Born Criteria Research Articles

FACTORS CONTROLLING THE STRONGEST SIZES IN THE INVERSE HALL–PETCH RELATIONSHIP

Incorporating the bond–order–length–strength correlation mechanism [C. Q. Sun, Prog. Solid State Chem.35, 1 (2007)] and Born's criterion for melting [J. Chem. Phys.7, 591 (1939)] into the conventional Hall–Petch relationship has turned out an analytical expression for the size and temperature dependence of the mechanical strength of nanograins, known as the inverse Hall–Petch relationship (IHPR). Reproduction of the measured IHPR of Ni , NiP , and TiO 2 nanocrystals revealed that: (i) the competition between the size-induced energy–density gain and atomic cohesive energy loss in the surface skins of nanograins originate from the IHPR; (ii) the competition between the activation and inhibition of atomic dislocations motion activate the entire IHPR behavior; (iii) the bond nature involved and the T/Tm ratio between the temperature of operating and the temperature of melting dictate the measured strongest sizes of a given specimen; (iv) a quasimolten phase present before melting determines the size-induced softening and the superplasticity of nanostructures.

Theoretical strength and structural response of Cu crystal

The theoretical strength and structural response of FCC crystal Cu under uniaxial loading along one edge of a unit cell have been investigated with MAEAM. The stability criteria of the tetragonal system are extended to the generally stressed orthogonal system. The results show that, even if an orthorhombic path is applied, the deformation is spontaneously along the tetragonal Bain path till tensile elastic limit at stretch λ1=1.085 where the Born criterion is violated. The branched orthogonal path is preferred over the conventional tetragonal Bain path from minimization of the stress σ1 or the energy E. Although a stress-free BCC phase with the local maximum energy of −3.460eV appearing either in compressive region or in tensile region is unstable and would slip spontaneously into a near neighbor stress-free mBCT phase with the local minimum energy of −3.461eV, the initial FCC phase with the lowest energy of −3.489eV is the most stable in correspondence with the actual behavior of pure Cu. Furthermore, the calculated elastic moduli of FCC phase are in good agreement with the experimental values. The stable region ranges from −2.89GPa to 7.252GPa in the theoretical strength or from 0.912 to 1.085 in the stretch λ1 correspondingly.

Ab initio study of polymorphism in layered ternary carbide M 4AlC 3 (M = V, Nb and Ta)

The mechanical and thermodynamic stabilities of M(4)AlC(3) (M = V, Nb and Ta) and Ti(4)AlN(3) polymorphs were investigated by means of the first-principles pseudopotential total energy method. All compounds satisfied the Born criteria for mechanical stability, but had different thermodynamic stabilities. Only Ta(4)AlC(3) showed a possible polymorphic phase transformation driven by thermodynamic competition. The present theoretical results support the polymorphism of Ta(4)AlC(3) in experimental synthesis and explain the underlying thermodynamic mechanism. Polymorphism was excluded from V(4)AlC(3), Nb(4)AlC(3) and Ti(4)AlN(3). (C) 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Melting behavior of single two-dimensional crystal

In an experimental system millimeter-sized steel balls repel each other through the Coulomb force to imitate a two-dimensional (2D) atomic lattice in a vacuum both topologically and dynamically. Care has been taken to avoid the formation of grain boundaries. This 2D single crystal melts into a liquid via the hexatic state consistent with the Kosterlitz-Thouless-Halperin-Nelson-Young scenario. Initially in the melting process defects of the 2D lattice tend to emerge from the edge of the crystal. These defects are found to be close to the liquid state according to the Lindemann and Born criteria, confirming the idea of edge melting.

Bridging Born and Lindemann criteria: The role of interstitial defects

We investigate theoretically the crucial role of the interstitial defects that trigger the melting of a surface-free elemental crystal. Based on a modified ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ lattice model, we have studied the quasistatic and dynamic properties of a face-centered-cubic elemental crystal. The avalanche of interstitial defects due to cooperation proves to be the instability mechanism at the melting point that bridges Lindemann and Born criteria.

Structural and mechanical stability of La3Ga5.5Ta0.5O14single crystal under hydrostatic pressure

The high pressure structural and elastic behaviour of LGT single crystal has been explored. The phase diagram has been investigated up to 20 GPa by x-ray diffraction and the equation of state deduced. The complete set of second order elastic constants (S.O.E.C.) and their pressure dependence have been determined by bulk acoustic wave measurements up to 8 GPa. The bulk and linear moduli and their pressure variations have been calculated and compared to values obtained by x-ray diffraction. The stability Born criteria have been deduced.

Molecular dynamics study of melting of the bcc metal vanadium. I. Mechanical melting

We present molecular dynamics simulations of the homogeneous (mechanical) melting transition of a bcc metal, vanadium. We study both the nominally perfect crystal and one that includes point defects. According to the Born criterion, a solid cannot be expanded above a critical volume, at which a ``rigidity catastrophe'' occurs. This catastrophe is caused by the vanishing of the elastic shear modulus. We found that this critical volume is independent of the route by which it is reached, whether by heating the crystal or by adding interstitials at a constant temperature which expand the lattice. Overall, these results are similar to what was found previously for an fcc metal, copper. The simulations establish a phase diagram of the mechanical melting temperature as a function of the concentration of interstitials. Our results show that the Born model of melting applies to bcc metals in both the nominally perfect state and the case where point defects are present.

Structural and electronic properties of diamond with hypothetical vacancies stabilized by nitrogen or boron atoms

In an effort to understand the behavior of hard materials in the presence of vacancies or impurities, we study the structural and electronic properties of diamond with specified vacancies. In our model, five carbon atoms in the shape of a centered tetrahedron are missing and 12 carbon atoms surrounding the vacancy are replaced by nitrogen or boron atoms. The bulk modulus of the material with nitrogen substitution is greater than that without substitution around vacancy, although it is still slightly smaller than that of vacancy-free diamond. Boron substitution results in a substantial relaxation of atomic positions and a large reduction in bulk modulus. Born's criterion for mechanical stability is satisfied in both cases. The calculated electronic structures suggest that the doping of N atoms around the vacancy as modeled here fails to generate conduction electrons while that of B atoms successfully produces conducting holes.

Structural Phase Transformation (F. C. C. \ra B. C. C.) in F. C. C. Metals and Their Stability on the Path of Transformation

Because of its importance in Solid-State Physics, Metalurgy, Solid Mechanics and geophysics, theoretical strength calculations are performed to locate the stress-free b.c.c phase on three f.c.c metals (Ca, Pb, Ir). Internal energies correspponding to the unstresed b.c.c. and f.c.c. phase, and the required stress and energy changes for f.c.c.\rab.c.c. transformation for these crystals are computed. To determine the range of stability (G stability), the Born criterion is used by calculating the values of deformation connecting the stress-free b.c.c. and stress-free f.c.c. phases of Ca, Pb and Ir. The studied crystals are subjected to unconstrained (100) uniaxial tension in all computations, and E.G.E.P. (Extended generalized exponantial potential) model is used to carry out these calculations.

Theoretical strength and phase stability of Cu and Ni under [100] uniaxial loading

Based on Born's criteria we studied phase stability and theoretical strength of fcc crystals of copper and nickel under [100] uniaxial loading. The calculation was carried out using a simple and completely analytical embedded atom method (EAM) potential proposed by the present authors. For Cu, the calculated value of its theoretical strength (0.33 ? 1011 dyn?cm-2) agrees well with the experimental value (0.30 ? 1011 dyn?cm-2), while the calculated strain (9.76%) is somewhat larger than the experimental one (2.8%). For Ni, its theoretical strength and strain predicted using the EAM potential are found smaller than those predicted using a pair potential. It is worthy to note that unlike previous calculations, in which pair potentials were used and three unstressed fcc, bcc, and fct structures included (for Ni only fcc state is found stable, while for Cu both fcc and bcc states are predicted stable), in present calculations using EAM potential the [100] primary loading path passes through only two zeroes (a stable unstressed fcc structure and an unstable stress-free bcc structure) either for Cu or for Ni.

Mechanical instability in ice I h. A mechanism for pressure-induced amorphization

The phenomenon of pressure-induced crystalline →amorphous phase transformation in ice Ih has been studied with constant-pressure molecular dynamics calculations. It was found that although the temperature dependency of the transformation pressure resembles that of the extrapolated melting curve, the actual mechanism is due to a mechanical instability in the water framework. The elastic moduli calculated at the transition show a violation of Born criteria for mechanical stability.

Structural Phase Transformation (F.C.C. → B.C.C.) in Ag, Pt, and Sr and Their Stability on the Path of Transformation

AbstractTheoretical calculations are made to locate the stress‐free b.c.c. phase in three f.c.c. metals (Ag, Pt, and Sr). Internal energies corresponding to the unstressed b.c.c. and f.c.c. phases, and the required stress and energy changes for f.c.c. → b.c.c. transformation in these crystals are presented. To determine the range of stability (G stability) the Born criterion is used by calculating the values of corresponding elastic moduli on the line of deformation connecting the stress‐free b.c.c. and stress‐free f.c.c. phases of the crystal. The Morse potential model is used to carry out these calculations.

Mechanical stability of iron under hydrostatic stresses

A comprehensive investigation of the mechanics of iron subjected to arbitrary fluid pressure has been carried out. Apart from the classical elastic moduli (k, μ, and μ′) and conventional elastic moduli (Green and stretch moduli) computations are carried out for a family of generalised moduli of which the conventional moduli are just specific members. With the generalised moduli the mechanical stability of iron is investigated through Born criteria. It is found that classical stability, Green stability and stretch stability are all represented uniquely by the present generalised scheme. The definition of effective classical moduli under stresses enabled the amalgamation of the Born criteria of lattice stability into the single classical criteria of lattice stability of cubic crystal under hydrostatic loading environment. Computations are also carried out to investigate the coordinate and stress dependence of Young's modulus of elasticity, Poisson's ratio, mean velocity of elastic wave, and Debye temperature. Surprisingly, it is found that all these properties of solids play an important role in representing the mechanical stability of the solid. The path of uniaxial loading of iron is also investigated along with its internal energy variation on this path. This indicated the existance of stress-free fcc phase of iron on the path of uniaxial deformation at cell length a=3.6444 A giving enthalpy of transformation (bcc→fcc) of 1.1 kJ/mol in good agreement with experimental results.

Pressure dependences of the elastic constants and structural stability of Zn1-xMnxSe with x=0.37 and 0.53.

We have measured the transit times of 30-MHz ultrasonic waves in wurtzite-structured ${\mathrm{Zn}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Mn}}_{\mathit{x}}$Se, with x=0.37 and 0.53, at 296 K using hydrostatic pressures up to 4.8 kbar. Our results indicate that pressure causes increases in the longitudinal-wave elastic constants and decreases in the shear-wave elastic constants. The changes are larger the higher the Mn concentration and indicate that pressure reduces structural stability more the higher the Mn content. Using a modified Born criterion and data obtained by others on ZnSe we deduce a lower structural-transition pressure the higher the Mn concentration. We attribute this to pressure causing a greater increase in the degree of hybridization of Mn 3d(${\mathit{t}}_{2}$) with Se 4p orbitals, and thereby a greater reduction in the availability of Se 4p orbitals for tetrahedral bond formation the higher the Mn concentration. Our results are compared with those on ${\mathrm{Cd}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Mn}}_{\mathit{x}}$Te.

Elastic constants and their pressure dependences in Cd1-xMnxTe with 0

We have measured ultrasonic transit times to determine the elastic constants of ${\mathrm{Cd}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{Mn}}_{\mathrm{x}}$Te crystals with 0\ensuremath{\le}x\ensuremath{\le}0.52 and of ${\mathrm{Cd}}_{0.52}$${\mathrm{Zn}}_{0.48}$Te at 296 K using hydrostatic pressures up to 4 kbar. It is found that Mn, but not Zn, weakens the zinc-blende crystal structure and makes it less stable under pressure. Applying a modified Born criterion to our data, we deduce the pressure expected to cause the transition to the rocksalt structure in each compound. The influence of Mn we attribute to Mn 3d orbitals hybridizing into the tetrahedral bonds.

Pressure dependence of the elastic moduli of semimetallic(Ti1−xVx)2O3at 296 K. Softening of a slow shear mode and two other elastic eigenvalues whenx≈0.04

Ultrasonic 30-MHz-wave travel times have been measured at 296 K for semimetallic, single-crystal ${({\mathrm{Ti}}_{1\ensuremath{-}x}{\mathrm{V}}_{x})}_{2}{\mathrm{O}}_{3}$ samples with $0.02\ensuremath{\le}x\ensuremath{\le}0.09$ as a function of hydrostatic pressure up to 4 kbar. The moduli of all waves except the slow transverse (ST) mode propagating in the [100] direction in the $x=0.04$ sample stiffen with increasing pressure. This ST mode is a symmetry-breaking one, and its modulus, ${\mathrm{C}}_{\mathrm{ST}}$, is an eigenvalue of the elastic constant matrix. Linear extrapolation of ${\mathrm{C}}_{\mathrm{ST}}$ implies a structural transition at very high pressure (1000 kbar using the Born criterion ${\mathrm{C}}_{\mathrm{ST}}$=0, and 275 kbar using a Demarest et al.-type criterion $\frac{{\mathrm{C}}_{\mathrm{ST}}}{B}=0.2$, where $B$ is the bulk modulus). The pressure dependences of all six independent elastic constants, ${C}_{11}$, ${C}_{12}$, ${C}_{13}$, ${C}_{14}$, ${C}_{33}$, and ${C}_{44}$ are determined for samples with $x=0.02$ and $0.04$ or $0.05$ and of all ${C}_{\mathrm{ij}}$'s except ${C}_{13}$ for $x=0.09$ samples. Comparison with data for semiconducting ${\mathrm{Ti}}_{2}$${\mathrm{O}}_{3}$ does not reveal any simple correlation between the $\frac{{\ensuremath{\Delta}C}_{\mathrm{ij}}}{\ensuremath{\Delta}P}$ values with electrical resistivity, lattice parameters, interatomic spacings, or the values of the ${C}_{\mathrm{ij}}$'s. It is found that the other eigenvalues of the ${C}_{\mathrm{ij}}$ matrix usually increase with pressure. Samples with $x=0.04$ or $0.05$ are exceptional again in that two other eigenvalues are decreased by pressure. One of them is associated with a symmetry-breaking mode and is approximately equal to ${C}_{11}\ensuremath{-}{C}_{12}$ because ${C}_{14}$ is very small.

Temperature dependence of the ideal fracture strength of a b.c.c. crystal

Abstract The temperature dependence of ideal strength is studied for a b.c.c. crystal in [001] and [001] loadings. Born's criterion, with Milstein's variables, is employed to determine the strength. The Helmholtz free energy is calculated under the self-consistent Einstein model devised by Matsubara. For mathematical convenience, a Gaussian type pair-wise central potential is employed to represent atomic interactions. We attempt to fit both elastic constants at 0 K and the thermal expansion for α-Fe, but both cannot be fitted simultaneously. Three different potentials are used for calculation and the results are compared. The calculated value of the strength is closely related to the predicted value of C 11 – C 12 or Young's modulus. With a potential which is approximately fitted to the experimental Young's modulus at 0 K, the calculated strength and the fracture strain are found to be in fair agreement with the experimental results for the α-Fe whiskers. Tensile strength in [011] loading is about five ti...

Elastic anharmonicity of InP: Its relationship to the high pressure transition

Ultrasonic wave transit times have been measured in n-type InP at room temperature using hydrostatic pressures up to 4 kbar. Linear pressure dependences are found for the elastic stiffness moduli implying that at the high pressure structural-electrical transition the shear-to-bulk modulus ratio (C 11−C 12) 2B has a (fractional) value which fits the modified Born criterion for stability developed by Demarest et al . The anharmonic force constants and some of the third order elastic constants are found to be smaller the higher the transition pressure for indium III-V compounds.

Elastic moduli and mode gammas of GaP: Their relationship to those of other isomorphic crystals and the high pressure structural-electrical transition

The transit times of ultrasonic waves associated with piezoelectrically inactive modes have been measured in sulfur-dope, n-type GaP using hydrostatic pressures up to 5 kbar. The velocities and elastic constants of most modes increase linearly with pressure but C S = 1 2 ( C 11− C 12) decreases slightly. The behavior of the elastic constants and their associated mode gammas are compared to those of other III–V compounds and of Si and Ge. We suggest that the behavior of C S is related to the occurrence of the structural-electrical transition which occurs at high pressure. A modified Born criterion for lattice stability developed by Demarest et al . is used to explain this relationship. The contribution of thermal dilatation to the temperature dependence of each elastic constant is deduced. Using it and data from the literature we find that the explicit thermal contribution to each elastic constant is significant and in the case of C 12 has an anomalous sign.

The high pressure phase transitions in KF and RbF

The high pressure B1 ⇹ B2 transitions in KF at ~ 40 kbar and RbF at ~ 30 kbar have been studied using hydrostatic X-ray diffraction. No other transitions have been observed. The addition of the ΔV V for these transitions to the already existing body of literature on B1–B2 transitions in alkali halides permits an extension of Pauling's theory to larger values of radius ratios. It also permits the modified Born criterion for predicting phase transitions to be further verified. Values of ionic radii for 8 coordination we suggest are 1.33 Å for F −, 1.84 Å for Cr −, 2.00 Å for Br − and 2.27 Å for I −.