The Energetics of the Hydrogenation of a Single-Walled Carbon Nanotube. Janet Ryu Nicola Marzari May 13, J

Transcription

1 The Energetics of the Hydrogenation of a Single-Walled Carbon Nanotube Janet Ryu Nicola Marzari May 13, J

2 Janet Ryu May 13, 2005 Final Paper 3.021J Carbon Nanotubes The Energetics of the Hydrogenation of a Single-Walled Carbon Nanotube Carbon nanotubes have brought about much excitement in the scientific community. Their unique electrical, thermal, and mechanical properties make nanotubes an appealing new option in material choice. For starters, nanotubes are extremely small; they are on the order of a few nanometers in diameter and can be on the order of a few millimeters in length. At the same 2 time, they exhibit incredible strength, because they are comprised completely of sp bonds. There are many different kinds of carbon nanotubes which vary in diameter and chirality. The chirality, or twist, is an important factor in characterizing the properties of a particular nanotubes. The chirality determines whether the nanotube is superconducting, insulating, semiconducting, or conducting. They can have as high a conduction as copper, at much lower resistence. This is another property of carbon nanotubes that has spurred research for using them in electronic devices, specifically computer hardware. Carbon nanotubes can also be functionalized with different macromolecules for even greater applicability. This paper discusses the energetics of functionalizing a single-walled carbon nanotube with two hydrogens. Carbon nanotubes are made by rolling a sheet of graphite into a cylinder. A carbon atom A is rolled such that it is in the same location of another carbon atom B on the lattice. The location of every atom on a graphite lattice can be expressed as a linear combination of two vectors. This means that a carbon nanotubes can vary immensely in how they are rolled, which gives them their variations in characteristics.

3 Quantum Mechanics The energetics calculations done in this project were based on quantum mechanics. The energies found were of infinite systems that were described using periodic boundary conditions. Energies are calculated over a unit cell, which is then repeatedly periodically. A unit cell is defined as the smallest number of atoms necessary to create a uniform crystal when regularly repeated. Fig. 1 below shows a unit cell and what it looks like when repeated periodically. Figure 1. The left is a unit cell, and the image on the right shows it repeated. Image taken from Density Functional Theory (DFT) Density functional theory is a powerful tool for performing many-body electronic structure calculations. It finds a very good approximate solution to the quantum mechanical many-body problem. With DFT, it is no longer nececssary to solve the practically unsolvable many-body wavefunction which is dependent on 3N variables, where N is the number of electrons in the structure. The wavefunction is replaced by the charge density over all occupied states, reducing the number of variables down to 3. It calculates the energy as a function of atomic position for a given number of electrons. DFT gives good results when compared to experimental data, at relatively low computational costs.

4 PWscf DFT is the basis of how PWscf (Plane-Wave Self-Consistent Field), the set of codes used in this project, calculates the energies. In a PWscf input file, the system is described by a number of parameters including the atomic positions in the unit cell. PWscf calculates the energy using a Plane-Wave basis set and pseudopotentials. An Pwscf input file for a 5,0 carbon nanotube is attacched in Appendix A. By calculating the energies, I found the equilibrium geometry for a given set of atoms by seeing which geometry had the minimum energy. Diamond The first structure I did energy calculations was on diamond. The bravais lattice of diamond is face-centered cubic (FCC). There are a total of 2 carbon atoms in a unit cell of diamond. The relatively simple structure of a diamond crystal allowed me to learn how to control various parameters in the input file and make sure I could properly specify the coordinates of the atomic positions in the unit cell. Below is a section of the input file and descriptions of what each line means. (1) ibrav= 2, celldm(1) =6.6, nat= 2, ntyp= 1 (2) ecutwfc =40 Line (1) specifies parts of the unit cell. The first part, ibrav, specifies the bravais lattice to be FCC, because 2" corresponds to that type of lattice in PWscf. The part celldm(1) refers to a length of the unit cell. In this case, it is 6.6 bohr. There are 2 atoms in this unit cell, so nat is 2. Diamond only has carbon atoms; it only has the carbon element in its structure. Therefore, ntyp is 1, because it only has 1 type of atom. The atomic positions could be specified in units of lattice vectors, bohrradius, crystal coordinates, or angstrom. For my project, I used crystal coordinates to specify atomic positions. But for diamond, it was simplest to use lattice vectors to specify the positions of the two atoms in the unit cell.

5 Figure 2. This plot shows energy as a function of lattice parameter. The ground state energy was calculated for diamond lattices of varying size. The lattice parameter that produces a structure with the lowest energy is the equilibrium lattice parameter. The plot above shows the relation between energy and the lattice parameter of diamond. Diamond has a lattice parameter of somewhere around 6.68 bohr. This is quite similar to the value 6.73 bohr, which has been found experimentally at 300 K. 5,0 Carbon Nanotube The specific carbon nanotube hydrogenated in my project was a 5,0 carbon nanotube. The unit cell contained 20 atoms in 2 rings of 10 carbons each. All of the C-C bond lengths are the same in a carbon nanotube. This length was varied and energy calculations were performed to find the equilibrium C-C bond length in a 5,0 carbon nanotube. The full input file is in Appendix A.

6 Figure 3. Energy calculations were done on the 5,0 carbon nanotube with varying C-C bond lengths. The minimum point of the energy curve corresponds to the structure with the equilibrium bond length. The 5,0" in the specification of the carbon nanotube refers to the values of n and m in the formula na 1 + na 2 = R, where R is the vector from atom A and B in the graphite lattice. 5,0 indicates that 5 hexagons of graphite are traversed by vector R. Therefore, we would expect the ring of this nanotube to have 5 hexagons of the graphite. XCrySDen All of the crystal and molecular structures specified by the PWscf input files can be modeled by a program called XCrySDen. This application allowed me to make sure that my input files described the geometry of the structures correctly. Fig. 4 shows the concepts of unit cell and periodicity. It has a unit cell of 20 carbon atoms in the 2 ring geometry, and when it is

7 repeated an infinite number of times, we get an infinitely long 5,0 carbon nanotube. XCrysden had other useful functions such as telling me the distance between two atoms in my structure or the coordinate of a particular atom. Figure 4. The upper two images are both of 1 unit cell of a 5,0 carbon nanotube, which contains 20 carbon atoms. The bottom image shows how each unit cell is repeated, creating a long carbon nanotube. Hydrogenated Carbon Nanotube Once the coordinates of the nanotube were found, I needed to find the coordinates of the hydrogen if they were to be attached to the wall of the nanotube. I did this using a code that a grad student, Young-Su Lee, wrote in fortran. Running the code gave me coordinates of the

8 hydrogens, such that they were a distance of 1.09 angstroms, the length of a H-C bond, away from the wall of the nanotube. Modeling the structure with XCrySDen showed that the nanotube was fully hydrogenated. Figure 5. A fully hydrogenated 5,0 carbon nanotube. A unit cell containing 40 atoms total: 20 carbon atoms and 20 hydrogen atoms. Once I had the coordinates of all 20 hydrogens, I removed all of them except for 2. I fixed one of the hydrogen atoms in position and moved the other hydrogen atom around the 2 rings. This created a total of 19 different hydrogenated nanotube structures, all of them containing a unit cell of 20 carbon atoms and 2 hydrogen atoms. The only difference between each of these structures was the position of the second hydrogen atom attached to the nanotube. I performed energy calculations on all 19 structures to find which configuration had the lowest energy. This gave me the structure that is thermodynamically favored. Results The purpose of the project was to calculate the energy required to attach two hydrogens to the surface of a carbon nanotube and determine whether it was energetically favorable. Before this could be done, I needed to calculate the energy of a hydrogen molecule, H 2. Once I had that energy, I could use the following equation to calculate the energies of atttachment: E f=e(cnt + 2H) - E(CNT) - E(H 2)

9 Table 1 Some of the Results of the Energy Calculations Structure Energy (ryd) CNT H CNT + 2H CNT + 2H CNT + 2H CNT + 2H CNT + 2H CNT + 2H CNT + 2H CNT + 2H CNT + 2H The lowest energy found for the hydrogenated carbon nanotube was used, because that is the most energetically favorable configuration for the hydrogens to be attached to the surface of the nanotube. By using that value along with the energy of the carbon nanotube and the hydrogen molecule, I found the energy of attachment, E f = ryd. Conclusions Since E f < 0, this indicates that hydrogenation of the nanotube is energetically favorable. E(CNT + 2H) is lower than the sum of E(CNT) and E(H 2); it requires less energy for hydrogens to break apart and attach to the surface of the nanotube, than to stay as a hydrogen molecule. Future Work There will be further research done on hydrogenated carbon nanotubes in order to learn more about its physical properties. I will study how it changes with temperature, study its phase transitions, and this will allow me to construct a thermodynamical model of the system.

10 Appendix A &control calculation = 'relax' restart_mode='from_scratch' prefix='cnt2.rlx' tstress =.true. tprnfor =.true. outdir = '/state/partition1/janetryu' pseudo_dir = '/home/janetryu/cnt/' / &system ibrav= 6 celldm(1) = 31 celldm(3) = nat= 20 ntyp= 1 ecutwfc = 30 ecutrho = 240 occupations = 'smearing' degauss = 0.03 smearing = 'cold' / &ions / ATOMIC_SPECIES C C.pbe-rrkjus.UPF ATOMIC_POSITIONS {crystal} C C C C C C C C C C C C C C C C C C C C K_POINTS {automatic}

11