Search NASA⌕ Search

SEARCH · Search NASA

Results for “coherent elastic”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 181 records · Page 10

An exactly solvable model for calculating critical misfit and thickness in epitaxial superlattices - Layers of equal elastic constants and thicknesses

A parabolic interaction potential has been used to develop a model for calculating the misfit dislocation (MD) energy in the case of a superlattice of alternating layers of materials with equal elastic constants and thicknesses. The model, which is believed to be a good one for small misfits and to have some merit for covalent bonded materials, is exactly solvable for the critical thickness above which it is energetically favorable to lose coherency by the introduction of MDs into the interfaces. It was found, for a given misfit f, that the critical thickness for epitaxial superlattices free from their substrate is somewhat more than four times that for a single epilayer on a thick substrate. Furthermore, the critical thickness varies almost inversely with misfit to the power 1.22 when Poisson's ratio is 1/3. It was also shown that the critical misfit f(c) obtained by equating maximal misfit strain and MD energies is a significant overestimate of f(c). The results for a superlattice are compared with those of a thin layer on a thick substrate.

Van Der Merwe, Jan H.↗

Potential Vorticity Mixing in a Tangled Magnetic Field

A theory of potential vorticity (PV) mixing in a disordered (tangled) magnetic field is introduced. The analysis is in the context of β -plane MHD, with a special focus on the physics of momentum transport in the stably stratified, quasi-2D solar tachocline. A physical picture of mean PV evolution by vorticity advection and tilting of magnetic fields is proposed. In the case of weak field perturbations, quasi-linear theory predicts that the Reynolds and magnetic stresses balance as turbulence Alfvénizes for a larger mean magnetic field. Jet formation is explored quantitatively in the mean field–resistivity parameter space. However, since even a modest mean magnetic field leads to large magnetic perturbations for large magnetic Reynolds number, the physically relevant case is that of a strong but disordered field. We show that numerical calculations indicate that the Reynolds stress is modified well before Alfvénization—i.e., before fluid and magnetic energies balance. To understand these trends, a double-average model of PV mixing in a stochastic magnetic field is developed. Calculations indicate that mean-square fields strongly modify Reynolds stress phase coherence and also induce a magnetic drag on zonal flows. The physics of transport reduction by tangled fields is elucidated and linked to the related quench of turbulent resistivity. We propose here a physical picture of the system as a resisto-elastic medium threaded by a tangled magnetic network. Applications of the theory to momentum transport in the tachocline and other systems are discussed in detail.

79 ASTRONOMY AND ASTROPHYSICS↗

Rock elastic properties and near-surface structure at Taurus-Littrow

Linear strain measurements are presented for two lunar basalts, 14310,82 and 71055,15 and one breccia, 15498,23 to 5 kb hydrostatic pressure. Compressional and shear acoustic velocities to 5 kb are also presented for the basalts, 14310,82 and 71055,15. These elastic properties, along with geological, seismological and rock mechanics considerations are consistent with a model of the structure of the Taurus-Littrow valley as follows, a thin surface regolith overlying a fractured mixture of basalt flows and ejecta material which in turn overlies a coherent breccia of highland ejecta debris.

Trice, R.↗

Molecular Dynamics Simulations of Austenite-Martensite Interfaces in NiTi Shape Memory Alloys

The unique properties of shape memory alloys (SMAs) arise from a reversible martensitic transformation. The nucleation and migration of austenite-martensite interfaces are the key to understanding the SMA properties. Molecular dynamics (MD) simulations can provide important atomic-scale information about these aspects of the transformation, but their time scales prevent the interface formation under near-equilibrium conditions relevant to experiment. We present a new MD methodology which allows for the natural formation of energetically preferred austenite-martensite interfaces under near-equilibrium conditions. Our simulation demonstrates that the interfaces in NiTi are semi-coherent, composed of a series of terrace planes and structural disconnections, and they migrate rapidly through single crystals with only a small thermodynamic driving force. In bi-crystals and polycrystals, the migration of these same interfaces is significantly impeded by grain boundaries and stored elastic energy. This behavior can result in SMA hysteresis via several mechanisms associated with nucleation and non-elastic strain accommodation.

Gabriel Plummer↗

Precipitation Model Validation in 3rd Generation Aeroturbine Disc Alloys

In support of application of the DARPA-AIM methodology to the accelerated hybrid thermal process optimization of 3rd generation aeroturbine disc alloys with quantified uncertainty, equilibrium and diffusion couple experiments have identified available fundamental thermodynamic and mobility databases of sufficient accuracy. Using coherent interfacial energies quantified by Single-Sensor DTA nucleation undercooling measurements, PrecipiCalc(TM) simulations of nonisothermal precipitation in both supersolvus and subsolvus treated samples show good agreement with measured gamma particle sizes and compositions. Observed longterm isothermal coarsening behavior defines requirements for further refinement of elastic misfit energy and treatment of the parallel evolution of incoherent precipitation at grain boundaries.

Olson, G. B.↗

Quasi-elastic nuclear scattering at high energies

The quasi-elastic scattering of two nuclei is considered in the high-energy optical model. Energy loss and momentum transfer spectra for projectile ions are evaluated in terms of an inelastic multiple-scattering series corresponding to multiple knockout of target nucleons. The leading-order correction to the coherent projectile approximation is evaluated. Calculations are compared with experiments.

Cucinotta, Francis A.↗

In Situ Nano-Indentation of a Gold Sub-Micrometric Particle Imaged by Multi-Wavelength Bragg Coherent X-ray Diffraction

The microstructure of a sub-micrometric gold crystal during nanoindentation is visualized by in situ multi-wavelength Bragg coherent X-ray diffraction imaging. The gold crystal is indented using a custom-built atomic force microscope. A band of deformation attributed to a shear band oriented along the (221) lattice plane is nucleated at the lower left corner of the crystal and propagates towards the crystal center with increasing applied mechanical load. After complete unloading, an almost strain-free and defect-free crystal is left behind, demonstrating a pseudo-elastic behavior that can only be studied by in situ imaging while it is invisible to ex situ examinations. The recovery is probably associated with reversible dislocations nucleation/annihilation at the side surface of the particle and at the particle-substrate interface, a behavior that has been predicted by atomistic simulations. The full recovery of the particle upon unloading sheds new light on extraordinary mechanical properties of metal nanoparticles obtained by solid-state dewetting.

36 MATERIALS SCIENCE↗

Molecular Dynamics Simulations of Microstructural Effects on Austenite-Martensite Interfaces in NiTi

Formation and migration of austenite-martensite interfaces plays the key role in reversible martensitic transformations of shape memory alloys (SMAs). How these interfaces interact with the SMA microstructure is a primary determining factor in important functional properties such as hysteresis and transformation span. As such, successful microstructural engineering of SMAs requires in-depth knowledge of interface behavior. The rapid nature of martensitic transformations makes experimental observations of moving austenite-martensite interfaces challenging. Molecular dynamics (MD) simulation is a unique tool which can probe the atomic-scale details of austenite-martensite interfaces as they migrate through different microstructures. However, in focusing on the entire transformation process, including the nucleation of new phases, MD studies are usually performed so far from equilibrium that their relevance to experiment is questionable. Here, we demonstrate new MD techniques to generate energetically preferred austenite-martensite interfaces in NiTi under near-equilibrium conditions. The interfaces are semi-coherent, exhibiting a series of structural disconnections, and they can migrate rapidly through single crystals under only small thermodynamic driving forces. In contrast, when interfaces migrate in polycrystals, their motion is impeded by thermoelastic effects as well as changes in orientation relationships at grain boundaries. Microstructures which accumulate large amounts of elastic energy tend to release some fraction through irreversible, hysteresis-inducing mechanisms. We demonstrate that engineering microstructures with less constraints is a viable strategy to produce SMAs with reduced hysteresis and transformation span. Similar thermoelastic and hysteresis-inducing mechanisms also arise when austenite-martensite interfaces encounter precipitates and can be controlled by tuning characteristics of the precipitate distribution.

Gabriel Plummer↗

Molecular Dynamics Simulations of Microstructural Effects on Austenite-Martensite Interfaces in NiTi

The formation and migration of austenite-martensite interfaces plays the key role in reversible martensitic transformations of shape memory alloys (SMAs). How these interfaces interact with the SMA microstructure is a primary determining factor in important functional properties such as hysteresis and transformation span. As such, successful microstructural engineering of SMAs requires in-depth knowledge of interface behavior. The rapid nature of martensitic transformations makes experimental observations of moving austenite-martensite interfaces challenging. Molecular dynamics (MD) simulation is a unique tool which can probe the atomic-scale details of austenite-martensite interfaces as they migrate through different microstructures. However, in focusing on the entire transformation process, including the nucleation of new phases, MD studies are usually performed so far from equilibrium that their relevance to experiment is questionable. Here, we demonstrate new MD techniques to generate energetically preferred austenite-martensite interfaces in NiTi under near-equilibrium conditions. The interfaces are semi-coherent, exhibiting a series of structural disconnections, and they can migrate rapidly through single crystals under only small thermodynamic driving forces. In contrast, when interfaces migrate in polycrystals, their motion is impeded by thermoelastic effects as well as changes in orientation relationships at grain boundaries. Microstructures which accumulate large amounts of elastic energy tend to release some fraction through irreversible, hysteresis-inducing mechanisms. We demonstrate that engineering microstructures with less constraints is a viable strategy to produce SMAs with reduced hysteresis and transformation span. Similar thermoelastic and hysteresis-inducing mechanisms also arise when austenite-martensite interfaces encounter precipitates and can be controlled by tuning characteristics of the precipitate distribution.

Gabriel Plummer↗

Molecular Dynamics Simulations of Microstructural Effects on Austenite-Martensite Interfaces in NiTi

Formation and migration of austenite-martensite interfaces plays the key role in reversible martensitic transformations of shape memory alloys (SMAs). How these interfaces interact with the SMA microstructure is a primary determining factor in important functional properties such as hysteresis and transformation span. As such, successful microstructural engineering of SMAs requires in-depth knowledge of interface behavior. The rapid nature of martensitic transformations makes experimental observations of moving austenite-martensite interfaces challenging. Molecular dynamics (MD) simulation is a unique tool which can probe the atomic-scale details of austenite-martensite interfaces as they migrate through different microstructures. However, in focusing on the entire transformation process, including the nucleation of new phases, MD studies are usually performed so far from equilibrium that their relevance to experiment is questionable. Here, we demonstrate new MD techniques to generate energetically preferred austenite-martensite interfaces in NiTi under near-equilibrium conditions. The interfaces are semi-coherent, exhibiting a series of structural disconnections, and they can migrate rapidly through single crystals under only small thermodynamic driving forces. In contrast, when interfaces migrate in polycrystals, their motion is impeded by thermoelastic effects as well as changes in orientation relationships at grain boundaries. Microstructures which accumulate large amounts of elastic energy tend to release some fraction through irreversible, hysteresis-inducing mechanisms. We demonstrate that engineering microstructures with less constraints is a viable strategy to produce SMAs with reduced hysteresis and transformation span. Similar thermoelastic and hysteresis-inducing mechanisms also arise when austenite-martensite interfaces encounter precipitates and can be controlled by tuning characteristics of the precipitate distribution.

Gabriel Plummer↗

Instrument for in situ hard x-ray nanobeam characterization during epitaxial crystallization and materials transformations

Solid-phase epitaxy (SPE) and other three-dimensional epitaxial crystallization processes pose challenging structural and chemical characterization problems. The concentration of defects, the spatial distribution of elastic strain, and the chemical state of ions each vary with nanoscale characteristic length scales and depend sensitively on the gas environment and elastic boundary conditions during growth. The lateral or three-dimensional propagation of crystalline interfaces in SPE has nanoscale or submicrometer characteristic distances during typical crystallization times. Here, an in situ synchrotron hard x-ray instrument allows these features to be studied during deposition and crystallization using diffraction, resonant scattering, nanobeam and coherent diffraction imaging, and reflectivity. The instrument incorporates a compact deposition system allowing the use of short-working-distance x-ray focusing optics. Layers are deposited using radio-frequency magnetron sputtering and evaporation sources. The deposition system provides control of the gas atmosphere and sample temperature. The sample is positioned using a stable mechanical design to minimize vibration and drift and employs precise translation stages to enable nanobeam experiments. Results of in situ x-ray characterization of the amorphous thin film deposition process for a SrTiO 3 /BaTiO 3 multilayer illustrate implementation of this instrument.

47 OTHER INSTRUMENTATION↗

Shadowgraph Study of Gradient Driven Fluctuations

A fluid or fluid mixture, subjected to a vertical temperature and/or concentration gradient in a gravitational field, exhibits greatly enhanced light scattering at small angles. This effect is caused by coupling between the vertical velocity fluctuations due to thermal energy and the vertically varying refractive index. Physically, small upward or downward moving regions will be displaced into fluid having a refractive index different from that of the moving region, thus giving rise to the enhanced scattering. The scattered intensity is predicted to vary with scattering wave vector q, as q(sup -4), for sufficiently large q, but the divergence is quenched by gravity at small q. In the absence of gravity, the long wavelength fluctuations responsible for the enhanced scattering are predicted to grow until limited by the sample dimensions. It is thus of interest to measure the mean-squared amplitude of such fluctuations in the microgravity environment for comparison with existing theory and ground based measurements. The relevant wave vectors are extremely small, making traditional low-angle light scattering difficult or impossible because of stray elastically scattered light generated by optical surfaces. An alternative technique is offered by the shadowgraph method, which is normally used to visualize fluid flows, but which can also serve as a quantitative tool to measure fluctuations. A somewhat novel shadowgraph apparatus and the necessary data analysis methods will be described. The apparatus uses a spatially coherent, but temporally incoherent, light source consisting of a super-luminescent diode coupled to a single-mode optical fiber in order to achieve extremely high spatial resolution, while avoiding effects caused by interference of light reflected from the various optical surfaces that are present when using laser sources. Results obtained for a critical mixture of aniline and cyclohexane subjected to a vertical temperature gradient will be presented. The sample was confined between two horizontal parallel sapphire plates with a vertical spacing of 1 mm. The temperatures of the sapphire plates were controlled by independent circulating water loops that used Peltier devices to add or remove heat from the room air as required. For a mixture with a temperature gradient, two effects are involved in generating the vertical refractive index gradient, namely thermal expansion and the Soret effect, which generates a concentration gradient in response to the applied temperature gradient. For the aniline/cyclohexane system, the denser component (aniline) migrates toward the colder surface. Consequently, when heating from above, both effects result in the sample density decreasing with altitude and are stabilizing in the sense that no convective motion occurs regardless of the magnitude of the applied temperature gradient. The Soret effect is strong near a binary liquid critical point, and thus the dominant effect is due to the induced concentration gradient. The results clearly show the divergence at low q and the predicted gravitational quenching. Results obtained for different applied temperature gradients at varying temperature differences from the critical temperature, clearly demonstrate the predicted divergence of the thermal diffusion ratio. Thus, the more closely the critical point is approached, the smaller becomes the temperature gradient required to generate the same signal. Two different methods have been used to generate pure concentration gradients. In the first, a sample cell was filled with a single fluid, ethylene glycol, and a denser miscible fluid, water, was added from below thus establishing a sharp interface to begin the experiment. As time went on the two fluids diffused into each other, and large amplitude fluctuations were clearly observed at low q. The effects of gravitational quenching were also evident. In the second method, the aniline/cyclohexane sample was used, and after applying a vertical temperature gradient for several hours, the top and bottom temperatures were set equal and the thermal gradient died on a time scale of seconds, leaving the Soret induced concentration gradient in place. Again, large-scale fluctuations were observed and died away slowly in amplitude as diffusion destroyed the initial concentration gradient.

Cannell, David↗

Visualization of shocked material instabilities using a fast-framing camera and XFEL four-pulse train

Many questions regarding dynamic materials could be answered by using time-resolved ultra-fast imaging techniques to characterize the physical and chemical behavior of materials in extreme conditions and their evolution on the nanosecond scale. In this work, we perform multi-frame phase-contrast imaging (PCI) of micro-voids in low density polymers under laser-driven shock compression. At the Matter in Extreme Conditions (MEC) Instrument at the Linac Coherent Light Source (LCLS), we used a train of four x-ray free electron laser (XFEL) pulses to probe the evolution of the samples. To visualize the void and shock wave interaction, here, we deployed the Icarus V2 detector to record up to four XFEL pulses, separated by 1-3 nanoseconds. In this work, we image elastic waves interacting with the micro-voids at a pressure of several GPa. Monitoring how the material’s heterogeneities, like micro-voids, dictate its response to a compressive wave is important for benchmarking the performances of inertial confinement fusion energy materials. For the first time in a single sample, we have combined an ultrafast x-ray framing camera and four XFEL pulse train to create an ultrafast movie of micro-void evolution under laser-driven shock compression. Eventually, we hope this technique will resolve the material density as it evolves dynamically under laser shock compression.

fusion↗

Direct Imaging of Hydrogen‐Driven Dislocation and Strain Field Evolution in a Stainless Steel Grain

Hydrogen embrittlement (HE) poses a significant challenge to the durability of materials used in hydrogen production and utilization. Disentangling the competing nanoscale mechanisms driving HE often relies on simulations and electron-transparent sample techniques, limiting experimental insights into hydrogen-induced dislocation behavior in bulk materials. This study employs in situ Bragg coherent X-ray diffraction imaging to track three-dimensional (3D) dislocation and strain field evolution during hydrogen charging in a bulk grain of austenitic 316 stainless steel. Tracking a single dislocation reveals hydrogen-enhanced mobility and relaxation, consistent with dislocation dynamics simulations. Subsequent observations reveal dislocation unpinning and climb processes, likely driven by osmotic forces. Additionally, nanoscale strain analysis around the dislocation core directly measures hydrogen-induced elastic shielding. These findings experimentally validate theoretical predictions and offer mechanistic insights into hydrogen-driven dislocation behavior. The quantified nanoscale phenomena serve as critical inputs for multiscale modeling frameworks to predict bulk material responses and accelerate the development of HE-resistant alloys.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Signatures of electronic and nuclear coherences in ultrafast molecular x-ray and electron diffraction

Femtosecond x-ray and electron diffraction hold promise to image the evolving structures of single molecules. We present a unified quantum-electrodynamical formulation of diffraction signals, based on the exact many-body nuclear + electronic wavefunction that can be extracted from quantum chemistry simulations. This gives a framework for analyzing various approximate molecular dynamics simulations. We show that the complete description of ultrafast diffraction signals contains interesting contributions involving mixed elastic and inelastic scattered photons that are usually masked by other larger contributions and are neglected. These terms include overlaps of nuclear wavepackets between different electronic states that provide an electronic decoherence mechanism and are important for the time-resolved imaging of conical intersections.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

STRUCTURE AND PROPERTIES STUDY ON ENERGY MATERIALS: THERMOELECTRIC MATERIAL TETRAHEDRITE AND LITHIUM ION CONDUCTOR LiPON

Development of efficient energy materials is critical in order to ease the energy demand and reduce our dependence of fossil fuel. Thermoelectric materials are promising due to their capability of generating electrical power by recovering waste heat. The performance of thermoelectric materials is quantified by a dimensionless figure of merit zT, which depends on their properties such as electrical conductivity, Seebeck coefficient and thermal conductivity. Tetrahedrites, a copper antimony sulfosalt mineral, typified by Cu12-xMxSb4S13, where M is a transition metal element such as Ni, Zn, Fe or Mn, have great potential for thermoelectric application due to their relatively high zT (close to 1 at 700 K), earth-abundance, environmental friendliness, favorable electrical properties, and most importantly intrinsic low lattice thermal conductivity (less than 1 W m-1 K-1) in wide temperature. In addition to energy recovery, reliable energy storage devices are also emerging to relieve the energy demand and improve the efficiency of consuming energy resources. Lithium-ion batteries are known to be reliable and successful electrochemical energy storage devices and appliable in various aspects, including laptops, smartphones and electrical vehicles. Lithium phosphorous oxynitride (LiPON) are widely used as thin-film solid-state electrolytes in Li-ion battery, which is the only demonstrated solid-state electrolyte that is quite stable in direct contact with Li metal at potentials from 0-5 V. However, the structure of LiPON, the effects of N doping, and the origin of its good electrochemical stability remains inconclusive.In this thesis, reliable modeling techniques accompanied with experimental tools, are applied to study the thermoelectric material tetrahedrite and the ionic conductor LiPON, in order to study their structural and dynamical properties. Accurate and efficient density-functional theory (DFT) and density-functional tight-binding (DFTB) methods, combined with molecular dynamics (MD) simulations are utilized in order to investigate the structures and properties of these energy materials. The incoherent and coherent atomic dynamics study of tetrahedrite Cu10.5NiZn0.5Sb4S13 provides the origin of softening upon cooling by investigate the motion of Cu12e at different temperatures. The dynamic structure factors in the longitudinal and transverse direction will also be discussed. The Cu movement of Cu-rich tetrahedrite Cu14Sb4S13 is revealed by Cu self-diffusivity, nuclear density map and "nudged elastic band" (NEB). Moreover, we investigate the effect of simulation cell size and basis sets on the DFT-based MD simulation results using tetrahedrite Cu10Zn2Sb4S13 thermoelectric as a model material, showing the advantage of larger cell by accessing smaller Q range. In addition, the low-temperature structural properties of Cu12Sb4S13 is measured by neutron diffraction, which indicates that no cubic to tetragonal transition occurs at metal-semiconductor transition (MST) temperature. Thermoelectric properties such as Seebeck coefficient, electrical resistivity and electrical thermal conductivity will also be investigated. DFTB method is implemented to study the structure and transport properties of Li3PO4 and LiPON, while the exploration of N doping effect is included. Lastly, the LiPON/Li interphase will be revealed in order to study the origin of electrochemical stability

25 ENERGY STORAGE↗

In‐Operando Spatiotemporal Imaging of Coupled Film‐Substrate Elastodynamics During an Insulator‐to‐Metal Transition

Abstract The drive toward non‐von Neumann device architectures has led to an intense focus on insulator‐to‐metal (IMT) and the converse metal‐to‐insulator (MIT) transitions. Studies of electric field‐driven IMT in the prototypical VO 2 thin‐film channel devices are largely focused on the electrical and elastic responses of the films, but the response of the corresponding TiO 2 substrate is often overlooked, since it is nominally expected to be electrically passive and elastically rigid. Here, in‐operando spatiotemporal imaging of the coupled elastodynamics using X‐ray diffraction microscopy of a VO 2 film channel device on TiO 2 substrate reveals two new surprises. First, the film channel bulges during the IMT, the opposite of the expected shrinking in the film undergoing IMT. Second, a microns thick proximal layer in the substrate also coherently bulges accompanying the IMT in the film, which is completely unexpected. Phase‐field simulations of coupled IMT, oxygen vacancy electronic dynamics, and electronic carrier diffusion incorporating thermal and strain effects suggest that the observed elastodynamics can be explained by the known naturally occurring oxygen vacancies that rapidly ionize (and deionize) in concert with the IMT (MIT). Fast electrical‐triggering of the IMT via ionizing defects and an active “IMT‐like” substrate layer are critical aspects to consider in device applications.

36 MATERIALS SCIENCE↗

Estimating lithospheric properties at Atla Regio, Venus

Magellan spehrical harmonic gravity and topography models are used to estimate lithospheric properties at Alta Regio, Venus, a proposed hotspot with dynamic support from mantle plume(s). Global spherical harmonic and local representations of the gravity field share common properties in the Atla region interms of their spectral behavior over a wavelength band from approximately 2100 to approximately 700 km. The estimated free-air admittance spectrum displays a rather featureless long-wavelength portion followed by a sharp rise at wavelengths shorter than about 1000 km. This sharp rise requires significant flexural support of short-wavelength structures. The Bouguer coherence also displays a sharp drop in this wavelength band, indicating a finite flexural rigidity of the lithosphere. A simple model for lithospheric loading from above and below is introduced (D. W. Forsyth, 1985) with four parameters: f, the ratio of bottom loading to top loading; z(sub m), crustal thickness; z(sub l) depth to bottom loading source; and T(sub e) elastic lithosphere thickness. A dual-mode compensation model is introduced in which the shorter wavelengths (lambda approximately less than 1000 km) might be explained best by a predominance of top loading by the large shield volcanoes Maat Mons, Ozza Mons, and Sapas Mons, and the longer wavelengths (lambda approximately greater than 1500 km) might be explained best by a deep depth of compensation, possibly representing bottom loading by a dynamic source. A Monte Carlo inversion technique is introduced to thoroughly search out the four-space of the model parameters and to examine parameter correlation in the solutions. Venus either is a considerabe deficient in heat sources relative to Earth, or the thermal lithosphere is overthickened in response to an earlier episode of significant heat loss from the planet.

Phillips, Roger J.↗