Comparison of transport models in dense plasmas

Johnson, Zachary A.; Silvestri, Luciano G.; Petrov, George M.; Stanton, Liam G.; Murillo, Michael S.

doi:10.1063/5.0204226

We compare a variety of models used for the calculation of transport coefficients in dense plasmas, including average-atom models, models based on kinetic theory, structure matching effective potentials, and pair-potential molecular dynamics. In particular, we focus on the parameter space investigated in the second charged-particle transport coefficient code comparison workshop [Stanek et al., Phys. Plasmas 31, 052104 (2024)]. Each model is based on the self-consistent output of our average-atom calculations. Ionic transport properties are generated from implicit electron pair matched molecular dynamics simulations, bypassing the need for either dynamical electron simulations or on-the-fly electronic structure calculations. These matched pair potentials are generated in a nonlinear way using a classical mapping procedure, further avoiding an expensive force-matching procedure. We compare these results with the density functional theory data presented at the workshop, as well as a set of widely used parametric models, which we have modified to enhance accuracy, especially at the low- and high-temperature extremes of the parameter space. We also detail the non-trivial statistical aspect of converging ionic transport coefficients.

I. INTRODUCTION

Dense plasma physics is characterized by a spectrum of scientific and technological applications composed of matter under extreme conditions of density and temperature.^1,2 Dense plasmas are not only prevalent in astrophysical settings such as stars,^3,4 giant planets, and exoplanets,⁵ but also play a crucial role in a range of technological applications, pulsed power systems,⁶ laser-produced plasmas,^7,8 laser-material interactions,⁹ beam-generated plasmas,^10,11 heavy ion fusion,^12,13 shock-produced plasmas,¹⁴ and inertial fusion energy schemes.¹⁵ The scientific challenges in this field are manifold, encompassing the need to accurately describe phenomena such as continuum lowering in compressed atoms,^1,16 electron degeneracy,² strong scattering,^17,18 and strong coupling.^2,19 These phenomena in dense plasmas present a formidable task for current computational methods, highlighting the need for advanced and precise modeling techniques to capture their unique characteristics.

When local equilibria exist within the plasma, the microscopic electronic and ionic structures determine the transport of energy and matter in terms of a set of transport coefficients—such as viscosities,²⁰ diffusion,²¹ thermal and electrical conductivities,^22,23 temperature relaxation,¹⁷ and fast-particle stopping,²⁴ which are essential inputs for designing experiments and developing predictive codes. Traditional kinetic theory frameworks, such as Boltzmann, Fokker–Planck, and Lenard–Balescu, are useful in certain contexts,^18,25 but they are based around binary collision models that are not applicable to dense plasmas without significant modification. Similarly, traditional condensed matter theory methods are also not applicable at the high temperatures of the dense plasma regime. These limitations underscore the need for innovative approaches and methodologies in plasma physics. The Transport Coefficient Code Comparison Workshops (TCCW) exemplify efforts to address these challenges,^26,27 aiming to evaluate and improve the methods used for calculating these critical transport coefficients. This paper describes our contribution to the second transport coefficient code comparison workshop (TCCW-2), shedding light on the latest advancements and ongoing research in the field of dense plasma physics.

Any non-phenomenological approach to the generation of transport coefficients in dense plasmas necessitates a many-body electronic structure calculation. Ab initio methods such as density functional theory molecular dynamics (DFT-MD), which incorporates ionic correlations and the ground state electronic structure, have seen significant success in the computation of these transport coefficients,^28–32 but at great computational expense. An alternative way to obtain the electronic structure of dense plasmas at lower computational cost is to employ the average-atom (AA) model.^33–39 It computes the atom's electronic structure, including the wave functions and number of free and bound electrons, density of states, mean ionization state, bound-state radius, and pressure. It is fast (a few minutes of run time) and covers a wide range of densities and temperatures. The AA model comes in different flavors. Starrett and Saumon model both the electronic and ionic structures of the plasma by including ion correlation effects that account for the penetration of adjacent ions into the ion sphere under consideration.³⁹ Dharma-wardana and Perrot developed the neutral-pseudo-atom approach (NPA), in which the electron wave functions are no longer confined to the Wigner–Seitz sphere, but allowed to extend beyond its boundary into a large correlation sphere.^40,41 In this work, the AA model follows Ref. 36.

Many methods exist for the conversion of such an electronic structure calculation into a set of transport coefficients. For example, the AA model can generate an electron–ion collision cross section to obtain electron transport coefficients. The AA model additionally provides atomic information that can be used to generate effective ion–ion potentials for use in ionic pair-potential molecular dynamics (MD) simulations or in kinetic theory.⁴² Multiple paths exist for generating these effective interactions, including linear response theory, the quantum Ornstein–Zernike hypernetted chain,^39,43 and the classical-map hypernetted chain (CHNC) method.⁴⁴

For simulations in which transport coefficients need to be calculated on-the-fly, tables or simple function calls are required. As an example of the latter case, the Stanton–Murillo transport (SMT) model generates a class of transport coefficients using algebraic fits that have been verified with MD simulations and recover known asymptotic limits.¹⁸ The SMT model determines transport processes by using effective, binary interactions to numerically construct a cross section, with the primary inputs being the charges of the interacting particles along with the effective screening length of the surrounding medium. The original model assumed interactions between classical particles, making it ideal for transport dominated by ionic physics for the systems of interest; however, the SMT model has also been successfully applied to electronic transport.

Another example is the Yukawa viscosity model (YVM)²⁰ that uses mean ionization to map cold-to-warm dense matter conditions onto an effective Yukawa model for viscosities from MD. For cases with small screening parameter, other semi-analytic fits are quite accurate.^45,46 Fits such as these are additionally useful since each problem can be mapped onto an effective OCP.⁴⁷ Finally, Lee and More (LM) developed a wide-ranging electronic transport model²² applicable across several phases in temperature, density, and element space and was improved upon in Refs. 23 and 48.

In this work, we generate basic electronic structure information with the AA approach based on the model developed in Refs. 9 and 49, hereby referred to as AA-NRL. The main distinction from similar AA models is the introduction of quasi-bound electrons, a feature that is important for transition metals such as Cu and Fe. From there, we generate electronic transport from the cross-sectional output of the AA-NRL code, using the Ziman formalism. For ionic transport, we employ molecular dynamics based on an effective potential scheme. The effective ion–ion potentials are generated in a pair-matching scheme based on a modification of the CHNC method for use in higher Z elements. We modify the form of the electron–ion pseudopotential to incorporate the effects of bound states, and we replace the Ornstein–Zernike equation with another thermodynamic closure⁵⁰ more suited for the semi-classical modeling of the degenerate regime. The output of this calculation is in the form of a set of radial distribution functions (RDF). We then remove the electrons and invert the integral equations to generate an effective ion–ion potential from the ion–ion RDF. We refer to this two-step model as potential-matching hypernetted chain (PM-HNC). The effective ion–ion pair potential is then used as an input in pair-potential MD (PP-MD) simulations for the calculation of the ionic transport coefficients.

We additionally make several improvements to the on-the-fly transport models mentioned above. The inter-particle plasma parameter in the SMT model is modified to include the appropriate scaling of diffraction physics in the high-temperature limit. Furthermore, we demonstrate the appropriate extensions of the model to incorporate exchange-correlation (XC) effects of the background electron–electron interactions, which become increasingly important at lower temperatures.

We also modify the original YVM to apply over a much wider temperature range. This is done by integrating a larger MD dataset with high-temperature data from the SMT model mentioned above. The AA generated ionization state is entered into a fit for the combined MD and SMT data. This “integrated YVM,” or IYVM, covers the much smaller couplings and screening parameters relevant to inertial fusion energy (IFE) conditions.

Each of the models mentioned thus far exploits certain approximations that hold under specific conditions like radial symmetry, weak scattering, low degeneracy, or linear screening. However, the complexity of plasma behavior necessitates models with broader applicability, capable of spanning extensive ranges in temperature, density, and species. We adopt the modified Lee–More model²³ (MLM), a wide-ranging phenomenological model that allows us to compute electron electrical and thermal conductivities across various elements. This model is comprehensive, encompassing the solid phase, liquid metal phase, a minimum mean-free-path phase, and both strongly and weakly coupled plasma states. By constructing an experimental dataset, we benchmark the MLM model against these transport coefficients, finding reasonable agreement between MLM and data. Notably, at elevated temperatures, our MLM model demonstrates a smooth convergence with other models discussed in this study, underscoring its versatility and robustness in handling a wide spectrum of plasma conditions. In Sec. II, we outline the parameter space that our calculations will explore, which includes the cases considered by the TCCW-2.

II. WORKSHOP CASES

The parameter space covered by the TCCW-2 workshop spans several orders of magnitude in density and temperature, ranging from condensed matter to the plasma regime. In this section, we introduce several parameters used to characterize the electrons and the ion physics over these regimes. The ionic physics is generally understood through the Coulomb coupling parameter, the ratio of the average (Coulomb) potential energy to the average kinetic energy. This is given explicitly as

Γ_{i i} = \frac{Q^{2}}{a_{i} T},

(1)

where

a_{i} = {(3 / (4 π n_{i}))}^{1 / 3}

is the Wigner–Seitz (or ion sphere) radius in terms of the ion density

n_{i}

⁠, T is the temperature in units of energy,

Q = \bar{Z} e

is the charge of the ion for mean ionization state

\bar{Z}

⁠, and e is the elementary charge. The electronic physics of a dense plasma, with electronic density

n_{e}

⁠, is typically characterized by the degeneracy parameter

Θ = \frac{T}{E_{F}}, E_{F} = \frac{ℏ^{2}}{2 m_{e}} {(3 π^{2} n_{e})}^{2 / 3},

(2)

with

E_{F}

being the Fermi energy, as well as a degeneracy-corrected electron–electron Coulomb coupling parameter

Γ_{e e} = \frac{e^{2}}{a_{e} T} \frac{1}{{(1 + {(\frac{3}{2} Θ)}^{- 9 / 5})}^{5 / 9}},

(3)

where

Q = e

is the fundamental charge, and the electron sphere radius is

a_{e} = {(3 / (4 π n_{e}))}^{1 / 3}

⁠. In Eq. , the kinetic energy is estimated by T at high temperature and by the Fermi temperature

T_{F} = \frac{2}{3} E_{F}

at low temperatures, where the exponent of 5/9 comes from a fit to Thomas-Fermi integrals.¹⁸ A region of dense plasma parameter space that is of particular interest is the region with both strong coupling and degeneracy effects, the warm dense matter regime (WDM). Many of the TCCW-2 cases are in this regime, with both

Γ_{e e}

and

Θ

approximately unity. This regime can be quantified using the WDM parameter

W = S (Γ_{e e}) S (Θ), S (x) = \frac{2}{x + x^{- 1}},

(4)

which was introduced in Ref. 51. Note that this parameter has a maximum value of 1 when

Γ_{e e} = 1

and

Θ = 1

⁠. Figure 1 shows a contour plot of

W (Γ_{e e}, Θ)

for all the TCCW-2 cases. In this figure, the electron number density,

n_{e} = \bar{Z} n_{i}

⁠, has been obtained from a

\bar{Z}

calculated using the Thomas-Fermi model.^52,53 Priority 1 cases are shown as circles, priority 2 cases are shown as crosses, and priority 3 cases that span temperature ranges are shown as squares. The three diamonds indicate the priority 1 cases for which we ran molecular dynamics simulations. The

\bar{Z}

values for these cases were obtained from the average-atom code described in Sec. III A. The white dashed lines indicate the temperature ranges explored in this paper for the elements hydrogen, carbon, and aluminum. Figure 1 shows that most of the priority 1 and priority 2 cases fall into the WDM regime.

FIG. 1.

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Contour plot of the WDM parameter $W (n, T)$ with TCCW-2 cases. TCCW-2 priority levels are identified by different markers, elements by different colors. The electron density was obtained from the effective ionic charge $\bar{Z}$ calculated using the Thomas-Fermi model⁵³ for all cases except the three diamonds. The three diamonds indicate the cases for which molecular dynamics simulations were run, with $\bar{Z}$ obtained from the AA-NRL code. White dashed lines indicate the temperature ranges explored in this paper for the elements hydrogen, carbon, and aluminum.

We generate a phase diagram for the ions using the well-studied Yukawa one-component plasma (YOCP). In the YOCP, the ions interact via the radial potential

u (r) = \frac{Q^{2}}{r} e^{- r / λ},

(5)

where r is the distance between two ions and

λ

is the screening length. Both

\bar{Z}

and

λ

are determined by the properties of the electronic background. The phase diagram of the YOCP is completely determined by two dimensionless parameters, the ion–ion coupling parameter

Γ_{i i}

and the screening parameter

κ = a_{i} / λ

⁠.

In Fig. 2, we show the TCCW-2 cases overlayed onto the YOCP phase diagram. The solid black line represents the melting line

Γ_{m} (κ)

⁠, given by²⁰

Γ_{m} (κ) = 171.8 + 82.8 (e^{0.565 κ^{1.38}} - 1) .

(6)

The dashed black line represents the Kirkwood line, approximated as

Γ_{K} (κ) = 1.21 e^{κ / 3.7},

(7)

which separates the region where the radial distribution function (RDF) increases monotonically from 0 to 1 with increasing r (⁠

Γ_{i i} < Γ_{K}

⁠) with the region in which the RDF admits oscillations (⁠

Γ_{i i} > Γ_{K}

⁠).^51,54 Similarly to Fig. 1, the diamonds indicate the cases which we investigate using MD simulations and the isochores, indicating the temperature ranges explored in this paper are shown as solid colored lines in Fig. 2.

FIG. 2.

Yukawa (Γ,κ) phase diagram with TCCW-2 cases. The parameters Z¯ and λ have been calculated using the Thomas-Fermi model,53 hence λTF. The marker indicates the priority level, while the color indicates the element. The solid black line indicates the melting line as defined in the text. The dashed black line indicates the line above which the ionic pair correlation function, h(r), changes from a monotonic behavior to an oscillatory behavior.

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Yukawa $(Γ, κ)$ phase diagram with TCCW-2 cases. The parameters $\bar{Z}$ and $λ$ have been calculated using the Thomas-Fermi model,⁵³ hence $λ_{TF}$ ⁠. The marker indicates the priority level, while the color indicates the element. The solid black line indicates the melting line as defined in the text. The dashed black line indicates the line above which the ionic pair correlation function, $h (r)$ ⁠, changes from a monotonic behavior to an oscillatory behavior.

Most of the priority 1 and priority 2 cases of the workshop lie below the YOCP melting line and above the Kirkwood line, meaning the workshop is focused on the liquid regime where strong coupling generates a non-trivial structure within the plasma. The only priority 2 cases in the solid phase correspond to the four cases in the bottom left corner of Fig. 1 for which $W ≪ 1$ ⁠.

For the sake of simplicity, in Table I, we report the values of all the above parameters for the cases investigated in this work.

TABLE I.

Table of densities, temperatures, and electron parameters for priority 1 cases considered in the effective ionic charge $\bar{Z}$ has been calculated using the AA-NRL code described in Sec. III A.

	$ρ$ (g/cm³)	T (eV)	$\bar{Z}$	$Γ_{i i}$	$κ$	$Γ_{e e}$	$Θ$	$W$
H1	1	2	1	9.78	1.84	1.12	0.08	0.148
C1	10	2	4	147.49	2.39	0.76	0.03	0.07
Al1	2.7	1	3	81.92	3.24	1.67	0.09	0.15

	$ρ$ (g/cm³)	T (eV)	$\bar{Z}$	$Γ_{i i}$	$κ$	$Γ_{e e}$	$Θ$	$W$
H1	1	2	1	9.78	1.84	1.12	0.08	0.148
C1	10	2	4	147.49	2.39	0.76	0.03	0.07
Al1	2.7	1	3	81.92	3.24	1.67	0.09	0.15

We generate transport coefficients for the points along the isochores, based on a set of models diagrammatically outlined in Fig. 3. Details on the generation of the plasma structure as input to the transport models are in Sec. III. The subsequent transport coefficient models are detailed in Sec. IV. We then compare the output of these transport models in Sec. V.

FIG. 3.

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Schematic of the models used to generate the transport coefficients in this paper, with relevant section numbers included in parentheses after each model. The primary inputs all models are the densities (⁠ $ρ$ ⁠), temperatures (T), and nuclear charges (Z) of a given system, which in turn, inform our NRL average-atom (AA-NRL) model detailed in Sec. III A. The AA-NRL model is then used to generate mean ionization states (⁠ $\bar{Z}$ ⁠), core radii (⁠ $R_{c}$ ⁠), phase shifts [ $δ_{ℓ} (ϵ)$ ], and densities of states [ $g (ϵ$ ⁠)] that serve as inputs to other models for calculating transport coefficients. In addition, $\bar{Z}$ and $R_{c}$ are used within a potential-matching hypernetted-chain (PM-HNC) theory to generate effective pair potentials $u_{i j}^{eff} (r)$ (Sec. III B) to be used in pair-potential molecular dynamics (PP-MD) simulations (Sec. IV D). The remaining models used are the Ziman model (Sec. IV C), the modified Lee–More (MLM) model (Sec. IV A), the Stanton–Murillo transport (SMT) model (Sec. IV B), and the integrated Yukawa viscosity model (Sec. IV E).

III. ELECTRONIC AND IONIC STRUCTURE

In this section, we discuss the models used to calculate equilibrium electronic and ionic structure properties to generate input quantities for use in transport coefficient models. In Sec. III A, we give an overview of our average-atom model developed at the Naval Research Laboratory (AA-NRL), which is used to calculate electronic structure. In Sec. III B, we then detail our ionic structure calculations using classical mapping methods, in which quantum statistical potentials and the integral equation theory of liquids are used with the ultimate goal of constructing pair potentials for MD simulations. Note that Hartree atomic units are used within this section (⁠ $e = m_{e} = ℏ = 4 π ϵ_{0} = 1$ ⁠).

A. Electronic structure—The average-atom model

In the AA approach, an atom is carved from the plasma, and a sphere surrounding the atom is defined with radius

a_{i}

⁠. An ion with nuclear charge Z, equal to the atomic number, is placed in the center of the sphere. A cloud of electrons with a total number equal to Z fills the sphere. The AA model is based on solving the Kohn–Sham equation,⁵⁵^,

[- \frac{1}{2} Δ + W (r)] ψ_{α} (r) = ϵ_{α} ψ_{α} (r),

(8)

for atomic orbitals

ψ_{a} (r)

⁠.

W (r)

is the effective potential energy that includes the Coulomb potential of electrons and ions and an exchange-correlation potential. The latter is treated in the local density approximation (LDA)

W (r) = W_{e} (r) + W_{x c} (r) .

(9)

The first term includes the electrostatic energy as

W_{e} (r) = - \frac{Z}{r} + \int_{0}^{a_{i}} \frac{n_{e} (r^{'})}{| r - r^{'} |} d^{3} r^{'} .

(10)

The remaining exchange-correlation component is approximated to only include the exchange contribution as

W_{xc} \approx W_{x}

⁠,⁵⁶ where

W_{x} (r) = - {(\frac{3 n_{e} (r)}{π})}^{1 / 3} {(1 + \frac{3}{2 Θ})}^{- 1} .

(11)

The potential energy is shifted by a factor

- W_{x} (a_{i})

so that

W (a_{i}) = 0

⁠.³⁸

By setting

ψ_{α} (r) = \frac{u_{α} (r)}{r} Y_{ℓ m} (θ, ϕ)

⁠, where

Y_{ℓ m}

is a spherical harmonic, n is the principal, and

ℓ

is the orbital quantum number, the Schrödinger equation is solved for the radial part of the wave function,

u_{α}

⁠:

[- \frac{1}{2} \frac{d^{2}}{d r^{2}} + \frac{ℓ (ℓ + 1)}{2 r^{2}} + W (r)] u_{α} (r) = ϵ_{α} u_{α} (r)

(12)

for both bound states

α = {n, ℓ}

with negative energy

ϵ < 0

and continuum states

α = {ϵ, ℓ}

with positive energy

ϵ > 0

⁠. For bound states, the wave function is normalized via

\int_{0}^{a_{i}} u_{α}^{2} (r) d r = 1

⁠. For continuum states, the wave function is normalized by matching it to the analytical solution just outside the Wigner–Seitz radius

\begin{matrix} u_{ℓ} (ϵ, r) = \sqrt{\frac{2 k (ϵ)}{π}} r [j_{ℓ} (k (ϵ) r) cos (δ_{ℓ} (ϵ)) - n_{ℓ} (k (ϵ) r) sin (δ_{ℓ} (ϵ))], \end{matrix}

(13)

where the potential energy is zero. Here,

j_{ℓ}

and

n_{ℓ}

denote the spherical Bessel functions of the first and second kind,

k (ϵ) = \sqrt{2 ϵ}

is the wave vector, and

δ_{ℓ} (ϵ)

is the phase shift.

The electron density is calculated from the wave functions of bound and continuum states,

\begin{matrix} n_{e} (r) = \frac{1}{4 π r^{2}} [\sum_{n, ℓ} 2 (2 ℓ + 1) f (ϵ_{n ℓ}) u_{n ℓ}^{2} (r) \\ + \sum_{ℓ = 0}^{ℓ_{max}} 2 (2 ℓ + 1) \int_{0}^{\infty} f (ϵ) u_{ℓ}^{2} (r, ϵ) d ϵ], \end{matrix}

(14)

by summation (bound states) and integration (continuum states) over the Fermi–Dirac distribution

f (ϵ) = {[1 + exp ((ϵ - μ) / T_{e})]}^{- 1}

⁠. The chemical potential

μ

is determined by enforcing quasineutrality in the form of

Z = \int_{0}^{a_{i}} 4 π r^{2} n_{e} (r) d r .

(15)

The wave function boundary condition for bound and free states at the origin is

u_{α} (0) = 0

⁠. At the Wigner–Seitz radius,

\frac{d}{d r} (\frac{u_{α} (a_{i})}{r}) = 0

for bound electrons³³ and Eq. for free states is used. Another option is to match the bound-state wave functions and their derivatives at the ion-sphere radius to solutions outside the Wigner–Seitz sphere that vanish exponentially.³⁷

The number of partial waves used in Eqs. (12) and (A) is important for convergence of the free electron density. At low temperatures, below the Fermi energy, $ℓ_{max}$ = 8–10 is sufficient. However, as temperature increases a larger number of $ℓ_{max}$ is needed, e.g., $ℓ_{max}$ = 80–100 for Al at $T_{e}$ = 1 keV. The number of partial waves is determined on-the-fly by requiring that the last contribution to the electron density becomes smaller than a preset value (typically $10^{- 5}$ ⁠).

By solving the above set of equations, we obtain the wave functions for the bound and free electrons

u_{n, ℓ} (r)

and

u_{ℓ} (r, ϵ)

⁠, respectively, the electron density distribution

n_{e} (r)

⁠, and the chemical potential

μ

⁠. Derived from these quantities is the primary output for use in transport coefficients, the mean ionization state,

\bar{Z}

⁠, defined as the number of free electrons, i.e.,

\bar{Z} = \sum_{ℓ = 0}^{ℓ_{max}} 2 (2 ℓ + 1) \int_{0}^{a_{i}} \int_{0}^{\infty} f (ϵ) u_{ℓ}^{2} (r, ϵ) d ϵ d r .

(16)

In Fig. 4, we compare

\bar{Z}

obtained with several other models for the TCCW-2 cases listed in Table I. At density

ρ = 1

g/cm³, H is fully ionized at any temperature due to pressure ionization, according to the AA-NRL. In contrast, the Thomas-Fermi model shows

\bar{Z} < 1

up to temperatures of 1 keV. For C, in the limit

T_{e} \to 0

⁠, the AA-NRL calculates

\bar{Z} = 4

⁠, while the Thomas-Fermi model yields a lower value,

\bar{Z} = 3

⁠. For Al, the AA-NRL predicts

\bar{Z} = 3

in the limit

T_{e} \to 0

⁠. It gradually increases with temperature, and at

T_{e} = 1

keV, Al is fully ionized. The aluminum

\bar{Z}

⁠, calculated from the AA-NRL, is in good agreement with other published data.^56–58

FIG. 4.

Mean ionization state vs electron temperature. Left panel: H at density ρ=1 g/cm3, middle panel: C at density ρ=10 g/cm3, right panel: Al at density ρ=2.7 g/cm3. Solid lines: AA-NRL calculations, dashed lines: More53 fit to the Thomas-Fermi AA, symbols: data from literature.16,57,58

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Mean ionization state vs electron temperature. Left panel: H at density $ρ = 1$ g/cm³, middle panel: C at density $ρ = 10$ g/cm³, right panel: Al at density $ρ = 2.7$ g/cm³. Solid lines: AA-NRL calculations, dashed lines: More⁵³ fit to the Thomas-Fermi AA, symbols: data from literature.^16,57,58

B. Ionic structure—Classical mapping methods

The ionic physics in a dense plasma can be modeled by an effective potential approach, in which the electrons are “integrated-out” from the simulation. In the Born–Oppenheimer picture of equilibrated electrons, this can be done exactly with an N-body potential. We approximate this physics with a pair potential, something that is inexact, and even technically problematic,⁵⁹ and yet has been found to be a good approximation in this regime.⁶⁰

The generation of effective potentials has been considered before;^61,62 our methodology is similar in flavor, but differs in many details. The first step consists in calculating the equilibrium ion–ion RDF $g_{i i} (r)$ from an electron–ion liquid using quantum statistical potentials (QSP) in conjunction with classical liquid integral theory. The second step consists in inverting $g_{i i} (r)$ via a one-component HNC calculation to obtain an effective ion–ion pair potential. Sections III B 1 and III B 2 describe the first step of our procedure; Sec. III B 3 describes the second step.

1. Quantum statistical potentials

Mapping the quantum mechanical properties of electrons onto classical properties is approximately done via modification of the Coulomb potential to include Pauli repulsion and finite-sized wavefunctions. To account for the non-zero Fermi velocity of degenerate electrons, an effective electron temperature,

T_{e}^{c}

⁠, is defined following the fit to diffusion Monte Carlo data developed in Ref. 63. This results in an effective two-temperature system for which an interspecies temperature must be defined,

T_{e i}

⁠, of which multiple possibilities have been considered.^64,65 In this work, it is defined as

T_{e i} = μ_{e i} (\frac{T_{i}}{m_{i}} + \frac{T_{e}}{m_{e}}),

(17)

where

μ_{i j} = m_{i} m_{j} / (m_{i} + m_{j})

is the reduced mass. Additionally, the superscript c is dropped from the electron temperature; it will be assumed that any electron temperature or inverse temperature is the classical mapped temperature for the rest of this section. The interactions between species are modeled using a modified Deutsch potential to account for core electrons. Better potentials exist, most notably the Kelbg potential,⁶⁶ but are less easily modifiable to take into account core electrons. The Deutsch potential is

u_{e i} (r) = - \frac{\bar{Z}}{r} (1 - exp (- r / Λ_{e i})),

(18)

u_{e e} (r) = \frac{1}{r} (1 - exp (- r / Λ_{e e})) + u^{P} (r),

(19)

where

Λ_{i j} = {(2 π μ_{i j} T_{i j})}^{- 1 / 2}

is the inter-particle thermal de Broglie wavelength, and

u^{P} (r)

is the Pauli potential that models electron–electron Fermi repulsion, here using the fit to Lado's model given in Ref. 67. For the case of hydrogen, this model is sufficient to model the electronic structure. We show the potentials taken as input to the PM-HNC calculation in Fig. 5(a), normalized by the pair charges involved. We see that the effect of the Deutsch potential is to reduce the Coulomb peak significantly.

FIG. 5.

Hydrogen at 2 eV and 1 g/cm3. (a) The potentials normalized by their charges Qi={e,Z¯e} taken as input to the HNC calculation, Coulombic for the ions with Z¯=1, and Eq. (18) in Hartree atomic units. (b) The resulting effective ion–ion potentials, normalized by the inverse temperature. (c) A comparison between the electron–ion correlation function of the AA-NRL and PM-HNC calculations. (d) The set of three RDFs determining the plasma structure, gii, gie, and gee, are shown. See the text for descriptions of the lines. The electron–ion case is the same as in the lower left panel, with the AA cutoff at the ion-sphere radius.

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Hydrogen at 2 eV and 1 g/cm³. (a) The potentials normalized by their charges $Q_{i} = {e, \bar{Z} e}$ taken as input to the HNC calculation, Coulombic for the ions with $\bar{Z} = 1$ ⁠, and Eq. (18) in Hartree atomic units. (b) The resulting effective ion–ion potentials, normalized by the inverse temperature. (c) A comparison between the electron–ion correlation function of the AA-NRL and PM-HNC calculations. (d) The set of three RDFs determining the plasma structure, $g_{i i}$ ⁠, $g_{i e}$ ⁠, and $g_{e e}$ ⁠, are shown. See the text for descriptions of the lines. The electron–ion case is the same as in the lower left panel, with the AA cutoff at the ion-sphere radius.

For ions with bound states, the QSPs fail to capture the complicated bound-state structure;⁶⁸ the wavefunction orthogonalization needed for this physics is only very approximately replicated by the uniform density Pauli potential. To remedy this, we generate a pseudopotential version of the QSP. The ion charge is fixed at the

\bar{Z}

generated from the AA-NRL code. Modification to the QSPs is then needed to prevent collapse of the free electrons onto the bound states. The Ashcroft pseudopotential is an often invoked analytic model, the general version of which is

u_{e i}^{A} (r) = {\begin{array}{l} - c_{0} \bar{Z} / R_{c}, & r \leq R_{c}, \\ - \bar{Z} / r, & r > R_{c}, \end{array}

(20)

for tunable critical radius

R_{c}

⁠, and core height parameter

c_{0}

⁠, for which

c_{0} = 1

is continuous, and

c_{0} = 0

the empty-core model. Motivated by the form of the Deutsch potential, which can be written in k-space with a tilde as

{\tilde{u}}_{e i} (k) = - 4 π \bar{Z} (\frac{1}{k^{2}} - \frac{1}{k^{2} + Λ_{e i}^{- 2}}),

(21)

we implement the smoothed Ashcroft (SA) model by Fourier transforming Eq. and cutting off modes higher than the de Broglie wavelength

\begin{matrix} {\tilde{u}}_{e i}^{SA} (k) = - 4 π \bar{Z} (\frac{1}{k^{2}} - \frac{1}{k^{2} + Λ_{e i}^{- 2}}) \\ \times ((1 - c_{0}) cos (k R_{c}) + \frac{c_{0}}{k R_{c}} sin (k R_{c})) . \end{matrix}

(22)

The real space potential is obtained via numerical Fourier transform. We determine the

R_{c}

numerically from the average bound radius of the AA-NRL computation. We adopted the empty-core Ashcroft potential, letting

c_{0} = 0

⁠, which is used for atoms with

Z > 1

⁠. The SA potentials are depicted in the panel a of Fig. 6 for carbon and Fig. 7 for aluminum.

FIG. 6.

Carbon at 2 eV and 10 g/cm3. (a) The potentials taken as input to the HNC calculation, Coulombic for the ions with Z¯=4.0, and Eq. (18), in Hartree atomic units. (b) The resulting effective ion–ion potentials, normalized by the inverse temperature. (c) A comparison between the electron–ion correlation function of the AA-NRL and PM-HNC calculations. (d) All radial distribution functions outputted by PM-HNC. The electron–ion case is the same as in the lower left panel. The ion–ion RDF is compared with the HNC informed Coulomb and Yukawa model results.

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Carbon at 2 eV and 10 g/cm³. (a) The potentials taken as input to the HNC calculation, Coulombic for the ions with $\bar{Z} = 4.0$ ⁠, and Eq. (18), in Hartree atomic units. (b) The resulting effective ion–ion potentials, normalized by the inverse temperature. (c) A comparison between the electron–ion correlation function of the AA-NRL and PM-HNC calculations. (d) All radial distribution functions outputted by PM-HNC. The electron–ion case is the same as in the lower left panel. The ion–ion RDF is compared with the HNC informed Coulomb and Yukawa model results.

FIG. 7.

Aluminum at 1 eV and 2.7 g/cm3. (a) The potentials taken as input to the HNC calculation, Coulombic for the ions with Z¯=3.0, and Eq. (18), in Hartree atomic units. (b) The resulting effective ion–ion potentials, normalized by the inverse temperature. (c) A comparison between the electron–ion correlation function of the AA-NRL and PM-HNC calculations. (d) All radial distribution functions outputted by PM-HNC. The electron–ion case is the same as in the lower left panel. The ion–ion RDF is compared with the effective OCP and Yukawa model results.

View large Download slide

Aluminum at 1 eV and 2.7 g/cm³. (a) The potentials taken as input to the HNC calculation, Coulombic for the ions with $\bar{Z} = 3.0$ ⁠, and Eq. (18), in Hartree atomic units. (b) The resulting effective ion–ion potentials, normalized by the inverse temperature. (c) A comparison between the electron–ion correlation function of the AA-NRL and PM-HNC calculations. (d) All radial distribution functions outputted by PM-HNC. The electron–ion case is the same as in the lower left panel. The ion–ion RDF is compared with the effective OCP and Yukawa model results.

2. Integral equation theory for ionic structure

Modern classical liquid theory, applied here at the semi-classical level to the dense plasma regime, typically involves solving the exact Ornstein–Zernike (OZ) equation, in conjunction with a closure. In our case, the hypernetted-chain (HNC) closure is used, known to be extremely accurate at moderate coupling;⁶⁹ modification is required for the WDM regime. For

Θ ≪ 1

⁠, the degenerate electrons are at an effective temperature

T_{e}^{c} ≫ T_{i}

⁠, necessitating a two-temperature treatment. Thus, we employ a thermodynamic closure denoted as SVT,⁵⁰ which has shown to be a good model for strongly coupled two-temperature plasmas.⁷⁰ These two equations, HNC and SVT, are

h_{i j} = exp (- β_{i j} u_{i j} + h_{i j} - c_{i j} + B_{i j}) - 1,

(23)

{\tilde{h}}_{i j} = {\tilde{c}}_{i j} + \sum_{k} n_{k} (\frac{μ_{i j} T_{i k}}{m_{i} T_{i j}} {\tilde{c}}_{i k} {\tilde{h}}_{k j} + \frac{μ_{i j} T_{k j}}{m_{j} T_{i j}} {\tilde{h}}_{i k} {\tilde{c}}_{k j}),

(24)

for inverse temperatures

β_{i j} = 1 / T_{i j}

⁠, direct correlation function

c_{i j}

⁠, where i, j iterate over electrons and all ionic species, and all functional dependence on r and k is suppressed for brevity.

From Fig. 2 and Table I, we can see all cases have expected oscillatory strongly coupled radial distribution functions; thus, extension of the HNC approximation is desired, which we improve via the addition of a bridge function to the ion–ion component, $B_{i j} (r) = δ_{i, ion} δ_{j, ion} B_{Y} (r)$ ⁠, fit for the Yukawa system in Refs. 71 and 72. The Yukawa parameters used for this bridge function as well as for the Yukawa pair potential are given by the AA-NRL mean ionization, and the Thomas-Fermi screening length as approximated in Eq. (3). This would be exact if the PM-HNC method generated a Yukawa potential; here, it is only an approximation, whose accuracy is parameterized by the closeness of the PM-HNC ion–ion RDF to that of the Yukawa ion–ion RDF, shown in panel d of Figs. 5–7; later the accuracy of this method is shown by comparison to MD in Fig. 9. The lack of a known bridge function for the strongly coupled electron–ion component is potentially a major source of uncertainty in this model. Semi-classical electron–electron bridge functions exist in the literature,⁷³ but are not important in the high density limit. We solve Eqs. (23) and (24) self-consistently, following Refs. 74 and 69.

The resulting electronic distribution can be compared to that of the AA-NRL calculation, with results shown in panel c of Figs. 5–7 for H, C, and Al, respectively. The normalized electronic density is shown with the mean density subtracted. In the case of hydrogen, the distribution exhibits a very similar structure to the AA-NRL calculation, up until $r \sim a_{i}$ where the AA-NRL calculation hits the free-wavefunction boundary conditions at $a_{i}$ ⁠.

In Fig. 5(d), we see the full set of RDFs describing the structure of the PM-HNC plasma model. For comparison, we have ion–ion RDFs in magenta dotted and blue dashed lines corresponding to the solutions to the HNC equations with Coulomb (OCP) and Yukawa potentials. The Yukawa model is known to overscreen, leading to reduced peak heights in the RDF, so it would be expected that the true solution lies somewhat intermediate between these cases. This is indeed the case; however, the exact result is discrepant with the KS-DFT workshop results,²⁷ the full spread of which is plotted as the grayed-out region. Interestingly, even the Yukawa model appears more strongly coupled than the results from DFT, with both higher peaks and a larger repulsive hole. This potentially indicates either that hydrogen is not completely ionized as expected from the AA-NRL model, or non-trivial phase behavior such as the formation of hydrogen molecules is happening, which would not be well modeled by the spherically symmetric pair potential.

The cases of carbon and aluminum can be seen in Figs. 6 and 7. In both cases, we can see in panel c that the SA QSP predicts a much smoother profile than the AA-NRL computation; this is due to the lack of explicit bound states in the PM-HNC computation, which are instead incorporated through the smooth psuedopotential in Eq. (1). We again show in the grayed region the spread in KS-DFT results from the workshop submissions.²⁷ The PM-HNC aluminum RDF is very similar in structure to the DFT results, particularly in terms of peak placement if the heights are slightly smoothed out. In the case of carbon, Fig. 6, which is both very strongly coupled and quite degenerate, there is clearly a qualitatively different structure.

3. Effective potentials from the radial distribution function

Finally, we generate the effective pair potential for use in MD as in Refs. 61 and 62, formed from the inversion of Eqs. and with input the ion–ion

g_{i i}

⁠, equivalently

h_{i i}

⁠, and output an effective ion–ion potential

β u_{i i}^{eff}

that implicitly incorporates the QSP electron–ion potentials. This can be stated as a simple matrix inversion,

{\tilde{c}}_{i k}^{eff} = \sum_{j} {(1 + n_{i} {\tilde{h}}_{i j})}^{- 1} {\tilde{h}}_{j k},

(25)

β_{i i} u_{i i}^{eff} = h_{i i} - c_{i i}^{eff} - B_{Y} + ln (h_{i i} + 1),

(26)

and is purely algebraic; any inexactness is a result of the HNC approximation with the earlier Yukawa bridge function,

B_{Y}

⁠. Here,

c_{i k}^{eff}

is the direct correlation function with superscript, indicating the electron has been removed. Note Eq. is the multi-species Ornstein–Zernike equation, which is what SVT reduces to when the ions are all at the same temperature. The cases we consider contain only one ion species, so the effective ion description of the plasma is single component, simplifying this equation from a matrix to an algebraic equation.

The resulting effective pair potentials for use in the ionic MD simulation for transport are shown in panel b of Figs. 5–7 for H, C, and Al, respectively. Note the y axis of the plot is symlog, a useful combination of linear at small values and logarithmic at large values to capture the full behavior of the pair potential. In all three cases, there is a region of exponential screening with screening length slightly larger than the over-screened Yukawa potential. At small distances, the screening is weak; the potentials all go as $1 / r$ ⁠. The addition of core–core repulsion effects, not modeled here, would cause a stronger low r repulsion, potentially bringing these effective potentials closer to the non-parametric fits in Ref. 60. The overall change in these cases from Yukawa to the nonlinear PM-HNC model is a steeper repulsion and then for carbon and aluminum, an attractive dip in the range $a_{i} ≲ r ≲ 2 a_{i}$ ⁠.

IV. ELECTRONIC AND IONIC TRANSPORT

In this section, we detail five models, each with different physics, that generate transport coefficients based on some of the output of the electronic and ionic structure calculations above. The models that encapsulate the most physical output are the Ziman and PP-MD models for electronic and ionic transport coefficients, respectively. These are contrasted with a set of models, MLM, SMT, and IVYM, which can be computed on-the-fly, dependent only on ionization input. For these last three, we detail improvements from the corresponding model in the literature.

A. Modified Lee–More (MLM) model

Lee and More (LM) developed a wide-ranging electronic transport model²² applicable across several phases over many orders of magnitude in temperature and density space for each element, though it was chiefly designed for metals. This is accomplished by providing separate models appropriate to each phase that seamlessly span the ${Z, n_{i}, T}$ space. Here,²³ we use a modified version of this model (MLM = modified Lee–More) to predict electronic transport coefficients for the TCCW-2 cases reported in this paper. Such a model allows us to predict properties for all points shown in Fig. 2. As shown in Fig. 3, the MLM model inputs are the same atomic physics as the other models we employ; units in the sections are chosen to match the original LM²² and MLM²³ papers.

LM models seek to decompose the ${Z, n_{i}, T}$ space into phases and limiting forms in those phases. Here, phases refer to solid, liquid, and plasma; these are separated by melting and vaporization boundaries in the ${Z, n_{i}, T}$ space. Each of these phases is associated with an electron–ion collision time $τ$ or, equivalently, a mean-free path (MFP) $Λ$ ⁠. Within some of the phases, subphases are defined in terms of limiting values of parameters (e.g., choices for the Coulomb logarithm within the plasma phase), which creates five separate regimes. We summarize each of these regimes here, beginning with the solid phase. The regions indicated refer to the “patchwork quilt” diagram in LM's Fig. 6.²²

In the solid phase, the MFP is given the value

Λ_{solid} = 50 a_{i} \frac{T_{m}}{T}, T < T_{m},

(27)

where

a_{i}

and T are the usual ion-sphere radius and temperature (in energy units), and

T_{m}

is the density-dependent melt temperature. The value 50 is empirical relative to the choice of length scale (given here as

a_{i}

⁠, and described in the original LM as

R_{i}

⁠, the average spacing between particles); in the original MLM model,²³ a value of 100 was used and is decreased here to better match experimental data for Al. This functional form originated from the Ziman formulation of conduction. The melting temperature

T_{m}

is extremely difficult to compute at any

{Z, n_{i}, T}

point. In the LM model, a parameterization by Cowan was used that gives the density dependence of melting. Note that the value of 50 and

T_{m}

in Eq. is empirical and attempt to capture the trend over all elements. For a given element, we expect to see deviations; this is not a weakness of the model, but rather an attempt to retain both simplicity and generality. For this reason, a more detailed fit to experimental data for

T_{m}

has been used here.²³

In the liquid phase, the MFP is given the value

Λ_{liquid} = \frac{Λ_{solid}}{γ}, T \geq T_{m},

(28)

where

γ = 2.5

in the MLM; this is a generic-species value and element-specific values could be used.⁷⁵ The solid and liquid phases are region 5.

Note that

Λ_{liquid}

is monotonically decreasing with increasing temperature. At some temperature, the MFP drops below

a_{i}

⁠, a “minimum MFP phase” in which the MFP is taken to have the value

Λ_{min} = a_{i},

(29)

or, equivalently, the relaxation time is

τ = a_{i} / \bar{v}

⁠, which in turn requires an opinion for the mean velocity. In the MLM, this velocity is taken to be

\bar{v} = \sqrt{\frac{3 T_{eff}}{m_{e}}},

(30)

T_{eff} = 2 T_{e} \frac{F_{1 / 2} (η)}{F_{- 1 / 2} (η)},

(31)

where

η = μ / T

in this section and

F_{x}

is the unnormalized Fermi–Dirac integral, Eq. (26) in Ref. 22. The form Eq. interpolates between the Fermi energy and temperature. This is region 4.

In the plasma regimes, the collision time $τ$ increases with temperature and the MFP is no longer smaller than $a_{i}$ ⁠; such large MFPs allow one to use a Coulomb logarithm $L$ for $τ$ that is derived in the usual way from Rutherford scattering, although with the form $\frac{1}{2} ln (1 + \frac{b_{max}^{2}}{b_{min}^{2}})$ that is positive definite. In addition, we enforce a lower bound on the Coulomb logarithm such that $L \geq 2$ ⁠. When this is saturated, we have the minimum- $L$ regime (region 3).

In the dense plasma regime (region 2), the Debye–Hückel screening lengths are smaller than the inter-particle spacing. In the MLM, following the spirit of the SMT discussed in Sec. IV B, clamps on the screening lengths are imposed at the electron and ion-sphere radii, $a_{e}$ and $a_{i}$ ⁠, respectively.

At the highest temperatures, the MFP is computed using standard plasma theory (region 1), including the use of Debye–Hückel values for screening lengths; this recovers the Spitzer–Härm limit.⁷⁶ The minimum impact parameter at high temperature is given by the de Broglie wavelength. As in the dense plasma case, these quantities are used in a Coulomb logarithm for $τ$ ⁠; together, regions $1 - 3$ form a Coulomb logarithm-based plasma model free of divergences.

The procedure is then to begin with

{Z, n_{i}, T}

⁠, compute the ionization level

\bar{Z}

⁠, compare the MFPs to obtain the appropriate

τ

and use the expressions

σ_{e} = \frac{n_{e} e^{2} A^{α} (η)}{m_{e}} τ,

(32)

κ_{e} = \frac{n_{e} k_{B} T A^{β} (η)}{m_{e}} τ,

(33)

where

k_{B}

is the Boltzmann constant.⁷⁷ Here, the functions

A^{α, β} (η)

are calculated in terms of Fermi integrals, as found in Eq. (25) in Ref. 22. Note that these expressions predict, with significant cancellations including

τ

itself, for the Wiedemann–Franz law a Lorenz number

L = \frac{κ_{e}}{T σ_{e}}

value of

L = \frac{A^{β}}{e^{2} A^{α}} .

(34)

In the extreme limits, we find

L_{cold} = (π^{2} / 3) / e^{2}

and

L_{hot} = (128 / 32) / e^{2}

⁠.

We summarize the MLM as

captures solid, liquid, and plasma phases,
has density-dependent melting temperature,
breaks plasma phase into three regimes: minimum- $L$ ⁠, minimum screening, and Spitzer–Härm limit,
each regime accounts for degeneracy.

The MLM method provides a great backdrop designed to give good electronic transport estimates in the many regions of parameter space. It does not, however, compute ionic transport, nor does it perfectly model the Coulomb logarithm in the plasma regime. We now turn to another model which computes ionic and electronic transport using fits to precise numerical integrals of the binary collision cross sections.

B. Stanton–Murillo transport (SMT)

The primary input to the Stanton–Murillo transport (SMT) model is an effective screening length, which attempts to capture the influence of many-body effects on a binary interaction through a single parameter at the atomic scale. This screening length is incorporated into the screened Coulomb interaction between two scattering particles

u_{i j} (r) = \frac{Q_{i} Q_{j}}{r} e^{- r / λ_{eff}},

(35)

where

Q_{i} = {\bar{Z}}_{i} e

is obtained from an AA calculation, as seen in Fig. 3, and the effective screening length in a system composed of electrons and

N_{species}

ionic species is expressed through the relation

λ_{eff}^{- 2} = λ_{e}^{- 2} + \sum_{i = 1}^{N_{species}} \frac{λ_{i}^{- 2}}{1 + 3 Γ_{i}^{IS}} .

(36)

Here, the screening length of each ionic species is given by the relation

λ_{i}^{- 2} = 4 π Q_{i}^{2} n_{i} / T

⁠, where

n_{i}

is the number density of that species, and T is the temperature in units of energy. The term

λ_{e}

is the electron screening length, and approximate formulas can be found in Ref. 18, though a simple approximation with minimal error is given by

λ_{e}^{- 2} \approx \frac{4 π e^{2} n_{e}}{{(T^{9 / 5} + {(\frac{2}{3} E_{F})}^{9 / 5})}^{5 / 9}} .

(37)

Furthermore, each ionic screening length is modified through an ion-sphere coupling parameter

Γ_{i}^{IS} = \frac{Q_{i}^{2}}{A_{i} T},

(38)

which is expressed in terms of an ion-sphere radius that has been modified for mixtures as

A_{i} = {(\frac{3 Q_{i}}{4 π ρ_{ion}})}^{1 / 3}, ρ_{ion} = \sum_{j = 1}^{N_{species}} Q_{j} n_{j} .

(39)

It should be noted that

A_{i}

is not the commonly used ion-sphere radius and was specifically designed for the SMT model.

Once the effective screening length (36) is known, the plasma parameter $g_{i j} = | Q_{i} Q_{j} | / (λ_{eff} T)$ is used to calculate the relevant collision integrals $K_{n m} (g_{i j})$ associated with each transport process, with fits found in Appendix C-3 of Ref. 18. Note that the same notation is used for the RDF's as well as the plasma parameter. In this section, $g_{i j}$ always refers to the plasma parameter.

However, it was shown in Ref. 78 that the SMT model fails to recover the high-temperature limit for transport processes dominated by electron scattering, which is due to the absence of diffraction physics. This physics becomes increasingly important at higher temperatures as the thermal de Broglie wavelength becomes comparable to the classical distance of closest approach

Λ_{i j} \sim r_{i j}^{ca}

⁠, where

r_{i j}^{ca} = | Q_{i} Q_{j} | / T

⁠, and

Λ_{i j}

is given in Sec. III B 1. To help ameliorate this diffraction issue and recover the expected scaling, we introduce an empirical modification to the plasma parameter in the form

g_{i j} \approx \frac{| Q_{i} Q_{j} |}{λ_{eff} T} f_{i j},

(40)

where the correction for diffraction,

f_{i j}

⁠, is given by

f_{i j} \approx {(1 + {(\frac{Λ_{i j}}{r_{i j}^{ca}})}^{2})}^{1 / 2}

(41)

= {(1 + \frac{T}{2 π | Q_{i} Q_{j} |^{2} μ_{i j}})}^{1 / 2} .

(42)

The motivation for this modification to the plasma parameter

g_{i j}

is to smoothly transition to the de Broglie wavelength being the dominant length scale in the high-temperature limit, where diffraction physics is expected to be most relevant. It can be seen in Fig. 8 that this correction is most relevant for electron–electron and electron–ion scattering, in order of the reduced masses. Note the electron–electron scattering corrections (solid lines) overlap exactly for the three elements, since the ion mass does not enter in there. We can see this correction is largely irrelevant for ion–ion scattering (dotted), even for hydrogen.

FIG. 8.

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In this plot, we show the mod SMT dispersion correction from Eq. (41) that multiplies the effective screening length to correct high-temperature dispersion effects.

The SMT model can also be modified to more rigorously include the leading-order exchange-correlation (XC) effects in the background electrons; however, we found contributions to have a negligible effect on the transport processes considered in this work. As such, XC effects were not included in the results in Sec. V. Furthermore, details of incorporating XC effects into the SMT model can be found in Appendix.

C. Ziman transport model

The above models derived electronic transport coefficients through combining the AA based mean ionization state with kinetic theory results based on approximations such as Debye–Hückel screening and analytic cross-sectional formulas. In this section, we take output from the AA-NRL model, specifically the electron–ion cross section and the density of states, and use them as an input to the Ziman model (Fig. 3). Throughout this sub-section, atomic units are used. The cross section,

σ_{mom} (ϵ) = \frac{4 π}{k^{2} (ϵ)} \sum_{ℓ = 0}^{\infty} (ℓ + 1) {sin}^{2} (δ_{ℓ + 1} (ϵ) - δ_{ℓ} (ϵ)),

(43)

is used to derive the collisional and transport coefficients in this section. The electron–ion momentum-transfer collision rate,

ν (T_{e}) = \frac{\int_{0}^{\infty} σ_{mom} (ϵ) v (ϵ) f (ϵ, T_{e}) g (ϵ) d ϵ}{\int_{0}^{\infty} f (ϵ, T_{e}) g (ϵ) d ϵ} n_{i},

(44)

is the momentum-transfer cross section

σ_{mom} (ϵ)

times electron velocity

v (ϵ) = \sqrt{2 ϵ}

averaged over the Fermi–Dirac distribution and density of states of free electrons, the latter calculated according to

g (ϵ) = \sum_{ℓ = 0}^{\infty} 2 (2 ℓ + 1) \int_{0}^{a_{i}} u_{ℓ}^{2} (r, ϵ) d r .

(45)

The direct-current (dc) electrical and thermal conductivity are

σ_{e} = K_{0},

(46)

κ_{e} = \frac{K_{2} - K_{1}^{2} / K_{0}}{T_{e}},

(47)

respectively, where

K_{n} = - \frac{1}{3} \int_{0}^{\infty} τ (ϵ) v^{2} (ϵ) ϵ^{n} g (ϵ) \frac{\partial f (ϵ, T_{e})}{\partial ϵ} d ϵ

(48)

are collision integrals with

τ (ϵ) = {[σ_{mom} (ϵ) v (ϵ) n_{i}]}^{- 1}

being the energy-dependent collision time.²² More elaborate expressions, accounting for ion correlations, can be found in Refs. 57 and 79. The alternating-current (ac) conductivity is given by an analogous expression.³⁶ The electron–ion coupling coefficient, derived in Ref. 80 for the WDM regime, reads

G_{e i} = \frac{n_{i}}{π^{2} m_{i} T_{e}} \int_{0}^{\infty} k^{5} (ϵ) f (ϵ) (1 - f (ϵ)) σ_{mom} (ϵ) d k (ϵ),

(49)

where

k (ϵ) = \sqrt{2 ϵ}

is the wave vector and

T_{e}

is the electron temperature in the case

T_{i} \neq T_{e}

⁠, though the ion temperature does not come into this formula for the relaxation time.

D. Ionic transport properties from molecular dynamics

We now turn to the model furthest to the right in Fig. 3: the PM-HNC informed PP-MD. In this model, we generate ionic transport properties from MD simulation and Green–Kubo relations, using the pair potential generated from PM-HNC.

We investigated three priority 1 cases from TCCW-2 listed in Table I and shown as diamonds in Figs. 1 and 2. For each case, we ran $N_{runs} = 10$ parallel MD simulations, using the code described in Refs. 81 and 82. Each MD run consisted of $N = 5000$ particles interacting via the effective ion–ion pair-potential cutoff at $r = 8.0 a_{i}$ ⁠. Each run used a time step of $Δ t = 0.126 ω_{p}^{- 1}$ ⁠, where $ω_{p} = {(4 π {\bar{Z}}^{2} e^{2} n_{i} / m_{i})}^{1 / 2}$ is the ionic plasma frequency with $\bar{Z}$ the ionic charge obtained from the AA-NRL code. Each run consisted of two distinct phases: an equilibration phase lasting between 300 to 600 inverse plasma frequencies, and a subsequent production phase extending for 37 700 inverse plasma frequencies (⁠ $t_{prod} = 37700 ω_{p}^{- 1}$ ⁠). To guarantee statistical independence across these simulations, we initialized the positions and velocities of the particles using unique random seeds for each of the $N_{runs}$ parallel runs. The equilibration phase spanned a randomly chosen number between 300 and 600 $ω_{p}^{- 1}$ ⁠, in order to enhance the statistical independence between runs. A BBK Langevin integrator⁸³ with damping coefficient $γ_{Lang} = 5.03 ω_{p}^{- 1}$ was used for the equilibration phase, while a velocity Verlet integrator was used in the production phase. Fluctuations of the total energy during the production phase did not exceed a relative standard deviation of 0.0025% for hydrogen, 0.002% for aluminum, and 0.0005% for carbon.

Figure 9 shows a comparison of the ionic pair distribution function, $g_{i i} (r)$ ⁠, obtained from the PM-HNC method and the MD for the three cases under consideration. We find very good agreement between the two methods across all three cases. As mentioned above, the slight discrepancies between the curves in Fig. 9 are due to the inaccurate approximation of using a Yukawa bridge function for the more strongly coupled non-Yukawa PM-HNC potential.

FIG. 9.

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Plots comparing the ionic pair distribution function as obtained from the potential-matching HNC (PM-HNC) method described in Sec. III B 3 and pair-potential MD (PP-MD). (a) hydrogen at 1 g/cm³ and $T = 2$ eV. (b) carbon at 10 g/cm³ and $T = 2$ eV. (c) Aluminum at solid density, 2.7 g/cm³ and $T = 1$ eV.

The ionic transport coefficients were obtained using Green–Kubo relations. The ionic shear viscosity is obtained from the stress autocorrelation function (SACF):

η_{i} = lim_{t \to \infty} η_{i} (t), η_{i} (t) = \frac{V}{T} \int_{0}^{t} d s C_{P} (t),

(50)

C_{P} (t) = \frac{1}{3} \sum_{μ} \sum_{ν \neq μ} ⟨ P_{μ ν} (s) P_{μ ν} (0) ⟩,

(51)

where

μ, ν = {x, y, z}

⁠,

C_{P} (t)

is the SACF, and

P_{μ ν} (t)

is the element of the stress tensor defined by

P_{μ ν} (t) = \frac{1}{V} \sum_{i}^{N} [m_{i} v_{i}^{μ} v_{i}^{ν} - \sum_{j \neq i}^{N} \frac{r_{i j}^{μ} r_{i j}^{ν}}{r_{i j}} \frac{d}{d r} u_{i j} (r)],

(52)

where

u_{i j}

is the effective ion pair potential, and

r_{i j}^{μ}

is the

μ

component of the distance

r_{i j} = | r_{i} - r_{j} |

between particles i and j. Note that the double sum in Eq. considers only the three off-diagonal elements

P_{x y}

⁠,

P_{x z}

⁠, and

P_{y z}

⁠.

The validity of each simulation has been tested by checking the bulk modulus sum rule. The isothermal bulk modulus,

G_{\infty}

⁠, can be calculated either via

G_{\infty} = n T + \frac{2 π}{15} n^{2} \int_{0}^{\infty} d r > g (r) \frac{d}{d r} [r^{4} \frac{d}{d r} u (r)]

(53)

or via

G_{\infty} = {(V T)}^{- 1} C_{P} (0)

⁠.⁸⁴ We found that the two methods agree within 0.02%.

The ionic thermal conductivity is obtained from the heat flux autocorrelation function (HACF):

κ_{i} = lim_{t \to \infty} κ_{i} (t), κ_{i} (t) = \frac{V}{T^{2}} \int_{0}^{t} d s > C_{J} (s),

(54)

where

C_{J} (s) = ⟨ J (s) \cdot J (0) ⟩

is the HACF and

J

is the heat flux vector defined as

J (t) = \sum_{i = 1}^{N} (\sum_{j \neq i}^{N} J_{i j} + \frac{1}{2} m_{i} | v_{i} |^{2} v_{i}),

(55)

J_{i j} (t) = \frac{1}{4} [v_{i j} u (r_{i j}) + r_{i j} (v_{i j} \cdot \nabla u (r_{i j}))],

(56)

and

v_{i j} = v_{i} + v_{j}

the sum of the velocities of particles i and j.

In Eqs. (50) and (54), we have highlighted the fact that the real transport coefficients are obtained from the infinite time integral of the autocorrelation functions, but the calculated values are given by a finite time integral, hence $η (t)$ and $κ (t)$ ⁠.

The choice of the integration time t is crucial in determining the value of transport coefficients. The value of t should be large enough that the ACFs has decayed to zero; however, as t increases so does the statistical noise inherent in the calculation of the ACF. Furthermore, in estimating the variance of the transport coefficient, it is important to consider the correlation between timesteps. We address these two issues in the following way: each simulation is divided into

N_{blocks}

blocks of data, and for each block, we calculate the SACF and HACF and the respective transport coefficients. The final reported ACFs and transport coefficients, generically listed as

O = {C_{O}, η_{i}, κ_{i}}

⁠, are obtained by averaging over all blocks and over the independent MD runs, i.e., the average of observable

O (t)

is

⟨ O (t) ⟩ = \frac{1}{N_{runs}} \sum_{r = 1}^{N_{runs}} \frac{1}{N_{blocks}} \sum_{b = 1}^{N_{blocks}} O^{(r, b)} (t),

(57)

and its standard deviation is

Δ O (t) = {[\sum_{r = 1}^{N_{runs}} \sum_{b = 1}^{N_{blocks}} \frac{{(O^{(r, b)} (t) - ⟨ O (t) ⟩)}^{2}}{N_{runs} N_{blocks}}]}^{1 / 2} .

(58)

A diagram illustrating the above procedure is shown in Fig. 10. As shown in the figure, the blocks of each run are of different lengths. The first block contains all the data produced in one run, the second block uses only

t_{prod} - t_{shift}

timesteps of data, the third block

t_{prod} - 2 t_{shift}

timesteps of data, etc. In this way, the ACFs at time

t = 0

are obtained from the average over

N_{runs} N_{blocks}

samples, while the ACFs at time

t = t_{prod} - t_{shift}

only over

N_{runs}

samples.

FIG. 10.

View large Download slide

Diagram illustrating the calculation of the ACF from a single MD run. Each simulation is divided into blocks of data of variable length. The small gray box in each block represents the data from one time step. The total number of blocks is determined by the choice of $t_{shift}$ which, here, is equal to three timesteps.

As seen in Fig. 10, the number of blocks per simulation is determined by the choice of

t_{shift}

⁠. In order to maximize

N_{blocks}

⁠, one may feel compelled to use every time step as the starting point for a new data block, i.e.,

t_{shift} = Δ t

⁠. However, this approach is flawed as successive timesteps are highly correlated leading to an underestimation of the standard deviation in Eq. .^85–87 To circumvent this issue, we calculate the correlation time,

τ_{corr}

⁠, which represents the time interval beyond which two blocks are sufficiently uncorrelated to be considered independent. The correlation time

τ_{corr}

can be calculated as^86,87

τ_{corr} (t) = 2 \int_{0}^{t} d s > \frac{C_{O}^{2} (s)}{C_{O}^{2} (0)},

(59)

where

C_{O} (t)

is the ACF of observable O. In Fig. 11, we show the correlation times of the SACF and HACF as a function of time on a log –log scale. Note that each curve is the average over

N_{runs}

and

N_{blocks}

⁠. The plateau indicates the minimum number of timesteps (in units of inverse plasma frequency) needed to ensure statistical independence between consecutive blocks. The slight increase visible at late times, i.e.,

ω_{p} t > 300

is due to noise in the tail of the ACFs. The plateau suggests statistical independence at

τ_{corr} = 6 ω_{p}^{- 1}

⁠; as such, we chose

t_{shift} = 7 ω_{p}^{- 1}

for all three cases. This choice leads to

N_{runs} N_{blocks} = 54540

samples for the first

t_{shift} / Δ t

timesteps,

N_{runs} N_{blocks} = 54530

samples for the next

t_{shift} / Δ t

timesteps, etc.

FIG. 11.

View large Download slide

Log-linear plots of the correlation time $τ_{corr} (t)$ of the stress ACF (SACF) (solid green line) and heat flux ACF (HACF) (solid magenta line). (a) Hydrogen at 1 g/cm³ and $T = 2.0$ eV. (b) Carbon at 10 g/cm³ and $T = 2.0$ eV. (c) Aluminum at solid density, 2.7 g/cm³ and $T = 1.0$ eV. The left y axis shows the value of $τ_{corr} (t)$ in units of plasma periods.

Before showing results, we summarize our method as follows: the effective PP-MD, as compared to QMD simulations, allows for a larger number of particles and longer simulation times, which in turn allows for a large upper limit in the integral of $η_{i} (t)$ and $κ_{i} (t)$ ⁠. The block division allows for smooth well-defined ACFs that extend for times longer than those obtained by dividing a single run into consecutive blocks of equal length.⁸⁶ The statistical independence of block ACFs is ensured by choosing $t_{shift} > τ_{corr}$ ⁠. Finally, the statistical noise is further reduced by running $N_{runs}$ parallel simulations.

Figure 12 shows plots of the normalized SACF (top panels) and HACF (bottom panels), averaged over the number of runs and number of blocks, on log-linear scales for the three cases. The log-linear scale emphasizes the Gaussian-like behavior of the ACFs at short times $ω_{p} t < 10$ ⁠. This behavior becomes more pronounced as the system becomes strongly coupled; compare Fig. 12(a) hydrogen at $Γ_{i i} \sim 10$ with Fig. 12(b) carbon at $Γ_{i i} \sim 150$ ⁠. For longer times, i.e., $ω_{p} t > 10$ ⁠, the ACFs show the non-trivial behavior of the slowly decaying tail, characteristic of strongly coupled systems.¹⁸ For comparison, in the inset we show these ACFs in a log –log plots compared with the $t^{- 3 / 2}$ decay (black dashed lines) of spontaneous hydrodynamic modes. The fast oscillations for $ω_{p} t ≳ 100$ are due to statistical noise and finite-size effects restricting the appearance of clear low wavenumber hydrodynamic modes. The insets of Fig. 12 further highlight the need of long simulation times for better resolving the tail of the SACF and HACF.

FIG. 12.

Log-linear plots of the normalized stress ACF (top panels) and heat flux ACF (bottom panels) (a) and (d) hydrogen at 1 g/cm3 and T=2.0 eV, (b) and (e) carbon at 10 g/cm3 and T=2.0 eV, and (c) and (f) aluminum at solid density, 2.7 g/cm3 and T=1.0 eV. The insets show the same curves of the main plots but on a log –log scale in order to emphasize the slowly decaying tails. Note that each curve is the average over independent runs and blocks.

View large Download slide

Log-linear plots of the normalized stress ACF (top panels) and heat flux ACF (bottom panels) (a) and (d) hydrogen at 1 g/cm³ and $T = 2.0$ eV, (b) and (e) carbon at 10 g/cm³ and $T = 2.0$ eV, and (c) and (f) aluminum at solid density, 2.7 g/cm³ and $T = 1.0$ eV. The insets show the same curves of the main plots but on a log –log scale in order to emphasize the slowly decaying tails. Note that each curve is the average over independent runs and blocks.

The running integral of the ACF provides time-dependent curves of the transport coefficient. The integrals $η_{i} (t)$ and $κ_{i} (t)$ are shown as solid green lines in Fig. 13; top panels show the viscosity $⟨ η_{i} (t) ⟩$ and bottom panels the ionic thermal conductivity $⟨ κ_{i} (t) ⟩$ ⁠. The right y axis of the figure illustrates the relative percent deviation of the transport coefficient, shown as dashed magenta lines. The value of the true transport coefficients, $η_{i}$ and $κ_{i}$ ⁠, is given by the plateau. An ideal scenario would be one with indefinitely long simulation times that result in a well-defined plateau. In the real scenario, however, only a short plateau is present and the final reported value of the transport coefficient is decided by visual inspection. In our case, we chose to report the value of $η_{i}$ and $κ_{i}$ at $t = 100 ω_{p} t$ ⁠. These values are shown as orange dots in Fig. 13 and reported in Table II.⁸⁸

FIG. 13.

View large Download slide

Log–log plots of the integrals $η_{i} (t)$ (top panels) and $κ_{i} (t)$ (bottom panels) averaged over the number of runs and number of blocks. The right y axis of the plots, cut at 100 for easier viewing, indicates the relative percent deviation of the transport coefficients, depicted as dashed magenta lines. The orange dot in the plots indicates the value of the transport coefficients reported in Table II. (a) and (d) hydrogen at 1 g/cm³ and $T = 2.0$ eV, (b) and (e) carbon at 10 g/cm³ and $T = 2.0$ eV, and (c) and (f) aluminum at solid density, 2.7 g/cm³ and $T = 1.0$ eV.

TABLE II.

Table of the reported values of transport coefficients for each of the three cases.

	$ρ$ (g/cm³)	T (eV)	$⟨ η_{i} ⟩$ (mPa s)	$⟨ κ_{i} ⟩ \times 10^{6}$ [W/(cm keV)]
H	1	2	$6.750 \pm 0.002$	$9.928 \pm 0.017$
C	10	2	$10.41 \pm 0.025$	$10.776 \pm 0.019$
Al	2.7	1	$0.715 \pm 0.001$	$0.508 \pm 0.001$

	$ρ$ (g/cm³)	T (eV)	$⟨ η_{i} ⟩$ (mPa s)	$⟨ κ_{i} ⟩ \times 10^{6}$ [W/(cm keV)]
H	1	2	$6.750 \pm 0.002$	$9.928 \pm 0.017$
C	10	2	$10.41 \pm 0.025$	$10.776 \pm 0.019$
Al	2.7	1	$0.715 \pm 0.001$	$0.508 \pm 0.001$

E. Integrated Yukawa viscosity model (IYVM)

The Yukawa viscosity model (YVM)²⁰ provides a simple method for computing the viscosity of WDM and liquid metals. The YVM was developed for and adapted to liquid metals and WDM because of the data (experimental and/or from high-fidelity physics codes) that are available for those regimes, and while comparisons with liquid metal data have revealed modest errors, accuracy appears to improve with increasing temperature in these regimes. However, it is less accurate in the high-energy-density plasma (HEDP) regime where the viscosity is greatest and therefore most important to hydrodynamics. This necessitates a modification of the YVM at high temperatures, particularly important in IFE applications. Here, the YVM is extended to HEDP with little loss of accuracy in the WDM and liquid metal regimes. The spirit of the YVM is retained in that MD data for the Yukawa model is combined with results from the SMT model.¹⁸ Emphasis is placed on behavior at high temperatures, with small $Γ_{i i}$ and $κ$ ⁠.

In the original YVM, the viscosity was parametrized as

\frac{η}{η_{0}} = 0.0051 \frac{Γ_{m} (κ)}{Γ_{i i}} + 0.374 \frac{Γ_{i i}}{Γ_{m} (κ)} + 0.022,

(60)

where

η_{0} = \sqrt{3} ω_{E} m_{i} n_{i} a_{i}^{2}

⁠, and

Γ_{m} (κ)

is the melt boundary as approximated in Eq. , and

ω_{E}

is the Einstein frequency. Note that the viscosity is also commonly reduced to dimensionless units using

η_{p} = ω_{p} m_{i} n_{i} a_{i}^{2}

instead, where

ω_{p}

is the usual ionic plasma frequency. The two frequencies can be connected to each other through the approximate relation

\frac{ω_{E}}{ω_{p}} \approx \frac{1}{\sqrt{3}} e^{- 0.2 κ^{1.62}} .

(61)

The coefficients in Eq. were obtained through a fit to Yukawa MD data for

κ = 2

and

κ = 3

(to emphasize the physics of WDM and liquid metals). The MD data were obtained from Saigo and Hamaguchi (SH),⁸⁹ which is restricted to

Γ_{i i} > 2

⁠. Two essential points are that the MD data used to generate the YVM only represent strongly coupled plasmas with

Γ_{i i} > 2

and that the values of the coefficients in Eq. were obtained only for the two cases of

κ = 2, 3

⁠; thus, the YVM should not be extrapolated to lower values of these parameters. As will be shown below, the YVM scales incorrectly in the HEDP regime. For this reason, an integrated YVM (IYVM) is developed here.

The IYVM integrates data from both MD and the SMT model and takes on a new functional form. In the original YVM, emphasis was placed on larger $κ$ because of the application to liquid metals and WDM, and the model was validated with very accurate liquid metal data. For HEDP, screening becomes considerably weaker, and fitting over a much larger range of $κ$ is needed. For this reason, the SH MD data are now used over the entire range of $κ$ ⁠; this reveals that the coefficients vary substantially with $κ$ ⁠. Furthermore, MD data for $Γ = 2$ were used from Donkó and Hartmann⁹⁰ because of its improved accuracy. In addition to MD data, the SMT model was used, where the fits to the relevant collision integrals can be found in Eqs. (C22)–(C24) of Ref. 18. Because that expansion is increasingly valid at lower coupling, we augment the SH MD data with SMT results in the range $0.1 < Γ_{i i} < 1$ for $κ = 1, 2, 3$ ⁠.

With the new dataset, we generalize the functional form for the YVM to the new IYVM form:

\frac{η}{η_{0}} = A {(\frac{Γ_{m}}{Γ_{i i}})}^{a} + B {(\frac{Γ_{i i}}{Γ_{m}})}^{b} + C,

(62)

where the coefficients are now

κ

-dependent and the power laws given by a and b are fit parameters. The resulting fit coefficients, using the expanded dataset, are given by

\begin{matrix} A (κ) = 1.45 \cdot 10^{- 4} - 1.04 \cdot 10^{- 4} κ + 3.69 \cdot 10^{- 5} κ^{2}, \end{matrix}

(63)

a (κ) = 1.78 + 0.13 κ - 0.062 κ^{2},

(64)

B (κ) = 0.3 + 0.86 κ - 0.69 κ^{2} + 0.138 κ^{3},

(65)

b (κ) = 1.63 - 0.325 κ + 0.24 κ^{2},

(66)

C (κ) = 0.015 + 0.048 κ^{0.754} .

(67)

Note that though this fit incorporates SMT data, the lack of a logarithm in Eq. implies an inexact scaling behavior in the very weakly coupled regime; the user who primarily cares about this limit can model the viscosity through an explicit transition to SMT viscosities for

Γ_{i i} ≪ 1

⁠. Results for the YVM and IYVM are discussed below in Sec. V C and are shown in Fig. 18.

F. Low-temperature data

We compare the transport models described above to experimental values of electrical and thermal conductivity and viscosity for liquid and solid Al for which there are very accurate measurements. These data, shown in Table III and included here for convenience, validate our models in the low-temperature limit. In the table, a quantity is assumed to fill the rows below it until a new quantity appears, to avoid clutter.

TABLE III.

Experimental Al transport properties in the condensed phase used to validate our models in the low-temperature limit. Single values or units are implied to continue down the column they are in. Here, $σ_{exp}$ is the experimental electrical conductivity, $κ_{exp}$ is the experimental thermal conductivity, and $η_{exp}$ is the experimental viscosity; that is, these are all total, not species, values. Where $D_{liq} (T)$ appears, the linear form of Eq. (68) should be applied to that row and rows below it using the temperature for that row. For the solid phase, we assume a constant density listed at the first row where that phase appears.

Coefficient	Source	Citation	$ρ$ (g/cm³)	T (eV)	Value	Unit
$σ_{exp}$
	Smithells (SMRB)	91	2.70	0.025 3	374 531.8	S/cm
			$↓$	0.032 2	281 690.1	$↓$
				0.040 8	209 205.0
				0.049 4	166 944.9
				0.058 0	136 986.3
			$D_{liq} (T)$	0.080 4	41 237.1
			$↓$	0.083 9	40 273.9
				0.092 5	38 022.8
				0.101 1	36 010.1
				0.109 7	34 199.7
	Leitner	92	$D_{liq} (T)$	0.128 5	28 634.3	S/cm
				0.103 8	33 276.5
				0.092 0	36 250.7
				0.086 7	38 427.0
				0.083 7	41 749.1
			2.67	0.080 6	58 843.2
				0.078 0	83 644.9
				0.072 2	92 742.3
				0.066 7	103 094.0
				0.062 7	110 644.8
$κ_{exp} / 10^{7}$
	Brandt and Neuer	75	$D_{liq} (T)$	0.087 9	1.1277	W/cm keV
				0.081 5	1.099
				0.080 3	1.201
				0.080 4	1.673
			2.698	0.080 0	2.3655
				0.080 1	2.423
				0.0675	2.5253
				0.051 5	2.6278
				0.040 7	2.7344
				0.031 9	2.8983
$η_{exp}$
	CRC	93
			$D_{liq} (T)$	0.083 86	1.24	mPa-s (cP)
				0.881 6	1.13
				0.092 5	1.04
				0.096 8	0.96
				0.101 1	0.90
				0.105 4	0.84

Coefficient	Source	Citation	$ρ$ (g/cm³)	T (eV)	Value	Unit
$σ_{exp}$
	Smithells (SMRB)	91	2.70	0.025 3	374 531.8	S/cm
			$↓$	0.032 2	281 690.1	$↓$
				0.040 8	209 205.0
				0.049 4	166 944.9
				0.058 0	136 986.3
			$D_{liq} (T)$	0.080 4	41 237.1
			$↓$	0.083 9	40 273.9
				0.092 5	38 022.8
				0.101 1	36 010.1
				0.109 7	34 199.7
	Leitner	92	$D_{liq} (T)$	0.128 5	28 634.3	S/cm
				0.103 8	33 276.5
				0.092 0	36 250.7
				0.086 7	38 427.0
				0.083 7	41 749.1
			2.67	0.080 6	58 843.2
				0.078 0	83 644.9
				0.072 2	92 742.3
				0.066 7	103 094.0
				0.062 7	110 644.8
$κ_{exp} / 10^{7}$
	Brandt and Neuer	75	$D_{liq} (T)$	0.087 9	1.1277	W/cm keV
				0.081 5	1.099
				0.080 3	1.201
				0.080 4	1.673
			2.698	0.080 0	2.3655
				0.080 1	2.423
				0.0675	2.5253
				0.051 5	2.6278
				0.040 7	2.7344
				0.031 9	2.8983
$η_{exp}$
	CRC	93
			$D_{liq} (T)$	0.083 86	1.24	mPa-s (cP)
				0.881 6	1.13
				0.092 5	1.04
				0.096 8	0.96
				0.101 1	0.90
				0.105 4	0.84

Comparing to experimental data in this way introduces three issues not addressable within the TCCW-2 cases. First, experiments typically measure the total transport; experimentally separating electronic and ionic contributions can be extremely difficult. Whether having species-resolved coefficients is important for HED modeling depends on the choices made for separating the hydrodynamic conservation laws by species or not; here, we simply note that the comparisons with our models and experiments are not purely “apples-to-apples.” Second, the TCCW-2 cases follow paths in parameter space different from most experiments. While the density of solid is roughly constant as a function of temperature, the liquid metal state is not: there is a discontinuity in the density (to a lower, liquid density) with a linear density dependence. In the table, we have given the nominal (approximately constant) solid density and labeled the liquid metal state as

D_{liq} (T)

⁠, where one uses at each T,⁹¹^,

\begin{matrix} D_{liq} (T) = D_{0} + (T - T_{0}) \frac{d D}{d T}, \\ D_{0} = 2.385 {g / cm}^{3}, \\ T_{0} = 0.0804 eV, \\ \frac{d D}{d T} = - 3.25 {g / eV - cm}^{3} . \end{matrix}

(68)

A much more detailed representation of the Al density in these phases is given in Fig. 1 of Leitner et al.⁹² In some cases,⁷⁵ the density is not given and we assume in Table III the form given above. Third, transport can be strongly impacted by impurities that are always present in experimental samples, and precise values vary from sample to sample; currently, little is known about the role of impurities at elevated temperatures⁷⁵ characteristic of HED matter.

The electrical and thermal conductivities of aluminum are very well known. We compare several measurements, given in Table III, in our results below. The different sources of experimental data agree well among each other. Most of the data crosses the melting transition, which serves to validate the solid and liquid phases of the MLM model, including the ratio at the melting transition.⁷⁵

We compare three datasets for Al viscosity, two of which are given by fits with the other given in the table. The Al viscosity from the Smithells reference is given by⁹¹

η_{S} = η_{0} exp (0.171 / T),

(69)

η_{0} = 0.1492 mPa - s .

(70)

Assael et al. also provide a fit to their data in the form⁹⁴

η_{A} = 0.185 \cdot 10^{0.0692 / T} mPa - s,

(71)

where T is in eV. We also include tabulated values from the CRC handbook, as given in the table. From the figure, we see good agreement among the experimental values for aluminum viscosity.

V. RESULTS

In this section, we compare the results of each of the models described above, including the available experimental data. We compute the electrical and thermal conductivities, noting the electron and ion contributions, as well as the viscosity, for the hydrogen, carbon, and aluminum cases shown in Fig. 1 as dashed white lines and colored diamonds. We compare with the DFT-MD predictions from the workshop;²⁷ these points are shown in black with error bars (where they are known).

A. Electrical conductivity

The electrical conductivities computed for these models are shown in the three panels of Fig. 14. The Lee–More-based MLM (solid green) model described in Sec. IV A contains the most detailed phase behavior. In each plot, the conductivity increases with decreasing temperature in the liquid and solid regimes and increases with increasing temperature in the weakly coupled limit. One can additionally see discontinuities in the derivative, corresponding to switches between MLM regions.

FIG. 14.

Electrical conductivity vs temperature for the cases in Fig. 1. (a) Hydrogen at 1 g/cm3. (b) Carbon at 10 g/cm3. (c) Aluminum at density, 2.7 g/cm3. In this case, data are available, see Table III. Note that only MLM captures well the phase behavior, with moderate agreement with the aluminum experimental data. Note that the experimental comparison is approximate as there are small density variations in the liquid metal data. Also shown is a prediction from an alternate AA formulation .57

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Electrical conductivity vs temperature for the cases in Fig. 1. (a) Hydrogen at 1 g/cm³. (b) Carbon at 10 g/cm³. (c) Aluminum at density, 2.7 g/cm³. In this case, data are available, see Table III. Note that only MLM captures well the phase behavior, with moderate agreement with the aluminum experimental data. Note that the experimental comparison is approximate as there are small density variations in the liquid metal data. Also shown is a prediction from an alternate AA formulation .⁵⁷

The high-T limit is in agreement with Spitzer theory.⁷⁶ One can see where this limit is expected to kick in by finding the point on the isotherms in Fig. 1 beyond which $W ≪ 1$ ⁠. This limit is additionally assuming the Lorentz gas approximation of no explicit electron–electron scattering. Generally, each of these models accounts for this electron–electron scattering in a limited sense and is a possible source of the discrepancy between these theories in the high-T limit. In particular, we note that the effects of this scattering can be thought of as coming in three forms, screening, electron distribution function reshaping, and the effects of explicit scattering on the current;²⁵ in the case of electrical conductivity, this last term vanishes. The first term is incorporated in every theory here, but the effect of electron–electron scattering on the thermal distribution of electrons is not accounted for in any of these theories, see Ref. 95 for an updated analysis over.²⁵

We note that of the many computations tabulated in Refs. 26 and 27 (not shown), most lie intermediate between the MLM and mod SMT (dashed blue) lines in the high-T region. The Ziman model (solid purple) is cut off at high temperature due to the large number of partial waves needed. Similarly, the Rozsnyai curve (dashed dotted red) in panel c) of Figs. 14 and 15 is likely too low due to only using 30 partial waves, which based on the discussion in Sec. III A is too low for $≳ 100$ eV. In comparison, we have DFT-MD points from the workshop (black errorbars), which lie intermediate between the models.

FIG. 15.

Electron thermal conductivity vs temperature for the cases in Fig. 1. (a) hydrogen at 1 g/cm3, (b) carbon at 10 g/cm3, (c) aluminum at 2.7 g/cm3. For the experimental data, see Table III.

View large Download slide

Electron thermal conductivity vs temperature for the cases in Fig. 1. (a) hydrogen at 1 g/cm³, (b) carbon at 10 g/cm³, (c) aluminum at 2.7 g/cm³. For the experimental data, see Table III.

In the low-temperature limit, these models are in qualitative disagreement. MLM, consistent with many of the findings in Refs. 26 and 27, predicts a decreasing conductivity with increasing temperature in the liquid phase. In the case of aluminum, Fig. 14(c) the melting transition of MLM predicted is in close agreement with experimental data described in Table III, shown here as various markers. These data are limited to the solid and near-solid regime, and we remind the reader about the non-constant density described in Sec. IV F. In the case of hydrogen, obtaining experimental data is much more difficult, but based on path integral Monte Carlo and other calculations,⁹⁶ a phase transition is not expected in these region; we expect the MLM phase transition is spurious in this case. In comparison, the SMT, mod SMT, and Ziman models do not distinguish between the solid and liquid phases, showing no phase transition. In particular, SMT is based on a binary collision approximation and so it is expected to disagree wherever strong coupling effects are relevant. On the other hand, we expect the Ziman theory to account for strong coupling, but we do not observe an increase in conductivity at low temperatures in any of the elements computed. In contrast, in panel c), we plot another AA based model by Rozsnyai⁵⁷ (dashed blue), which shows such an increase. It is unclear what the cause of the difference is between the two AA models.

B. Thermal conductivity

Here, we describe the results of our models for thermal conductivity. The Wiedemann–Franz law relating thermal and electrical conductivities implies that much of the discussion above holds if a factor of temperature is added. An additional complication is the fact that the heat current, unlike the electrical, is explicitly affected by electron–electron scattering. Generally, the effect of incorporating these collisions is to significantly lower the thermal conductivity.²⁵ The existence of such effects also complicates the issue of defining separate electron and ion thermal conductivities, which, unlike the electrical case, can both be relevant.

There is no unique way to separate the total thermal conductivity into that of the individual electron and ion species in a fully interacting plasma. In the case of experimental data, it is the total thermal conductivity that is measured, but the electronic physics typically dominates; to benchmark our models, we assume the ion conductivity is subdominant and compare the experimental data with the electronic component of the thermal conductivity only, bearing in mind that this is only an approximation.

In the SMT model, we define the ionic thermal conductivity as $κ_{i} = κ_{i i}^{eSMT}$ ⁠, Eq. (15) in Ref. 78, and the electronic as $κ_{e} = κ^{eSMT}$ ⁠, specifically in the form of the approximation in Eq. (17) of Ref. 78. The ionic component is thus defined here in terms of ion–ion scattering only; the only impact of the electrons is through the screening length. The electronic component is taken as this approximate form of the full conductivity, accurate to $O (\sqrt{m_{e} / m_{i}})$ ⁠, subject to temperature differences and order one factors; this is also achieved by taking the limit where all ion–ion collisions go to zero. One must be careful in the application of these conductivities as the relation $κ = κ_{e} + κ_{i}$ does not hold for these definitions, with error equal to the difference of Eq. (14) and Eqs. (15) and (17) in Ref. 78, and is smaller by an additional square root of the electron–ion mass ratio.

In contrast, the Ziman picture of electronic transport is more subtle. Electron–ion scattering is the bulk of the input from the AA-NRL model to Ziman transport coefficients, but explicit electron–electron scattering is absent, see the discussion in the previous discussion. Similarly, in the case of MLM, the only direct collisional input is in the form of an electron–ion cross section. We thus consider the output of MLM (solid green), Ziman (solid magenta) as entirely electronic, with the caveat that electron–electron corrections are absent. These are plotted in Fig. 15. Much of the features carry over from the electrical conductivity discussion. In this case, the rapid growth of $κ_{e}$ with temperature hides much of the disagreement.

Significantly less is known about the ionic component of the thermal conductivity, despite its importance near the boundary of shocks⁹⁷ where temperature ratios can be large. In Fig. 16, we show the comparison between the mod SMT model (solid cyan line), our MD simulations (orange error bars), and the workshop DFT-MD points (black error bars). In this case, we note that no discernible difference is found for the ionic transport at high temperature due to the improved $g_{i j}$ ⁠. In comparison, the predicted PP-MD conductivity is larger than expected for hydrogen, but remarkably consistent in the carbon case, though here SMT overpredicts the conductivity from DFT-MD. In the case of aluminum, there were no workshop data.

FIG. 16.

Ionic thermal conductivity vs temperature for the cases in Fig. 1. (a) hydrogen at 1 g/cm3, (b) carbon at 10 g/cm3, (c) aluminum at 2.7 g/cm3. For the experimental data, see Table III. Here, the SMT and mod SMT values are overlapping.

View large Download slide

Ionic thermal conductivity vs temperature for the cases in Fig. 1. (a) hydrogen at 1 g/cm³, (b) carbon at 10 g/cm³, (c) aluminum at 2.7 g/cm³. For the experimental data, see Table III. Here, the SMT and mod SMT values are overlapping.

Typically, the ionic contribution to the thermal conductivity is subdominant to the electronic contribution; in the kinetic theory of gases, this is due to the large mass ratio, but in the moderate-temperature limit, the situation is less clear. We show the results of our models in Fig. 17, which generally corroborate our intuition that the electronic contribution is larger; however, overall the disagreement between the models is qualitative. Complicating this issue is that the SMT model is not designed for the $Γ_{i i} ≳ 1$ regime, where the binary collision approximation is no longer good. For this reason, we show the ratio of the SMT electronic to ionic models, as well as the MLM to the ionic SMT model, under the assumption that MLM will give a better low-T prediction than SMT. The points that self-consistently incorporate the most physics are the ratio of the ionic component obtained from PP-MD over the electronic component obtained from the Ziman model (sky blue error bar).

FIG. 17.

View large Download slide

The ratio of the ionic to the electronic thermal conductivity vs temperature for the cases in Fig. 1. In solid cyan is the ratio of mod SMT for the electronic component (eSMT) to that of the ionic component (iSMT), computed using the corrections in Sec. IV B. In green, the ratio of ionic mod SMT to the electronic MLM. In solid magenta, the ratio of iSMT to Ziman. The sky blue error bar is the ratio of the Ziman transport model to the effective potential MD point. (a) hydrogen at 1 g/cm³, (b) carbon at 10 g/cm³, (c) aluminum at 2.7 g/cm³.

We also show the thermal conductivity ratio predicted by DFT-MD (black error bars) in Fig. 17. Since no group submitted both ionic and electronic components, the single data point shown is shown assuming each quoted uncertainty,⁹⁸

σ_{j}

⁠, for jth dataset in Ref. 27, is interpreted as a standard deviation. From this, we create an estimate for the mean and overall uncertainty of the thermal conductivity ratio, weighted with

w_{j} = σ_{j}^{- 2} / \sum σ_{j}^{- 2}

⁠, as

{\bar{κ}}_{e} = \sum_{j} w_{j} κ_{e}_{j},

(72)

σ_{e} = \sqrt{\sum_{j} w_{j} {(κ_{e}_{j} - {\bar{κ}}_{e})}^{2}} .

(73)

Except for the very low-temperature region of panel (b), where we should not trust the ionic SMT-based results anyway, all of the models corroborate our picture that the ionic contribution is subdominant; however, in many cases, the effect still appears to be of order

1 - 10 %

and would be even more significant in cases where the ionic temperature is much larger than the electronic temperature.

C. Viscosity

In this section, we detail the results of our transport models for ionic viscosity, shown in Fig. 18. As in the prior cases, we benefit from experimental data taken in the liquid metal regime for aluminum, shown by symbols in panel c). In this case, we no longer have the MLM model, which only computes electron transport. Instead, we can compare mod SMT (cyan), which for viscosity is degenerate with the original SMT, to a fit to Yukawa MD data given by the YVM model (red),²⁰ and its correction the IYVM model (red dashed).

FIG. 18.

Viscosity vs temperature from the SMT model and the YVM20 (red), and IYVM model of Sec. IV E (dashed red). Here, the SMT and mod SMT values are overlapping. These are compared to our MD simulation (orange errorbar). (a) Hydrogen at 1 g/cm3. The phase transition from a metallic liquid to a plasma is indicated with a vertical line. (b) Carbon at 10 g/cm3. (c) Aluminum at 2.7 g/cm3. We also show data taken in the case of aluminum for temperatures just above the melting point.

View large Download slide

Viscosity vs temperature from the SMT model and the YVM²⁰ (red), and IYVM model of Sec. IV E (dashed red). Here, the SMT and mod SMT values are overlapping. These are compared to our MD simulation (orange errorbar). (a) Hydrogen at 1 g/cm³. The phase transition from a metallic liquid to a plasma is indicated with a vertical line. (b) Carbon at 10 g/cm³. (c) Aluminum at 2.7 g/cm³. We also show data taken in the case of aluminum for temperatures just above the melting point.

When a richer $κ$ dependence is used, the IYVM results can differ considerably from the YVM, especially for small $Γ$ and/or $κ$ ⁠, manifesting as a large difference in the high-temperature range for hydrogen, panel a, and the low-temperature range for carbon, panel b, and aluminum, panel c. The original YVM is otherwise accurate in the original, intended parameter space of moderate to large $Γ$ and large $κ$ seen for intermediate temperatures in the above plots. The use of the SMT results in the fit for IYVM greatly improves the fit of the IYVM over the YVM in the high-temperature region most important for IFE applications.

In all panels, we show the results down to a temperature of 0.05, but for aluminum, panel c, where the liquid–solid transition is known, we draw a vertical line to indicate where the viscosity computed loses much of its meaning (shaded region), as the solid would be better described by a stiffness tensor.

Our PP-MD simulation (orange error bar) aligns very well with both YVM and SMT-based models; in particular, it is within 5 $%$ of IYVM in the case of hydrogen, panel a). For the carbon and aluminum cases, the PP-MD predicted viscosity is larger than the on-the-fly models, likely reflecting the attractive dip in the PM-HNC pair potential, not present in the Yukawa based YVM and SMT models. In each case, TCCW-2 DFT-MD, data (black error bars) are particularly consistent with both YVM and IYVM models. Interestingly, though in Figs. 5–7, the hydrogen and carbon cases are both overly strongly peaked, the resulting DFT-MD coefficients agree well for the hydrogen viscosity, and the carbon thermal conductivity, but not vice versa.

VI. CONCLUSIONS AND FUTURE OUTLOOK

We have compared many methods of generating transport properties in dense plasmas. For each of the on-the-fly models described in this paper—SMT, LM, and YVM—some improvement was made, generating the SMT+dispersion correction, the MLM, and the IYVM models. We generally find good consistency between these models at high temperature, and when applicable, the IYVM and MLM models agree well with experimental measurement. The high-T improvements to IYVM make applications to IFE significantly more accurate. SMT now better handles the WDM physics, below where it was originally designed, and MLM now yields even better agreement with experimental data. A significant remaining question lies in understanding what role electron–electron scattering plays in these models. We leave these corrections to future work.

These models, however, are either largely phenomenological in nature (MLM), or are approximated by linear screening (SMT, IYVM). We compare this to two higher fidelity models, the AA-NRL with the Ziman formalism for electronic transport and PP-MD using PM-HNC potentials for ionic transport. We find good agreement overall, though the discrepancy can be up to an order of magnitude.

Significant improvement could be made to the PM-HNC method with the inclusion of better pseudopotentials. In Ref. 99, an inversion of the HNC computation was proposed using $h_{e i}$ as an input to generate $u_{e i}$ ⁠, but such an inversion requires an extended-sphere neutral pseudo-atom calculation. Instead, one might take a simple pseudopotential such as the smoothed Ashcroft potential we proposed and fit it to output from such a neutral pseudo-atom or QMD calculation. Additionally, a significant difficulty in the modeling of dense plasmas with attractive electron–ion HNC-based methods is convergence. It was found that a combination of Picard-based methods, including Refs. 69 and 74, was insufficient for $θ ≲ 0.1$ and $Γ_{i i} ≳ 100$ when electrons and ions were both considered. Simple root finding algorithms were able to extend this approach, albeit with a significantly longer computation time. Future improvements could include multi-scale newton methods,¹⁰⁰ or wavelet transform methods.¹⁰¹

The resulting PP-MD was limited in scope partly due to the current implementation of Green–Kubo as a post-processing step; in the future, we intend to make this an on-the-fly computation, reducing the amount of data needing to be saved.

Future work on AA-NRL will address transport properties at low temperatures (<1 eV) by including the structure factor $S (k)$ in the electron–ion momentum-transfer cross section.

In this paper, we compared our results against aluminum data and found promising agreement. A caveat is that each calculation was done along an isochore, rather than the density-varying isobar over which the experimental data were collected. A severe test of the models then is to formulate them at constant pressure and compare with the solid and liquid metal data; this establishes the low-T limit of the models.

The multi-species limit of the models discussed in this paper is of interest for ICF and astrophysical processes, given that the combination of single-species transport coefficients is not unique.⁹⁷ Each of the models we present here can be generalized to mixtures with varying levels of success and difficulty. In the AA approach, one can in principle treat only one species at a time, but one can still attempt multiple calculations, which self-consistently agree on the pressure, see Ref. 102. The PM-HNC approach, given AA ionization input, is already in a multi-species format and only needs to be modified through the choice of bridge function for each species, and from this point, MD is straightforward. The parametric models can also be modified; a prescription for YVM was developed in Ref. 20, and in SMT, some of the coefficients generalize trivially, such as the electrical conductivity,⁷⁸ but others require a matrix inversion whose resulting form grows rapidly in complexity. Instead, one can simplify the ion description by invoking averaged quantities such as $\bar{Z}$ ⁠, which is one way to generalize MLM.

Looking forward to the next workshop, TCCW-3, we hope to address some of the limitations and challenges of this workshop. A significant potential benefit of these workshops lies in the generation of large datasets for use in the machine learning and fitting of transport coefficients. One issue, inherent in the workshop itself, is the lack of known theoretical uncertainties for each of the many methods presented, something likely much larger than the statistical errors typically presented. For the next workshop, we hope to quantify such issues, including sensitivity analyses to important model inputs. Additional challenges lie in the extrapolation, rather than interpolation, of data points. Fitting over the large parameter spaces in many real applications may benefit significantly from the computation of additional isotherms and outlying points in parameter space to better constrain possible results. Furthermore, real experiments rarely involve single-species plasmas. A focus in the next workshop can be on the effect of known mixtures, something that was included to a limited extent in this workshop, as well as on the effects of small impurities. The consistency between the transport coefficients generated and those that would be used in actual hydrodynamic simulation was another issue that presented itself in this workshop. In the next workshop, we hope to ensure this is consistent, as well as check for the relative importance of potential hydrodynamic inputs, such as electronic and ionic-based coefficients, something we began detailing in the case of thermal conductivity. One highly motivated application where all of these ideas may be implemented in a focused way is in the computation of transport coefficients specifically for the densities, temperatures, and compositions in IFE applications. Such a focused TCCW-3 may help significantly in gaining a full understanding of both the values and uncertainties of transport coefficients in IFE relevant dense plasmas.

ACKNOWLEDGMENTS

We thank the participants of the TCCW-2 workshop for their feedback. The work of Z.A.J. and M.S.M. was supported in part by the National Science Foundation under Grant No. NSF-2108505. G.M.P. was supported by the NRL Base Program. Additionally, this work was supported in part through computational resources and services provided by the Institute for Cyber-Enabled Research at Michigan State University.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Zachary A. Johnson: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (equal); Project administration (lead); Software (lead); Supervision (lead); Validation (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (lead). Luciano G. Silvestri: Conceptualization (supporting); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). George M. Petrov: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Visualization (supporting); Writing – original draft (equal); Writing – review & editing (supporting). Liam G. Stanton: Formal analysis (equal); Investigation (supporting); Methodology (equal); Visualization (supporting); Writing – original draft (equal); Writing – review & editing (supporting). Michael S. Murillo: Conceptualization (equal); Data curation (supporting); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (supporting); Software (supporting); Supervision (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

NOMENCLATURE

$a_{e}$: Electron sphere radius
$a_{i}$: Ion-sphere (Wigner–Seitz) radius
$A_{i}$: Ion-sphere radius for mixtures
$B (r)$ ⁠, $B_{i j} (r)$: Bridge function
$c (r)$ ⁠, $c_{i j} (r)$: Direct correlation function
e: Elementary charge
$E_{F}$: Fermi energy
$f_{i j}$: DISPERSION correction factor
$F_{xc}$: Exchange-correlation energy functional
$f (ϵ)$: Fermi–Dirac distribution
$F_{x} (η)$: Fermi–Dirac integral of order x
$\tilde{f} (k)$: Fourier transform of function $f (r)$
$g (ϵ)$: Density of states
$G (k)$: Local field correction
$g_{i j}$: SMT plasma parameter
$g (r)$ ⁠, $g_{i j} (r)$: Radial (pair) distribution functions
$h (r)$ ⁠, $h_{i j} (r)$: Total correlation function
$J$: Heat flux vector
k: Fourier space variable
$L$: Coulomb logarithm
L: Lorenz number
$ℓ$: Orbital (azimuthal) quantum number
$m_{i}$: Particle mass of species i
n: Principal quantum number
$n_{e}$: Electron number density
$n_{i}$: Ion number density of species i
$P_{μ ν}$: Stress tensor
$Q_{i}$: Charge of species i
$R_{c}$: Atomic core radius
T: System temperature (in energy units)
$T_{i}$: Temperature of species i
$T_{i j}$: Cross-temperature
$T_{m}$: Melt temperature
$t_{corr}$: Correlation time
$t_{prod}$: Production phase period
$t_{ω_{p}}$: Plasma period
$W$: WDM parameter
$Z_{i}$: Nuclear charge number of species i
${\bar{Z}}_{i}$: Mean ionization state (“Z bar”)
$β$: Inverse temperature
$Γ_{e e}$: Electron–electron coupling parameter
$γ_{0}$: Exchange-correlation coefficient
$Γ_{i i}$: Ion–ion coupling parameter
$Γ_{i}^{IS}$: Ion-sphere coupling parameter
$Γ_{K} (κ)$: Kirkwood curve in $(Γ, κ)$ space
$γ_{Lang}$: Langevin damping coefficient
$Γ_{m} (κ)$: Melt curve in $(Γ, κ)$ space
$ϵ$: Energy variable
$η$: Dimensionless chemical potential
$η_{i}$: Ionic viscosity
$Θ$: Degeneracy parameter
$κ$: Screening parameter
$κ_{i}$: Thermal conductivity of species i
$λ_{eff}$: Effective screening length
$Λ$: Mean free path
$Λ_{i j}$: Thermal de Broglie wavelength
$λ_{i}$: Screening length of species i
$μ$: Chemical potential
$μ_{i j}$: Reduced mass of species i and j
$ρ$: Mass density
$ρ_{ion}$: Charge density
$σ_{i}$: Electrical conductivity of species i
$σ_{mom}$: Momentum-transfer cross section
$τ$: Electron–ion collision time
$ψ_{α} (r)$: Kohn–Sham orbital
$ω_{p}$: Plasma frequency

NOMENCLATURE

$a_{e}$: Electron sphere radius
$a_{i}$: Ion-sphere (Wigner–Seitz) radius
$A_{i}$: Ion-sphere radius for mixtures
$B (r)$ ⁠, $B_{i j} (r)$: Bridge function
$c (r)$ ⁠, $c_{i j} (r)$: Direct correlation function
e: Elementary charge
$E_{F}$: Fermi energy
$f_{i j}$: DISPERSION correction factor
$F_{xc}$: Exchange-correlation energy functional
$f (ϵ)$: Fermi–Dirac distribution
$F_{x} (η)$: Fermi–Dirac integral of order x
$\tilde{f} (k)$: Fourier transform of function $f (r)$
$g (ϵ)$: Density of states
$G (k)$: Local field correction
$g_{i j}$: SMT plasma parameter
$g (r)$ ⁠, $g_{i j} (r)$: Radial (pair) distribution functions
$h (r)$ ⁠, $h_{i j} (r)$: Total correlation function
$J$: Heat flux vector
k: Fourier space variable
$L$: Coulomb logarithm
L: Lorenz number
$ℓ$: Orbital (azimuthal) quantum number
$m_{i}$: Particle mass of species i
n: Principal quantum number
$n_{e}$: Electron number density
$n_{i}$: Ion number density of species i
$P_{μ ν}$: Stress tensor
$Q_{i}$: Charge of species i
$R_{c}$: Atomic core radius
T: System temperature (in energy units)
$T_{i}$: Temperature of species i
$T_{i j}$: Cross-temperature
$T_{m}$: Melt temperature
$t_{corr}$: Correlation time
$t_{prod}$: Production phase period
$t_{ω_{p}}$: Plasma period
$W$: WDM parameter
$Z_{i}$: Nuclear charge number of species i
${\bar{Z}}_{i}$: Mean ionization state (“Z bar”)
$β$: Inverse temperature
$Γ_{e e}$: Electron–electron coupling parameter
$γ_{0}$: Exchange-correlation coefficient
$Γ_{i i}$: Ion–ion coupling parameter
$Γ_{i}^{IS}$: Ion-sphere coupling parameter
$Γ_{K} (κ)$: Kirkwood curve in $(Γ, κ)$ space
$γ_{Lang}$: Langevin damping coefficient
$Γ_{m} (κ)$: Melt curve in $(Γ, κ)$ space
$ϵ$: Energy variable
$η$: Dimensionless chemical potential
$η_{i}$: Ionic viscosity
$Θ$: Degeneracy parameter
$κ$: Screening parameter
$κ_{i}$: Thermal conductivity of species i
$λ_{eff}$: Effective screening length
$Λ$: Mean free path
$Λ_{i j}$: Thermal de Broglie wavelength
$λ_{i}$: Screening length of species i
$μ$: Chemical potential
$μ_{i j}$: Reduced mass of species i and j
$ρ$: Mass density
$ρ_{ion}$: Charge density
$σ_{i}$: Electrical conductivity of species i
$σ_{mom}$: Momentum-transfer cross section
$τ$: Electron–ion collision time
$ψ_{α} (r)$: Kohn–Sham orbital
$ω_{p}$: Plasma frequency

APPENDIX: XC CONTRIBUTIONS TO SMT MODEL

As shown in Ref. 21, the leading-order exchange-correlation (XC) effects can be incorporated into the electron screening length through the modification

λ_{e}^{2} \to λ_{e}^{2} - γ_{0} .

(A1)

The parameter

γ_{0}

⁠, the effect of which is shown in Fig. 8, can be defined through the local field correction in the long-wave limit as

γ_{0} = lim_{k \to 0} \frac{1}{k^{2}} G (k),

(A2)

G (k) \equiv - \frac{k^{2}}{4 π e^{2}} \frac{δ^{2} F_{xc} [n_{e}]}{δ n_{e} (k^{'}) δ n_{e} (k^{″})},

(A3)

where

k = | k^{'} - k^{″} |

⁠, and

F_{xc} [n_{e}]

is the XC free energy functional associated with the electrons.

The XC free energy

F_{xc} [n_{e}]

is often approximated by decomposing the functional into independent exchange and correlation contributions,

F_{xc} [n_{e}] = \int d r f_{xc} (n_{e})

(A4)

\approx \int d r (f_{x} (n_{e}) + f_{c} (n_{e})),

(A5)

where

f_{x}

and

f_{c}

are the exchange and correlation components of the free energy density, respectively. Using the XC model of Yonei et al.,¹⁰³ the approximate free energy densities are given here in atomic units by

f_{x} (n_{e}, T) \approx - \frac{{(3 n_{e})}^{4 / 3}}{4 π^{1 / 3}} tanh (\frac{4}{9 Θ}),

(A6)

\begin{matrix} f_{c} (n_{e}, T) \approx - \frac{2}{3} {(π β n_{e}^{3})}^{1 / 2} tanh (\frac{c_{3} β^{c_{1}} n_{e}^{c_{2}}}{1 + β^{c_{4}}}) - c_{5} n_{e} log (1 + n_{e}), \end{matrix}

(A7)

where the coefficients are given by

c_{1} = - 1.51,

(A8)

c_{2} = - 0.840,

(A9)

c_{3} = 0.275,

(A10)

c_{4} = - 0.553,

(A11)

c_{5} = 0.0103 .

(A12)

We can thus express

γ_{0} = γ_{x} + γ_{c}

⁠, where the exchange component is

γ_{x} (n_{e}, T) = - \frac{1}{4 π} \frac{\partial f_{x}}{\partial n_{e}}

(A13)

\begin{matrix} = \frac{1}{2 π} {(\frac{3 n_{e}}{8 π})}^{1 / 3} tanh (\frac{4}{9 Θ}) + \frac{n_{e}}{12 T} {sech}^{2} (\frac{4}{9 Θ}), \end{matrix}

(A14)

and the correlation component is

γ_{c} (n_{e}, T) = - \frac{1}{4 π} \frac{\partial f_{c}}{\partial n_{e}}

(A15)

\begin{matrix} = \frac{1}{4} {(\frac{β n_{e}}{π})}^{1 / 2} [tanh (\frac{c_{3} β^{c_{1}} n_{e}^{c_{2}}}{1 + β^{c_{4}}}) \\ + \frac{2}{3} \frac{c_{2} c_{3} β^{c_{1}} n_{e}^{c_{2} + 1 / 2}}{1 + β^{c_{4}}} {sech}^{2} (\frac{c_{3} β^{c_{1}} n_{e}^{c_{2}}}{1 + β^{c_{4}}})] \\ - c_{5} (log (1 + n_{e}) + \frac{n_{e}}{1 + n_{e}}) . \end{matrix}

(A16)

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Comparison of transport models in dense plasmas

I. INTRODUCTION

II. WORKSHOP CASES

III. ELECTRONIC AND IONIC STRUCTURE

A. Electronic structure—The average-atom model

B. Ionic structure—Classical mapping methods

1. Quantum statistical potentials

2. Integral equation theory for ionic structure

3. Effective potentials from the radial distribution function

IV. ELECTRONIC AND IONIC TRANSPORT

A. Modified Lee–More (MLM) model

B. Stanton–Murillo transport (SMT)

C. Ziman transport model

D. Ionic transport properties from molecular dynamics

E. Integrated Yukawa viscosity model (IYVM)

F. Low-temperature data

V. RESULTS

A. Electrical conductivity

B. Thermal conductivity

C. Viscosity

VI. CONCLUSIONS AND FUTURE OUTLOOK

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

NOMENCLATURE

NOMENCLATURE

APPENDIX: XC CONTRIBUTIONS TO SMT MODEL

REFERENCES

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Comparison of transport models in dense plasmas Open Access

I. INTRODUCTION

II. WORKSHOP CASES

III. ELECTRONIC AND IONIC STRUCTURE

A. Electronic structure—The average-atom model

B. Ionic structure—Classical mapping methods

1. Quantum statistical potentials

2. Integral equation theory for ionic structure

3. Effective potentials from the radial distribution function

IV. ELECTRONIC AND IONIC TRANSPORT

A. Modified Lee–More (MLM) model

B. Stanton–Murillo transport (SMT)

C. Ziman transport model

D. Ionic transport properties from molecular dynamics

E. Integrated Yukawa viscosity model (IYVM)

F. Low-temperature data

V. RESULTS

A. Electrical conductivity

B. Thermal conductivity

C. Viscosity

VI. CONCLUSIONS AND FUTURE OUTLOOK

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

NOMENCLATURE

NOMENCLATURE

APPENDIX: XC CONTRIBUTIONS TO SMT MODEL

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Related Content

Resources

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Comparison of transport models in dense plasmas