Quotidian Equation Of State Research Articles

RHDLPP: A multigroup radiation hydrodynamics code for laser-produced plasmas

In this paper, we introduce the RHDLPP, a flux-limited multigroup radiation hydrodynamics numerical code designed for simulating laser-produced plasmas in diverse environments. The code bifurcates into two packages: RHDLPP-LTP for low-temperature plasmas generated by moderate-intensity nanosecond lasers, and RHDLPP-HTP for high-temperature, high-density plasmas formed by high-intensity laser pulses. The core radiation hydrodynamic equations are resolved in the Eulerian frame, employing an operator-split method. This method decomposes the solution into two substeps: first, the explicit resolution of the hyperbolic subsystems integrating radiation and fluid dynamics; second, the implicit treatment of the parabolic part comprising stiff radiation diffusion, heat conduction, and energy exchange. Laser propagation and energy deposition are modeled through a hybrid approach, combining geometrical-optics ray-tracing in sub-critical plasma regions with a one-dimensional solution of the Helmholtz wave equation in super-critical areas. The thermodynamic states are ascertained using an equation of state, based on either the real gas approximation or the quotidian equation of state (QEOS). For ionization calculations, the code employs a steady-state collisional-radiation (CR) model using the screened-hydrogenic approximation. Additionally, RHDLPP includes RHDLPP-SpeIma3D, a three-dimensional spectral simulation post-processing module, for generating both temporally-spatially resolved and time-integrated spectra and imaging, facilitating direct comparisons with experimental data. The paper showcases a series of verification tests to establish the code's accuracy and efficiency, followed by application cases, including simulations of laser-produced aluminium (Al) plasmas, pre-pulse-induced target deformation of tin (Sn) microdroplets relevant to extreme ultraviolet lithography light sources, and varied imaging and spectroscopic simulations. These simulations highlight RHDLPP's effectiveness and applicability in fields such as laser-induced breakdown spectroscopy, extreme ultraviolet lithography sources, and high-energy-density physics.

The ion equation of state of plasmas in the warm dense matter regime

The existing “quotidian equation of state (QEOS)” model [More et al., Phys. Fluids 31, 3059 (1988)] has been revised, and an alternative set of formulas is provided for the Helmholtz free energy, internal energy, heat capacity, and pressure. A novel scheme for constructing the ion equation-of-state is proposed based on the additive of solid and fluid components that act throughout the temperature range, in contrast to the QEOS that matches the thermodynamic parameters at selected interfaces. These components are continuous along with their first and second derivatives and obey limiting cases and thermodynamics laws (Lindemann melting, Dulong–Petit law, Grüneisen pressure law, and ideal gas law). Thus, the new scheme eliminates discontinuities in thermodynamics parameters across interfaces and ensures that the thermodynamics parameters are consistent with each other. The Helmholtz free energy, internal energy, heat capacity, and pressure have been approximated with computationally efficient formulas that can be used as parts of other models, e.g., hydro-simulations.

The equation of state package FEOS for high energy density matter

The equation of state package FEOS for high energy density matter

Numerical model for electrical explosion of copper wires in water

This paper presents a simple but quite accurate numerical model for analyzing electrical explosion of copper wires in water. The numerical model solves a circuit equation coupled with one-dimensional magneto-hydrodynamic (MHD) equations with the help of appropriate wide-range equation of state (EOS) and electrical conductivity for copper. The MHD equations are formulated in a Lagrangian form to identify the interface between the wire and surrounding water clearly. A quotidian EOS (QEOS) that is known as the simplest form of EOS is utilized to build wide-range EOS for copper. In the QEOS, we consider the liquid-vapor phase transition, which is critical in analyzing the wire explosion system. For the electrical conductivity of copper, a semi-empirical set of equations covering from solid state to partially ionized plasma state are employed. Experimental validation has been performed with copper wires of various diameters, which are exploded by a microsecond timescale pulsed capacitive discharge. The simulation results show excellent agreements with the experimental results in terms of temporal motions of a plasma channel boundary and a shock front as well as current and voltage waveforms. It is found that the wire explodes (vaporizes) along the liquid branch of a binodal curve irrespective of wire dimension and operating voltage. After the explosion, the wire becomes a plasma state right away or after the current pause (dwell), depending on the operating conditions. It is worth noting that such a peculiar characteristic of wire explosion, i.e., current pause and restrike, is well simulated with the present numerical model. In particular, it is shown that the wire cools down along the vapor branch of the binodal curve during the current dwell, due to a significant difference of thermodynamic characteristics across the binodal curve. The influence of radiation for studying nonideal plasmas with a wire explosion technique and a physical process for shock wave formation by an exploding wire is discussed as well.

BADGER v1.0: A Fortran equation of state library

The BADGER equation of state library was developed to enable inertial confinement fusion plasma codes to more accurately model plasmas in the high-density, low-temperature regime. The code had the capability to calculate 1- and 2-T plasmas using the Thomas–Fermi model and an individual electron accounting model. Ion equation of state data can be calculated using an ideal gas model or via a quotidian equation of state with scaled binding energies. Electron equation of state data can be calculated via the ideal gas model or with an adaptation of the screened hydrogenic model with ℓ-splitting. The ionization and equation of state calculations can be done in local thermodynamic equilibrium or in a non-LTE mode using a variant of the Busquet equivalent temperature method. The code was written as a stand-alone Fortran library for ease of implementation by external codes. EOS results for aluminum are presented that show good agreement with the SESAME library and ionization calculations show good agreement with the FLYCHK code. Program summaryProgram title: BADGERLIB v1.0Catalogue identifier: AEND_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEND_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 41480No. of bytes in distributed program, including test data, etc.: 2904451Distribution format: tar.gzProgramming language: Fortran 90.Computer: 32- or 64-bit PC, or Mac.Operating system: Windows, Linux, MacOS X.RAM: 249.496 kB plus 195.630 kB per isotope record in memoryClassification: 19.1, 19.7.Nature of problem:Equation of State (EOS) calculations are necessary for the accurate simulation of high energy density plasmas. Historically, most EOS codes used in these simulations have relied on an ideal gas model. This model is inadequate for low-temperature, high-density plasma conditions; the gaseous and liquid phases; and the solid phase. The BADGER code was developed to give more realistic EOS data in these regimes.Solution method:BADGER has multiple, user-selectable models to treat the ions, average-atom ionization state and electrons. Ion models are ideal gas and quotidian equation of state (QEOS), ionization models are Thomas–Fermi and individual accounting method (IEM) formulation of the screened hydrogenic model (SHM) with l-splitting, electron ionization models are ideal gas and a Helmholtz free energy minimization method derived from the SHM. The default equation of state and ionization models are appropriate for plasmas in local thermodynamic equilibrium (LTE). The code can calculate non-LTE equation of state (EOS) and ionization data using a simplified form of the Busquet equivalent-temperature method.Restrictions:Physical data are only provided for elements Z=1 to Z=86. Multiple solid phases are not currently supported. Liquid, gas and plasma phases are combined into a generalized “fluid” phase.Unusual features:BADGER divorces the calculation of average-atom ionization from the electron equation of state model, allowing the user to select ionization and electron EOS models that are most appropriate to the simulation. The included ion ideal gas model uses ground-state nuclear spin data to differentiate between isotopes of a given element.Running time:Example provided only takes a few seconds to run.

Influence of real gas effects on ablative Rayleigh-Taylor instability in plastic target

In this research, real gas effects on ablative Rayleigh-Taylor instability are investigated in a plastic target. The real gas effects are included by adopting the quotidian equation of state (QEOS) model. Theoretical solutions for both QEOS and ideal gas EOS are obtained and compared, based on a same set of ablation parameters. It is found that when real gas effects are considered, the density gradient becomes less steep than that of ideal gas assumption, even though this cannot be used directly to draw a stabilization conclusion for the real gas effects. Further analysis shows that when real gas effects are considered, lower ∂p/∂T in the dense shell region has the effect of stabilization, whereas the dependence of the internal energy on the density, lower specific heat (at constant volume) in the dense shell region, and higher specific heat in the low-density ablation region contribute to stronger destabilization effects. Overall, when real gas effects are considered, the destabilization effects are dominant for long wavelength perturbations, and the growth rates become much higher than the results of ideal gas assumption. In our specific case, the maximum relative error reaches 18%.

Equation of State for laboratory astrophysics applications

New improvements on the calculation of Equation of State (EOS) for laboratory astrophysics applications are presented. A new empirical multiplier for the EOS is included to the original quotidian equation of state (QEOS) model developed by More et al. (Phys. Fluids 31:3059, 1988) to adapt it to the available experimental data and ab initio molecular dynamics simulation. This model is used to obtain EOS tables suited for an adaptive mesh refinement hydrodynamic code with radiation transport for high energy density plasmas simulations called ARWEN introduced by Ogando and Velarde (J. Quant. Spectrosc. Radiat. Transf. 71(2–6):541, 2001).

Effects of bound electrons and radiation on shock Hugoniot

Hydrodynamic flow of materials, initiated by shock waves, is determined by their equation of state (EOS). For comparatively weak shocks, the zero-temperature isotherm and thermal motion of ions, mainly, determine the EOS. However, in processes involving high energy density, as in inertial confinement fusion, astrophysical phenomena, nuclear explosion, etc., very strong shocks $(Pg\text{few}\text{ }\text{megabars},\phantom{\rule{0.3em}{0ex}}Tg\text{few}\text{ }\text{eV})$ are encountered. Such shocks give rise to many thermal effects leading to dissociation of molecules, ionization of electrons, radiation emission, etc., in addition to the quantum-mechanical pressure ionization in materials. Therefore, hydrodynamics due to strong shocks crucially depend on the behavior of electrons and the radiation emitted by the electrons. This paper aims at developing a simple but quantitative model of electronic binding in plasmas and its effects on compressibility of materials. An improved version of the screened hydrogenic model is developed for this purpose. The effect of radiation emission is incorporated using Stefan-Boltzmann law. The Hugoniot of various elements such as Al, Be, Fe, etc., are, then, computed. These are in excellent agreement with those obtained using sophisticated self-consistent field calculations, and the oscillations in Hugoniot are shown to be due to ionization of electrons from different shells. Shell effects are also reflected in the variation of electronic specific heat with temperature and the relation between shock velocity and fluid velocity. Further, at very high temperature and pressure, equilibration between radiation and matter increases the compressibility of materials. The limiting compression of all materials, via a strong shock, is found to be 7 unlike 4, which is the limit for free-electron gas or ideal monoatomic gas. The model reported here can be employed in lieu of Thomas-Fermi-type theories used in global EOS packages such as quotidian equation of state (QEOS).

Equation of state using scaled binding energy model

Global equation of state models are currently used for hydrodynamic simulations of inertial confinement fusion systems. The quotidian equation of state (QEOS) is one of such models. In spite of a sound theoretical basis, QEOS uses two empirical corrections for obtaining agreement with measured shock wave data. While the first is used for all materials, the second is to account for structural phase transitions. Evaluation of these corrections requires a priori knowledge of experimental data. Two improvements to QEOS proposed in this article obviate the use of both corrections. First, a modified version of the universal scaled binding energy is used for the zero-temperature isotherm. Then, an expression is derived for the Grüneisen parameter γ(ρ), which also includes the noncentral features of interparticle potential. The Debye temperature and melting temperature, deduced from γ(ρ), show excellent agreement with experimental results or electronic structure calculations. Predictions of the improved EOS model also compare very well with measured Hugoniot for normal metals such as Al, Cu, Pd, Pt, etc., and Fe and Zr, which undergo shock induced phase transitions.

Possibility of an unequivocal test of different models of the equation of state of aluminum in the coupling regime Γ∼1–50

The equation of state (EOS) in regimes of density (rho) and temperature (T) which are inaccessible to experiment has to be determined using theories which may themselves be out of their range of validity. Even for Al, the EOS in the region 0.1<rho<2 g cm(-3), and 1<T<50 eV, where the ionic coupling parameter Gamma ranges from 1- approximately 50, is not unknown. We present results for the EOS using the Thomas-Fermi model (TF), the quotidian equation of state, the Sesame tabulation, and using the density-functional neutral-pseudoatom (NPA) model which is a first-principles theory applicable at strong coupling. It is found that the NPA predictions are very different from the other models, and experiments could provide an unequivocal test of the validity of the different EOS models. We report theoretical results for the Hugoniot, and the electrical conductivity in the regime of interest.

A new quotidian equation of state (QEOS) for hot dense matter

The quotidian equation of state (QEOS) is a general-purpose equation of state model for use in hydrodynamic simulation of high-pressure phenomena. Electronic properties are obtained from a modified Thomas–Fermi statistical model, while ion thermal motion is described by a multiphase equation of state combining Debye, Grüneisen, Lindemann, and fluid-scaling laws. The theory gives smooth and usable predictions for ionization state, pressure, energy, entropy, and Helmholtz free energy. When necessary, the results may be modified by a temperature-dependent pressure multiplier which greatly extends the class of materials that can be treated with reasonable accuracy. In this paper a comprehensive evaluation of the resulting thermodynamic data is given including comparison with other theories and shock-wave data.