**6.3 Contracting volume model**

$$\frac{\mathbf{d}\_{\mathbf{r}}}{\mathbf{d}\_{\mathbf{t}}} = -\mathbf{k}\_{\text{int}} \tag{5}$$

$$(\mathbf{1} - (\mathbf{1} - \varepsilon)^\ddagger = \frac{k\_{int}}{r\_0} t = kt \tag{6}$$

Where kint= rate constant (interface controlled reaction), k = generalized rate constant.

### **6.4 Jander model**

$$\frac{\rho d\_r}{d\_t} = \frac{D \Delta \mathbf{C}}{r\_0 - r} \tag{7}$$

$$\left[\mathbf{1} - (\mathbf{1} - \boldsymbol{\varepsilon})^{\frac{1}{2}}\right]^2 = \frac{2D\Delta C}{r\_0^2 \rho} t = kt \tag{8}$$

Where *ΔC* = concentration difference, D = diffusion constant, *ρ* = density.

#### **6.5 Ginstling-Brounshtein model**

$$\frac{\rho d\_r}{d\_t} = \frac{D \Delta \mathcal{C}}{r \ln\left(\frac{r\_0}{r}\right)}\tag{9}$$

$$\ln\left(1-\varepsilon\right)\ln\left(1-\varepsilon\right)+\varepsilon = \frac{4\text{D\'AC}}{r\_0^2\rho}t = kt \tag{10}$$

For two dimensional cylinder:

$$\frac{\rho d\_r}{d\_t} = \frac{D \Delta \mathbf{C} r\_0}{(r\_0 - r)r} \tag{11}$$

$$\mathbf{1} - \frac{2}{3}\boldsymbol{\varepsilon} - (\mathbf{1} - \boldsymbol{\varepsilon})^\dagger = \frac{2\mathbf{D}\Delta\mathbf{C}}{r\_0^2 \rho}\mathbf{t} = \mathbf{k}\mathbf{t} \tag{12}$$

*For three dimensional cylinder:*

#### **6.6 Valensi-Carter model**

$$\frac{\rho \mathbf{d\_r}}{\mathbf{d\_t}} = \frac{\mathbf{D} \Delta \mathbf{C}}{\mathbf{r} - \frac{\mathbf{r^2}}{\left(\mathbf{Zr\_0^3} + \mathbf{r^3}(1 - \mathbf{z})\right)^{\frac{1}{3}}}} \tag{13}$$

$$\frac{z - (\mathbf{1} + (z - \mathbf{1})\varepsilon)^{\frac{2}{5}} - (z - \mathbf{1})(\mathbf{1} - \varepsilon)^{\frac{2}{5}}}{z - \mathbf{1}} = \frac{2\mathbf{D}\Delta\mathbf{C}}{r\_0^2 \rho} t = kt \tag{14}$$

Where z = volume ratio of product to reactant.

The *contracting volume model* assumes that rate of hydrogenation is regulated by the interface process. This model is deemed as the simplest method in geometrical contraction volume models because no other assumptions are made. Through this method the dimensionality and a generalized constant can be obtained through fitting a simplified isothermal curve [58, 60]. Bösenberg et al. proved that LiH-MgB2 with transition metal deposits such as Titanium and Vanadium follow the CV model very well because it is interface control [65].

The *Jander Model* however is readily used for diffusion regulated reactions. The two important assumptions made in this model are: 1. the volume of hydrogen storage materials remains constant before and after absorption and desorption reactions 2. Two and three dimensional diffusion are deemed as one meaning the interface area remains constant for diffusion [59, 66]. However these assumptions are not always applicable for hydrogen storage materials because the volume of some materials increases during the uptake of hydrogen and decrease with the

release of it. In contrast Shao et al. found that nanocrystalline Mg doped with Ti under 1Mpa of H2 follow this model very well [67].

According to Ginstling and Crank the Ginstling-Brounshtein model develops the Jander Model using Fick's Law for radial diffusion for 2D and 3D spheres [61, 68]. Although intricate; this method is regarded as a more accurate model due to its consideration of the variance in interface area due to diffusion. Chaudhary et al. reported that MgH2-Si synthesized by ball milling, cryomilling and ultrasonicating produces desorption curves which follow this model well [69]. The Valensi-Carter model is an extension of the G-B model which regards the change in volume in hydrogen storage materials during adsorption and desorption. This model is reportedly the most accurate in geometrical contraction model however it has not yet been applied to hydrogen storage materials [70–72].

#### **6.7 Nucleation-growth impingement models**

The Nucleation growth impingement models are generally described as the Johnson-Mehl-Avrami-Kolomogorov (JMAK) models. The JMAK models define hydrogenation and dehydrogenation reactions as three synchronized processes: nucleation, growth and impingement ([73]; Melvin [72, 74–78]). **Figure 6** is a diagram showing the three processes:

This model is founded on the basis of; 1. Enhancing nucleation, growth and impingement 2. Solving analytical problems [78]. This model is given as:

$$\varepsilon = F\left[\int\_0^t I(r)V(r)d\_r\right] \tag{15}$$

Where I = nucleation module, r = nucleation rate, F = impingement module defining the relationship between real reaction fraction and extended reaction fraction, V = Growth module [79].

A variation of this equation, classical JMAK (C-JMAK), which takes into consideration interface controlled growth and diffusion controlled is given by:

$$W\_{(r)} = \left[ G\_0 \int\_r^t e \exp\left(\frac{-\Delta E\_\mathrm{g}}{RT}\right) d\eta \right]^\frac{d}{m} \tag{16}$$

Where *ΔEg*= activation energy for growth, *G*0= intrinsic growth rate, m = growth mode parameter, *<sup>d</sup> <sup>m</sup>* = growth inde.

**Figure 6.** *Nucleation, growth and impingement (Adapted from [58]).*

The C-JMAK model defines absorption and desorption of nucleation, growth and impingement very well. Nucleation in this regard refers to site saturation, growth mode to interface and diffusion controlled sites and impingement to randomly scattered nuclei isotropic growth [80]. The Avrami exponent of the desorption reaction of NaAlH4 doped with Titanium was determined using this model and was found to be 3. This proved that the three dimensional interface process is the rate controlling step. Pang et al. however discovered that the actual nucleation modes propagate continuous nucleation and are not restricted to linear continuous nucleation and site saturation an assumption which this model is founded on [78, 81]. This inhibits the practical application of this model for progressive hydrogen storage applications.

Analysis methods are used to obtain kinetic mechanism by determining and comparing the best fit models and kinetic parameters that go along with it. The two most used analysis methods is isothermal fitting and non-isothermal fitting [82]. However based on literature reviewed understanding the kinetic mechanisms for hydrogen storage materials can be difficult because the analysis have downfalls which may result in misunderstanding of the kinetic mechanisms entirely [58].

## **7. Conclusion**

Nanocomposites have proven to be materials of the 21st century with their flexibility and wide range application. These materials bridge the gap between heterogenous and homogenous catalysis through the combination of different nanomaterials. In this chapter, graphene based nanocomposites were reviewed and discussed intensely for hydrogen storage applications. The kinetic models of hydrogen storage materials was also reviewed. More work needs to be done in terms of the practical application of these material for energy storage, because to date, no material meets the standards for practical application in energy storage.
