Hexagonal Close-packed Crystals Research Articles

Phase behavior of polarizable spherocylinders in external fields

Applied electric fields are known to induce significant changes in the properties of systems of polarizable molecules or particles. For rod-shaped molecules, the field-induced behavior can be rather surprising, as in the case of the negative electric birefringence of concentrated solutions of rodlike polyelectrolytes. We have investigated the interplay of shape anisotropy and field-induced anisotropy in molecular dynamics simulations of systems of polarizable soft spherocylinders in an electric field, in the limit of infinitely anisotropic polarizability, taking full account of mutual induction effects. We find a novel crystalline structure (K(2)) in the high-field limit, whose formation is driven by interactions between induced dipoles. For high pressures, the phase diagram exhibits a polar nematic phase between the hexagonal close-packed crystal phase and the K(2) phase. We also compare this system with an analogous system of spherocylinders with permanent electric dipoles and find that qualitatively similar behavior is obtained in the limit of strong coupling of the permanent dipoles to the external field.

Optical properties of inverted opal photonic band gap crystals with stacking disorder.

We present a detailed study of the effect of planar stacking disorder on optical properties of inverted opal photonic crystals. Systems with periodic stacking sequences are first studied that include face centered cubic, hexagonal close-packed, and doubly hexagonal close-packed photonic crystals. For the structures with periodic stacking order, we evaluate the band structure followed by calculation of transmission spectrum along the direction perpendicular to the hexagonal close-packing plane of the structures. Inverted opal photonic crystals with random stacking sequences are then studied by calculating average transmittance of the photonic crystal slabs over various random stacking configurations. The position and width of the lowest stop gap along the direction normal to the hexagonal close-packing plane is found to be invariant, regardless of the stacking sequence in the photonic crystal. We show how the propagation properties at higher frequencies are affected by the stacking configurations (both periodic and disordered stacking sequences), particularly those near the edges of the absolute band gap. The obtained results are directly relevant to transmission/reflection experiments on inverted opal photonic crystals with complete band gap.

Colloidal epitaxy: playing with the boundary conditions of colloidal crystallization.

We have studied, with quantitative confocal microscopy, epitaxial colloidal crystal growth of particles interacting with an almost hard-sphere (HS) potential in a gravitational field and density matched colloids interacting with a long-range (LR) repulsive potential with a body-centred cubic (BCC) equilibrium crystal phase. We show that in both cases it is possible to grow thick, stacking fault-free metastable crystals: close-packed crystals with any stacking sequence, including hexagonal close packed (HCP), for the HS particles and face-centred cubic (FCC) in the case of the LR colloids. In accordance with recent computer simulations done for HS particles it was found that the optimal lattice constant to grow HS HCP crystals was larger than that of equilibrium FCC crystals. In addition, because of the absence of gravity, pre-freezing could be observed for the particles with the LR potential on a template of charged lines. We also argue that the ability to manipulate colloids with highly focused light, optical traps or tweezers, will become an important tool in both the study of colloidal crystallization and in making new structures. We show how cheap 2D and 3D templates can be made with optical tweezers and demonstrate, in proof of principle experiments with core-shell colloids, how light fields can generate crystal nuclei and other structures in the bulk of concentrated dispersions and how the effect of these structures on the rest of a dispersion can be studied quantitatively in 3D.

Structural aging of crystals of hard-sphere colloids.

We report a detailed experimental study of the aging of the (initial) random hexagonal close-packed (rhcp) crystals formed in suspensions of hard-sphere colloids near the melting point. By suspending the same colloidal particles in two different mixtures of solvents we are able to tune the strength of the gravitational forces acting on the particles. The crystal structure is deduced from diffraction patterns measured by the light scattering equivalent of powder x-ray crystallography. A spontaneous aging of the structure is observed over long periods of time, consisting of a fraction of pure face-centered cubic (fcc) crystals growing at the expense of the randomly stacked crystallites. The rate of growth of the new crystals is small and consistent with the predictions by Pronk and Frenkel [13]. Gravity is also revealed to affect the crystals and favor fcc order but through a slow gradual rearrangement of the random stacking. An important new observation is that small mechanical perturbations can strongly affect the structure of the colloidal crystals, promoting fcc growth and interfering with the spontaneous aging process. Previous experimental results are also discussed in the light of these new findings.

Mobility of interstitial clusters in alpha-zirconium

Clusters of self-interstitial atoms (SIAs) formed in displacement cascades in metals irradiated with energetic particles play an important role in microstructure evolution under irradiation. They have been studied in the fcc and bcc metals by atomic-scale computer simulation, and in this article, we present the results of a similar study in a hexagonal close-packed (hcp) crystal. Static and dynamic properties of clusters of up to 30 SIAs were studied using a many-body Finnis-Sinclair type interatomic potential for Zr. The results show a qualitative similarity of some properties of clusters to those for cubic metals. In particular, all clusters larger than four SIAs exhibit fast thermally activated one-dimensional (1-D) glide, which is in a 〈11\(\bar 2\)0〉 direction in the hcp lattice. Due to the structure of the hcp lattice, this mechanism leads to two-dimensional mass transport in basal planes. Some clusters exhibit behavior peculiar to the hcp structure, for they can migrate two-dimensionally (2-D) in the basal plane. The jump frequency, activation energy, and correlation factors of clusters have been estimated, and comparisons drawn between the behavior of SIA clusters in different structures.

Analytical and Numerical (Monte Carlo) Studies of Point and Space Group Symmetry-Breaking in the Liquid–Solid Phase Transition

We present a complete classification of symmetry-breaking terms in the one- and two-body densities for crystallographic space group P63/mmc according to irreducible representations (IR) of the extended Euclidean space group E(|R3) in three dimensions. We show that these IRs of E(|R3) are the order parameter symmetries of the liquid–solid phase transition from a spatially uniform liquid, with space group E(|R3), to a crystal of space group P63/mmc. The analytical group-theoretical methods derived in this paper furnish mathematically rigorous criteria at the level of the microscopic spatial structure of materials for identifying the occurrence of a spontaneous solidification of a liquid and the low-symmetry crystallographic subgroup of E(|R3) in which the liquid crystallizes. We prove in particular that all of the irreducible order parameter symmetries of the liquid–solid phase transition appear in the two-body density, whereas the vast majority of symmetry-breaking IRs of E(|R3) is absent from the one body density. The novel aspects arising from this finding for analyzing phase transitions are discussed. We present numerical results of an exact Fourier path integral Monte Carlo simulation of spontaneous crystallization of liquid natural argon at the experimental triple point temperature into a hexagonal close-packed crystal, with space group P63/mmc. The experimental relevance of this system in particular is addressed. We mention the relation of the theory presented here to experimental X-ray diffraction and neutron scattering analysis of the microstructure of materials. We outline the applicability of formal and/or computational techniques derived and employed in this research to (envisaged) Monte Carlo investigations of adsorbed solid and liquid films, phase transitions in liquid crystals, computation of the local electron number density for electronic energy band structure calculations, and variational shadow wave functions Monte Carlo studies of spontaneous crystallization of superfluid 4He.

The relative surface energy of hexagonal close-packed crystals

Surface energies of hexagonal close-packed (hcp) crystals for three families of lattice site planes have been computed and their angular correlations have been determined. The broken bond method and the model of stepped microprojections of crystal faces were employed. The atomic interactions were taken into account in two coordinate spheres. Relative surface energies have been computed for Be, Cd, Co, Hf, Mg, Ti, Zn, Zr. The theoretical results are compared with experimental data.

On modelling the elasto-viscoplastic response of metals using polycrystal plasticity

A constitutive framework for the elasto-viscoplastic response of metals that utilizes polycrystal plasticity is presented together with a corresponding numerical integration procedure. The single crystal equations are written in an intermediate configuration obtained by elastically unloading the deformed crystal without rotation from the current configuration to a stress-free state. The elastic strains are assumed always to be small. The accompanying numerical integration is implicit and proceeds by decoupling the volumetric and the deviatoric crystal responses. The extended Taylor hypothesis is used to relate the response of individual crystals to that of the polycrystal. Various homogeneous deformations have been simulated using the constitutive model and the integration scheme to compute the stress response and texture development. Aggregates of either face centered cubic (FCC) or hexagonal close-packed (HCP) crystals are subjected to both monotonic and non-monotonic loading histories. Numerical results demonstrate the performance of the model as well as show the stability and accuracy of the integration procedure. The present constitutive model and corresponding numerical procedures can be used to predict elastic effects (e.g. residual stresses) during the large deformation of polycrystalline materials while accounting for texture development and the associated anisotropy.

Pressure and entropy of hard-sphere crystals

The pressure and entropy of the face-centred cubic and hexagonal close-packed hard-sphere crystals were measured. At the melting density, the hcp crystal has a higher pressure. A precise equation of state is reported for the crystals. The equation extrapolates to a spinodal instability in the metastable low-density crystal.

Anisotropy of Interfaces in an Ordered HCP Binary Alloy

A multiple-order-parameter mean-field theory of ordering on a binary hexagonal- close-packed (HCP) crystal structure is developed, and adapted to provide a continuum formulation that incorporates the underlying symmetries of the HCP crystal in both the bulk and gradient energy terms of the free energy. The work is an extension of the previous treatment by Braun et al. [Phil. Trans. Roy. Soc. Lond. A355:1787 (1997)] of order–disorder transitions on a face-centered-cubic crystal (FCC) lattice. The theory is used to compute the orientation dependence of the structure and energy of interphase and antiphase boundaries in ordering to the Cd3Mg and CdMg structures, which are the HCP analogs of Cu3Au and CuAu structures in FCC. As in the corresponding FCC case, the multiple order parameters do not form a vector. Anisotropy is a natural consequence of the underlying crystal symmetries and the multiple-order-parameter continuum formation presented here. The isotropy transverse to the sixfold axis expected for a scalar order parameter is not found.

Rovibrational states of a tetrahedral molecule in a hexagonal close-packed crystal

The vibration-rotation states of a tetrahedral molecule in a field of D3h symmetry are studied by an extended group theory. Symmetry adapted rotational wave functions for the ground and the triply degenerate excited vibrational states are constructed. The crystal field potential is derived explicitly for a tetrahedral molecule in a hexagonal close-packed crystal. As an example, the rovibrational energy states of methane in solid parahydrogen are computed. Optical selection rules are also derived from the theory.

Expanded Heat Treatment to Form Residual Compressive Hoop Stress on Inner Surface of Zirconium Alloy Tubing.

A specific heat treatment process that introduces hoop stress has been developed. This technique can produce zirconium alloy tubing with a residual compressive hoop stress near the inner surface by taking advantage of the mechanical anisotropy in hexagonal close-packed zirconium crystal. Since a crystal having its basal pole parallel to the tangential direction of the tubing is easier to exhibit plastic elongation under the hoop stress than that having its basal pole parallel to the radial direction, the plastic and elastic elongation can coexist under a certain set of temperature and hoop stress conditions. The mechanical anisotropy plays a role to extend the coexistent stress range. Thus, residual compressive hoop stress is formed at the inner surface where more plastic elongation occurs during the heat treatment. This process is referred to as expanded heat treatment. Since this is a fundamental crystallographic principle, it has various applications. The application to improve PCI/SCC (pellet cladding...

Aggregate size effect on the predicted plastic response of hexagonal close-packed polycrystals

The effect of aggregate size (number of crystals per aggregate) on texture development and mechanical response of hexagonal close-packed (HCP) polycrystals has been studied numerically. The single crystal deforms only by basal and prismatic slip, and, hence, has an inextensible hexagonal direction (c-axis). The polycrystal is modeled using the hybrid approach developed by Parks and Azhi, where a fourth-order projection tensor depending on the average of the c-axis orientation plays a key role in the formulation. In this model, the deformation applied to the crystals of the aggregate is determined by this projection tensor, which depends on aggregate size, and the imposed macroscopic deformation. The dependence of the average projection tensor on aggregate size is studied by simulating plane strain compression tests on aggregates of different size comprised of inextensible HCP crystals. Both material point and finite element simulations are used. Numerical results show that (i) the average projection tensor is very sensitive to aggregate size, resulting in predictions of sharper texture and stronger hardening for smaller aggregates, and (ii) the spatially non-uniform deformation among aggregates within a finite element discretization increases as the aggregate size is reduced, tending to diffuse texture. Based on this study, a minimum number of 250 crystals per aggregate is suggested to minimize the aggregate size effect in numerical simulations of large-scale HCP metal deformation processes using Parks and Azhi's hybrid model.

Pair model of magnetostriction for hexagonal close-packed crystals.

The magnetostriction of ferromagnets which crystallize in the hcp structure is discussed phenomenologically. It is shown that, in addition to the coupling of the magnetization to uniform deformations, there is a linear coupling to an internal distortion, i.e., one which does not alter the shape of the unit cell. This distortion transforms in the same manner as the orthorhombic (\ensuremath{\gamma}) strain under the operations of the space group of the hcp lattice; hence the two are also coupled linearly. Taking these couplings into account, the pair model of magnetostriction is extended and applied to an analysis of the magnetostriction of hcp Co and Co-Fe alloys. It is shown that, when the effect of the internal distortion is included, a reasonable account can be given for the strengths and signs of all the modes of magnetostriction in this series.

Orientational Disorder in Solvent-Free Solid C 70

The high-temperature structure of solvent-free C(70) has been determined with high-resolution x-ray powder difraction and electron microscopy. Samples crystallized from solution form hexagonal close-packed crystals that retain an appreciable amount of residual toluene, even after prolonged heating. Samples prepared by sublimation, which contain no detectable solvent, are primarily face-centered cubic with some admixture of a hexagonal phase. The relative volume of the hexagonal phase can be further reduced by annealing. The structures of both phases are described by a model of complete orientational disorder. The cubic phase contains an appreciable density of stacking faults along the [111] direction.

The roughening transition of a growing 4He crystal

We present a careful study of the growth of hexagonal close-packed 4He crystals near the roughening transition of their (0001) surfaces, at TR = 1.28 K. We confirm that the dominant growth mechanism in the smooth state (T < TR) is the two-dimensional nucleation of terraces of atomic height. Moreover, we discovered a new regime of growth, which takes place slightly below TR, for a surface state intermediate between smooth and rough. It exists in a temperature domain whose width increases with the applied driving force. All our experimental results are consistent with a new calculation based on the critical theory of roughening, which predicts a completely continuous or infinite-order transition. Precise values are deduced for the roughening temperature (TR = 1.28 ± 0.01 K) and the relative amplitude of the lattice potential (tc = 0.63 ± 0.13).

The further geometry of grain boundaries in hexagonal close-packed metals

A technique is given for finding partial DSC vectors appropriate to crystals with more than one atom per lattice site. The DSC lattice is made up of vectors that represent displacements of one crystal with * Now at Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. respect to the other that leave the boundary structure shifted, but not complete. A new, rapid method for finding the step vectors associated with perfect DSC dislocations is described. Partial DSC vectors and step vectors for perfect DSC dislocations in hexagonal close-packed crystals are determined. The availability of reactions between lattice partial dislocations and grain boundaries in hexagonal closepacked crystals is also assessed. 0108-7681/87/050416-07501.50 (~ 1987 International Union of Crystallography FU-RONG CHEN AND A. H. KING 417

Helium crystals

Abstract Hexagonal close-packed helium crystals in equilibrium with superfluid have been found to be one of the few systems in which an anisotropic solid comes into true thermodynamic equilibrium with its melt. The discovety of roughening transitions at the liquid-solid interface have shown this system to be ideal for the study of the statistical mechanics of interface structures. We describe the effect of roughening on the shape and growth of macroscopic crystals from both the theoretical and experimental points of view.

Monte Carlo simulation of the crystal-melt interface

Abstract Initial results are reported on Monte Carlo calculations on the crystal-molt interface of Lennard-Jones particles, using much larger assemblies over much longer Monte Carlo chains than hoe been used previously. In particular, ten crystal layers are included in calculations on the (0001) close-packed face of a hexagonal close-packed crystal. The major result concerns changes in the crystal close to the interface, which becomes significantly defective. These layers near the interface begin to show some disordering, and an overall expansion in the direction of the liquid. This crystal disordering, together with the softness of the potential, is thought to he responsible for the apparent, disappearance of the density deficit obtained in previous calculations at the interface between the melt and the perfect crystal surface. Whether this disordering is an equilibrium effect or reflects the onset of melting is as yet unclear.

Stability of fcc and hcp hard-sphere crystals

The face-centered cubic (fcc) and hexagonal close-packed (hcp) hard-sphere crystals are known to have almost the same entropy. In this paper, the entropy of both crystals close to dense packing is calculated by direct evaluation of the configurational partition function. This is done using an improved Monte Carlo integration technique. The result is that the entropy difference per particle is 0.02 k ( k is Boltzmann's constant) favouring the fcc structure. This value is in accordance with the estimate based on the supposed relationship between the free volume and ( R 2), ( R 2) being the mean square displacement of a sphere from its lattice site.