Critical Length Lc Research Articles

The dichotomy property in stabilizability of 2 × 2 linear hyperbolic systems

This paper is devoted to discuss the stabilizability of a class of 2 × 2 non-homogeneous hyperbolic systems. Motivated by the example in the Section 5.6 of Bastin and Coron’s book in 2016, we analyze the influence of the interval length L on stabilizability of the system. By spectral analysis, we prove that either the system is stabilizable for all L > 0 or it possesses the dichotomy property: there exists a critical length Lc > 0 such that the system is stabilizable for L ∈ (0, Lc) but unstabilizable for L ∈ [Lc,+∞). In addition, for L ∈ [Lc, +∞), we obtain that the system can reach equilibrium state in finite time by backstepping control combined with observer. Finally, we also provide some numerical simulations to confirm our developed analytical criteria.

Peculiar electronic properties of wider armchair graphene nanoribbons: Multiple topological end-states and “phase transitions”

In contrast to the thinner 5,6,7-AGNRs in which one pair of topological end-states is generated at a critical length Lc in a form of a metal-insulator-like “phase transition”, in wider AGNRs (with Z = 3n, 3n±1, n = ,2, 3 … zigzag rings) n ≥ 2 pairs of end-states (associated with n ≥ 2 transitions) can be produced, with very peculiar properties, such as quantum interference (QI), stemming from the interaction between (multiple) pairs of end states localized in the same narrow region. At the level of density functional theory such AGNRs at large enough lengths would be misleadingly appear as highly spin polarized with spin s = n. Our results are fully supported by experiment, wherever available. We verify that the Z = 3n-1 AGNRs are always metallic with near zero bandgap, as was originally believed, contrary to some recent suggestions, and do not strictly follow the above pattern. It is shown that in the 13-AGNRs (Z = 3n, n = 2) a unique and intense (close to the theoretical limit) constructive QI (CQI) appears, which covers a relatively wide region of lengths from about L∼ 50 Å to L∼80 Å. This peculiar L-dependent CQI is strongly related to their aromaticity pattern consisting of migrating Clar sextets. The rest of AGNRs exhibit only small kinks (for n = 2) or dips (for n = 3) in conductivity, which could be considered as weak and narrow CQI, or DQI (destructive QI), respectively.

Prioritizing Design Parameters for Stepped Chutes and Shear Stress Distribution

Stepped chutes offer high efficiency in decreasing flow velocity due to roughness; however, negative impacts may still be experienced by the receiving water body into downstream. These effects might be mitigated if geometric and hydraulic parameters governing the structure are well addressed. Herein, five influential parameters were developed, i.e., longitudinal slope S (S = tan θ), discharge (Q), pool height above steps (hp), chute width (W), and chute height (H), employing a three dimensional (3D) numerical model. Through 600 simulations, two regression models were developed for predicting depth-averaged velocity at the last step Vd (m/s) and critical length Lc (cm) at the downstream where the maximum velocity occurs, using response surface methodology (RSM) based on the mixed-level full factorial design. The prediction data obtained by developed regression models were agreeable with actual data with coefficient determination (R2) of about 0.95, highlighting the accuracy and ability of the models for the prediction of Vd and Lc. Additionally, the analysis of variance (ANOVA) was used to prioritize the impact of the studied parameters on Vd and Lc. Results highlighted that among geometric parameters, W and S had a significant influence on Vd and Lc; however, the impact of W was more pronounced. Using a regression model for Lc, a cross section was obtained, and the shear stress distribution of the downstream was compared with that of the last step and sidewalls. The shear stress patterns showed that the maximum value shifted from the side walls to the downstream between the lower and higher slopes. Further, the longitudinal distribution of shear stress at the downstream revealed that geometric and hydraulic characteristics played a negligible role in the changing pattern of shear stress. The results of this study reveal the dynamic behavior of the given structure where different geometric options are available for structure design.

Topological metal-insulator transition in narrow graphene nanoribbons?

We show that very narrow armchair graphene nanoribbons of length L and width of 2 zigzag-rings undergo a metal-insulator-like transition at a critical length Lc ≈ 10 nm, where the energy gap drops rather abruptly and topological “end” states appear, linked to dramatically transformed aromaticity. At Lc the conductivity, estimated through an invoked computational scheme, also rises almost discontinuously to a value near the nominal minimum conductivity of grapheneσmin=4e2h. The end states (at the zigzag edges) generate sharp peaks in the density of states around the Fermi level at the Dirac points, coinciding with charge-neutrality points, associated with σmin. This suggests metallic-like behaviour, which however is an uncommon combination of interrelated “short-long”, (or “bulk”-“edge”) topological-aromatic transition(s) due to strong quantum confinement, combined with inversion symmetry conflict. This “multi-transition” is rather universal occurring also for wider AGNRs but for a much smaller Lc and in a less sharp way. The assumed lower total energy of such open states is an artefact of the mean-field treatment of the electron-electron interaction. In contrast, the topological “end” states reported here are unique and not spin, but rather pseudospin polarized. Thus, any observed magnetism should be considered non-conventional or of questionable origin.

The formation of satellite droplets in micro-devices due to the rupture of neck filament

Formation of satellite droplets during drop formation or breakup leads to an increment of polydispersity of emulsion and runs contrary to the original intention of microfluidics. Thus, more attention is needed to shed light on the mechanism of satellite droplets formation to alleviate polydispersity. In this work, we studied the satellite droplet formation process in microfluidic flow-focusing junction and T-junction with capillary number Ca=μcuc/σ in a range of 0.039–0.32. The formation of satellite droplets during the thinning of neck filament was recorded by high speed camera. Using self-written programs, the evolution of the minimal width, the volume of and the length of neck filament were analyzed to understand the satellite formation process. It was found that before final pinch-off, liquid neck expriences four different thinning regimes. There is a critical length Lc=μcμd/σρd during the evolution of the neck thinning processes. When the neck width is less than the critical length, the thinning processes always experience the transition from rapid shrinking to slow shrinking facilitating the formation of satellite droplets.

Phase transitions in finite size systems

We follow recent developments of a finite size thermal field theoretic formalism, which is mainly relevant to the investigation of small size corrections to the thermodynamical properties of the quark-gluon plasma. In doing so, we rely on statistical mechanics and large deviation theory arguments, and investigate geometrically confined systems of infinite volume V with a finite spatial extent L. More specifically, we focus on systems in contact with a heat bath as well as on isolated ones. We find and present in detail the characteristic behaviors of a first and a second order phase transition, respectively. In the former case, we establish a connection between the phase transition and the mechanical support of the system. In the latter one, we identify a critical length Lc~(V/EL)1/3, where E is the total energy of the system which is fixed. We also briefly comment on the relevance of our findings, for respectively both the Casimir experiments and the future quark-gluon plasma phenomenology.

Inertial migration of single particle in a square microchannel over wide ranges of Re and particle sizes

Inertial migration of particles has been widely used in inertial microfluidic systems to passively manipulate cells/particles. However, the migration behaviors and the underlying mechanisms, especially in a square microchannel, are still not very clear. In this paper, the immersed boundary-lattice Boltzmann method (IB-LBM) was introduced and validated to explore the migration characteristics and the underlying mechanisms of an inertial focusing single particle in a square microchannel. The grid-independence analysis was made first to highlight that the grid number across the thin liquid film (between a particle and its neighboring channel wall) was of significant importance in accurately capturing the migrating particle’s dynamics. Then, the inertial migration of a single particle was numerically investigated over wide ranges of Reynolds number (Re, from 10 to 500) and particle sizes (diameter-to-height ratio a/H, from 0.16 to 0.5). It was interesting to find that as Re increased, the channel face equilibrium (CFE) position moved outward to channel walls at first, and then inflected inwards to the channel center at high Re (Re > 200). To account for the physical mechanisms behind this behavior, the secondary flow induced by the inertial focusing single particle was further investigated. It was found that as Re increased, two vortices appeared around the particle and grew gradually, which pushed the particle away from the channel wall at high Re. Finally, a correlation was proposed based on the numerical data to predict the critical length Lc (defined to describe the size of fluid domain that was strongly influenced by the particle) according to the particle size a/H and Re.

Critical Nucleation Length for Accelerating Frictional Slip

AbstractThe spontaneous nucleation of accelerating slip along slowly driven frictional interfaces is central to a broad range of geophysical, physical, and engineering systems, with particularly far‐reaching implications for earthquake physics. A common approach to this problem associates nucleation with an instability of an expanding creep patch upon surpassing a critical length Lc. The critical nucleation length Lc is conventionally obtained from a spring‐block linear stability analysis extended to interfaces separating elastically deformable bodies using model‐dependent fracture mechanics estimates. We propose an alternative approach in which the critical nucleation length is obtained from a related linear stability analysis of homogeneous sliding along interfaces separating elastically deformable bodies. For elastically identical half‐spaces and rate‐and‐state friction, the two approaches are shown to yield Lc that features the same scaling structure, but with substantially different numerical prefactors, resulting in a significantly larger Lc in our approach. The proposed approach is also shown to be naturally applicable to finite‐size systems and bimaterial interfaces, for which various analytic results are derived. To quantitatively test the proposed approach, we performed inertial Finite‐Element‐Method calculations for a finite‐size two‐dimensional elastically deformable body in rate‐and‐state frictional contact with a rigid body under sideway loading. We show that the theoretically predicted Lc and its finite‐size dependence are in reasonably good quantitative agreement with the full numerical solutions, lending support to the proposed approach. These results offer a theoretical framework for predicting rapid slip nucleation along frictional interfaces.

Instabilities in granular gas–solid flows

A linear stability analysis of the hydrodynamic equations with respect to the homogeneous cooling state is performed to study the conditions for stability of a suspension of solid particles immersed in a viscous gas. The dissipation in such systems arises from two different sources: inelasticity in particle collisions and viscous friction dissipation due to the influence of the gas phase on the solid particles. The starting point is a suspension model based on the (inelastic) Enskog kinetic equation. The effect of the interstitial gas phase on the dynamics of grains is modeled though a viscous drag force. The study is carried out in two different steps. First, the transport coefficients of the system are obtained by solving the Enskog equation by means of the Chapman–Enskog method up to first order in spatial gradients. Explicit expressions for the Navier–Stokes transport coefficients are obtained in terms of the volume fraction, the coefficient of restitution and the friction coefficient characterizing the amplitude of the external force. Once the transport properties are known, then the corresponding linearized hydrodynamic equations are solved to get the dispersion relations. In contrast to previous studies (Garzó et al 2016 Phys. Rev. E 93 012905), the hydrodynamic modes are analytically obtained as functions of the parameter space of the system. For a d-dimensional system, as expected linear stability shows d − 1 transversal (shear) modes and a longitudinal ‘heat’ mode to be unstable with respect to long enough wavelength excitations. The results also show that the main effect of the gas phase is to decrease the value of the critical length Lc (beyond which the system becomes unstable) with respect to its value for a dry granular fluid. Comparison with direct numerical simulations for Lc shows a qualitative good agreement for conditions of practical interest.

Giant magnetoimpedance intrinsic impedance and voltage sensitivity of rapidly solidified Co66Fe2Cr4Si13B15 amorphous wire for highly sensitive sensors applications

We report a systematic study of the influence of wire length, L, dependence of giant magneto-impedance (GMI) sensitivity of Co66Fe2Cr4Si13B15 soft magnetic amorphous wire of diameter ~100 µm developed by in-water quenching technique. The magnetization behaviour (hysteresis loops) of the wire with different length (L = 1, 2, 3, 5, 8 and 10 cm) has been evaluated by fuxmetric induction method. It was observed that the behaviour of the hysteresis loops change drastically with the wire length, being attributed to the existence of a critical length, L C, found to be around 3 cm. GMI measurements have been taken using automated GMI measurement system and the GMI sensitivities in terms of intrinsic impedance sensitivity (S Ω/Am −1) and voltage sensitivity (S V/Am −1) of the wire have been evaluated under optimal bias field and excitation current. It was found that the maximum (S Ω/Am −1) max ≈ 0.63 Ω/kAm−1/cm and (S V/Am −1) max ≈ 3.10 V/kAm−1/cm were achieved at a critical length L C ~ 3 cm of the wire for an AC current of 5 mA and a frequency of 5 MHz. These findings provide crucial insights for optimization of the geometrical dimensions of magnetic sensing elements and important practical guidance for designing high sensitive GMI sensors. The relevant combinations of magnetic material parameters and operating conditions that optimize the sensitivity are highlighted.

Application of dynamic scaling theory for growth kinetic studies of AlN-thin films deposited by ion beam sputtering in reactive assistance of nitrogen plasma

Ion beam sputter deposition of AlN thin films to different time scales was carried out in reactive assistance of N+/N2+ ions. The incipient stages of the growth morphology were characterized using atomic force microscopy. Dynamic scaling theory was invoked to analyze the evolution of surface roughness and the growth mechanism therein. Two distinct exponents ‘α’ (static) and ‘β’ (dynamic) were used to unravel the film growth characteristics. Our results show that as the deposition time (t) increases, ‘α’ decreases gradually and substrate surface coverage increases indicated by a decrease in critical length Lc. Dynamic scaling exponent ‘β’ was estimated to be 0.36 for the deposition from isolated nuclei to full surface coverage of the substrate. During the growth, rms roughness of the film was increased from 1.99 to 3.42nm as the deposition time was increased from 3min to 15min. Surface diffusion becomes the major roughening phenomenon while bulk diffusion subside it at each stage by smoothening to yield corresponding rms roughness.

FinFET Mismatch in Subthreshold Region: Theory and Experiments

In this paper, we study the drain-current mismatch of FinFETs in subthreshold, from both modeling and experimental point of view. We propose a simple model that takes into account the effect of threshold voltage and subthreshold swing fluctuations and their correlation. For long-channel devices (longer than a critical length LC), characterized by a subthreshold swing close to the ideal value, the overall current mismatch is dominated by threshold voltage fluctuations and, therefore, is gate voltage independent. The subthreshold swing fluctuations give a negligible effect on the drain-current mismatch and are uncorrelated with the threshold voltage fluctuations. For short-channel devices (shorter than a critical length LC), characterized by a strong dependence of subthreshold swing on the channel length, the overall current mismatch presents an additional relevant contribution associated with the subthreshold swing fluctuations. This component depends on the gate voltage overdrive and is ascribed to the gate line edge roughness, resulting in a partial correlation between threshold voltage and subthreshold swing fluctuations.

Inverse Supersymmetry Breaking in S1 × R3

In this paper, we study the influence of hard supersymmetry breaking terms in a N = 1, d = 4 supersymmetric model, in S1 × R3 spacetime topology. It is shown that when the radius of the compact dimension is large supersymmetry is unbroken, and dynamically breaks as the radius decreases. We point out that this resembles the inverse symmetry breaking of continuous symmetries at finite temperature (however, in the case of supersymmetry, the role of the temperature is played by the compact dimension’s radius). Furthermore, we also find a universality in the dependence of the critical length Lc as a function of a coupling g3, after comparing all cases.

Effects of line tying on resistive tearing instability in slab geometry

The effects of line tying on resistive tearing instability in slab geometry are studied within the framework of reduced magnetohydrodynamics [B. B. Kadomtsev and O. P. Pogutse, Sov. Phys. JETP 38, 283 (1974); H. R. Strauss, Phys. Fluids 19, 134 (1976)]. It is found that line tying has a stabilizing effect. The tearing mode is stabilized when the system length L is shorter than a critical length Lc, which is independent of the resistivity η. When L is not too much longer than Lc, the growth rate γ is proportional to η. When L is sufficiently long, the tearing mode scaling γ∼η3/5 is recovered. The transition from γ∼η to γ∼η3/5 occurs at a transition length Lt∼η−2/5.

Theoretical analysis of polydispersity in the nematic phase of self-assembled semiflexible chains

In recent simulations of semiflexible equilibrium polymers [X. Lu and J. T. Kindt, J. Chem. Phys. 120, 10328 (2004)] a roughly biexponential distribution of chain lengths was observed in the nematic phase. In this study, we show that a theory representing the nematic phase as an equilibrium mixture of randomly oriented chains below a critical length Lc and chains above Lc obeying a Gaussian orientational distribution reproduces this distribution qualitatively. The agreement between predicted and simulated phase boundaries is improved, with a narrower isotropic-nematic coexistence region, compared with a monoexponential chain distribution. We find, furthermore, that the critical length Lc scales as the inverse of monomer concentration, irrespective of the bond strength of the chains, but that at the phase boundary, the fraction of monomers found in disordered chains peaks at a certain bond strength.

Analysis of the Thermal Activation of High-Mobility Dislocation Loops

We report thermally activated transport of highly mobile dislocation loops in terms of a line tension model where the dislocation loops are assumed to be a flexible string. The activation energy and transition rate are calculated on the basis of a classical rate theory. The activation energy merely increases with the length of the dislocation loops. However, the activation process and temperature dependence of the transition rate qualitatively change at a critical length Lc. If the dislocation loops are longer than the critical length, the thermal activation occurs through the conventional double-kink formation process on the dislocation lines. On the other hand, if the dislocation loops are shorter than that, the saddle point configuration is not the double-kink type but non-deformation one. Therefore, the critical length Lc is a plausible criterion for the dislocation loops to distinguish dislocation like from point-defect like in size.

Glass transition in systems without static correlations: a microscopictheory

We present a first step toward a microscopic theory for the glass transition insystems with trivial static correlations. As an example we have chosenN infinitely thin hardrods with length L,fixed with their centres on a periodic lattice with lattice constanta. Startingfrom the N-rodSmoluchowski equation we derive a coupled set of equations for fluctuations of reducedk-roddensities. We approximate the influence of the surrounding rods on the dynamics of a pairof rods by introduction of an effective rotational diffusion tensor D(Ω1, Ω2)and in this way we obtain a self-consistent equation for D.This equation exhibits a feedback mechanism leading to a slowing down ofthe relaxation. It involves as an input the Laplace transform υ0 (l/r)at z = 0,l = L/a,of a torque–torque correlator of an isolated pair of rods with distanceR = ar. Ourequation predicts the existence of a continuous ergodicity-breaking transition at a criticallength lc = Lc /a.To estimate the critical length we perform an approximate analytical calculation of υ0 (l/r)based on a variational approach and obtain lcvar ∼ = 5.68,4.84 and 3.96 for an sc, bcc and fcc lattice. We also evaluate υ0 (l/r) numericallyexactly from a two-rod simulation. The latter calculation leads to lcnum ∼ = 3.45,2.78 and 2.20 for the corresponding lattices. Close tolcthe rotational diffusion constant decreases as D(l) ∼ (lc − l)γ withγ = 1and a diverging timescale tϵ ∼ |lc − l|−δ, δ = 2, appears. Onthis timescale the t-and l-dependenceof the one-rod density is determined by a master function depending only on t/tϵ. Incontrast to present microscopic theories our approach predicts a glass transitiondespite the absence of any static correlations.

Computational analysis of the steady states of the Sivashinsky model of hydrodynamic flame instability

In this work a numerical analysis of an integro-differential equation modelling the Darrieus–Landau instability of plane flame fronts was undertaken. A relatively new computational method based on saturated asymptotic approximations was used. Within the considered computational times a steady limiting shape of the flame front was not reached in large enough computational domains of size L > Lc . Instead, a smooth surface of an almost steadily shaped flame is repeatedly disturbed by small perturbations, resembling small cusps, appearing and disappearing randomly in time. The nature of these small cusps as well as of the steady limiting shape of the flame front was studied with a relatively new computational method. The correlation between the critical length Lc and parameters of the computational algorithm and the computer precision was investigated. The calculations confirmed that, unlike the round-off errors, there is no significant link between the approximation accuracy of the algorithm and Lc . The obtained dependence of the critical length Lc on the magnitude of the round-off errors, considered as an external noise, was compared with the predictions given by other researchers. The agreement supports the idea of high sensitivity of solutions of the Sivashinsky equation to the external noise. A similarity between the appearance of small cusps on the surface of large enough flames governed by the Sivashinsky equation and streamwise streaks accompanying the loss of stability of the classic Hagen–Poiseuille flow was noted.

Phase transition in a magnetic superlattice: high-temperature series expansions approach

The phase transition of a ferromagnet spin-S Heisenberg superlattice consisting of two materials is examined using the high-temperature series expansions technique. The critical reduced temperature of the system τc=kBTc/S(S+1)Jaa is studied as function of the thickness of the constituents in the unit cell and the exchange interactions in each material. A critical value of interface exchange interaction above which the interface magnetism appears is obtained. A number of characteristic behaviours such as the possibility of existence of a critical length Lc above which the temperature of the system remains insensitive to the exchange coupling within interface are reported.

Transmission and reflection studies of periodic and random systems with gain

The transmission (T) and reflection (R) coefficients are studied in periodic systems and random systems with gain. For both the periodic electronic tight-binding model and the periodic classical many-layered model, we obtain numerically and theoretically the dependence of T and R. The critical length of periodic system L[sub c][sup 0], above which T decreases with the size of the system L while R approaches a constant value, is obtained to be inversely proportional to the imaginary part cursive-epsilon (double-prime) of the dielectric function cursive-epsilon . For the random system, T and R also show a nonmonotonic behavior versus L. For short systems (L < Lc) with gain (left-angle)ln T(right-angle) = (l[sub g][sup -1] - xi [sub 0][sup -1])L. For large systems (L(very-much-greater-than)Lc) with gain (left-angle)ln T(right-angle) = - (l[sub g][sup -1] + xi [sub 0][sup -1])L. Lc, lg, and xi 0 are the critical, gain, and localization lengths, respectively. The dependence of the critical length Lc on cursive-epsilon (double-prime) and disorder strength W are also given. Finally, the probability distribution of the reflection R for random systems with gain is also examined. Some very interesting behaviors are observed.