point defects, lattice structure and melting slava sorkin

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  • POINT DEFECTS, LATTICE

    STRUCTURE AND MELTING

    SLAVA SORKIN

  • POINT DEFECTS, LATTICE STRUCTURE

    AND MELTING

    RESEARCH THESIS

    SUBMITTED IN PARTIAL FULFILLMENT OF THE

    REQUIREMENTS

    FOR THE DEGREE OF MASTER OF SCIENCE

    IN PHYSICS

    SLAVA SORKIN

    SUBMITTED TO THE SENATE OF THE TECHNION ISRAEL INSTITUTE OF TECHNOLOGY

    ELUL, 5765 HAIFA SEPTEMBER, 2005

  • THIS RESEARCH THESIS WAS SUPERVISED BY MY LOVELY ADVISORS

    DR. JOAN ADLER AND PROF. EMIL POLTURAK UNDER THE AUSPICES

    OF THE PHYSICS DEPARTMENT

    ACKNOWLEDGMENT

    I wish to express my gratitude to Dr. J. Adler and Prof. E. Polturak

    for the excellent guidance and support during this research.

    I am grateful to Dr. A. Hashibon, Dr. G. Wagner and Dr. Z. Salman,

    and to my fellow students A. Kanigel, N. Schreiber and to my wife A.

    Sorkin for their help during the research period.

    I would like to express my gratitude to J. Tall and M. Goldberg for the

    help on the IUCC machines and parallel computing. I also thank the

    IUCC for allowing me CPU time on their linux-cluster, the Cray T90

    and Cray SV1 supercomputers of the Ben Gurion University, and the

    Silicon Graphics supercomputer of the Technion.

    THE GENEROUS FINANCIAL HELP OF THE TECHNION AND THE

    GERMAN ISRAELI FOUNDATION (GIF) IS GRATEFULLY ACKNOWLEDGED

  • Contents

    iii

  • List of Figures

    iv

  • List of Tables

    v

  • Abstract

    Melting is a fundamental process in which a crystal undergoes a phase transition from

    a solid to a melt. Despite its common occurrence, understanding this process still a

    challenge.

    A number of theories, which consider melting as a process occurring homoge-

    neously throughout the crystal have been proposed during the past century. For

    example, according to Lindemann, melting is triggered by a mechanical instability of

    the solid, which caused by enhanced vibration of the atoms. Solids liquefy when the

    amplitude of atomic thermal vibrations exceeds some fraction of interatomic spac-

    ing. According to Born, melting arises from the onset of a mechanical instability of

    the crystal lattice, which manifests itself in an imaginary phonon frequency and the

    vanishing shear elastic moduli, accompanied by the collapse of the crystal lattice.

    Other models are based on spontaneous thermal production of the intrinsic lattice

    defects (vacancies, interstitials and dislocations) near the melting point and this leads

    to break-down of the long-range crystalline order and melting transition. However,

    the extrinsic defects (free surfaces and grain boundaries) were not considered as an

    important ingredient of a melting scenario. Those models are based on the concept

    of one-phase melting or continuous melting, i.e. they imply that the phase change

    might be continuous or nearly so; given sufficient resolution it should be possible to

    vi

  • ABSTRACT vii

    track the breakdown of the solid throughout its transition to the liquid state. These

    models are capable of calculate the melting temperature Tm, but in the most it is

    overestimated.

    The existing theories of melting are still far from being complete and raise new

    questions. Hence, the purpose of the present research is to gain a better understanding

    of the mechanism of melting transition, and especially to investigate the role of point

    defects and the surface of the solid in the melting transition. Despite the fact that

    we learned very much recently about the melting of fcc metals, it is not clear if those

    results were specific to fcc structure of these metals, so we decided to study melting

    of a bcc metal, vanadium, by means of computer simulations.

    An interatomic potential proposed by Finnis and Sinclair [42], was chosen for

    our simulations. The potential was tested by calculation of various properties of

    a perfect crystal of vanadium. The results are in good agreement with available

    experimental data. Afterwards point defects were introduced into the bulk either by

    the removal an atom (vacancy) or by the addition one (self-interstitial). The most

    stable configuration of defects at low temperatures was found to be a dumb-bell, the

    < 110 > split-interstitial. Point defects change the physical properties of the solid.

    Interstitials expand the sample, while vacancies decrease its volume. The change of

    the volume is less noticeable for vanadium than for copper which is attributed to the

    less close-packed structure of its bcc lattice. We found also that the shear moduli are

    softened as a result of the volume expansion of the solid which is associated either

    with an increase in temperature or interstitial concentration. This softening of the

    moduli is less pronounced for vanadium in comparison with copper.

    There is a strong evidence that Born instability is the trigger for bulk melting.

    The instability is set in by interstitials which expand the solid up to a critical volume,

  • ABSTRACT viii

    at which the lattice of the crystal becomes mechanically unstable and collapses. This

    defect-mediated mechanical melting occurs at the temperatures below the melting

    temperature of the perfect crystal. We verified that the critical volume at which the

    crystal melts is independent of the path thru the phase space by which it reached, i.e.

    either by heating the perfect crystal or by adding defects at a constant temperature.

    We performed simulations with various concentrations of point defects and found

    that bulk melting temperature Tb is lowered by interstitials, but this effect is less

    pronounced in comparison with the same effect for copper.

    The mechanical melting can not be observed directly in the laboratory, because

    a real crystal will eventually melt at Tm which is lower than Tb via thermodynamic

    melting process that nucleates at its surface. The process can be suppressed ex-

    perimentally if the surface is eliminated, for example, by coating one material with

    another one, with larger Tm. In this way silver coated by gold was superheated by

    25K above Tm [93]. In computer experiments we are able to eliminate the surface

    using periodic boundary conditions in all directions and thus can investigate bulk

    melting.

    In order to study surface melting we use periodic boundary conditions only in two

    directions and create a free surface in the third one. The thermodynamic melting

    temperature was found to be Tm = 2220 10K by means of the method proposed

    by Lutchko et al. [77] (the bulk melting temperature of a perfect sample is Tb =

    2550 5). Melting of crystals begins at the surface, because the activation energy for

    formation of a liquid phase is lower at the surface, than in the bulk. The liquid layer

    at the surface eliminates the barrier for nucleation of the liquid phase, and thus no

    metastability effects (superheating) exist.

    Most of the theoretical models of surface melting phenomenological in nature,

  • ABSTRACT ix

    and therefore neglect the atomistic details of the phenomenon. Only recently the

    first microscopic theory has been proposed [36], which is capable of describing static

    properties of the rare-gas crystals. The microscopic description of surface melting

    phenomena emerged mainly from computer simulations.

    We studied surface premelting of vanadium using molecular dynamics. The struc-

    tural, transport and energetic properties of the various low-index surfaces of vana-

    dium, namely Va(001), Va(011) and Va(111), at different temperatures were inves-

    tigated. We found that upon increasing the temperature the vibrations of atoms at

    the surface region becomes so large, that they disturb each other. As a result,

    point defects are generated which begin to migrate between the surface layers and an

    adlayer appears on the top of the first surface layer. The disorder begins to spread

    from the topmost layer to a deeper ones. At higher temperatures a thin quasiliquid

    film appears in the surface region. The observed premelting phenomena are most

    pronounced in the surface region of the least packed Va(111) face, and is less notice-

    able for the closest packed Va(011) face. Similar results were obtained in simulations

    of fcc metals, where the least packed face (011) exhibits premelting, while the closest

    packed face (111) remains ordered almost up to the melting point.

    In order to understand the relation between the bulk and surface melting, we

    applied the Born criterion of melting to the surface region and found a linear relation

    between the activation energy of surface defects and the melting temperature. This

    relation was confirmed by results of experiments and computer simulations of metals

    with fcc structure [92]. In order to test the model for metals with bcc structure, we

    calculated the activation energy of the surface defects for the least packed Va(111)

    and compared it with theoretical prediction. The agreement between the theory and

    simulations was found to be reasonable. A general conclusion was made that the

  • ABSTRACT x

    Born criterion correctly describes both surface and bulk melting, and may provide

    the missing link which will finally tie together these two scenarios for melting

    transitions.

  • List of symbols

    T temperature

    P pressure

    V volume

    N number of atoms

    S entropy

    E eenrgy