**3. Proposed exfoliation and principle**

#### **3.1 Exfoliation model**

When two materials (atoms, material made of atoms) are brought together, we can distinguish two types of energy: adhesive and interactive energy. The adhesive energy describes the forces when two materials are in contact (when they adhere) while the interaction energy describes the forces exerted by each material on the other. The interactive energy can be described by the Lennard-Jones (LJ) potential in **Figure 4**. When we look at the interactive energy curve, there is a minimum which is known as the binding energy *ε:* It is obtained at equilibrium. The broader and deeper it is, the stronger is the bond between the materials (mostly for noncovalent bonds). *σ* is the distance at which the potential is 0. It can be taken as the repulsive core.

The working principle of our exfoliation model is shown in the next figure and can be viewed here (**Video 1**: https://drive.google.com/file/d/1eE48rBa0de\_vzHd\_zfxd QN940se1UqBj/view?usp=drivesdk). It is based on the interaction energy between three materials: graphite (HOPG), graphene (the bottom layer of HOPG) and a substrate. As we can see in the **Figure 5**, we consider HOPG as graphite and graphene. So our goal is to study the interaction between these materials.

**Figure 4.** *Lennard Jones potential curve.*

**Figure 5.**

*Exfoliation of graphene based on interaction energy.*

Here is a small description of the working principle.


We simulated first the interaction energy (binding energy) between graphene and graphite. Then we determine the interaction energy between graphene and a couple of materials to see what materials can exfoliate and finally a study of the results to see how we can make the process more effective.

*Graphene Exfoliation from HOPG Using the Difference in Binding Energy between Graphite… DOI: http://dx.doi.org/10.5772/intechopen.107142*

#### **3.2 Calculation of the interaction energy using the LJ potential**

To calculate the interaction energy between materials in this project, we used the standard LJ 6–12 potential expressed as:

$$\mathbf{V}(\mathbf{r}) = 4\mathbf{e} \left[ \left( \frac{\sigma}{\mathbf{r}} \right)^{12} - \left( \frac{\sigma}{\mathbf{r}} \right)^{6} \right] \tag{1}$$

where *V r*ð Þ is the interaction energy between two atoms. The first term of the potential *<sup>σ</sup> r* � �<sup>12</sup> is the repulsive term, and the second term *<sup>σ</sup> r* � �<sup>6</sup> is the attractive term. *ε and σ* are the minimum value of the potential, and the distance at which the potential is zero respectively. *r* is the distance between the atoms interacting. We opted for this potential because it is easy to manipulate, good for non-covalent bonds like VDW forces and more used for qualitative purpose.

#### *3.2.1 Potential in terms of material*

The potential energy is the sum of a long range attractive contribution and short range repulsion as we saw in the potential curve. One thing about this LJ potential is that the bond strength increase with the atomic number/atomic mass (molecular weight). This is because atoms with large atomic numbers have high number of electrons that will then increase the VDW forces to get stronger. Second, the shape of the molecules also plays a role in the VDW bond strength. The shape will dictate the spread of the electrons, whether they are widely distributed or centered. For example, molecules of the same molecular weight have greater bond strength if they have larger aspect ratio shapes.

These understanding helps us in the designing or the choice of the materials that we are going to use to exfoliate graphene from graphite. From the previous paragraphs, we asked the following questions:


#### *3.2.2 Parameters ε, σ*

The LJ potential is a pairwise potential. Atoms parameters of *ε and σ* are mostly defined by experiments. The parameters values *ε*,*σ* for different atoms combinations were obtained through Rappe et al. [21], and the references [22, 23] by using the formulas:

$$
\sigma\_{A-B} = \frac{\sigma\_{A-A} + \sigma\_{B-B}}{2} \tag{2}
$$

$$
\varepsilon\_{A-B} = \sqrt{\varepsilon\_{A-A}\varepsilon\_{B-B}} \tag{3}
$$

*Graphene - A Wonder Material for Scientists and Engineers*


#### **Table 1.**

*Lennard Jones parameters of atoms and pair of atoms.*

**Table 1** shows the parameters values calculated for atoms combination we will use for simulations.

#### *3.2.3 Computation process (pairwise summation)*

The LJ potential *V r*ð Þ as described in Eq. (1) is the interaction between two atoms a and b. When we talk about the interaction energy of graphene with a substrate, we are referring to more than two atoms (100, 1000 atoms for example). In general, to determine the interaction energy between two bodies G (graphene or graphite) and S (substrate), we do the summation of the LJ potential *V r*ð Þ over all the possible combinations of atoms in the bodies. For instance, the interaction energy between G and S is given by:

$$U(a) = \sum\_{i} \sum\_{j>i} V(r\_{\vec{\eta}}) \tag{4}$$

where *rij* is the separation distance between atoms i and j in the bodies G and S respectively. To compute the previous formula Eq. (4), we use the same method described by Giralfco and co-workers in their paper [14]:

• We calculate first the interaction between a carbon atom in graphene and all the atoms in the substrate. Since there is two type of carbon atoms in graphene, we have the following expressions

$$U\_{\mathbb{N}}(a) = \sum\_{j} V(r\_{\mathbb{N}}) \tag{5}$$

$$U\_{2\circ}(a) = \sum\_{j} V(r\_{2\circ}) \tag{6}$$

where *a* is the distance between the graphene surface and the substrate surface as we can see in **Figure 6**; *r*1*<sup>j</sup>* and *r*2*<sup>j</sup>* are the distance between a carbon atom 1 (or 2) and the j atom in the substrate.

*Graphene Exfoliation from HOPG Using the Difference in Binding Energy between Graphite… DOI: http://dx.doi.org/10.5772/intechopen.107142*

**Figure 6.**

*Left – xy view of graphene/silicon dioxide interaction; right – xz view of graphene/silicon dioxide separated. The yellow, blue and red points are carbon, silicon and oxygen atoms respectively.*

• After, to determine the interaction energy between a graphene sheet and the substrate, we integrate the previous terms Eqs. (5) and (6) over all the graphene sheet. Since here we are interested in general results, we are dividing the previous terms by the surface of the atoms (*δ*: surface of one carbon atom).

$$U\_{\rm GS}(a) = \frac{1}{2\delta} \left( U\_{\mathbb{1}\circ}(a) + U\_{\mathbb{1}\circ}(a) \right) \tag{7}$$

where *δ* is the area occupied by a carbon atom in a graphene sheet. <sup>1</sup> <sup>2</sup>*<sup>δ</sup>* is the number of atom of each type per square unit area in the monolayer.

#### *3.2.4 Simulation set-up*

For the simulation, we are using 2–3 supercells depending on the length of the lattice of the substrate. For example, silicon has a lattice parameter of 5.43 ̊*A*. So, we are using two supercells for a total dimensions of 10.86\*10.86\*10.86. Based on those dimensions, we define the dimensions of graphite 11\*11 ̊*A*<sup>2</sup> for this case. We choose such a small set up to gain in simulation time and performance of the algorithm.

There are 5 simulations. Cupper, silicon, gold and silver have a face-centered cubic (fcc) lattice structure. So we will not show all the figures. By showing only one system we can guess how the other ones will look like since they have the same lattice structure with different lattice parameter. For instance, we have: Silicon (5.43 ̊*A*), Cupper (3.625 ̊*A*), Silver (4.09 ̊*A*) and Gold (4.08 ̊*A*).

The other simulation is with silicon dioxide (**Figure 6**). SiO2 is a complex structure that can be found in many structures. In this project, we consider silicon dioxide as a crystal in a simple tetragonal structure. Here the dimensions of the system are 10.5\*10.5\*14 ̊*A*<sup>3</sup> .

There are five simulations. The principle of our algorithm is to calculate the interaction energy as a function of the separation distance *a* between the substrate and the graphene/graphite.

## **4. Results and discussions**

Most of the researches that has been done between graphene and a substrate is more concerned about the adhesive energy. Few papers are actually about the interaction energy or binding energy. For this reason, we first verified the acceptance of our algorithm by calculating the cohesive energy of graphite (cleavage energy) and compare the result to see if it is similar to others in the literature. From the computed interaction potential between graphite and graphene, we found the cohesive energy of graphite or the binding energy between graphite and graphene to be –0.3060 *J=m*<sup>2</sup> at 3.6 ̊*A* (**Figure 7**). This result is closed to those got by Girifalco [14] and some practical experiments [24]. Now that we know our algorithm is acceptable, our goal is to compare the interaction between graphite – graphene and graphene – substrate. In the next figures, the blue curve represents the energy potential of graphene/graphite and the orange curve, the potential of graphene/substrate.

Using the same algorithm, we computed the interaction potential between silicon and graphene. We found the binding energy to be 0.1014 J*=*m2. This result agrees with the one obtained by Norio Inui and Sho Iwasaki [22]. As we can see from the **Figure 8**, the cohesive energy of graphite is much more superior to the binding energy between graphene and silicon. From this energy difference, it is clear that silicon cannot exfoliate graphene from graphite.

For silicon dioxide, we found the binding energy with graphene to be 0.090 *<sup>J</sup>=m*<sup>2</sup> at 3.45 ̊*A* (**Figure 9**) which is the value got by Ishigma et al. [25] in their work. By comparing it to the cohesive energy of graphite, silicon dioxide cannot exfoliate graphene using this principle. However, a study by Wei Gao and coworkers shows that the interaction strength is strongly influenced by changes in the silicon dioxide surface structure due to surface reaction with water [26]. By reconstructing a Si*O*<sup>2</sup> surface, they could increase the interaction energy. In this project, we used silicon dioxide simple

*Graphene Exfoliation from HOPG Using the Difference in Binding Energy between Graphite… DOI: http://dx.doi.org/10.5772/intechopen.107142*

**Figure 7.** *Graphite graphene interaction energy.*

**Figure 8.** *Graphene silicon interaction (orange), graphene/graphite interaction (blue).*

tetragonal structure without surface modification. This is to say that we can increase the binding energy by surface modification.

In the case of metals substrates, we find the binding energy between copper and graphene to be 0.3792 *<sup>J</sup>=m*<sup>2</sup> at 1.2 ̊*<sup>A</sup>* (**Figure 10**). Here, we can notice that the equilibrium distance at which we obtain the binding energy is small. At first, it is tempting to say that copper cannot exfoliate graphene since at 1.2 ̊*A*, the cohesive energy of graphite is infinity. But in practice, the interlayer distance between graphene layer is fixed, so by approaching graphite at 1.2 ̊*A* from the copper surface, we can see a weak exfoliation since they almost have the same energy. However, the distance is small.

The obtained binding energy between gold/graphene and silver/graphite is 0.5924 *<sup>J</sup>=m*2and 0.5426 *<sup>J</sup>=m*<sup>2</sup> both at 3.2 ̊*<sup>A</sup>* respectively (**Figures 11** and **<sup>12</sup>**). They

**Figure 9.** *Graphene/silicon dioxide interaction potential (orange curve), graphite/graphene interaction (blue curve).*

#### **Figure 10.**

*Graphene/cupper interaction potential (orange curve), graphite/graphene interaction (blue curve).*

both can clearly exfoliate graphene based on the principle described in **Figure 5**. These values of binding energy largely differ from copper and this may be because of the large atomic number and higher lattice parameter. From this result, we can agree that the atomic density has more influence on the binding energy.

## **5. Conclusion**

The graphene industry still has a lot of rooms to improve in terms of standardization, conservation, synthesis, etc. The lack of a reliable and cost effective synthesis method avoids graphene to fulfill its potential. In this project, we proposed for the

*Graphene Exfoliation from HOPG Using the Difference in Binding Energy between Graphite… DOI: http://dx.doi.org/10.5772/intechopen.107142*

**Figure 11.** *Graphene/gold interaction potential (orange curve), graphite/graphene interaction (blue curve).*

**Figure 12.** *Graphene/silver interaction potential (orange curve), graphite/graphene interaction (blue curve).*

first time an exfoliation model of graphene from HOPG graphite based on the cohesive energy of graphite and the binding energy of the substrate with graphene. Calculations using the Lennard Jones potential and lattice summation showed metal substrate like gold, silver and copper can exfoliate graphene from graphite at 3.2 Å. Conventional substrate like silicon and silicon dioxide showed a binding energy inferior to the cohesive energy of graphite and cannot therefore exfoliate graphene. This result opens a new range of possibilities and opportunities for the graphene industry, notably in applications requiring first higher graphene quality. However further studies are still going on. First the distance at which the exfoliation can happen is small. 3.2 Å from the graphite or substrate surface is difficult to achieve. So the

**Figure 13.** *Perfect substrate behavior (red curve).*

challenge is to find or design a material that can exfoliate at an achievable distance which will make the process easier and effective. The perfect material curve looks like the one in **Figure 13**. where the red curve would be the behavior of the "perfect material". Second we need to find substrate materials that are polyvalent and can be used anywhere. Finally, we need to put more accuracy on our algorithm and do some practical tests. The truth is that this method is working, simply because by touching softly graphite or a pencil, there will be graphene layers in our finger and our goal was to understand it and automatized it. This model as shown in **Figure 5** can be easily scaled up to industrial manufacturing if we can achieve a good exfoliation distance. One thing is sure, a reliable, reproducible and effective synthesis technique could have a significant impact in the graphene industry.

## **Acknowledgements**

I whole-hearted express gratitude to my guide Dr. Ashwath Narayana for his guidance and teachings.
