Vacancies and Free Surfaces in Iron
 
 

Introduction

    Iron is a silvery, lustrous metal that constitutes approximately 5% of the earth's crust.  Having both biological and technological significance, it is the cheapest and most abundant, useful, and important of all metals.   Iron is an essential component to plant and animal life.  It is also a critical ingredient in the automotive, aeronautics, chemical, construction, manufacturing, and naval industries as well as numerous others.  The pure metal is extremely chemically reactive corroding rapidly, particularly in moist air or high temperature conditions.  In the typical method of recovery of iron, hematite ore is fused to drive off oxygen, sulfur, and other impurities.  The ore is melted in a blast furnace in direct contact with coke and limestone.  The coke breaks down to form carbon monoxide which in turn reacts with the iron oxide (hematite) to produce molten iron and carbon dioxide.  The limestone decomposes into calcium oxide and carbon dioxide.  The calcium oxide reacts with impurities in the iron producing slag.  Because slag is less dense than molten iron, it forms a layer over the resulting iron which can easily be removed.  Thus, the resultants of the process include exhaust gases (mostly carbon dioxide), slag, and an iron called "pig iron".  "Pig iron" is still highly impure containing 3-4% carbon and trace amounts of silicon, manganese, and phosphorus.  It is hard, brittle, fairly fusible, and is used to produce other alloys such as steel.  When producing steel from "pig iron", the carbon content is lessened to approximately 2%.  A second class of iron, produced in a different manner, is wrought iron.  It contains a smaller percentage of carbon, typically a few tenths of a percent.  Wrought iron is tough, malleable, less fusible, and usually has a fibrous structure.  Drawing from this comparison, manipulation of atom-to-atom relationships between iron, carbon, and various alloying elements establishes specific properties of ferrous metals.

    Properties such as strength, toughness, impact resistance, hardness, and ductility also alter when atoms rearrange in a crystal lattice whether the crystal lattice is composed of various elements ( i.e. iron, carbon, and others) or even just iron atoms.  Rearrangement of atoms in a crystal lattice structure arises from defects such as vacancies and free surfaces.  A vacancy is a point defect where an atom is missing from its lattice site.  They form in crystal lattices during solidification or as a result of atomic vibrations.  Nearest neighboring atoms to the vacant site adjust positions due to the defect of energy in the structure.  This adjustment, often called "relaxation", occurs in shells concentric to the vacant site and typically has an oscillatory character decreasing in magnitude of change in the original position.  For example, the first shell of nearest neighboring atoms to the vacant site will move inwards some distance from their original locations while the next shell of atoms will move outwards some smaller distance from their original position.  A free surface is a plane of atoms at which a crystal lattice terminates.  Because the surface atoms are not bonded to the maximum number of nearest neighboring atoms, they have a higher energy state than the inner atoms.  The absence of the bonds creates a surface energy.  Just as in the vacancy defect, atoms adjust positions due to the defect energy.  Adjustment occurs in planes with movement perpendicular to the free surface.  Again, the adjustment has an oscillatory character with decreasing magnitude of change in the position of a plane of atoms as distance increases from the free surface.  Thus, the first plane away from the free surface may move inwards some distance while the second plane away from the free surface may move outwards some smaller distance.  Because rearrangement of atoms in a crystal lattice influences material properties, a thorough understanding of these defects discussed is critical in order to improve processing of iron and iron alloys and the developing of strong, useful materials in general.

    In the analysis to follow, computer simulations calculate defect energies and atom displacements resulting from vacancy defects and a free surface defect in iron crystal lattices.  Iron can exist as a face-centered cubic (fcc) crystal structure or a body-centered cubic (bcc) crystal structure.  The fcc structure has a lattice parameter, a, of 3.515 Angstroms and a cohesive energy of 4.196 eV (6.723 x 10-19 J).  The lattice parameter, a, for the bcc structure is 2.87 Angstroms, and the cohesive energy is 4.28 eV (6.86 x 10-19 J).  The analysis will encompass:  1)  a vacancy defect in a fcc structure, 2) a vacancy defect  in a bcc structure,  and 3) a free surface defect in a bcc structure.
 
 

A Vacancy Defect in Face-centered Cubic Iron

    The computer generated fcc iron crystal lattice contains 256 free atoms, 1792 buffer atoms, and 6740 fixed atoms resulting a total of 8788 atoms.  The inner block, holding the 256 free atoms, has dimensions 14.06 Angstroms by 14.06 Angstroms by 14.06 Angstroms.  The vacancy defect introduced is at the approximate center of the inner block.

A wire frame model of the block is at left.  Click on the image to view a larger image of the wire frame model.  The lines are representative of the bonds between iron atoms.  In the larger image, a red box surrounds the area of the vacancy.  Missing bonds distinguish this region from other areas of the block.  The regular pattern of bonds has been broken.

    A ball and stick model of the block is at right.  Click on this image to view a larger image of the ball and stick model.  Some atoms and bonds have been deleted from the block originally created in the computer simulation in order to make visualization less confusing.  In the larger image, the defected region is within the red circle.

    The absence of an atom is more clear in this image.  The structure truly appears incomplete.  Again, the vacancy has broken the regularity of the bonds.  A three-dimensional model may further aid in visualization of the vacancy defect within the block.  Click here  to view a vrml of the ball and stick model.

    Minimum energy of the block after introducing the vacancy defect is 8587.478 eV (1.376 x 10-15 J).  The defect energy is 5.858 eV (9.386 x 10-19 J).  The change in energy due to relaxation is 0.122 eV (1.955 x 10-20 J).  The maximum atomic displacement after relaxation is 0.03911 Angstroms (3.911 x 10-12 m).

    The image on the left is a part of the crystal structure towards one corner of the block generated.  The fcc structure is essentially preserved.  The lattice parameter for fcc iron of 3.515 is essentially upheld.  The image on the right is of the nearest neighboring atoms of the vacant lattice site.  The crystal structure has severely altered almost resembling a bcc type structure.  These atoms constitute the "first shell" to the vacancy and undergo the greatest amount of atomic displacement.
 
 

A Vacancy Defect in Body-centered Cubic Iron

    The bcc iron structure generated contains 9826 total atoms for study of a vacancy defect.  128 atoms are free; 1872 are buffer atoms.  The remaining 7826 atoms are fixed atoms.  The dimensions of the inner block of study are 11.48 Angstroms by 11.48 Angstroms by 11.48 Angstroms.  Again, the vacancy has been positioned in the approximate center of the inner block of free atoms.

The image to the left depicts a wire frame model of the bcc iron structure having a vacancy defect.  Click on the image to obtain a larger image of the wire frame model with the approximate location of the vacancy boxed in red.  Notice the missing bonds and the break from the regularity of line representation that is apparent in other parts of the simulation block.

The image to the right depicts a ball and stick model of the bcc iron structure having a vacancy.  Click on the image to view a larger image of the ball and stick model.  This is only a portion of the original simulation block.  The defect region is circled in red.

    The central cube is clearly missing its central atom.  Crystal structure transformation due to the vacancy is less extensive than in the fcc case.  The bcc structure appears mostly sustained, and atomic displacement is not really noticeable.  However, calculations performed in the computer simulation indicate a value of 0.08651 Angstroms (8.651 x 10-12 m) for maximum displacement of "first shell" nearest neighboring atoms to the vacant site.  Minimum energy of the block after introducing the vacancy defect is 8554.012 eV (1.371 x 10-15 J).  The defect energy is 5.988 eV (9.594 x 10-19 J).  The change in energy due to relaxation is 0.243 eV (3.893 x 10-20 J).  Click here for a vrml file of the ball and stick model.

    Away from the center of the simulation block, especially near the outside faces, the bcc structure is preserved.  The image displayed above on the left is a part of the crystal generated taken from a corner of the simulation block.  The bcc structure is in tact; the lattice parameter of 2.87 Angstroms has been maintained.  The image on the right depict the nearest neighboring atoms of the vacant site.  The missing atom may be viewed as a missing central atom for a bcc unit.  This collection of atoms may now be viewed as forming a simple cubic unit.  The bcc iron lattice parameter of 2.87 does not apply after atomic displacement of these "first shell" atoms.  Rather all interatomic distances, taken as edges of the cube, are 2.77 Angstroms.  Thus, the eight atoms essentially create a simple cubic unit with a lattice parameter of 2.77 Angstroms.
 
 

A Free Surface Defect in Body-centered Cubic Iron

    The image at left depicts the bcc iron crystal structure generated with a free surface in a computer simulation.  The lattice block contains 170 total atoms: 120 being free atoms, 24 being buffer atoms, and 26 being fixed atoms.  For discussion, an axes system shall be defined as follows:  x is horizontal to the right, y is vertically upwards, and z is directed perpendicularly out of the screen.  The free surface is at the bottom of the image in the xz plane containing the atom with the minimum y coordinate.  In the computer simulation setup, the free surface was established as a {1,2,0} surface.  Periodic boundary conditions were used in the x and z directions, and fixed boundary conditions were used in the y direction.

    Bonds shown between atoms are not consistent throughout the structure.  No pattern develops.  The bond angles for atoms in the first few planes above the free surface appear slightly distorted.  Significant atomic displacement is not easy to determine through visual inspection of the image.  However, calculations completed during the running of the computer simulation reveal a maximum atomic displacement of 0.24523 Angstroms (2.453 x 10-11 m) occurring in the first plane above the free surface.

    With introduction of the free surface, the minimum energy of the simulation block is 595.9352 eV (9.548 x 10-17 J).  The specific energy of the free surface defect, defined as the defect energy per unit area of the plane, is 0.1537 eV/Angstrom (2.459 J/m2).

    In an analysis of change of interplanar distances with progression away from the free surface, an oscillatory character was observed but amplitude did not decrease or converge on a particular interplanar distance.  Oscillation actually appears to result due to the orientation of the crystal structure not because of oscillation between approaching and recessing atomic displacement with respect to the free surface.  In calculation, interplanar distance oscillated between 6 and 7 Angstroms due to the orientation.  There are instances where subsequent plane sets have the same interplanar distance.  Convergence to a particular interplanar distance was not realized again due to the orientation of the crystal and the manner in which interplanar distance is measured.  Oscillatory behavior in atomic displacement with respect to the surface may have occurred, however, but may not have been found in our results due to tolerances in the calculations of coordinates of atoms. The plot below shows the behavior observed in change of interplanar distance with progression away from the free surface. Adjacent planes create a set and are counted up beginning with one corresponding to the set containing the free surface and the plane just above it.  This data is plotted on the x-axis.  Corresponding interplanar distance is on the y-axis.


 
 
 

Works Consulted

WebElements, http://www.shef.ac.uk/chemistry/web-elements/Fe/key.html

Machine Design Materials Reference,  http://www.penton.com/md/bde/rvmat5.html

Callister, William D., Jr.  Materials Science and Engineering an Introduction.  4th ed.  John Wiley & Sons:  New York, 1997.

Masterton, William L.  Chemistry:  Principles & Reactions. 3rd ed.  Saunders College Publishing:  Fort Worth, 1989.