Appendix VI

Hydrogen sorption, isotherms and van´t Hoff plots.

As discussed in Chapter II on storage of hydrogen, a large number of elements will react with hydrogen according to the formula

 

Me  +  x/2 H₂  =   MeH_x

 

We also discussed the dissociation on the metal surface and the diffusion of hydrogen atoms through the bulk metal even at room temperature to form a metal-hydride solid solution commonly referred to as the a–phase. We also discussed the hydrogen in many cases occupying interstitial sites, tetrahedral or octachedral, all of different crystal symmetries, face centred cubic, body centred cubic or hexagonal close packing structures. The occupation of hydrogen in these lattices most often leads to lattice expansion.

 

The hydrogen concentration in the a–phase, which we will call c_H can be written as a function of pressure and is found to obey Sievert´s law

 

c_H  =  k p ^!/2 

 

Here, k is a temperature dependent constant and p is the hydrogen pressure. As the hydrogen pressure is increased, hydrogen is absorbed and saturation occurs and formation of the metal hydride MeH_x starts.  In accordance with Gibbs´ phase rule, the conversion of the saturated solution phase to hydride takes place at a constant pressure. The Gibbs´phase rule now states that:

 

MeH_y  +  (x-y)/2  H₂   =  MeH_x

 

The plateau in pressure is temperature dependent and measures the equilibrium dissociation pressure of the hydride and is in many ways an indicator of the stability of the hydride. Figure AVI1 shows the pressure/composition isotherm in LaNi₅, a popular and well researched alloy. The alpha phase is the metal hydrogen solid solution. Then, as the terminal solubility of the alpha phase is reached, the precipitation of the beta-phase starts at the plateau pressure. At this point in the lifetime of a hydride, often there will be multiple plateaus possible. In fact such multiple plateaus are observed in composite materials consisting of tow hydride forming metals or alloys. The equilibrium dissociation pressure is a key property of a hydride storage material. This period is called the “α+β” phase and if anything, represents a significant crystal structure change in the lattice. Isotherms often bear other, usually undesirable characteristics like hysteresis and plateau slope.

 

The hydriding/dehydriding reaction that makes intermetallic compounds, IMCs interesting is highly endo/exothermic.  Therefore the temperature and pressure of the operating environment are major concerns to hydride chemistry. In the literature (e.g. J. Kapischke, J. Hapke 1998), a pressure-composition-temperature (pcT) isotherm is the most often used method to demonstrate these conditions as well as quantify the hydrogen sorption. 

 

Fig. AVI 1– PCT isotherm for a LaNi₅ derived storage alloy.  “alpha”, “beta” and “alpha+beta” regions refer to distinct phases of hydride behaviour (Jain et al 2002)

 

 

Fig AVI  2 - Hydride hysteresis demonstrated via a PCT isotherm (Sandrock 1999)

 

Fig. AVI 3 – Normal and atypical PCT isotherm forms (Sandrock 1995)

 

Figure AVI 2. gives an idealized version of hysteresis, the respective discrepancy between the plateau pressures during absorption and de-sorption reactions.  In many cases the plateau pressure during absorption is significantly higher (recall, the logarithmic scales of most PCT plots allow both cycles to be graphed together) then plateau pressure under desorption.  Hysteresis is usually measured when applicable and reported in literature

 

The formula for calculating hysteresis from pressure data is:

P_a is pressure of absorption plateau while P_d is the pressure of desorption plateau.  Hysteresis is a factor of energy loss through the physical transition to hydrided material.  The hydriding process is a rather abrupt change to a different crystal structure which results in lattice defects. These defects hinder the propagation of phonons through the material and ultimately result in energy loss along the way.  Some materials are more resistant to forming defects during the structure change and therefore tend to measure lower hysteresis values. Lastly, hysteresis is largely an unwanted characteristic of metal hydrides as it tends to upset the conditions the hydride is expected to function at, although it is something that can be planned for and finally reversed through reheating the material to loosen out the defects.

 

The second undesirable in pcT measurements is an exaggerated plateau slope. Most controlled (well formulated, well analyzed) hydrides have one type of interstitial site which corresponds to a given bond energy.  This then relates to the desired, flat(ter) plateau as the hydrogen only has one place to go at a single energy value. However, hydrides can certainly have many different interstitial sites with different bond strengths. This is often the case in materials that are made from rare-earth alloys such as the so-called mischmetal, or have been doped with metals to alter other properties.  Hydrides can also lose their uniform distribution and concentrate certain components in domains.  This usually happens after repeated cycling but can occur during preparation as well. This can lead to disproportionate reactions and therefore alter the plateau.

 

Illustrated in Figure AVI 3, are variations on the single plateau type that are often seen.  Some materials exhibit two or more separate plateaus while others do not show a plateau at all. Multiple plateaus usually signify that a material contains multiple distinct interstitial sites that are filled at considerably different energy values. Also multiple plateaus can be a result of several successive crystal structures forming during sorption. Lack of plateau usually suggests one of two things, either the material is above its critical temperature, in which the hydride phase does not occur, or the material is not a true hydride and the hydrogen simply goes into solution and no bonding or structure change occurs regardless of temperature or pressure. 

 

A very popular method of analyzing hydride data is through usage of a van’t Hoff plot. A van’t Hoff plot or van’t Hoff diagram is constructed by gathering pressure readings at the isotherm plateau centre for various temperatures. The collected readings are plotted, 1/T vs. P (P on a logarithmic scale), typically producing a straight line of negative slope (technically exponential decay but appears straight due to log scale).  Van’t Hoff plots are very useful towards better understanding the thermodynamics of a material, considering that most IMC systems follow the van’t Hoff equation:

 

 

The origins of the van’t Hoff relation and van’t Hoff diagram are provided as follows. Due to their high prevalence in metal hydride literature it is valuable to understand their basis. 

 

A characteristic value is K, the equilibrium constant which is derived from the partial pressures of the gas reaction i.e.

 

Assuming that A and B are reactants and C and D are the products of the chemical equation. K changes appropriately depending on the layout of the chemical reaction (i.e. A + B → C + D). 

 

Derived from:

 

 

 

This is another form of the van’t Hoff equation, substituting K in for P and taking the derivative of the previous with respect to temperature. This equation relates plateau pressure to temperature. R is the gas constant, ΔH is heat of formation and ΔS is entropy. Heat of formation (or enthalpy) and entropy are important values that can be calculated from the van’t Hoff plot with help from the equation. By extrapolating 1/T=0 one can determine the entropy of the material (although this is roughly constant for most hydrides, around -0.13 KJ/mole) and then subsequently the enthalpy (heat of formation). Figure AVI 4. shows a collection of LaNi₅ PCT isotherms plotted on the same graph at several temperatures (Sandrock 1995). Figure III 5. shows a van’t Hoff plot with data for many AB₅ type hydrides. The superimposed box on the van’t Hoff plot represents the room temperature operating range, 0-100 C and 1-10 atm.

 

 

 

 

 

 

 

 

 

 

Figure AVI4  Several temperatures of LaNi₅ PCT isotherms plotted on the same graph (Sandrock 1995)

 

Fig. AVI 5. - Van’t Hoff diagram showing the operating range of AB₅ type hydrides.  The central box confines temperatures 0-100 °C and pressures 1-10 bar – the geothermal window (Schaer and Sigfusson 2006, based on Sandrock 1999).