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At least 721 records · Page 40

Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems

In this work, we present Lift & Learn, a physics-informed method for learning low-dimensional models for large-scale dynamical systems. The method exploits knowledge of a system’s governing equations to identify a coordinate transformation in which the system dynamics have quadratic structure. This transformation is called a lifting map because it often adds auxiliary variables to the system state. The lifting map is applied to data obtained by evaluating a model for the original nonlinear system. This lifted data is projected onto its leading principal components, and low-dimensional linear and quadratic matrix operators are fit to the lifted reduced data using a least-squares operator inference procedure. Analysis of our method shows that the Lift & Learn models are able to capture the system physics in the lifted coordinates at least as accurately as traditional intrusive model reduction approaches. This preservation of system physics makes the Lift & Learn models robust to changes in inputs. Numerical experiments on the FitzHugh–Nagumo neuron activation model and the compressible Euler equations demonstrate the generalizability of our model.

97 MATHEMATICS AND COMPUTING↗

High-Performance Computing Optimization for Aladyn – Adaptive Neural Network Molecular Dynamics Mini-Application

This report provides a description and performance evaluation of the optimization techniques for high performance computing (HPC) implementation of the open source Computational Materials mini-application Aladyn (https://github.com/nasa/aladyn). Aladyn is a basic molecular dynamics code written in FORTRAN 2003, which is designed to demonstrate the use of adaptive neural networks (ANNs) in atomistic simulations. The role of ANNs is to efficiently reproduce the very complex energy landscape resulting from the atomic interactions in materials with the accuracy of the more expensive quantum mechanics-based calculations. The ANN is trained on a large set of atomic structures calculated using the density functional theory (DFT) method. While achieving orders of magnitude faster computational performance than DFT, the ANN-based approach was still very computationally demanding compared to the conventional approach of using empirically fitted energy functions. After its initial development, Aladyn was evaluated and optimized by experts at the NASA Advanced Supercomputing (NAS) division to exploit modern supercomputer architectures. The code has been optimized for execution on multicore central processing units (CPUs), including Intel® Skylake microarchitecture, and on graphic accelerators, such as Nvidia® V100 graphic processing units (GPUs), using Open Multi-Processing (OpenMP) and Open Accelerators (OpenACC) programming interfaces. The optimization achieved a speedup of 4.7 times the baseline version on CPU performance and an additional 2.4 times on CPU+GPU performance. Atomistic computer simulations are a fundamental tool in materials research to model material properties form physics-based first principles. Atomic interaction, governed by Quantum Mechanics (QM) require sophisticated and highly computationally demanding mathematical models to calculate [1]. Classical methods use approximate functional forms, empirically fitted through a set of variable parameters to emulate atomic energies as direct functions of atomic coordinates [2]. While empirical potentials are computationally much simpler, allowing simulations of large-scale systems of up to a trillion (1012) atoms [3], they are substantially less accurate compared to quantum calculations and applicable only to very specific atomic configurations or predefined crystallographic phases. A recently suggested approach is to use heuristic machine learning methods [4], such as those based on Adaptive Neural Networks (ANNs) to predict atomic energies, after being trained on a sufficiently large database of QM-calculated structures [5,6]. This approach reduces significantly the computational complexity, allowing for simulations of orders of magnitude larger systems compared to QM-based methods without compromising accuracy. Still, compared to classical methods using empirical energy functions, ANN methods remain two- to three orders of magnitude more computationally demanding. Hence, the computational cost of simulations, together with the need for extensive training of ANNs, still makes the practical implementation of ANN-based methods quite challenging. The purpose of the Aladyn mini-application software [7], available as open source at https://github.com/nasa/aladyn, is to be a testbed for exploring possible optimization strategies to develop highly scalable parallel algorithms for ANN-based atomistic simulations. Aladyn is aimed at utilizing the architecture of the high-end modern highperformance computing (HPC) hardware based on multicore central processing units (CPUs) equipped with graphic processing unit (GPU) accelerators. Specifically, the goal is to optimize the performance on a single HPC compute node, before implementing scaling to multi-node parallelization using message passing interface (MPI). At the same time, the open source code of Aladyn can serve as a training model for students and professors in academia.

Yamakov, Vesselin I.↗

Evolving Dark Sector and the Dark Dimension Scenario

String theory naturally leads to the expectation that dark energy is not stable, and may be evolving as captured by the Swampland de Sitter conjectures. Moreover, motivated by the distance conjecture, a unification of dark sector has been proposed, where the smallness of dark energy leads to one extra dimension of micron size with dark matter being the Kaluza--Klein graviton excitations in this extra dimension. We consider the natural possibility that the radius of the dark dimension varies as the dark energy decreases, leading to the variation of the dark matter mass. This correlates the decrease of the dark energy with the variation of the dark matter mass as they depend on the variations of a scalar field $ϕ$ controlling the radius of the extra dimension. A simple realization of this idea for small range of $ϕ$ is adequately captured by choosing a potential which is locally of the form $V=V_0\ {\rm exp}(-cϕ)$ and dark matter mass $m_{\rm DM}=m_0\ {\rm exp}(-c' ϕ)$ where the sign of $ϕ$ is chosen such that $c'\geq 0$ while we have two choices for the sign of $c$ depending on whether the dark dimension expands or shrinks when the dark energy dominates. We find excellent agreement with recent experimental data from DESI DR2 combined with SN measurements (from DES, Union3 or Pantheon+) and reproduces the same significance as CPL parametrization with the added benefit of providing a natural explanation for the apparent phantom behavior ($w<-1$) reported by DESI and DES based on a physical model. DESI and SN datasets independently favor non-zero values of $c'$ and $c$, respectively, both lying within the expected $\mathcal{O}(1)$ range suggested by the Swampland criteria. Moreover, our best fit value $c'\simeq 0.05 \pm 0.01$ is remarkably consistent with the experimental upper bound of $c'\lesssim 0.2$ demanded by the lack of detection of fifth force in the dark sector.

Cosmology and Nongalactic Astrophysics (astro-ph.C↗

Doppler Backscattering Data Analysis and Integrated Modeling with OMFIT

One Modeling Framework for Integrated Tasks (OMFIT) is a widely used software tool in the magnetic fusion research community. OMFIT provides magnetic fusion energy researchers with a framework for the development of special-purpose physics modules. This paper describes an OMFIT physics module pertaining to the Doppler Backscattering (DBS) fusion plasma diagnostic. DBS measures density fluctuations and flow velocity through plasma scattering of electromagnetic waves. The OMFIT DBS module was developed to analyze experimental DBS data and facilitate modeling of DBS systems installed on multiple tokamak devices. The OMFIT DBS module is designed to support several analysis workflows: detailed analysis of experimental data, experimental planning, and theory-based synthetic diagnostic modeling. The DBS module uses integrated modeling by leveraging other OMFIT physics modules to perform tasks related to DBS, e.g. ray/beam–tracing simulations, edge-localized mode–synchronized data analysis, magnetic equilibrium reconstruction, and fitting kinetic profile data. Furthermore, this paper describes several supported workflows and serves a reference for the OMFIT DBS module.

Doppler backscattering↗

Photometric redshifts and intrinsic alignments: Degeneracies and biases in the 3 × 2 pt analysis

We present a systematic study of cosmological parameter bias in weak lensing and large-scale structure analyses for upcoming imaging surveys induced by the interplay of intrinsic alignments (IA) and photometric redshift (photo-z) model misspecification error. We first examine the degeneracies between the parameters of the tidal alignment–tidal torquing (TATT) model for IA and of a photo-z model including a mean shift ( Δ z ¯ ) and variance ( σ z ) for each tomographic bin of lenses and sources, under a variety of underlying true IA behaviors. We identify strong degeneracies between: (1) the redshift scaling of the tidal alignment amplitude and the mean shift and variances of source bins, (2) the redshift scaling of the tidal torquing amplitude and the variance of the lowest- z source bin, and (3) the IA source density weighting and the mean shift and variance of several source bins. We then use this information to guide our exploration of the level of cosmological parameter bias which can be induced given incorrect modeling of IA, photo-z, or both. We find that marginalizing over all the parameters of TATT is generally sufficient to preclude cosmological parameter bias in the scenarios we consider. However, this does not necessarily mean that IA and photo-z parameters are themselves unbiased, nor does it mean that the best-fit model is a good fit to the data. We also find scenarios where the inferred parameters produce χ d . o . f . 2 values indicative of a good fit but cosmological parameter bias is significant, particularly when the IA source density weighting parameter is not marginalized over. Published by the American Physical Society 2024

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Exploring solar neutrino oscillation parameters with the liquid scintillator counter at Yemilab with a comparison to JUNO

We investigate the sensitivities of the liquid scintillator counter (LSC) at Yemilab and JUNO to solar neutrino oscillation parameters, focusing on θ 12 and Δ m 21 2 . We compare the potential of JUNO with LSC at Yemilab utilizing both reactor and solar data in determining those parameters. We find that the solar neutrino data of LSC at Yemilab is highly sensitive to θ 12 enabling its determination with a higher precision compared to reactor experiments. Our study also reveals that if Δ m 21 2 is larger, with a value close to the best-fit value of KamLAND, JUNO reactor data will have about two times better precision than the reactor LSC at Yemilab. On the other hand, if Δ m 21 2 is smaller and closer to the best-fit value of solar neutrino experiments, the precision of the reactor LSC at Yemilab will be better than JUNO. Published by the American Physical Society 2024

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Evidence of Mixed Scaling for Mean Profile Similarity in the Stable Atmospheric Surface Layer

A new mixed scaling parameter Z = z/(Lh)1/2 is proposed for similarity in the stable atmospheric surface layer, where z is the height, L is the Obukhov length, and h is the boundary layer depth. In comparison with the parameter ζ = z/L from Monin–Obukhov similarity theory (MOST), the new parameter Z leads to improved mean profile similarity for wind speed and air temperature in large-eddy simulations. It also yields the same linear similarity relation for CASES-99 field measurements, including in the strongly stable (but still turbulent) regime where large deviations from MOST are observed. Results further suggest that similarity for turbulent energy dissipation rate depends on both Z and ζ. The proposed mixed scaling of Z and relevance of h can be explained by physical arguments related to the limit of z-less stratification that is reached asymptotically above the surface layer. The presented evidence and fitted similarity relations are promising, but the results and arguments are limited to a small sample of idealized stationary stable boundary layers. Here, corroboration is needed from independent datasets and analyses, including for complex and transient conditions not tested here.

54 ENVIRONMENTAL SCIENCES↗

The broad band X-ray spectrum of SN 1978k and two other luminous X-ray sources in the spiral galaxy NGC 1313

We present preliminary results of our analysis of the Advanced Satellite for Cosmology and Astrophysics (ASCA) PV phase observation of the nearby spiral galaxy NGC 1313. ASCA cleanly resolves the three previously known luminous sources, one of which is the very luminous supernova, SN 1978k. The spectrum of SN 1978k is described by either a power law with a photon index gamma approximately 2.2 or a thermal model with temperature kT approximately 3.0 keV and abundances Z approximately 0.2 Z(sun). There is no evidence for strong line emission from it or from the other two sources. The spectrum of SN 1978k arises either in shocked gas in extreme departure from ionization equilibrium or from synchrotron processes associated with a newborn pulsar. A second source, near the galactic center, is well-fit by a power-law with a photon index of approximately 1.8. It is possibly an active nucleus-like source, but physically displaced from the optical nucleus of the galaxy. The spectrum of the third source, located 8 kpc south of the nucleus, along with the absence of an optical counterpart, suggests that it is a low mass X-ray binary; but its high X-ray luminosity clouds this interpretation. This observation demonstrates the ability of ASCA to perform effective broad band spectroscopic measurements of sources at a 2-10 keV flux level of 5 x 10(exp -13) erg cm(exp -2) s(exp -1).

Petre, Robert↗

Potential origin of the state-dependent hard tail in the black hole microquasar Cygnus X-1 as seen with INTEGRAL

Context.0.1–10 MeV observations of the black hole microquasar Cygnus X-1 have shown the presence of a spectral feature in the form of a power law in addition to the standard black body (0.1–10 keV) and Comptontonization (10–200 keV) components usually seen in all black hole X-ray binaries. This so-called “high-energy tail” has recently been shown to be strong in the hard spectral state and has been interpreted as high-energy part of the emission from a compact jet. Aims. This result was, however, obtained from a data set largely dominated by hard state observations. In the soft state, only upper limits on the presence and hence the potential parameters of a hard tail could be derived. Using an extended data set we aim at obtaining better constraints on the properties of this spectral component in both states. Methods. We make use of data obtained from about 15 years of observations with the INTEGRAL satellite. The data set is separated into the different states and we analyse stacked state-resolved spectra obtained from both the gamma-ray Imager and the Spectrometer onboard. Results. A high-energy component is detected in both states, confirming its earlier detection in the hard state and its suspected presence in the soft state as seen in a much smaller SPI data set. We first characterize the hard tail components in the two states through a model-independent, phenomenological analysis. We then apply physical models based on hybrid Comptonization (eqpair and belm). The spectra are well modeled in all cases, with a similar goodness of the fit. The spectral properties of the tails in the two states are, however, quite different. This might indicate that the emission originates from different media in the two cases. Our results are compatible with a compact jet origin in the hard state and hybrid Comptonization in the soft state.

F. Cangemi↗

Direct evaluation of the phase diagrams of dense multicomponent plasmas by integration of the Clapeyron equations

Accurate phase diagrams of multicomponent plasmas are required for the modeling of dense stellar plasmas, such as those found in the cores of white dwarf stars and the crusts of neutron stars. Those phase diagrams have been computed using a variety of standard techniques, which suffer from physical and computational limitations. Here we present an efficient and accurate method that overcomes the drawbacks of previously used approaches. In particular, finite-size effects are avoided as each phase is calculated separately; the plasma electrons and volume changes are explicitly taken into account; and arbitrary analytic fits to simulation data as well as particle insertions are avoided. Furthermore, no simulations at “uninteresting” state conditions, i.e., away from the phase coexistence curves, are required, which improves the efficiency of the technique. The method consists of an adaptation of the so-called Gibbs-Duhem integration approach to electron-ion plasmas, where the coexistence curve is determined by direct numerical integration of its underlying Clapeyron equation. The thermodynamics properties of the coexisting phases are evaluated separately using Monte Carlo simulations in the isobaric semigrand canonical ensemble (NPT Δ μ ). We describe this Monte Carlo-based Clapeyron integration method, including its basic physical and numerical principles, our extension to electron-ion plasmas, and our numerical implementation. We illustrate its applicability and benefits with the calculation of the melting curve of dense carbon-oxygen plasmas under conditions relevant for the cores of white dwarf stars and provide analytic fits to implement this new melting curve in white dwarf models. While this work focuses on the liquid-solid phase boundary of dense two-component plasmas, a wider range of physical systems and phase boundaries are within the scope of the Clapeyron integration method, which had until now only been applied to simple model systems of neutral particles.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Newton algorithm for fitting transfer functions to frequency response measurements

In this paper the problem of synthesizing transfer functions from frequency response measurements is considered. Given a complex vector representing the measured frequency response of a physical system, a transfer function of specified order is determined that minimizes the sum of the magnitude-squared of the frequency response errors. This nonlinear least squares minimization problem is solved by an iterative global descent algorithm of the Newton type that converges quadratically near the minimum. The unknown transfer function is expressed as a sum of second-order rational polynomials, a parameterization that facilitates a numerically robust computer implementation. The algorithm is developed for single-input, single-output, causal, stable transfer functions. Two numerical examples demonstrate the effectiveness of the algorithm.

Spanos, J. T.↗

Progenitors and explosion properties of supernova remnants hosting central compact objects: II. A global systematic study with a comparison to nucleosynthesis models

ABSTRACT Core-collapse explosions of massive stars leave behind neutron stars, with a known diversity that includes the ‘Central Compact Objects’ (CCOs). Typified by the neutron star discovered near the centre of the Cas A supernova remnant (SNR), CCOs have been observed to shine only in X-rays. To address their supernova progenitors, we perform a systematic study of SNRs that contain a CCO and display X-ray emission from their shock-heated ejecta. We make use of X-ray data primarily using the Chandra X-ray observatory, complemented with XMM–Newton. This study uses a systematic approach to the analysis of each SNR aimed at addressing the supernova progenitor as well as the explosion properties (energy and ambient density). After fitting for the ejecta abundances estimated from a spatially resolved spectroscopic study, we compare the data to six nucleosynthesis models making predictions on supernova ejecta yields in core-collapse explosions. We find that the explosion models commonly used by the astrophysics community do not match the ejecta yields for any of the SNRs, suggesting additional physics, for example multidimensional explosion models or updated progenitor structures, are required. Overall we find low-mass (≤25 solar masses) progenitors among the massive stars population and low-energy explosions (<1051 ergs). We discuss degeneracies in our model fitting, particularly how altering the explosion energy affects the estimate of the progenitor mass. Our systematic study highlights the need for improving on the theoretical models for nucleosynthesis predictions as well as for sensitive, high-resolution spectroscopy observations to be acquired with next generation X-ray missions.

79 ASTRONOMY AND ASTROPHYSICS↗

FastrSHWFS Analysis

This code analyzes the reflection geometry of modified Shack-Hartmann Wavefront Sensor (SHWFS) masks, with a focus on the fastrSHWFS designs. It takes a height map of a reflective mask, divides it into sub apertures, removes the global focus term by referencing a central block, and fits local surface planes to each sub aperture. From these fits it computes reflection angles and propagates the corresponding rays downstream to determine spot positions and the overall reflected beam footprint as a function of distance. The implementation is parameterized in sub aperture size, number of sub apertures, focal length, and mask dimensions, so it can be adapted to different mask designs beyond the two current fastrSHWFS masks. The code: Converts mask bitmaps into physical height in microns and reconstructs a focus subtracted mask, divides the mask into a grid of sub apertures and assigns pixel and micron coordinates to each block, excludes user specified unused sub apertures from the analysis, fits a plane to each sub aperture to obtain local surface normals, propagates reflected rays to a range of z positions to compute ray spots and the full reflected beam width, computes the effective mask tilt angle with respect to the incoming beam as a function of propagation distance, and provides helper routines for plotting the grid over the mask and for visualizing the beam geometry. These tools are intended for iterating on fastrSHWFS mask designs and for planning downstream optical layouts, for example choosing lens positions and apertures that capture the reflected beam given known focal plane distances and beam widths.

Gerard, BenjaminL [Lawrence Livermore National Lab↗

Two Views of the Radius Gap and the Role of Light Curve Fitting

Recently, several groups have resolved a gap that bifurcates planets between the size of Earth and Neptune into two populations. The location and depth of this feature is an important signature of the physical processes that form and sculpt planets. In particular, planets residing in the radius gap are valuable probes of these processes as they may be undergoing the final stages of envelope loss. Here, we discuss two views of the radius gap by Fulton & Petigura (F18) and Van Eylen et al. (V18). In V18, the gap is wider and more devoid of planets. This is due, in part, to V18's more precise measurements of planet radius R{sub p}. Thanks to Gaia, uncertainties in stellar radii R{sub ⋆} are no longer the limiting uncertainties in determining R{sub p} for the majority of Kepler planets; instead, errors in R{sub p}/R{sub ⋆} dominate. V18's analysis incorporated short-cadence photometry along with constraints on mean stellar density that enabled more accurate determinations of R{sub p}/R{sub ⋆}. In the F18 analysis, less accurate R{sub p}/R{sub ⋆} blurs the boundary the radius gap. The differences in R{sub p}/R{sub ⋆} are largest at high impact parameter (b≳0.8) and often exceed 10%. This motivates excluding high-b planets from demographic studies, but identifying such planets from long-cadence photometry alone is challenging. We show that transit duration can serve as an effective proxy, and we leverage this information to enhance the contrast between the super-Earth and sub-Neptune populations.

79 ASTRONOMY AND ASTROPHYSICS↗

Improved Infrared Optical Constants from Pressed Pellets: II. Ellipsometric n and k Values for Ammonium Sulfate with Variability Analysis

Infrared spectral imaging is facilitated via the comparison of spectra recorded in situ to a databank of actual or synthetic infrared reflectance spectra. It has recently been shown that reference spectra corresponding to the many different morphological forms of the same chemical can be generated synthetically using the imaginary, k, and real, n, components of the complex refractive index, n~ = n + ik. One method to obtain the n and k vectors is infrared ellipsometry, which measures the changes in amplitude, tan ?, and phase, ?, of polarized light reflected from the sample both as a function of wavenumber and angle of incidence. The method requires specularly reflected light so best results are usually obtained with polished planar samples of large surface area. Due to the difficulties of obtaining such samples, however, we investigate the possibility of pressing powders of neat materials and obtaining the corresponding optical constants from the pellets. In this paper, variability in the sample pellet and preparation method is investigated, as is variability in the fitting procedure for the derived optical constants. The n/k vectors are derived from the measured ellipsometric parameters, tan ? and ?, as they are fit by an oscillator model which yield n(?) and k(?) vectors as a function of wavenumber, ?. Construction of the oscillator model is not automatic and depends on significant input from the analyst as well as the sample’s physical characteristics. For the pellet pressing, the experimental variability was found to be minimized for size-selected powdered samples as gauged by the minimal variance in ? and ? for three different pellets; similarly, the analytical precision for multiple measurements of the same pellet was also quite good, suggesting that a pressed pellet is a viable sample preparation method. The greatest variability came in the analytic fitting procedure with differences up to 20% arising from four different fits to the same data set. The final ellipsometric n/k data are compared to literature values obtained from crystalline (NH4)2SO4 samples as well as single-angle reflectance measurements that also used pressed pellets. Comparison with the previous literature values shows generally good agreement although higher k-values are observed for the independent sets of data derived from pressed pellets. These data are suggested as an improved set of optical constants for (NH4)2SO4.

optical constants, ellipsometry, ammonium sulfate,↗

A statistical mechanics framework for polymer chain scission, based on the concepts of distorted bond potential and asymptotic matching

To design increasingly tough, resilient, and fatigue-resistant elastomers and hydrogels, the relationship between controllable network parameters at the molecular level (bond type, non-uniform chain length, entanglement density, etc.) to macroscopic quantities that govern damage and failure must be established. Many of the most successful constitutive models for elastomers have been rooted in statistical mechanical treatments of polymer chains. Typically, such constitutive models have used variants of the freely jointed chain model with rigid links. However, since the free energy state of a polymer chain is dominated by enthalpic bond distortion effects as the chain approaches its rupture point, bond extensibility ought to be accounted for if the model is intended to capture chain rupture. To that end, a new bond potential is supplemented to the freely jointed chain model (as derived in the u FJC framework of Buche and Silberstein (2021) and Buche et al. (2022)), which we have extended to yield a tractable, closed-form model of single chain behavior that should be amenable to continuum-level constitutive model development. Inspired by the asymptotically matched u FJC model response in both the low/intermediate chain force and high chain force regimes, a simple, quasi-polynomial bond potential energy function is derived. This bond potential exhibits harmonic behavior near the equilibrium state and anharmonic behavior for large bond stretches tending to a characteristic energy plateau (akin to the Lennard-Jones and Morse bond potentials). Using this bond potential, approximate yet highly-accurate analytical functions for bond stretch and chain force dependent upon chain stretch are established. Then, using this polymer chain model, a stochastic thermal fluctuation-driven chain rupture framework is developed. This framework is based upon a force-modified tilted bond potential that accounts for distortional bond potential energy, allowing for the derivation and subsequent calculation of the dissipated chain scission energy. Here, the cases of rate-dependent and rate-independent scission are accounted for throughout the rupture framework. The impact of Kuhn segment number on chain rupture behavior is also investigated. The model is fit to single chain mechanical response data collected from atomic force microscopy tensile tests for validation and to glean deeper insight into the molecular physics taking place. Due to their analytical nature, this polymer chain model and the associated rupture framework can in the future be implemented in finite element models accounting for fracture and fatigue in polydisperse elastomer networks.

36 MATERIALS SCIENCE↗

Temperature-dependent model of helium bubble expansion and bursting in tungsten

Continuum-scale models that can reliably predict the behavior of helium in tungsten are of interest to the fusion community due to the projected impact of these materials on fusion reactor operation. In this work, we perform molecular dynamics simulations of spherical helium bubbles of various sizes in tungsten at different temperatures and depths with the goal of determining a mathematical model of the pressure and density at which the bubbles initially expand or burst as a function of depth, size, temperature, and surface orientation. The bubbles are small enough that their loop-punching pressures cannot be accurately predicted with continuum mechanics, and their expansion behavior is important, as it appears to cause many of the features observed on helium-irradiated tungsten surfaces. We vary the temperature, bubble size, bubble depth, and surface orientation in each case, recording the bubble pressure and density that result in bubble expansion. An exponential function with three adjustable parameters is found to fit the results well; the parameters that best fit our results are provided.

36 MATERIALS SCIENCE↗