Calculating Surface Energies of Lead Magnesium Niobate Using Density Functional Theory
- PDF / 70,696 Bytes
- 6 Pages / 612 x 792 pts (letter) Page_size
- 9 Downloads / 175 Views
Calculating Surface Energies of Lead Magnesium Niobate Using Density Functional Theory George Kavarnos1 and Roger Richards Submarine Sonar Department, Naval Undersea Warfare Center Division, Newport, Rhode Island 02841 1 EGG, Inc., Groton, Connecticut 06340 ABSTRACT Computations involving density functional theory have been performed on lead magnesium niobate (PMN) single crystal models in an effort to calculate their surface energies, which are believed to play a role in brittle fracture mechanisms. To establish credibility of this approach, test calculations were performed on MgO and SiC single crystal models. The surface energy of MgO was determined to be 1.2 J/m2, which is in close agreement with the experimental value. Similarly, the value for SiC, 8.03 J/m2, supported a level of confidence with this methodology. Surface energies were calculated for several simple perovskites and several PMN models. The calculated values suggest that changes in the A-site ion of PMN do not result in any significant changes in the surface energies. INTRODUCTION Lead magnesium niobate (PMN) is well-known as a relaxor ceramic distinguished by its large strain and high energy density [1]. Addition of lead titanate, barium titanate, or strontium titanate to PMN leads to an improvement in relaxor properties [2]. However, these compositions are still relatively brittle materials, hindering attempts to collect reliable strain data. The brittleness of the ceramic has been attributed to processing conditions and impurities in the materials used in processing. In an attempt to understand the root origin of brittleness in PMN ceramics, we have used density functional theory to model the surface energies of model crystals. The objective of this work was to demonstrate that a quantum mechanical approach can be used to calculate the surface energy, which is an invaluable parameter for investigations of brittle fracture. THEORY The approach is based on Griffith’s theory where crack formation is viewed as a thermodynamic balance between the reduction in mechanical energy on crack extension and the resulting increase in surface energy of the surface created by the extension of the crack [3]. Crack propagation is a thermodynamic process, i. e., during propagation, energy stored in the material is constantly released [4,5]. When a crack is initiated in a material such as a ceramic or crystal, the area generated by crack propagation is approximately proportional to the square of length of the crack, which implies that the relief of strain energy should be proportional to the square of the crack length or depth. At the same time, the energy expended to form the surfaces of the newly-formed crack B6.10.1
(surface energy) is proportional to only the first power of the crack depth. Thus, a critical Griffith crack length, lg, is defined as the critical length of a crack beyond which more energy is released so that crack propagation is highly favorable. A simple formulation of the Griffith crack length is [4]: lg =
2GE πs 2
(1)
Where G is the free s
Data Loading...
