT2
The Clausius-Clapeyron equation is subject to the same restrictions for the assumption of ideal gas behavior as the van’t Hoff equation, discussed earlier. These assumptions hold well for the pressures specified in the USDOE targets. Higher pressure systems may need a fugacity correction to retain accuracy in non-ideal systems. Typically, ∆Hdesis a weak function of temperature but a relatively strong function of surface coverage. Because of this values of ∆Hdes are defined along isosteres, or constant values of surface coverage and are often defined in the literature as isosteric heat.56
An additional complication to the thermodynamic discussions arises in physisorption processes.57 The Clausius-Clapeyron equation is based on a liquid-vapor equilibrium model, in which the liquid and vapor are separate phases occupying their own distinct volumes. In the gas-adsorbate system, however, these boundaries are less distinct, and molecules from the vapor phase may penetrate or otherwise be included within the volume associated with the adsorbate phase due to the normal compression of the gas, as illustrated in Figure 1.7. This behavior leads to two descriptions of the adsorption: absolute adsorption, which treats the volumes as separate layers and includes some gas phase molecules within the adsorbate layer, and excess adsorption, which excludes these gas molecules from the adsorbate layer. The molar balance between absolute and excess quantities can be defined as
nexcess=nabs−ρVGibbs
Vapor-Liquid Equilibrium Gas-Adsorbate Equilibrium Substrate
Figure 1.7: Schematic diagram of the relationship between liquid-vapor equilibrium and gas- adsorbate equilibrium. The dashed line represents the Gibbs dividing line for both phases, with spheres below the line representing the absolute condensed phase. The liquid-vapor system has a well-defined interface, while the gas-adsorbate does not. Spheres in white below the line in the gas-adsorbate system are considered part of the gas phase density and are excluded from the excess adsorption calculation.
where ρis the density of the gas phase, andVGibbs is the volume of the adsorbate on the material surface defined by a second boundary layer called the Gibbs dividing surface. The location of this surface is defined somewhat arbitrarily as the endpoint of the extension of the potential well from the material surface. Defining the location of the Gibbs defining surface from experiment is currently beyond available imaging technologies, and the surface location is also likely a function of system conditions.
The separation of measurements into absolute and excess isotherms also result from these defini- tions. Idealized versions of these isotherms, based on the Langmuir adsorption isotherm, appear in Figure 1.8. Experimental measurements can only yield excess isotherms, as these represent the pres- sure drop from expected values. Absolute isotherms are only available from theoretical calculations, requiring a definition of the Gibbs dividing surface. Excess isosteric heats are typically reported for physisorption materials for both theoretical and experimental studies, to maintain consistency
Pressure
Absolute Uptake Excess Uptake Gas Density
Figure 1.8: Idealized adsorption curves showing the relationship between absolute adsorption, excess adsorption, and gas density and typical lineshapes for the three types. Curves are based on the Langmuir isotherm and the ideal gas equation of state.
within measurements. These values typically lie between 5 kJ/mol and 20 kJ/mol, representing the relatively weak binding in these materials.58