Author Archives: Charles Bigelow

Analysis of Platinum’s Lattice Structure in Respect to Minimal Energy

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By Charles Bigelow

Abstract

Platinum’s optimal lattice parameters were analytically derived using Density-Functional Theory (DFT) methods. The Cambridge Serial Total Energy Package (CASTEP) software package, which uses planewave basis sets in order to analyze crystal structures, was implemented to analyze the SC, HCP, and FCC lattices of platinum to determine the optimal parameters. Platinum’s optimal structure was calculated to be the FCC Lattice at a volume of 63.619 cubic Angstroms. Simple Cubic was found to be the least optimal structure of the three lattices constructed and optimized.

Introduction

As with all metals, platinum has a preferred lattice structure under Standard Temperature and Pressure (STP); however one must determine which structure is preferred over other possible structures. Instead of more traditional experimental methods to determine Platinum’s lattice constants, Density-Functional-Theory can be utilized in order to determine the optimal parameters. This enables chemists to perform analysese normally not readily accessible to the experimentalist due to the flexibility of the calculations. Using this flexibility, one can analyze several types of crystal structures which would normally not be present under normal circumstances, allowing predictions and further in-depth probing of lattice systems. This opens up the possibility to predict structures of substances with unknown crystal structures, or to predict if a certain crystal is not at an energetically favorable composition.

Methods

All calculations were run with the MaterialsStudio software package, using the CASTEP package to analyze FCC, HCP, and SC crystal structures. The On-The-Fly-Generation (OTFG) ultrasoft pseudo-potential was used in the calculations, in which allows the lowest cutoff energy for the given basis set to reduce computational cost. The platinum configuration used in this pseudo-potential was with a Bohr radii of 1.5, an electron configuration of 4f14 5s2 5p6 5d9 6s1 with 32 electrons, and 20 different energy bands.

A k-point set was chosen using a 3x3x2 grid for SC and HCP (18 k-points), while a 4x4x3 grid was used for FCC (60 k-points), both of which achieved reasonable convergence onto the calculations while minimizing cost.

 

The energy cutoff was optimized at 320eV. This parameter affects the smearing of the wave-function to minimize fluctuations in energy calculations, however also increasing computational cost; 320eV was found to have  the lowest reasonable convergence and efficiency.

The calculations implemented the generalized gradient approximation (GGA) using the Perdew–Burke–Ernzerhof functional.

The analysis of the Hexagonal Close-Packed (HCP) crystal structure consisted of determining both the optimal crystal volume as well as the optimal “c” to “a” vector ratio. For the volume, the optimization was done using a 1.6:1 a to c ratio, and four separate volume’s; 44, 40, 34, and 28 cubic Angstroms (A^3). Of these, 40, 34, and 28 were used to determine the polynomial equation for the energy with respect to volume.

The minimum of the polynomial was determined by setting its derivative to zero and evaluating for x; the optimum lattice volume was determined as 32.284 cubic Angstroms.

Further observation was done with varying the c/a ratio along with volume to observe the general trend of the preferred configuration; it was discovered that at smaller volumes, a 1.6:1 ratio is preferred, while at the optimal volume and above, the lower ratios were dominant.

The c/a ratio was then analyzed using the optimized volume of 32.284 cubic Angstroms.

It was discovered that at this lattice volume, the preferred ratio was determined by the derivative of the polynomial evaluated to zero. It was found to be 1.34368:1 and the corresponding energy was determined to be -26101.61 eV.

 

A Simple Cubic (SC) Lattice was generated from Platinum to observe its preferred lattice volume; this was done in a similar manner as HCP.

The optimal volume was determined to be 19.7596 cubic Angstroms by evaluating its polynomial at zero. The energy corresponding to this volume was -13049.77 eV.

 

A Face-Centered Cubic (FCC) crystal lattice of platinum was analyzed to determine its optimum volume in the same manner as the HCP and the SC systems, as well as evaluation of its polynomial at zero. the three points closest to the lowest value were used to generate the polynomial.

 

The optimal volume was determined to be 63.61879 cubic Angstroms. At this volume, the FCC structure of Platinum has an energy of -52203.63 eV.

 

Comparison of the three different configurations indicates that the FCC lattice structure is optimal for platinum, while also giving the largest optimized volume of the three analyzed. It has markedly higher stability than does the SC or the HCP configurations.

 

With the data provided, it is hypothesized that if pressure were to increase, HCP and SC would become the more stable configurations to the crystal structure of platinum.

Calculation of The Most Energetically Stable Surface of Copper

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By Charles Bigelow

Abstract

A copper crystal was analyzed using DFT to observe the relaxation effects of three of its Miller indices to predict the optimal surface structure under zero degree Kelvin conditions. The three miller indices observed were the 1 1 1, 1 1 0, and the 1 0 0 lattices. The overall result of the calculations indicated that 1 1 1 Miller plane is the most stable structure of the copper surface.

 

Introduction

Oftentimes, a metallic or crystal compound will have several different configurations depending on which plane of reference that is being analyzed; these are referred to as miller indices, which are planes which bisect a given crystal lattice structure at particular coordinates in reference to the base structure. These various indices are of crucial importance to chemical and physical properties of a substance and its surface.

One can use Miller indices to generate and calculate surface energies and orientations of a particular lattice using DFT. This gives the ability to calculate a particular material’s surface structure theoretically; this gives added flexibility as simulations are able to analyze systems normally outside of the capabilities of current experimental methods with relative ease, and allows prediction of structures that have not yet been synthesized. This is often accomplished by constructing a surface of the crystal at a specific plane, a certain number of layers selected, and then a vacuum slab generated to produce a surface with periodic boundaries. The vacuum must be large enough to prevent interactions between successive slabs in the periodic structure.

 

Methodology

In order to observe the surface energy of the various copper Miller planes, surface slabs were generated using the materials studio software. Various slabs needed to be generated, each having multiple iterations with different thicknesses. The significance in varying the slab thickness is to ensure that the inner portion of the slab is significant enough in size to properly simulate bulk properties; too small of a slab would produce unconverged energies, while too large of a slab would significantly increase computational costs.

Slab generation was done by taking a FCC single unit lattice of copper and cutting along the miller indices of 111, 110, and 100 using the cleave crystal function found in the build section of Materials Studio. All three lattices were using a single unit cell. 111 was generated in integer steps of thickness, each layer adding another atom into the lattice. for 110 and 100, the slabs were generated using half integer steps, as their layers consist of 2 atoms per layer, versus the 1 atom per layer of 111; this is most likely due to the structure repeating after two atom’s depth for both 110 and 100. The vacuum distance used for all slabs was 10 angstroms to prevent interactions between the periodic slabs in the C axis of the lattice structure.

Figure 1: The 111 surface at a relative thickness of five layers

 

Figure 2: The 110 surface at a relative thickness of 2.5 layers

 

Figure 3: The 100 surface at a relative thickness of 2.5 layers

 

The cutoff energy was converged for a five layered slab of 111 copper; 360eV was found to be an effective value to ensure efficient and accurate calculations, as it only differed by roughly 0.1eV from a cutoff frequency of 380eV . Higher values were observed at  400eV and 440eV, however these oscillated around the same energy within a value of 0.1eV. Further optimization of this parameter was limited due to constraints of computational power. 360eV was used for all three miller indices.

All three indices had a basis set of 1s2 2s2 2p6 3s2 3p6 3d10 4s1. The SCF tolerance was set at 2.0e-6eV/atom, and a maximum of 130 SCF cycles to allow for more precise convergence. a 4x4x1 k-point lattice was used for all three of the indices and was kept constant for each system; it would be more optimal to individually optimize the k-point mesh for each lattice type, however this was not pursued due to lack of computational capabilities. The pseudopotential which was used was on-the-fly-generation ultrasoft [1]. In these calculations, a GGA PBE functional was used; These parameters were used for all systems[2].

The surface energy was calculated using equation 4.2 in chapter four of the “Density Functional Theory, A Practical Introduction” book by David S. Sholl.

1/2A*[E(slab) – n*E(bulk)]

where 2A is the surface area in angstroms (both top and bottom surface), E(slab) is the energy of the slab, n the number of atoms present, and E(bulk) is the FCC copper lattice energy for one atom.

Results

Observation of the 111 energies displayed a convergence at a slab thickness of four and five; it should be noted that beyond five, convergence issues were present, but due to the limits in computational capability and time allotted, these were unable to be addressed, and would require further investigation to the cause of the notable deviation present; even with higher level calculations, the deviations remained in the calculations. These observations were also noted in the 110 and 100 slabs above 3.5 layers (7 atoms).

For the 110 and 100 slabs, convergence was achieved at 3-3.5 and 3.5-4 layers, respectively, with 3.5 being used as energy reference for both 100 and 110. Larger slabs ran into the issues described above. Before the energy per surface area was calculated, the gross difference between the bulk and the surface energy was roughly

Before the energy per surface area was calculated, the gross difference between the bulk and the surface energy ranged between two to three electron volts in difference, which is a factor of 10 greater than the 0.1eV deviations observed with the various cutoff values.

 

The energy for the chosen slabs of 111, 110, and 100 were

0.0946eV/Angstrom^2,      0.146eV/Angstrom^2,   and      0.116eV/Angstrom^2     respectively.

 

Conclusion

By comparing the three surfaces, it is predicted that the 111 surface is the most probable surface present in a sample of copper metal; it is theorized that this is due to the larger density of atoms on the surface of the 111 miller plane that causes this lower energy; this prediction is consistent between the 100 and 110 surfaces as well. This analysis was consistent with what was seen in experimental data.[3,4]

 

URLs for References

  1. https://www.tcm.phy.cam.ac.uk/castep/documentation/WebHelp/content/modules/castep/tskcastepsetelecpotentials.htm
  2. https://www.tcm.phy.cam.ac.uk/castep/documentation/WebHelp/content/modules/castep/tskcastepsetelecxc.htm
  3. D. E. Fowler and J. V. Barth, Structure and Dynamics of the Cu(001) surface Investigated by Medium-Energy Ion Scattering, Phys. Rev. B 52 (1995), 2117.
  4. F. R. de Boer, R. Boom, W. C. M. Mattens, A. R. Miedema, and A. K. Niessen, Cohesion in Metals , North-Holland, Amsterdam, 1988.

Diffusion Pathways for Platinum Adatoms on a Platinum Surface

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By Charles Bigelow

Abstract

Analysis of two possible methods of diffusion of platinum  on a platinum metal surface in the forfold site. The initial and final conditions were optimized and both the hopping and the substitution pathways were analyzed. It was discovered that the hopping mechanism between forfold sites was more energetically favorable than the concerted substitution between the adatom and a surface platinum.

 

Introduction

     Propagation of adsorbed atoms on a metal surface is a chemical phenomena which is of great significance to surface chemistry. Knowledge of this diffusion helps the scientific community to design better catalysts, substrates, and give a better understanding of the underlying mechanics and reaction pathways of a given process.

Metals, in particular, have various ways of migrating across a metallic surface. This is achieved through hopping between different sites on the metal, or through substitution with an atom in the crystal structure itself. These pathways will be analyzed to observe their barrier energy, which will indicate which process is more energetically favorable under normal conditions. This information can be used to help aid in designing pathways for a new catalyst to be produced, as well as to observe how likely catalyst poisoning may occur through substitution.

Figure 1: Platinum Surface with adatom (Top)

Figure 2: Platinum Surface with adatom (side)

Method

Figure 3: 1.5 Layer Slab

     A 1.5 layer slab of platinum was generated by cleaving the surface of a bulk sample of platinum along its 100 miller plane. The size of the unit cell was doubled to minimize errors due to periodic boundary conditions; normally the lattice would be tested at various unit cell sizes, however due to computational limitations a larger surface area would not be feasible for the given machine.

A vacuum slab of 13 angstroms was used to ensure that the adatom would not have interactions with the slab above; this is an increase from previous vacuums used earlier in the semester. After generation of the surface, the adatom was generated on the slab at one of the fourfold sites, and its distance optimized from the surface of the lattice manually to save on computational cost. The adatom’s optimal distance from the surface was calculated to be 3.9 angstroms. This was accomplished by varying the distance of the adatom from the surface in the z direction and taking the derivative of the resulting polynomial equation.

An initial and a final slab were generated by moving the adatom to an adjacent site on the platinum surface and placing at the optimal distance. A reaction pathway was generated and 5 steps between initial and final were used to calculate the activation energy. The LDA PWC functional was used to run the calculation using a DND basis set and a basis file of 3.5 [1].

The concerted substitution was done similarly, the only difference was that the initial adatom was associated with the final center lattice atom, and the final adatom associated with the initial center lattice atom. Smearing of 0.005 hartree was implemented for the electron transition of the adatom substitution to remove the discontinuity of the calculation; the hopping mechanism did not require any smearing to run to completion [2].

 

Results

     The most optimal distance was found to be at 3.965 angstroms above the fourfold site, which achieved the lowest equilibrium energy for both the initial and final structures.

 

When the two methods of diffusion were analyzed, it was discovered that the hopping substitution had an activation energy of 0.002471 hartree, while the substitution was at 0.007148 hartree.

Conclusion

The hopping mechanism was the lowest energy process in terms of diffusion across the surface of the metal surface,  indicating it is the primary method of movement for adatoms on the surface. The substitution method was over double the required activation energy compared to diffusion via hopping; this indicates that this is less likely to occur at a given temperature, however, the energy difference was small enough to allow adatom substitution at a notable rate over time, indicating that the surface would have atoms replaced over an extended time frame. This mechanism is observed in various cases of catalyst poisoning, contributing to the difficulty of effective catalyst design.

 

URL References

  1. http://nees.sci.upc.edu.cn/_upload/article/files/39/f5/5460e8894554bd75148145ba414e/188a6221-e993-431c-bf9e-28e3051fd772.pdf
  2. http://nees.sci.upc.edu.cn/_upload/article/files/39/f5/5460e8894554bd75148145ba414e/188a6221-e993-431c-bf9e-28e3051fd772.pdf