Semi-infinite Geometry Research Articles

Peculiarities of the Landau level collapse in graphene ribbons in crossed magnetic and in-plane electric fields

Employing the low-energy effective theory alongside a combination of analytical and numerical techniques, we explore the Landau level collapse phenomenon, uncovering previously undisclosed features. We consider both finite-width graphene ribbons and semi-infinite geometries subjected to a perpendicular magnetic field and an in-plane electric field, applied perpendicular to both zigzag and armchair edges. In the semi-infinite geometry the hole (electron)-like Landau levels collapse as the ratio of electric and magnetic fields reaches the critical value +(−)1. On the other hand, the energies of the electron (hole)-like levels remain distinct near the edge and deeply within the bulk, approaching each other asymptotically for the same critical value. In the finite geometry, we show that the electron (hole)-like levels become denser and merge, forming a band. Published by the American Physical Society 2024

Computationally-efficient linear scheme for overlap time-gating spatial frequency domain diffuse optical tomography using an analytical diffusion model.

Time-domain (TD) spatial frequency domain (SFD) diffuse optical tomography (DOT) potentially enables laminar tomography of both the absorption and scattering coefficients. Its full time-resolved-data scheme is expected to enhance performances of the image reconstruction but poses heavy computational costs and also susceptible signal-to-noise ratio (SNR) limits, as compared to the featured-data one. We herein propose a computationally-efficient linear scheme of TD-SFD-DOT, where an analytical solution to the TD phasor diffusion equation for semi-infinite geometry is derived and used to formulate the Jacobian matrices with regard to overlap time-gating data of the time-resolved measurement for improved SNR and reduced redundancy. For better contrasting the absorption and scattering and widely adapted to practically-available resources, we develop an algebraic-reconstruction-technique-based two-step linear inversion procedure with support of a balanced memory-speed strategy and multi-core parallel computation. Both simulations and phantom experiments are performed to validate the effectiveness of the proposed TD-SFD-DOT method and show an achieved tomographic reconstruction at a relative depth resolution of ∼4 mm.

Transient Thermal Spreading From a Circular Heat Source in Polygonal Flux Tubes

Abstract This study develops a numerical simulation to assess transient constriction resistance in various semi-infinite flux channel geometries, including circle on circle, triangle, square, pentagon, and hexagon, which are derived from various heat source arrangements in a large domain. Using both isothermal and isoflux circular heat sources in polygonal flux channels, and employing a finite volume method, the study evaluates transient constriction resistance. The research confirms that for different geometries, similar nondimensionalized constriction resistance results are obtained, particularly when using the square root of the source area as the characteristic length and the square root of the constriction area ratio. The study reveals that flux tube shape has a minimal impact on thermal spreading resistance, with the circle-on-triangle configuration displaying the largest deviation from a simple circle-on-circle model. These insights advance our understanding of thermal spreading resistance in polygonal flux channels and their applications in thermal engineering, especially in contact heat transfer problems.

A plane defect in the 3d O(N) model

It was recently found that the classical 3d O(N)(N) model in the semi-infinite geometry can exhibit an “extraordinary-log” boundary universality class, where the spin-spin correlation function on the boundary falls off as \langle \vec{S}(x) \cdot \vec{S}(0)\rangle \sim \frac{1}{(\log x)^q}〈S→(x)⋅S→(0)⟩∼1(logx)q. This universality class exists for a range 2 ≤N <N_c2≤N<Nc and Monte-Carlo simulations and conformal bootstrap indicate N_c > 3Nc>3. In this work, we extend this result to the 3d O(N)(N) model in an infinite geometry with a plane defect. We use renormalization group (RG) to show that in this case the extraordinary-log universality class is present for any finite N ≥2N≥2. We additionally show, in agreement with our RG analysis, that the line of defect fixed points which is present at N = ∞N=∞ is lifted to the ordinary, special (no defect) and extraordinary-log universality classes by 1/N1/N corrections. We study the “central charge” aa for the O(N)O(N) model in the boundary and interface geometries and provide a non-trivial detailed check of an aa-theorem by Jensen and O’Bannon. Finally, we revisit the problem of the O(N)(N) model in the semi-infinite geometry. We find evidence that at N = N_cN=Nc the extraordinary and special fixed points annihilate and only the ordinary fixed point is left for N > N_cN>Nc.

Convectively cooled solidification in phase change materials in different configurations subject to internal heat generation: Quasi-steady analysis

Phase change phenomena involving internal heat generation are of central interest in various applications, including freeze drying, preservation of biological tissues, microwave/Joule heating processes, materials processing using lasers, and nuclear power generation systems. In this work, mathematical models are developed to study the solidification processes driven by convective cooling effects on the exterior surface of phase change materials (PCM) in different geometric configurations, where both the solid and the liquid phases are subject to volumetric heating. By invoking a quasi-steady assumption and performing a simplified unified analysis, new semi-analytical approximate solutions for this unique Stefan problem in planar, cylindrical, spherical, and semi-infinite geometries of the PCM are obtained and studied. The interface evolution depends on the characteristic dimensionless parameters, such as the Biot number, Bi, internal heat generation parameter, Q, Stefan number, St, and the thermal conductivity ratio, κ. The effects of all the dimensionless parameters, viz., Bi, Q, St, and κ are systematically investigated for each geometrical configuration of the PCM. In particular, the results show the significant role of the Biot number on the transient evolution as well as the steady-state thickness of the solidifying front in the presence of an inhibiting internal heat generation. In addition, a simple and unified condition relating the dimensionless internal heating rate (Q) and the external convective cooling rate (Bi) for the onset of remelting for the special case of κ = 1 applicable for all the above geometries of the PCM is deduced.

Two-layer analytical model for estimation of layer thickness and flow using Diffuse Correlation Spectroscopy.

Diffuse correlation spectroscopy (DCS) has been widely explored for its ability to measure cerebral blood flow (CBF), however, mostly under the assumption that the human head is homogenous. In addition to CBF, knowledge of extracerebral layers, such as skull thickness, can be informative and crucial for patient with brain complications such as traumatic brain injuries. To bridge the gap, this study explored the feasibility of simultaneously extracting skull thickness and flow in the cortex layer using DCS. We validated a two-layer analytical model that assumed the skull as top layer with a finite thickness and the brain cortex as bottom layer with semi-infinite geometry. The model fitted for thickness of the top layer and flow of the bottom layer, while assumed other parameters as constant. The accuracy of the two-layer model was tested against the conventional single-layer model using measurements from custom made two-layer phantoms mimicking skull and brain. We found that the fitted top layer thickness at each source detector (SD) distance is correlated with the expected thickness. For the fitted bottom layer flow, the two-layer model fits relatively consistent flow across all top layer thicknesses. In comparison, the conventional one-layer model increasingly underestimates the bottom layer flow as top layer thickness increases. The overall accuracy of estimating first layer thickness and flow depends on the SD distance in relationship to first layer thickness. Lastly, we quantified the influence of uncertainties in the optical properties of each layer. We found that uncertainties in the optical properties only mildly influence the fitted thickness and flow. In this work we demonstrate the feasibility of simultaneously extracting of layer thickness and flow using a two-layer DCS model. Findings from this work may introduce a robust and cost-effective approach towards simultaneous bedside assessment of skull thickness and cerebral blood flow.

Phonon thermal transport in silicon thin films with nanoscale constrictions and expansions

In patterned thin film devices, abrupt geometric changes can introduce thermal constriction and expansion resistances whose magnitude and relative importance depends on the device's size and geometry as well as the dominant heat carrier mean free path spectra of the comprising material. Existing analytical models of thermal constriction and expansion resistances at the nanoscale have focused primarily on semi-infinite geometries or other situations which are quite different from those encountered in modern nanopatterned thin film devices. In this work, Monte Carlo methods are used to simulate phonon transport in silicon thin films patterned with a commonly utilized source-channel-drain geometry. The length, width, and thickness of the channel region were varied, and the dependence of the thermal constriction–expansion resistance on these parameters was determined. Results show that thin film source–drain reservoirs with diffuse boundary scattering do not behave as semi-infinite reservoirs for feature sizes smaller than approximately 100 nm in silicon near 300 K, and that existing analytical models cannot be readily applied to such systems. In addition, our results support the case that ballistic phonon effects in silicon nanowires at room temperature, if present, are small and not easily observable. Finally, we provide guidance and perspective for Si nanowire measurements near room temperature as to what scenarios may lead to a non-negligible amount of error if constriction–expansion geometry effects are ignored.

Analytical treatment on singularities for 2-D elastoplastic dynamics by time domain boundary element method using Hadamard principle integral

A fully analytical method by adopting the concept of the finite part of an integral, Hadamard principle integral, is proposed to treat the singularities in both time and space in the plastic strain-related coefficients matrices Q and F in the TD-BEM formulation for elastoplastic analysis. Comparing with the treatment on singularity by using the method of rigid body displacement, the novel analytical treatment on the singularity is from the viewpoint of dynamics, without any numerical component of the elastostatic concept, meaning that the treatment is theoretically suitable for elastoplastic analysis. Meanwhile, comparing with the treatment on singularity by the initial stress expansion technique and the method of the constant strain fields, the novel analytical treatment on the singularity retains all of the inherent advantages of the TD-BEM formulation for elastoplastic analysis, restricting the discretization on boundary and in the potential plastic region and allowing models with infinite or semi-infinite geometry extension without artificial boundaries. Finally, the novel analytical treatment on the singularity is strictly mathematically tenable, without any computational error. The proposed treatment on singularity for dynamic elastoplastic analysis in TD-BEM formulation is verified by three examples.

Hydrodynamic ejection caused by laser-induced optical breakdown

A focused laser can cause local optical breakdown of a gas, which leads to rapid deposition of energy into a high-temperature plasma kernel that expands and induces a complex flow. For some conditions, hot gas is rapidly ejected along the laser axis up to distances several times the kernel size, with a particularly curious feature: relatively small changes in, for example, initial pressure can cause the direction of this ejection to reverse. Detailed axisymmetric simulations of a model energy kernel in an inert gas provide a hydrodynamic description of this phenomenon, reproducing key observations in corresponding experiments, including the vortex-ring-like features that constitute the ejection. These simulations are analysed to show how changes in the early-time kernel can lead to ejection or its reversal via alteration in the relative strength and position of the vorticity produced. A corresponding semi-infinite geometry is used to isolate two mechanisms: vorticity production by the generated shock and by baroclinic torque at the kernel boundary. Dependence on the initial kernel asymmetry is quantified, as it ultimately determines whether the vorticity, upon its subsequent evolution, develops into the ring-like structure that ejects. Even simple elongation of the energy kernel alone can reverse the direction.

Biopolyelectrolyte Surface Diffusion Within a Planar Slit Geometry

The effective surface diffusion of biopolyelectrolytes within nanoscale confinement is integral to biosensing, biocatalysis, bioseparations (such as chromatography columns utilizing nanoporous media), as well as biophysical processes. Confinement influences the surface-mediated diffusion of biopolyelectrolyte chains due to the presence of electric double layers and the increased frequency of transient polymer/surface interactions. Here, the diffusion of poly-L-lysine (PLL) in a nanoslit was studied using Convex Lens-induced Confinement (CLiC) single-molecule tracking microscopy. Three surface chemistries were employed to understand and compare the effects of electrostatic and short-range interactions, respectively. In all cases, the effective surface diffusion coefficient increased rapidly with slit height until it saturated at its value for a semi-infinite interface for slit heights <30 nm. The evolution of the diffusion coefficient with slit height was faster for amine-functionalized surfaces with which PLL exhibited stronger short-range interactions. These hydrogen-bonding interactions influenced the intermittent random walk behavior by increasing the waiting times between flights and the re-adsorption probability (a.k.a. sticking coefficient) during flights. Intermittent random walks were simulated within a planar slit geometry, using the appropriate adsorption probabilities and waiting time distributions (obtained from diffusion at a single interface) to account for the characteristic short-range interactions between PLL and each type of surface chemistry. The simulated step-size distributions were in good agreement with experimental measurements, indicating that the fine details of diffusion in nanoscale confinement can be quantitatively described by a model that combines confined Brownian motion with transient surface adsorption and desorption, using parameters obtained only from the behavior at a single interface in a semi-infinite geometry. These results highlight the importance of short-range biopolyelectrolyte/surface interactions on diffusion within confined environments and moreover, demonstrate the complex relationship between electrostatics, confinement, and local polymer and surface chemistry on biopolyelectrolyte surface diffusion within a nanoslit.

Revisiting ‘penetration depth’ in falling film mass transfer

For the analysis of falling film mass transfer such as gas absorption in liquid phase flowing over a vertical wall, conventionally an infinite penetration depth for penetrating component has been assumed. This assumption is originated from the fact that magnitude of diffusion depth is too small relative to the film thickness as if the mass transfer into the liquid film is taking place in a semi-infinite geometry. Despite its validity in the most engineering applications, it is still an unanswered question that ‘to what physically meaningful extent the diffusion may occur or the mass transfer boundary layer into the liquid film could progress alongside the wall, and consequently how it might affect the mass transfer characteristics’. We addressed such concern here by considering a finite depth of penetration into the film, derived governing equations for the continuity of mass and solved it analytically and numerically. The progress of penetration depth over the vertical coordinate i.e. flowing direction of falling film is observable as it approaches the film thickness far down the wall. The method gives the opportunity to examine the progress of the penetration depth with mass transfer properties such as diffusivity, average liquid velocity and etc. which have not been touched before.

Kinetics of moisture adsorption during simulated storage of whole dry cocoa beans at various relative humidities

A simple combined model involving semi-infinite slab geometry with Fickian diffusion and the Harkin-Jura adsorption isotherm, adjusted to the final equilibrium experimental moisture content given by the model, was developed to predict the kinetics of moisture adsorption of dried cocoa beans under simulated external storage temperature and relative humidity. This model was able to predict accurately the moisture gain by individual beans during storage (R2 > 0.994). Experimental moisture adsorption kinetics of whole cocoa beans were determined for a relative humidity range of 64–93% at 25 °C (298 K). A diffusion coefficient at 298 K of 3.30 × 10−11 ± 0.56 × 10−11 m2/s was determined and found independent of moisture concentration in the experimental range considered. An operational storage map was proposed to predict the average moisture content of whole cocoa beans of any commercial thickness (0.600 ≤ l ≤ 1.200 cm) (6.00 × 10−3 ≤ l ≤ 1.200 × 10−2 m) during storage under atmospheres of constant relative humidity (64% ≤ RH ≤ 93%) and temperatures from 5 to 35 °C (278–308 K).

Short-Distance Symmetry of Pair Correlations in Two-Dimensional Jellium

We consider the two-dimensional one-component plasma (jellium) of mobile pointlike particles with the same charge $e$, interacting pairwisely by the logarithmic Coulomb potential and immersed in a fixed neutralizing background charge density. Particles are in thermal equilibrium at the inverse temperature $\beta$, the only relevant dimensionless parameter is the coupling constant $\Gamma\equiv\beta e^2$. In the bulk fluid regime and for any value of the coupling constant $\Gamma=2\times{\rm integer}$, \v{S}amaj and Percus [J. Stat. Phys. {\bf 80}, 811--824 (1995)] have derived an infinite sequence of sum rules for the coefficients of the short-distance expansion of particle pair correlation function. In the context of the equivalent fractional quantum Hall effect, by using specific methods of quantum geometry Haldane [PRL {\bf 107}, 116801 (2011) and arXiv:1112.0990v2] derived a self-dual relation for the Landau-level guiding-center structure factor. In this paper, we establish the relation between the guiding-center structure factor and the pair correlation function of jellium particles. It is shown that the self-dual formula, which provides an exact relation between the pair correlation function and its Fourier component, comes directly from the short-distance symmetry of the bulk jellium. The short-distant symmetry of pair correlations is extended to the semi-infinite geometry of a rectilinear plain hard wall with a fixed surface charge density, constraining particles to a half-space. The symmetry is derived for the original jellium model as well as its simplified version with no background charge (charged wall surface with ``counter-ions only''). The obtained results are checked at the exactly solvable free-fermion coupling $\Gamma=2$.

Extension of the Einstein molecule method for solid free energy calculation to non-periodic and semi-periodic systems

Free energy calculations on solid phases are important for understanding the phase behavior of various systems. For periodic crystalline solids, the Einstein molecule method can be used to determine the free energy difference between the solid of interest and an ideal crystal for which the free energy can be found analytically. In this work, we show how this method is extensible to systems which are nonperiodic or periodic in some dimensions but not in others. This allows for the calculation of exact absolute free energies of finite-sized crystals having specific shapes and surface geometries. We demonstrate this using the fcc Lennard-Jones solid and also illustrate how surface contributions to free energies can easily be extracted from simulations of this solid in semi-infinite slab geometries. We have developed a software package which interfaces with the LAMMPS molecular dynamics code to perform these calculations.

Hyper-spectral Recovery of Cerebral and Extra-Cerebral Tissue Properties Using Continuous Wave Near-Infrared Spectroscopic Data

Near-infrared spectroscopy (NIRS) is widely used as a non-invasive method to monitor the hemodynamics of biological tissue. A common approach of NIRS relies on continuous wave (CW) methodology, i.e. utilizing intensity-only measurements, and, in general, assumes homogeneity in the optical properties of the biological tissue. However, in monitoring cerebral hemodynamics within humans, this assumption is not valid due to complex layered structure of the head. The NIRS signal that contains information about cerebral blood hemoglobin levels is also contaminated with extra-cerebral tissue hemodynamics, and any recovery method based on such a priori homogenous approximation would lead to erroneous results. In this work, utilization of hyper-spectral intensity only measurements (i.e., CW) at multiple distances are presented and are shown to recover two-layered tissue properties along with the thickness of top layer, using an analytical solution for a two-layered semi-infinite geometry. It is demonstrated that the recovery of tissue oxygenation index (TOI) of both layers can be achieved with an error of 4.4%, with the recovered tissue thickness of 4% error. When the data is measured on a complex tissue such as the human head, it is shown that the semi-infinite recovery model can lead to uncertain results, whereas, when using an appropriate model accounting for the tissue-boundary structure, the tissue oxygenation levels are recovered with an error of 4.2%, and the extra-cerebral tissue thickness with an error of 11.8%. The algorithm is finally used together with human subject data, demonstrating robustness in application and repeatability in the recovered parameters that adhere well to expected published parameters.

Analytic Pade-like approximations of exp(–sqrt(s)) for simulations of diffusion processes in the semi-infinite geometry

Detailed transient behaviors of mass and heat transfer processes are required to solve partial differential equations. When those partial differential equations are coupled, they are still difficult to solve in time domain. For linear mass and heat transfer processes, their Laplace-domain solutions are obtainable and, when they are approximated by rational polynomials in the Laplace variable s, the problems can be transformed to a set of ordinary differential equations solved easily in time domain for various initial conditions. In this approximation, the conventional Pade method based on the Tayler series expansion of the Laplace-domain solutions has been well developed and effective. However, for some mass and heat transfer processes in the semi-infinite geometry, the Pade approximation is not applicable because the Laplace-domain solutions involving exp(–sqrt(s)) are not analytic at s = 0. Here, for such processes, analytical methods to approximate exp(–sqrt(s)) by rational polynomials are proposed. First it is expanded in series in terms of cosh–1(2ksqrt(s)) which converges fast. This series, when truncated, is analytic at s = 0 and its Pade approximations are available. The proposed method enables partial differential equations be replaced to a set of ordinary differential equations, reducing computations considerably for coupled partial differential equations. Performances of the proposed method are illustrated with several realistic mass and heat transfer processes.

Spatial frequency domain imaging using an analytical model for separation of surface and volume scattering

.A method to correct for surface scattering in spatial frequency domain imaging (SFDI) is presented. The use of a modified analytical solution of the radiative transfer equation allows calculation of the reflectance and the phase of a rough semi-infinite geometry so that both spatial frequency domain reflectance and phase can be applied for precise retrieval of the bulk optical properties and the surface scattering. For validation of the method, phantoms with different surface roughness were produced. Contrarily, with the modified theory, it was possible to dramatically reduce systematic errors due to surface scattering. The evaluation of these measurements with the state-of-the-art theory and measuring modality, i.e., using crossed linear polarizers, reveals large errors in the determined optical properties, depending on the surface roughness, of up to . These results were confirmed with SFDI measurements on a phantom that has a structured rough surface.

Self-Calibration Phenomenon for Near-Infrared Clinical Measurements: Theory, Simulation, and Experiments.

An irradiated turbid medium scatters the light in accordance to its optical properties. Near-infrared (NIR) clinical methods, which are based on spectral-dependent absorption, suffer from an inherent error due to spectral-dependent scattering. We present here a unique spatial point, that is, iso-pathlength (IPL) point, on the surface of a tissue at which the intensity of re-emitted light remains constant. This scattering-indifferent point depends solely on the medium geometry. On the basis of this natural phenomenon, we suggest a novel optical method for self-calibrated clinical measurements. We found that the IPL point exists in both cylindrical and semi-infinite tissue geometries (Supporting Information, Video file). Finally, in vivo human finger and mice measurements are used to validate the crossing point between the intensity profiles of two wavelengths. Hence, measurements at the IPL point yield an accurate absorption assessment while eliminating the scattering dependence. This finding can be useful for oxygen saturation determination, NIR spectroscopy, photoplethysmography measurements, and a wide range of optical sensing methods for physiological aims.

Mullins' Self-Similar Grooving Solution Revisited

In W.W. Mullins' classical 1957 paper on thermal grooving, motion by surface diffusion was proposed to describe the development of a thermal groove separating two grains in a simple semi-infinite planar geometry. After making a small slope approximation which is often realistic, Mullins' sought self-similar solutions, and obtained an explicit time series solution for the groove depth. In the years since, Mullins' grooving solution has become a standard tool; however it has yet to be rigorously demonstrated that self-similar solutions exist when the small slope approximation is not applicable. Here we demonstrate that reformulation of Mullins' nonlinear problem in arc-length variables yields a particularly simple fully nonlinear formulation, which is useful for verifying large slope grooving properties and which should aid in proving existence.

Instability in the self-similar motion of a planar solidification front

Understanding the solidification process of a binary alloy is important if one is to control the microstructure obtained during the casting of metals. While much work has been done on the steady state solidification problem, despite their relevance to metallurgical applications, there is less known about non-steady solidification problems and their stability. In the paper we shall consider the non-steady solidification problem in which the planar solidification front moves in a self-similar manner, in both infinite and semi-infinite planar one-dimensional geometries. For each geometry exact solutions are known for the resulting Stefan problem. We direct our attention to the stability of each solution, demonstrating that while the concentration and thermal solutions remain stable, the interface corresponding to the solidification front can develop instabilities. For each geometry, we find that there are always unstable perturbations, although we observe qualitative differences in the form of the unstable perturbations for each case. These results generalize and extend several existing studies in the literature, and throw light on the instability inherent in the non-steady solidification process.