Semi-infinite Models Research Articles

Atomistic and multiscale analyses of brittle fracture in crystal lattices

Applicability of the Griffith criterion [A. A. Griffith, Philos. Trans. R. Soc. London, Ser. 221, 163 (1920); S. Zhang, S. L. Meilke, R. Khare, D. Troya, R. S. Ruoff, G. C. Schatz, and T. Belytschko, Phys. Rev. B 71, 115403 (2005)] for predicting the onset of crack extension in crystal lattices is systematically evaluated using atomistic and multiscale simulations with a focus on the effects of crack size and lattice discreteness. An atomistic scheme is developed to determine the true Griffith load defined by the thermodynamic energy balance of crack extension for both finite-sized and semi-infinite crack models. For a model monolayer lattice, we identify a characteristic crack length (about ten lattice spacings) below which the Griffith fracture stress markedly overestimates the true Griffith load. Through a stability analysis of crack-tip bond separation, the athermal (nonthermally activated) loads of instantaneous fracture are determined, thereby yielding the estimated lattice trapping range. Our simulations show that the strength of lattice trapping depends on the interaction range of the interatomic force fields. Using the reaction pathway exploration method, we determine the minimum energy paths of bond breaking and healing at a crack tip, giving a more precise estimate of the lattice trapping range. The activation energy barriers governing the rate of kinetic crack extension are extracted from the minimum energy paths. Implications concerning the distinction between the athermal and Griffith fracture loads are discussed. Based on these results, a general criterion is established to predict the onset of crack growth in crystal lattices. In addition to taking into account the lattice trapping effect, this criterion is applicable to a large spectrum of crack sizes.

Effect of axial finiteness on electron heating in low-frequency inductively coupled plasmas

Total power absorption inside the plasma (by taking the thermal motion of the electrons into account) has been calculated using different inductively coupled plasma models. The comparison shows that in the low-frequency region the results of the semi-infinite plasma models are different from those of the finite-length plasma models. The semi-infinite plasma models show net reduction of heating in the low-frequency region, due to thermal motion of the electrons from inside the skin region to outside the skin region. The finite-length plasma models on the other hand (due to change in the skin depth owing to the boundary condition of E=0 at z=L, and reflection of electrons from the plasma boundary) show that the decrease in heating due to the motion of the electrons from inside the skin depth to outside the skin depth is recovered by the reflection of the electrons from the plasma boundary. Hence, it is concluded that the results of the semi-infinite plasma models presented by Tyshetskiy et al. [Phys Rev. Lett. 90, 255002 (2003)] can be misleading (in the low-frequency region), since they overlooked the effect of axial finiteness of the plasma.

Ovoid geometry of the Pacinian corpuscle is not the determining factor for mechanical excitation

Static displacements in Pacinian corpuscles (PCs) were measured using video microscopy. Mechanical stimuli of 10–40 µm steps were applied to the PC capsule surfaces using cylindrical contactors with different diameters. Displacements parallel to the stimulation axis were measured at various locations in the focal plane of the optical setup. In contrast to previous data in the literature, the displacements within the corpuscle were found to be linearly related to the indentation amplitude. Displacements decreased as a function of lamella depth, with a more negative slope close to the surface and less negative slope at deeper locations. The experimental data were compared to the predictions of a previous mechanical model, and to the results of two new models: () elastic semi-infinite continuum model; () ovoid isotropic finite-element model. Although the previous model did not specify displacement boundary conditions, it predicted the current experimental results well. On the other hand, the experimental displacements were found to be smaller than those predicted by the semi-infinite continuum and finite-element models. However, both semi-infinite continuum and finite-element models yielded close results, which show that the three-dimensional ovoid geometry of the corpuscle is not the primary factor for determining the displacements in physiological conditions. Furthermore, simulations with the finite-element model using a wide range of material properties yielded similar results. This supports the hypothesis that a homogeneous isotropic model for the PC cannot predict experimental results. The modeling analyses suggest that the experimental results are largely affected by the displacement of the incompressible interlamellar fluid and the layered structure of the corpuscle.

Model of Tellurium- and Zinc-Doped Indium Antimonide Solidification in Space

Numerical simulations were performed to determine the effect of residual microaccelerations on the distribution of dopants (Te and Zn) during solidification of InSb in space. A moving geometry model was developed and used to account for the reduction in melt size during growth. The model demonstrates that diffusion controlled segregation in doped InSb can be obtained at 10 -5 g 0 gravity for the considered growth parameters. The results for the moving geometry and semi-infinite melt domain models are compared to each other and to the analytical correlations for the case of diffusion-controlled segregation.

Solar prominences with Na and Mg emissions and centrally reversed Balmer lines

We study spectral lines in exceptionally bright solar limb prominences with pronounced sodium and magnesium emission. We find that most prominences with significant NaD2 and Mgb2 emission show centrally reversed profiles of H-alpha and occasionally even of H-beta, which are are well reproduced by semi-infinite models. The maximum H-alpha source function corresponds to an excitation temperature of 3950 K, for pronounced central reversions 4000 K; the related optical thickness exceeds 10.0. The narrow widths of the NaD2 and Mgb2 profiles yield a non-thermal broadening of 5 km/s.

Surface critical behaviour of semi-infinite systems with cubic anisotropy at the ordinary transition

The critical behavior at the ordinary transition in semi-infinite n-component anisotropic cubic models is investigated by applying the field theoretic approach in d=3 dimensions up to the two-loop approximation. Numerical estimates of the resulting two-loop series expansions for the critical exponents of the ordinary transition are computed by means of Pade resummation techniques. For $n<n_{c}$ the system belongs to the universality class of the isotropic n-component model, while for $n>n_{c}$ the cubic fixed point becomes stable, where $n_{c}<3$ is the marginal spin dimensionality of the cubic model. The obtained results indicate that the surface critical behavior of the semi-infinite systems with cubic anisotropy is characterized by a new set of surface critical exponents for $n>n_{c}$.

Magnetization and dimerization profiles of the cut two-leg spin ladder

The physical properties of the edge states of the cut two-leg spin ladder are investigated by means of the bosonization approach. By carefully treating boundary conditions, we derive the existence of spin-1/2 edge states in the spin ladder with a ferromagnetic rung exchange and for the open spin-1 Heisenberg chain. In contrast, such states are absent in the antiferromagnetic rung coupling case. The approach, based on a mapping onto decoupled semi-infinite off-critical Ising models, allows us to compute several physical quantities of interest. In particular, we determine the magnetization and dimerization profiles of the cut two-leg spin ladder and of the open biquadratic spin-1 chain in the vicinity of the $\mathrm{SU}{(2)}_{2}$ Wess-Zumino-Novikov-Witten critical point.

Surface critical exponents at a uniaxial Lifshitz point

Using Monte Carlo techniques, the surface critical behaviour of three-dimensional semi-infinite ANNNI models with different surface orientations with respect to the axis of competing interactions is investigated. Special attention is thereby paid to the surface criticality at the bulk uniaxial Lifshitz point encountered in this model. The presented Monte Carlo results show that the mean-field description of semi-infinite ANNNI models is qualitatively correct. Lifshitz point surface critical exponents at the ordinary transition are found to depend on the surface orientation. At the special transition point, however, no clear dependency of the critical exponents on the surface orientation is revealed. The values of the surface critical exponents presented in this study are the first estimates available beyond mean-field theory.

Balanced games arising from infinite linear models

Kalai and Zemel introduced a class of flow-games showing that these games have a non-empty core and that a minimum cut corresponds to a core allocation. We consider flow-games with a finite number of players on a network with infinitely many arcs: assuming that the total sum of the capacities is finite, we show the existence of a maximum flow and we prove that this flow can be obtained as limit of approximating flows on finite subnetworks. Similar results on the existence of core allocations and core elements are given also for minimum spanning network models (see Granot and Huberman) and semi-infinite linear production models (following the approach of Owen).

The dynamic character of the hemisphere-cylinder wake

The dynamic character of the hemisphere-cylinder wake was studied over the entire range of angles of attack, i.e. α=0° to 90°. The work was carried out in two wind tunnel facilities, using hot-wire anemometry. Velocity auto- and cross-spectra in the wake reveal that the leeward vortices exhibit periodic motions, with multiple dominant frequencies for 24°<α<42°, one single dominant frequency for 15°<α<22°, while no periodic activity is detected for α<15°. In the regime 15°<α<42°, a new periodic heaving motion of the vortices is documented. Vortex heaving transitions to the more classical vortex shedding periodicity in the neighborhood of α=45°. Above α=55°, vortex shedding occurred at a single Strouhal number of 0.15. The limiting case α=90° was investigated in detail for semi-infinite and finite hemisphere-cylinder models and comparisons were made to axisymmetric bodies with different nose shapes.

Finite thickness and semi-infinite photothermal radiometric models for the characterization of semiconductors

A comparative computational study of the finite thickness and semi-infinite photothermal radiometric model is carried out for silicon, germanium, and gallium arsenide. The sensitivity of the photothermal radiometric finite thickness model towards the existence of very thin amorphous layers on crystalline substrates is also explored. This study can be used as an important guide for experimentalists.

Surface magnetization and critical behavior of aperiodic Ising quantum chains.

We consider semi-infinite two-dimensional layered Ising models in the extreme anisotropic limit with an aperiodic modulation of the couplings. Using substitution rules to generate the aperiodic sequences, we derive functional equations for the surface magnetization. These equations are solved by iteration and the critical exponent ${\mathrm{\ensuremath{\beta}}}_{\mathit{s}}$ can be determined exactly. The method is applied to three specific aperiodic sequences, which represent different types of perturbation, according to a relevance-irrelevance criterion. On the Thue-Morse lattice, for which the modulation is an irrelevant perturbation, the surface magnetization vanishes with a square-root singularity, like in the homogeneous lattice. For the period-doubling sequence, the perturbation is marginal and ${\mathrm{\ensuremath{\beta}}}_{\mathit{s}}$ is a continuous function of the modulation amplitude. Finally, the Rudin-Shapiro sequence, which corresponds to the relevant case, displays an anomalous surface critical behavior which is analyzed via scaling considerations. Depending on the value of the modulation, the surface magnetization either vanishes with an essential singularity or remains finite at the bulk critical point, i.e., the surface phase transition is of first order.

Universality classes for the dynamic surface critical behavior of systems with relaxational dynamics.

We investigate the problem of which universality classes of dynamic surface critical behavior exist for semi-infinite systems whose dynamic bulk critical behavior and static surface critical behavior are representative of a given dynamic bulk universality class and a given static surface universality class. To this end systems whose bulk dynamics are described by either model B or model A of Halperin, Hohenberg, and Ma are considered, where the dynamics are allowed to be modified near the surface as follows: In the case of model B, surface terms that locally break the conservation law for the order-parameter density \ensuremath{\varphi} are allowed; in the case of model A, \ensuremath{\varphi} is assumed to be locally conserved near the surface. Semi-infinite field-theory models are constructed that are (i) compatible with the requested bulk dynamics and the requirements of causality, detailed balance, and relaxation to thermal equilibrium and (ii) minimal in the sense that no redundant or irrelevant surface terms are included. The boundary conditions to which the surface terms of the action correspond are worked out. For model B it is shown that nonconservative surface terms are relevant. Their strength can be parametrized by a coupling constant c${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$\ensuremath{\ge}0 whose inverse plays the role of an extrapolation length for the auxiliary (Martin-Siggia-Rose) field \ensuremath{\varphi}\ifmmode \tilde{}\else \~{}\fi{}. Thus two distinct semi-infinite extensions of model B exist\char22{}one with c${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$g0 called model ${\mathit{B}}_{\mathit{A}}$, and one with c${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$=0 called model ${\mathit{B}}_{\mathit{B}}$\char22{}and each static surface universality class splits up into two dynamic surface universality classes.

Two-dimensional semi-infinite multicritical Ising models

The order-parameter correlation functions of the Z2-invariant multicritical points in the unitary minimal series of conformal field theory are derived for a semi-infinite plane constrained to fixed or free boundary conditions. These yield the corresponding universal surface exponents which distinguish the behaviour between even- and odd-critical models. We also make explicit the interplay between duality and boundary conditions.

Surface reentrance in the semi-infinite spin-1 Ising models

The critical behavior of the semi-infinite Blume-Capel and Blume-Emery-Griffiths models is investigated in the pair approximation of the cluster variation method. Equations for bulk and surface order parameters and n.n. correlation functions are given, from which analytical expressions for the second order bulk and surface critical temperatures are derived. The phase diagrams of the Blume-Capel model are classified, and the existence of a surface first-order transition is discussed. This transition is shown to be, under certain conditions, slightly reentrant, and the behavior of the surface order parameters and correlation functions is given for such a case. The extension of our results to the Blume-Emery-Griffiths model is briefly discussed.

Marginally inhomogeneous semiinfinite two-dimensional Ising models

The scaling dimensions of the semiinfinite two-dimensional Ising model with smoothly inhomogeneous couplings are determined, in general, by the eigenvalues of a Schrodinger-type equation. The author obtains exact results for a marginal, two-parameter inhomogeneity that corresponds to a supersymmetric quantum mechanical potential. The results are compared with predictions of conformal invariance.

Phase diagrams of the semi-infiniteZ(q) models by real space renormalization-group method

We investigate the three-dimensional semi-infiniteZ(q) models using an infinitesimal Migdal-Kadanoff method. A rich variety of phase diagrams is obtained. The massless spin-wave phase which appears in the infinite two-dimensionalZ(q) models on the Clock line between the disordered and ferromagnetic phase forq≧qc is also present in the semi-infinite system on the surface when the bulk is disordered. We also observe that if the bulk is in the phase to which the symmetry is engendered by a subgroupZ(p), such thatp<q, the surface of the system is in the same phase or in a less symmetrical phase to which the symmetry is engendered by a subgroupZ(m) ofZ(p) such thatp=αm (m≦p) with α an integer number satisfying 1≦α≦p. The case α=p corresponds to the least symmetrical phase. Since the infiniteZ(q) models exhibit a rich variety of phase transitions and multicritical points, the semi-infinite models present new ordinary, extraordinary, surface and special phase transitions which do not occur in the semi-infinite Ising-like systems. As theZ(q) model transforms into theX−Y model whenq→∞, we have deduced the phase diagram of the semi-infiniteX−Y model. It is qualitatively similar to the phase diagram of the semi-infinite Ising model.

Phase transitions of dilute semi-infinite Ising models.

The magnetic properties of dilute semi-infinite Ising models are studied in the mean-field theory. We obtained the exact expression for the critical temperature (${\mathit{T}}_{\mathit{C}\mathit{S}}$) of these systems and found that it is dependent on both the spin coupling constants and the concentrations of magnetic atoms on different atomic layers. The diagrams of the dependence of the magnetization on the temperature are presented in different cases.

Externally excited semi-infinite one-dimensional models

The problem of a semi-infinite harmonic chain with the first mass subject to a forced periodic motion is solved exactly. The scattering of a quantum particle by a semi-infinite Kronig–Penney chain is analyzed analytically. The behavior of extended and localized solutions is analyzed in both problems and many analogies are found between them. Special attention is paid to the asymptotic behavior of localized solutions near the band edges.

Derivation of Landau theories and lattice mean-field theories for surface and wetting phenomena, from semi-infinite Ising models

Abstract The phenomenological aspects of surface and interfacial phenomena such as wetting phase transitions are commonly studied using the classification scheme provided by the Landau theory. Although this approach is very meaningful, it is also important to establish a firm connection between the phenomenology and the microscopic models of statistical mechanics. This is true, especially in view of the remarkable sensitivity of wetting phenomena to the details of the substrate-adsorbate interactions. We study such connections and derive a variety of lattice mean-field theories, which make a bridge between the Landau theory and the semi-infinite Ising model with a surface. We discuss standard mean-field approximations (MFA) and improvements thereof, renormalization group techniques, reaction-field approximations, and cluster variation methods. We pay special attention to the derivation of the newly introduced triplet surface field h 3 in the Landau theory for wetting.