Quantum molecular dynamics study on the proton exchange, ionic structures, and transport properties of warm dense hydrogen-deuterium mixtures

Comprehensive knowledge of physical properties such as equation of state (EOS), proton exchange, dynamic structures, diffusion coefficients, and viscosities of hydrogen-deuterium mixtures with densities from 0.1 to 5 $\mathrm{g}/{\mathrm{cm}}^3$ and temperatures from 1 to 50 kK has been presented via quantum molecular dynamics (QMD) simulations. The existing multi-shock experimental EOS provides an important benchmark to evaluate exchange-correlation functionals. The comparison of simulations with experiments indicates that a nonlocal van der Waals density functional (vdW-DF1) produces excellent results. Fraction analysis of molecules using a weighted integral over pair distribution functions was performed. A dissociation diagram together with a boundary where the proton exchange (${\mathrm{H}}_2+{\mathrm{D}}_2\rightleftharpoons 2\mathrm{HD}$) occurs was generated, which shows evidence that the HD molecules form as the ${\mathrm{H}}_2$ and ${\mathrm{D}}_2$ molecules are almost 50% dissociated. The mechanism of proton exchange can be interpreted as a process of dissociation followed by recombination. The ionic structures at extreme conditions were analyzed by the effective coordination number model. High-order cluster, circle, and chain structures can be founded in the strongly coupled warm dense regime. The present QMD diffusion coefficient and viscosity can be used to benchmark two analytical one-component plasma (OCP) models: the Coulomb and Yukawa OCP models.

Figure 1

Convergence tests on (a) the simulation cutoff energies, (b) particle numbers in the supercell for pressure and internal energy and (c) diffusion and viscosity, and (d) simulation steps at 1 $\mathrm{g}/{\mathrm{cm}}^3$.

Figure 2

The multi-shock EOS of Gu et al. [13] and our QMD results. The inset is an enlarged graph of the principal Hugoniot and the second-shock EOS.

Figure 3

The PDFs of H-H, D-D, and H-D at three densities and different temperatures. The insert in (h) is an enlarged picture of D-D PDFs.

Figure 4

Snapshots of the simulation box at (a) 4000 K and 0.5 $\mathrm{g}/{\mathrm{cm}}^3$ (b) 1500 K and 1 $\mathrm{g}/{\mathrm{cm}}^3$, visualized by the VESTA program [63] with depth cueing. Red balls represent hydrogen ions while green spheres are deuterium ions, and the electron density $n{}_e$ of isosurfaces (yellow) is set to be 0.18 $\mathrm{e}/\AA {}^3$.

Figure 5

The $\rho$–$T$ contour of dissociation fraction and the boundary line of DISREC.

Figure 6

The averaged values of ECN and effective bonding diameter $d_{\mathrm{av}}$ at three densities and different temperatures.

Figure 7

Fractions of different dynamic structures at 1 and 5 $\mathrm{g}/{\mathrm{cm}}^3$ and different temperatures. Single, Chain 2, and Chain 3 represent the structures with single-atom, two-atom, and three-atom, respectively. Circle represents the circle structure with every atom in the structure owns only two nearest atoms in a circle. Cluster represents a complex structure whose central atom has more than three nearest atoms.

Figure 8

The normalized VACFs of H and D at two representative densities and different temperatures.

Figure 9

VACFs and MCACFs of the H-D mixture at 1 $\mathrm{g}/{\mathrm{cm}}^3$ (left panel) and 5 $\mathrm{g}/{\mathrm{cm}}^3$ (right panel) and all at T = 10 kK. The present QMD data (dark cyan circles) are compared with the fitting results by a multi-time scale function (red solid line) and a single exponential fit (blue dashed line). D${}_{\mathrm{R}}$ represents the self-diffusion coefficients obtained by Eq. (1), while the values of D${}_V$ and D${}_{\exp }$ are extracted from the fitting of VACFs or MCACFs using Eq. (6) and the first term of Eq. (6), respectively. The VACFs and MCACFs are in atomic units.

Figure 10

Mutual diffusion coefficients at densities of 1 and 5 $\mathrm{g}/{\mathrm{cm}}^3$ and different temperatures.

Figure 11

Mutual-diffusion coefficients as a function of temperatures at densities of 1 $\mathrm{g}/{\mathrm{cm}}^3$ (top panel) and 5 $\mathrm{g}/{\mathrm{cm}}^3$ (bottom panel). Open red diamonds and open blue squares represent the results with PBE and vdW-DF1, respectively. Orange short-dashed and black solid curves are the OCP calculations of Hanson et al. [70] and Daligault [68], respectively. Green dash-dot-dotted and the red solid curves are the YOCP results based on the OCP models of Hanson et al. [70] and Daligault [68], respectively.

Figure 12

Viscosities as a function of temperatures at densities of (a) 1 $\mathrm{g}/{\mathrm{cm}}^3$ and (b) 5 $\mathrm{g}/{\mathrm{cm}}^3$. Open red diamonds and open blue squares represent the results with PBE and vdW-DF1, respectively. Black solid and green short-short-dashed curves are the OCP calculations of Bastea [75] and Daligault [68], respectively. Red solid, the orange dash-dot-dotted, and the olive dotted curves are the YOCP results based on the OCP models of Bastea [75], Daligault [68], and Wallenborn [78], respectively. The cyan short-dotted curve is the YVM calculation of Murillo.

Figure 13

The self-diffusion coefficients of H (a) and viscosities (b) in H-D mixtures, which are compared with those in pure hydrogen [33, 47] and H-He mixtures with different He mass concentrations (Y) [47, 79]. Colors indicate temperature (inset scale).