Reduced Ionic Diffusion by the Dynamic Electron-Ion Collisions in Warm Dense Hydrogen
Yunpeng Yao, Qiyu Zeng, Ke Chen, Dongdong Kang, Yong Hou, Qian Ma, Jiayu Dai
aa r X i v : . [ phy s i c s . c o m p - ph ] S e p Reduced Ionic Diffusion by the Dynamic Electron-Ion Collisions in WarmDense Hydrogen
Yunpeng Yao, Qiyu Zeng, Ke Chen, Dongdong Kang, Yong Hou, Qian Ma, and Jiayu Dai a) Department of Physics, National University of Defense Technology, Changsha, Hunan 410073,P. R. China (Dated: 10 September 2020)
The dynamic electron-ion collisions play an important role in determining the static and transport properties of warmdense matter (WDM). Electron force field (eFF) method is applied to study the ionic transport properties of warm densehydrogen. Compared with the results from quantum molecular dynamics and orbital-free molecular dynamics, the ionicdiffusions are largely reduced by involving the dynamic collisions of electrons and ions. This physics is verfied by thequantum Langevin molecular dynamics simulations, which includes electron-ion collisions induced friction into thedynamic equation of ions. Based on these new results, we proposed a model including the correction of collisionsinduced friction (CIF) of ionic diffusion. The CIF model has been verified to be valid at a wide range of density andtemperature. We also compare the results with the one component plasma (OCP), Yukawa OCP (YOCP) and EffectiveOCP (EOCP) models, showing the significant effect of non-adibatic dynamics.
I. INTRODUCTION
Warm dense matter (WDM) exists widely in the innercores of giant gas planets and the outer shells of brownand white dwarf stars . It is also an important stage inthe X-ray conversion zone and main fuel layer of implosioncompression during the inertial confinement fusion (ICF) process. Moreover, it is a transient state between coldcondensed matter and ideal gas plasma such that neithercondensed-matter nor ideal-plasma theory assumptions workwell. In the WDM regime, bound and free electrons, mul-tiple ions, atoms, molecules and clusters coexist, leadingstrong coupling, partial degeneracy and partial ionization allplay important roles in describing the structures and proper-ties of WDM. The Born-Oppenheimer (BO) approximation—decouples ions from electrons to the instantaneously adjustingpotential energy surface (PES) formed by fast electrons—hasachieved great success on both classical molecular dynamics(CMD) and first principle method such as density functiontheory (DFT) . It is possible to perform efficient simula-tions owing to BO approximation. However, at higher tem-perature and density such as in WDM regimes, the free elec-trons will collide with ions frequently, and the gap betweenPES of different eigenstates of electrons becomes narrow, sothat ions are possible to hop out of the current PES. Here, theBO approximation breaks down, and non-adiabatic dynamicelectron-ion collisions will exhibit significant effects on theequilibrium and the non-equilibrium processes . With theimprovement of diagnostic methods, especially the usage ofX-ray Thomson scattering techniques , electronic informa-tion of WDM can be obtained in the laboratory. To interpretthe experimental data, a more precise theory beyond BO ap-proximation is required on account of the complex environ-ment of WDM.The non-adiabatic effect has been considered by somemethods to get more accurate interactions between electrons a) Electronic mail: [email protected] and ions in WDM. Derived from time-dependent Kohn-Shamequation, time-dependent density function theory (TDDFT) gives the relatevly exact electronic structure information.Thanks to the coupling of the electrons and ions, TDDFT-Ehrenfest approach can give the results such as energy dis-sipation process, excitation energies and optical properties etc . However, TDDFT is extremely time-consuming,limited by finite time and size scale. Thus low frequencymodes can not be described well and the convergence ofscale is required to be verified carefully. Quantum Langevinmolecular dynamics (QLMD) holds a more efficient first prin-ciples computation efficiency, simultaneously regarding dy-namic electron-ion collisions as frictional forces in Langevindynamical equation of ions . Using the QLMD method, astronger ionic diffusive mode at low frequency has been foundwhen the selected friction parameter becomes larger, as wellas the decrease of the sound-speed . Nevertheless, the deter-mination of the friction parameter is a priori . Recently, Si-moni et al have provided ab-initio calculations of the frictiontensor in liquid metals and warm dense plasma . They ob-tain a non-diagonal friction tensor, reflecting the anisotropyof instantaneous dynamic electron-ion collisions. Electronforce field (EFF) expresses electrons as Gaussian wave pack-ets, so that it can include the non-adiabatic effect intrinsicallyin molecular dynamics simulation . Lately the method hasbeen applied to warm dense aluminum and found similar con-clusions that non-adiabatic effect enhances ion modes around ω = 0, however, the effect is not sensitive to the sound speed .Q. Ma et al have developed the EFF methodology to studywarm and hot dense hydrogen . They conclude that dy-namic electron-ion collisions reduce the electrical conductivi-ties and increase the electron-ion temperature relaxation timescompared with adiabatic and classical framing theories. Asanother approach, Bohmian trajectory formalism has been ap-plied by Larder et al recently. Constructing a thermally aver-aged, linearized Bohm potential, fast dynamical computationwith coupled electronic-ionic system is achieved . The resultalso reveals different phenomenon of dynamic structure fac-tor (DSF) and dispersion relation from DFT-MD simulation.All researches reflect that electron-ion collisions affect signif-icantly on the study of dynamic properties of WDM, for bothelectrons and ions. Nevertheless, the effect of non-adiabaticeffect on the ionic transport properties such as diffusion coef-ficient is few studied, in both numerical simulations and ana-lytical models.The transport properties such as self-diffusion, mutual-diffusion and viscosity of WDM and plasma systems areimportant for the study of astrophysics and ICF experi-ments. And they are pivotal input parameters to the radiation-hydrodynamics simulations . Diffusion leads to the sedimen-tation of heavy elements and the formation of white dwarfstars, as well as the complex phenomenon of giant planetslike phase separation . It also appears in the implosioncompression of ICF process because of the mixture of fuel .Transport properties can also affect the fluid instabilities likeRayleigh-Taylor instability (RTI) and Richtmyer-Meshkovinstability (RMI) which are crucial in ICF implosions. Toemphasize the non-adiabatic effect, the present work will fo-cus on the self-diffusion of the simplest element: hydrogen.Hydrogen and its isotopes are important in astrophyscs andICF, and it is a perfect system to compare the physics in dif-ferent models. For ideal gas plasma, kinetic theory has a gooddescription based on binary-collision approximation . Thediffusion coefficient can be obtained from Chapman-Enskogformulas deduced from Boltzmann equation. In parallel,Hansen et al have run massive molecular-dynamics simula-tions based on the classical one-component plasma (OCP)model, and given a series of transport properties over a widerange of thermodynamic states . Daligaut has systematicallystudied the strong collective behavior of OCP and given betterfitting formulas of the diffusion and viscosity coefficient . Itshould be noticed that all results should be examined in WDMbecause of the appearance of many-body interactions, electrondegeneracy and electron screening. There have been manyefforts on theories and simulations to valid the above mod-els in WDM. For example, effective potential theory (EPT) and effective screening potential have been developed to getmore accurate collision integrals in a more strongly coupledregion. Adding the screen parameter in OCP, Yukawa one-component plasma (YOCP) model is developed and the largerself-diffusion coefficients are found because of shieldingeffects . Deducing effective coupling parameter and effec-tive charge, effective one-component plasma (EOCP) model isconstructed , and it has been successfully applied in warmand hot dense plasma comparing with Kohn-Sham densityfunction molecular dynamics (KSDFT-MD) and orbital-freemolecular dynamics (OFMD) simulations . Binary ionicmixtures (BIMs) and multi-ionic mixtures are also commonin nature and laboratories. The researches on mixing processcan often be found in papers such as in Refs. 46–49. However,the influence of non-adiabatic electron-ion collisions on ionicdiffusion is still required to be checked, and we could imagethe existence of dynamic electron-ion collisions will inducenew effects such as dissipation or friction. In particular, forthe analytical models based on traditional BO methods, weshould study the non-adiabatic dynamic collisions effect onthe self-diffusion in warm dense matter, and propose a newmodel including collisions induced friction (CIF). The paper is organized as follows. Firstly, details of the-oretical methods and the computation of diffusion coefficientare introduced in Section II. Then, in Section III, the staticand transport results of QMD, OFMD, QLMD, and (C)EFFsimulations are showed and the the dynamic collisions effectis discussed. In section IV, we systematically study the col-ision frequency effect on ionic diffusions and the CIF modelis introduced to estimate the impact of electron-ion collisions.In section V, the results are compared with the OCP, YOCP,and EOCP models. Finally, the conclusions are given in sec-tion VI. All units are in atom unit if not emphasized. II. THEORETICAL METHODS AND COMPUTATIONALDETAILSA. (Constrained) electron force field methodology
EFF method is supposed to be originated from wave packetmolecular dynamics (WPMD) and floating spherical Gaus-sian orbital (FSGO) method . Considering each electronicwave function as a Gaussian wave packet, the excitation ofelectrons can be included with the evolution of positions andwavepacket radius. N-electrons wave functions are taken as aHartree product of single-electron Gaussian packet written as Ψ ( r ) = (cid:18) s π (cid:19) / exp (cid:18) − (cid:18) s − p s is ¯ h (cid:19) ( r − x ) (cid:19) · exp (cid:18) i ¯ h p x · r (cid:19) . (1)Where s and x are the radius and average positions of theelectron wave packet, respectively. p s and p x correspond tothe conjugate radial and translational momenta. Nuclei in EFFare treated as classical charged particles moving in the meanfield formed by electrons and other ions.Substituting simplified electronic wave function in thetime-dependent Schrödinger equation with a harmonic poten-tial, equation of motion for the wave packet can be derived ˙x = p x / m e , (2a) ˙p x = − ∇ x V , (2b)˙ s = ( / d ) p s / m e , (2c)˙ p s = − ∂ V / ∂ s . (2d)Where d is the dimensionality of wave packets. For a three-dimensional system, d is equal to 3, and it becomes 2 in 2Dsystems. V is the effective potential. Combining with ionicequations of motion, EFF MD simulations have been imple-mented in LAMMPS package .In addition to electrostatic interactions and electron kineticenergy, spin-dependent Pauli repulsion potential is added inthe Hamiltonian as the anti-symmetry compensation of elec-tronic wave functions. In EFF methodology, the exchange ef-fect is dominated by kinetic energy. All interaction potentialsare expressed respectively as E nuc − nuc = ∑ i < j Z i Z j / R i j , (3a) E nuc − elec = ∑ i < j − ( Z i / R i j ) erf (cid:16) √ R i j / s j (cid:17) , (3b) E elec − elec = ∑ i < j ( / r i j ) erf (cid:16) √ r i j / q s i + s j (cid:17) , (3c) E ke = ∑ i ( / ) (cid:0) / s i (cid:1) , (3d) E Pauli = ∑ σ i = σ i E ( ↑↑ ) i j + ∑ σ i = σ i E ( ↑↓ ) i j . (3e)Where Z is the charge of nucleus, r i j and R i j correspondto the relative positions of two particles (nuclei or elec-trons). erf ( x ) is error function, σ means the spin of elec-trons. Pauli potential is consists of same and opposite spinelectrons repulsive potentials. More details can be found inRefs. 21, 22, 52, and 53.However, EFF model also suffers from the limitation ofWPMD. The wave packets spread at high temperature . Toavoid excessive spreading of wave packets, the harmonic con-straints are often added. Recently, Constrained EFF (CEFF)method has been proposed using L = λ D + b as the boundaryof the wave packets , getting much lower electron-ion energyexchange rate agreeing with experimental data . B. Quantum molecular dynamics and orbital-free moleculardynamics
In QMD simulations, electrons are treated quantum me-chanically through the finite temperature DFT (FT-DFT).While ions evolve classically along the PES determined bythe electric density. Each electronic wave function is solvedby the Kohn-Sham equation (cid:18) − ∇ + V KS [ n e ( r )] (cid:19) ϕ i ( r ) = E i ϕ i ( r ) . (4)Where E i is the eigenenergy, − / ∇ is the kinetic energycontribution, and the Kohn-Sham potential V KS [ n e ( r )] is givenby V KS [ n e ( r )] = υ ( r ) + Z n e ( r ′ ) | r − r ′ | d r ′ + V xc [ n e ( r )] . (5)Where υ ( r ) is the external potential, the second term in theright hand of the above equation is the Hartree contribution, and V xc [ n e ( r )] respresents the exchange-correlation potential.The electronic density consists of single electronic wave func-tion n e ( r ) = ∑ i | ϕ i ( r ) | . (6)At high temperatures, the requirement of too many bandslimits the efficiency of QMD method. OFMD is a good choicewhen dealing with high temperature conditions. Withinorbital-free frame, the electronic free energy is expressed as F e [ R , n e ] = β Z d r n n e ( r ) Φ [ n e ( r )] − √ π β / I { Φ [ n e ( r )] } o + Z d r V ext ( r ) + ZZ d r d r ′ n e ( r ) n e ( r ′ ) | r − r ′ | + F xc [ n e ( r )] . (7)Where R is the ionic position, β = / k B T where T isthe temperature and k B is the Boltzmann constant. I ν is theFermi integral of order ν . V ext ( r ) represents the external orthe electron-ion interaction, and F xc [ n e ( r )] is the exchange-correlation potential. The electrostatic screening potential isrepresented by Φ [ n e ( r )] depend on electronic density n e ( r ) only ∇ Φ [ n e ( r )] = π n e ( r ) = √ π β / I { Φ [ n e ( r )] } . (8) C. Quantum Langevin molecular dynamics
QMD and OFMD are good tools in describing static prop-erties of warm dense matters. However, the information ofelectron-ion dynamical collisions is lost because of the as-sumption of BO approximation. In addition, electron-ion col-lisions are important for WDM in which electrons are excitedbecause of the increasing temperature and density. To de-scribe the dynamic process, QMD has been extended by con-sidering the electron-ion collision induced friction (EI-CIF) inLangevin equation, and corresponding to the QLMD model.In QLMD, ionic trajectory is performed using the Langevinequation . M I ¨ R I = F − γ M I ˙ R I + N I . (9)Where M I and R I is the mass and position of the ion re-spectively, F is the force calculated from DFT simulation, γ means the friction coefficient, and N I represents a Gaussianrandom noise. In QLMD, the force produced by real dynamicsof electron-ion collisions can be replaced by the friction on ac-count of less time scale of electronic motions comparing withthat of ions. The friction coefficient γ is the key parametershould be determined a-priori . Generally, at high temperaturesuch as WDM and HDM regimes, the EI-CIF dominates thefriction coeffeicient, and can be estimated from the Rayleighmodel γ = π m e M I Z ∗ ( π n i ) / r k B Tm e . (10)Where m e is the electronic mass, Z ∗ is the average ioniza-tion degree, and n i means the ionic number density. There isanother way to assess γ based on the Skupsky model , andin this work, we adopted Rayleigh model only considering thehydrogen we studied has high density and high temperature.To make sure that the particle velocity satisfies the Boltz-mann distribution, the Gaussian random noise N I should obeythe fluctuation-dissipation theorem h N I ( ) N I ( t ) i = γ M I k B T dt . (11)Where dt is the time step in the MD simulation. The anglebracket denotes the ensemble average. D. Self-diffusion coefficient
In MD simulations, the self-diffusion coefficient is oftencalculated from the velocity autocorrelation function (VACF)using Green-Kubo formula D = lim t → ∞ D ( t ) , (12a) D ( t ) = Z t dt h v i ( t ) · v i ( ) i . (12b)In which v i ( t ) is the center of mass velocity of the i th par-ticle at time t , and the angle bracket represents the ensembleaverage. Generally, the integral is computed in long enoughMD trajectories so that the VACF becomes nearly zero andhas less contribution to the integral. All the same species ofparticles are considered in the average to get faster convergentstatistical results.In practical, it is impossible to get a strict convergent resultbecause the infinite simulation is forbidden. Thus, we usuallyuse a exponential function h v ( t ) · v ( ) i = a exp ( − t / τ ) to fitthe VACF to get the self-diffusion coefficient D = a · τ . Where a and τ are fitting parameters determined by a least-squaresfit. τ is corresponds to the decay time. In moderate and strongcoupling regimes, a more sophisticated fitting expression isneed to be considered . In the exponential function fitting,the statistical error can be estimated by ε = s τ NT traj (13)Where N is the number of particles, T traj is the total time inthe MD simulation. III. RESULTS AND DISCUSSIONA. Computational details
We have performed (C)EFF, QMD, and OFMD simulationsto study the static and transport properties of warm dense hy-drogen at the density of ρ = and ρ = . The temperatures are from 50kK to 300kK. The ionic coupling pa-rameters Γ = Z i e / ( r i k B T ) is larger than 1 in all regimes westudied, here r i = ( / ( π n i )) / is the Wigner-Seitz radius.The electronic degeneracy parameter Θ = T / T F describes thequantum effect of electrons, here T F = (cid:0) π n e (cid:1) / / Θ is always less than 1, revealing that the quantumeffects play important roles in these states.QMD simulations have been performed using Quantum-Espresso (QE) open-source software , the electron-ion in-teraction is described as plane wave pseudopotential. Theelectron-electron exchange-correlation potential is repre-sented by the Perdew-Burke-Ernzerhof (PBE) functional (PBE) in the generalized-gradient approximation (GGA). Thevelocity Verlet algorithm is used to update positions and ve-locities of ions. In our simulations, we used supercells con-taining 256 H atoms. The time step is set from 0 . fs to 0 . fs at different temperature to ensure convergence of energy. Thecutoff energy is tested and set from 100 Ry to 150 Ry. Thenumber of bands is sufficient for the occupation of electrons.Only the Γ point ( k = ) is sampled in the Brillouin-zone.Each density and temperature point is performed for at least4000-10000 time steps in the canonical ensemble, and the en-semble information is picked up after the system reaches equi-librium. The OFMD simulations are performed with our lo-cally modified version of PROFESS . The PBE functional is also used to treat the exchange-correlation potential. 256atoms are also used in the supercell. The kinetic energy cut-off is 7000 eV when the density ρ = , and 10000 eV at ρ = . The time step is set from 0 . fs to 0 . fs withthe temperature increasing. The size effect has been tested inall MD simulations.In the EFF simulations, the real electron mass is used sothat we choose the time step as small as 0 . as . 1000 ions and1000 electrons are used in the simulation. 5 ps microcanonicalensemble with a fixed energy, volume, and number of par-ticles (NVE) has been performed to calculate statistical av-erage after 10 ps simulations with fixed temperature, volume,and number of particles (NVT). When temperature and den-sity become higher, CEFF is applied to avoid packets spread-ing. B. Static and transport properties
We firstly calculate the radial distribution function (RDF) g ( r ) of H-H, as shown in Fig. 1. It is shown that the RDFsfrom OFMD calculations agree well with RDFs of QMD re-sults. Moreover, the RDFs calculated from (C)EFF reflectsimilar microscopic characteristics with QMD and OFMD re-sults, especially when the temperature is relatively low, wherethe electron-ion collisions are not so important. For thesecases, it is appropriate to show the intrinsic different physicsbetween static and transport properties if the RDFs shown arevery close to each other. It should be noticed that the RDFsof (C)EFF model shows a little more gradual than QMD’s andOFMD’s with the increase of temperature. It is deduced thatthe non-adiabatic effect play little role in the static structures T=300kKT=50kKT=100kKT=150kKT=200kK g H - H (r) QMD OFMD (C)EFF a T=50kKT=100kKT=200kKT=150kK g H - H (r) r(a.u.) T=300kK b FIG. 1. The RDFs of H-H at 5g/cm (a) and 10g/cm (b). The or-dinate is differentiated by adding factors at different temperatures.Blue double dots lines represent the results from (C)EFF simulation.Black solid and red dashed lines are the QMD and OFMD results,respectively. of warm dense hydrogen shown here, which is similar to theeffects of Langevin dynamics on the static structures, inwhich the choice of friction coefficients has little effect on theRDFs.However, non-adiabatic effects on dynamic properties aresignificant . We calculated the self-diffusion coeffi-cients for warm dense hydrogen by integrating the VACF. Toget a convergent value, a simple exponential function men-tioned in Section II is applied. The self-diffusion coefficientvaries with temperature at 5g/cm and 10g/cm are shown inFig. 2 using different methods of (C)EFF, QMD, and OFMD.It is very interesting that three methods give consistant re-sults when temperature is relatively low. And the OFMDand QMD results have close values even with the increase oftemperature. However, the (C)EFF simulations have a dis-tinct reduce on the self-diffusion coefficients comparing withQMD and OFMD results. And the difference becomes moreobvious at higher temperature. We boil it down to the non-adiabatic electron-ion dynamic collisions, which is lost in theframework of BO approximation such as QMD and OFMD.Regarding the electron as a Gaussian wave packet, (C)EFFmethodology implements the electron-ion dynamics simula-tions, in which the dynamic coupling and collisions can be
50 100 150 200 250 300020406080100 a S e l f - d i ff u s i on ( - c m / s ) QMD OFMD (C)EFF QLMD
50 100 150 200 250 3000102030405060 b S e l f - d i ff u s i on ( - c m / s ) QMD OFMD (C)EFF QLMDTemperature(kK)
FIG. 2. The self-diffusion coefficients of H as a function of temper-ature at 5g/cm (a) and 10g/cm (b), calculated by QMD, OFMD,QLMD, and (C)EFF mthods. Black squares represent the QMD re-sults, red circles are the results of OFMD’s, and the (C)EFF resultsare represented by blue triangles. The QLMD results are repre-sented by purple diamonds naturally included. As shown in Fig. 2, with the temperatureincrease, more electrons are excited or ionized and becomefree electrons. These free electrons lead continual and non-negligible electron-ion collisions, supplying drag forces forthe motion of ions, and giving rise to much lower diffusioncoefficients. The collision rate increases with the tempera-ture, showing lower diffusive properties for ions, significantlyaffects the transport properties of WDM.The lost of dynamic collisions can be introduced into theQMD model by considering electron-ion collision inducedfriction in Langevin equation. Here, we use the Raylieghmodel to estimate the friction coefficient γ , and the QLMDsimulations have been performed. It is very exciting that theQLMD results, showed in Fig. 2, agree well with (C)EFFsimulations. The greatest difference between the two mod-els is 12%, but mostly within 6%. This suggests that the re-duction in ionic diffusion from (C)EFF simulations does in-deed come from electron-ion dynamic collisions. We believethe small difference belongs to the choice of friction coeffi-cient γ . Since the prior parameter should be determined arti-ficially in QLMD simulations, we are encouraged to do quan-titative analysis about the electron-ion collisions effect usingthe (C)EFF results as benchmark for the results of all adiabaticmethods and analytical models. IV. ELECTRON-ION COLLISIONS EFFECT ASSESSMENT
As shown above, we should figure out the mechanism howdoes the dynamic collisions work on the ionic transport? Wecan find a clue from the Landau-Spitzer (LS) electron-ion re-laxation rate ( ν ei ) ν ei = √ π n i Z e m e m i (cid:18) k B T e m e + k B T i m i (cid:19) − / ln Λ (14)Where m e ( m i ) , n e ( n i ) and T e ( T i ) are the mass, num-ber density and temperature of electrons(ions), respectively.The Coulomb logarithm ln Λ can be calculated by the GMSmodel . In Eq. 14, it is obvious that with the increase of den-sity and temperature (the Coulomb logarithm also varies withthe density and temperature), the collision frequency becomeshigher, leading the diffusion coefficients reduce more signifi-cantly. The results of QMD and (C)EFF simulations showedin Fig. 2 exhibit the same bahaviors.As shown in Eq. 14, the electron-ion relaxation rate ( ν ei )is the function of temperature and density. However, thethermodynamic state also changes with temperature and den-sity, therefore it is diffucult to distinguish the electron-ioncollisions effect. For this purpose, we can change the ef-fective mass of electrons in the (C)EFF simulation withoutaltering the instrinsic interactions in the Hamiltonian of thesystem . Since the mass of ions is much greater than thatof electrons, we can find a simple relationship between theelectron-ion collision frequency ν ei and the mass of the elec-tron m e from Eq. 14 ν ei = f ( ρ , T ) m / e (15)When the dynamic electron mass is larger, the motion ofeffective electrons exhibit more classical, and the collisionsbetween electrons and ions become stronger. By this way, wecan study the influence of electron-ion collisions by adjustingelectronic mass in (C)EFF simulations. The VACFs and self-diffusion coefficients of H at different dynamic electron massare showed in Fig. 3.From the VACF results we can see, The change of dynamicelectron mass does not alter the thermodynamic states of ions.While, dynamic collisions reduce the correlation of particles,showing lower decay time with the increase of dynamic elec-tron mass, as well as the electron-ion collision frequency. Dif-fusions reflect similar trends, and more interestingly, the dif-fusion of ions is inversely proportional to the log of electronicmass as showed in Fig. 3(b). The inverse ratio relation re-flects the reduction of the diffusion due to electron-ion col-lisions, and the slope determines the magnitude of this influ-ence. In Fig. 3(b), it is shown that the influence of electron-ioncollisions becomes stronger with the increase of temperature, VA C F ( a . u . ) time(a.u.) ·10 -3 T=50kKT=100kK VA C F ( a . u . ) time(a.u.) m e =100 m e =500 m e =1823 ·10 -3 T=200kK a
100 500 1000 200005101520 ·10 -3 b D=-6.10·10 -4 log m e +0.0055D=-6.30·10 -4 log m e +0.010 T=200kK T=100kK T=50kK fitting S e l f - d i ff u s i on ( a . u . ) m e (a.u.) D=-8.72·10 -4 log m e +0.022 FIG. 3. (a) VACFs of H for different dynamic electron mass at200kK, 100kK, and 50kK from top to bottom. The denisty is10g/cm . The black lines, red lines, and blue lines represent thedynamic electron mass of 100a.u., 500a.u., and 1823a.u., respec-tively. Deatils are showed in the insets. (b) The corresponding ionicself-diffusion coefficients as a function of dynamic electron mass.The squares, circles, and triangles represent the temperature at 5kK,10kK, and 20kK, respectively. The lines are the fitting results, andthe fitting functions are listed below the lines. since the electrons are more classical at higher temperature.To quantitatively describe the relationship between diffusionand collision frequency, we performed more intensive simula-tions on dynamic electron mass as showed in Fig. 4. Here, theQMD results are used as the value at reference point, corre-sponding to no dynamic electron-ion collisions, since m e cannot be zero.As showed in Fig. 4, the change of diffusion coeffi-cients decrease much steeper when the electron dynamicmass becomes smaller, revealing more significant effect ofelectron-ion collisions. Another decaying function as D = a log ( + bm ce ) + d can well describe this relation of diffusionvarying with the dynamic electron mass m e . This functioncan transit to the linear form when m e is large. Here, wehave found the approximate relationship between ionic dif-fusion coefficient D and electron-ion collision rate ν ei takingEq. 15 into the fitting function S e l f - d i ff u s i on ( a . u . ) m e +1(a.u.) raw data fitting ·10 -3 FIG. 4. Dynamic electron mass effects on ionic diffusion. We showthe (C)EFF simulation results with different dynamic electron massat 5g/cm and the temperature is 5kK. The mass of electrons has beenshifted to avoid infinity definition of log function at zero point. Thevalue at zero point is replaced by the QMD result. The fitting resultis respresent by the red line. D = f ( ρ , T ) log (cid:16) + f ( ρ , T ) ν f ( ρ , T ) ei (cid:17) + f ( ρ , T ) (16)Where f ( ρ , T ) , f ( ρ , T ) , f ( ρ , T ) , f ( ρ , T ) are the functionof the density ρ and temperature T . If ν ei is set to zero,the first term in the right hand of Eq. 16 vanishes, and D = f ( ρ , T ) = D . Here, the remaining term D represents thediffusion without electron-ion collisions. We call the first termas collisions induced friction (CIF) of the ionic diffusion D CIF .Within this consideration, the total diffusion coefficient can beobtained via D = f ( ρ , T ) log (cid:16) + f ( ρ , T ) ν f ( ρ , T ) ei (cid:17) + D = D CIF + D (17)For D , plenty of models have been developed to study onit, such as QMD and OFMD which are based on BO approxi-mation. In this paper, the diffusion coefficient including non-adiabatic effect has been calculated using (C)EFF method. Asthe collision frequency is a small term, the equation can besimplified as D CIF = D − D = f ( ρ , T ) ν f ′ ( ρ , T ) ei . D and D canbe obtained from (C)EFF and QMD simulations, respectively.We develop an empirical fitting function from the availabledata as the assessment of electron-ion collisions induced thedecrease of ionic diffusions D CIF = ν . ei a ρ / T / + b ρ + c / T / + d (18)where the fitting coefficient a = − . × − , b = . × − , c = . d = − .
50 100 150 200 250 300020406080100 a S e l f - d i ff u s i on ( - c m / s ) QMD (C)EFF QMD+CIF
50 100 150 200 250 3000102030405060 b S e l f - d i ff u s i on ( - c m / s ) QMD (C)EFF QMD+CIFTemperature(kK)
FIG. 5. Self-diffusion coefficients of H calculated by different meth-ods at 5g/cm (a) and 10g/cm (b). The solid black squares and redtriangles represent the QMD and (C)EFF results, respectively. TheCIF model is used to correct the QMD results as the consideration ofnon-adiabatic effect. The results are represented by the blue circles. by the CIF model, which are shown in Fig. 5, agree well with(C)EFF simulations. To verify the accuracy of the fitting func-tion, we calculate self-diffusion of H and He at some othertemperatures and densities. The results are listed in Table I.In Table I, the ionic self-diffusion coefficients obtainedfrom the QMD model can be corrected by adding the CIF fac-tor as the compensation of electron-ion collisions. The resultsare in good agreement with the (C)EFF and QLMD results,showing that our CIF correction can be applied to warm densematter. It needs to be emphasized that the CIF correction isindependent of other models, therefore, any model based onthe adiabatic framework can use it to offset the lost of theelectron-ion collisions. V. COMPARISION WITH ANALYTICAL MODELS
The expensive computational costs of first principles sim-ulations make it difficult to apply online or generate largeamount of data. On the contrary, some analytical modelsbased on numerical simulations have been proposed, supply-ing promising approaches to the establishment of database.
TABLE I. The self-diffusion coefficients calculated by QMD, QLMD and (C)EFF models. The QMD results corrected by the CIF model isalso listed in the table.species density ( g/cm ) temperature(K) D QMD ( cm / s ) D QLMD ( cm / s ) D (C)EFF ( cm / s ) D QMD + CIF ( cm / s ) H 8 100000 0.0155 0.0133 0.0122 0.0123H 8 200000 0.0386 0.029 0.0264 0.0284H 15 200000 0.0237 0.0181 0.0182 0.0183H 15 300000 0.0396 0.0289 0.03 0.0278He 10 100000 0.00757 0.0066 0.00598 0.00596He 10 200000 0.0181 0.016 0.0108 0.0128
The OCP is a widely used model to describe a single speciesof ion surrounded by a uniform, neutralizing background ofelectrons. All properties of OCP model are dependent on thecoupling parameter Γ . However, the validity of OCP model indescribing WDM should be examined as shown in the previ-ous work . Daligault has given a rational function to calcu-late the reduced self-diffusion and viscosity. The functionis fitted based on a mass of MD simulations on condensedmatter D ∗ = D / D = ∑ i = a i Γ i / ∑ i = b i Γ i (19)where D ∗ is the reduced self-diffusion and D = ω p a . ω p = (cid:0) π n i Z e / m i (cid:1) / is the plasma frequency and r i is ionicWigner-Seitz radius. a i and b i are fitting parameters as givenin Table II.In this work, we use the form proposed by Arnault to fixthe discontinuity of Daligault’s parametrizations at the thresh-old of Γ = D ∗ OCP = D ∗ D , if Γ < . , . Γ − . , if 1 . ≤ Γ < . , D ∗ D . if Γ ≥ . . (20)Here, the Yukawa potential has been applied replacing thebare Coulomb interactions . In this way, the screeningeffect is included in YOCP model as a development of OCPmodel. In the YOCP framework, the inverse screening length κ also affects the structures and properties of matters. Usu-ally, κ is defined as the reciprocal of the Thomas-Fermi (TF)screening length λ TF = (cid:16) π Z (cid:17) / √ r i (21)where Z is the ionic charge, and r i is the Wigner-Seitz ra-dius, respectively. The self-diffusion can be obtained usingYukawa potential u ( r ) = q e − κ r / r on MD simulations. Dali-gault has applied it in a wide range of κ and over the entirefluid region . In the gas-like small coupling regime, the re-duced self-diffusion coefficients model can be extended fromthe Chapman-Spitzer results as D ∗ ( κ , Γ ) = r π α ( κ ) Γ / ln Λ ( κ , Γ ) (22) The generalized Coulomb logarithm ln Λ ( κ , Γ ) is expressedasln Λ ( κ , Γ ) = ln (cid:18) + B ( κ ) λ D b c (cid:19) = ln (cid:18) + B ( κ ) √ Γ / (cid:19) (23)where λ D is the Debye length λ D = p π q n / k B T and b c is the classical distance of closest approach b c = Zq / k B T . α ( κ ) and B ( κ ) are fitting parameters dependent on κ only α ( κ ) = r π a + a κ a (24) B ( κ ) = b + b erf (cid:16) b κ b (cid:17) (25)All coefficients are listed in Table II. The self-diffusioncoefficent is obtained according to D = D ∗ ω a , and ω = (cid:0) π n i Z ∗ e / m i (cid:1) / , where Z ∗ is the average ionization de-gree. Z ∗ can be estimated using average atom (AA) model ,in which the energy level broadening effect is considered. Thecomparision between the results of different models are shownin Fig. 6. The QMD results and the CIF correction of QMD’sare also showed as the benchmarks.It is shown in Fig. 6 that the self-diffusion coefficients fromYCOP model are much larger than those of OCP model. It isbecause that when we consider the screening of electrons, therepulsive interactions between ions become weaker, so thations exhibit more free and diffusive. For warm dense hydro-gen at the density of 5g/cm , YOCP model can excellentlyreproduce results from the QMD simulations. However, athigher density, the YOCP model overestimates the diffusioncompared with QMD results. The reason for the phenomenonis that the Yukawa model underestimates the ionic interac-tions at short distances . When the density and tempera-ture are higher, the electronic charges around the ions over-lap, and the Pauli principle makes electrons more repulsive.A short-range repulsion (SRR) should be added to Yukawamodel as a correction. In the YOCP model, we attributethe lost of SRR to the over-shielding of Thomas-Fermi (TF)screening. An empirical parameter can be introduced to cor-rect the TF screening length. For 10g/cm hydrogen, we use κ ∗ = / λ ∗ TF = / ( . λ TF ) instead of the TF inverse screen-ing length and the results from the modified YOCP (MYOCP)model are in good agreement with the QMD results. Simi-lar situations are shown in Ref.78 and 79. There is anotherscheme to deal with larger ionic coupling systems, in whichwe can reduce effective volumes of particles so that the col-lision frequency can be increased and the ionic transportation TABLE II. Coeffieients from OCP and YOCP models as in Ref. 36 and 73. a a a a b b b b OCP ( Γ ≤ ) -0.732979 3.89667 0.489336 -0.271605 0.0426862 -0.501987 1.34111 0.741386OCP ( Γ > ) . × − − . × − -32.1103 56.2554 1.24087 0.0371926YOCP 1.55973 1.10941 1.36909 2.20689 1.351594 1.57138 3.34187
50 100 150 200 250 300020406080100
OCP YOCP MYOCP EOCP S e l f - d i ff u s i on ( - c m / s ) a
50 100 150 200 250 300010203040506070 S e l f - d i ff u s i on ( - c m / s ) Temperature(kK)
QMD QMD+CIF b FIG. 6. Comparison of QMD and modified QMD simulations withdifferent analytical models for self-diffusion coefficients of warmdense H at 5g/cm (a) and 10g/cm (b). The black solid squares andcircles represent the results calculated by QMD and QMD with CIFcorrection, respectively. The red and blue solid lines are the resultsof the OCP model and YOCP model . The modified YOCP(MYOCP) model, corrected by the effective screening parameter, isrepersented by the double dots line. The purple lines repersent theresults from the EOCP models . The effective coupling parametersof the EOCP model are obtained from the RDFs of QMD. can be draged or disspative. The model has successfully im-proved the transport properties of strongly coupled plasmas inthe range 1 ≤ Γ ≤ .However, all the interactions in YOCP models are staticthat dynamic electron-ion collisions are already lost. Thisis why the results obtained from modifed QMD are muchsmaller than the YOCP results, as well as the QMD’s. On thecontrary, the self-diffusion coefficients for OCP model lookscloser to the modified QMD results. We conclude that the co-incidence is the result of the competition between electronic screening and electron-ion collisions. In WDM regimes, theinfluence of dynamic electron-ion collisions matches screen-ing effects. The ions feel the friction caused by the electrons,leading changes in transport properties. With the increase ofdensity and temperature, the collision becomes more violentand this non-adiabatic effect will more dominate the trans-port of ions. This leads to the offset of screening by dynamicelectron-ion collisions, and the agreement between OCP andmodified QMD predictions. But we should also notice thatin OCP model, it is difficult to divide the dynamic collisionsfrom the screening effect. For example, for the case of Heat 10g/cm and 100000K, OCP gives lower diffusion coeffi-cients about 20% comparing with CEFF calculation becauseof stronger electronic screening comparing with electron-ioncollisions, and the model fails.It is shown in Fig. 6 that, the EOCP model has a better de-scription in all density and temperature range we studied withthe QMD simulations. In EOCP model, The effective cou-pling parameter Γ e and ionization Q e are introduced as thecorrection of the OCP model to reproduced the static struc-tures of the OFMD’s , the model also works well on trans-ports properties such as diffusion and viscosity . In this pa-per, we set Γ e by the procedure developed by Ott et al asis Γ e = .
238 exp (cid:16) . r / (cid:17) − . , (cid:0) r / < . (cid:1) (26)where r / is obtained from the RDFs g ( r ) at g ( r ) = .
5, Thedistance is expressed in the Wigner-Seitz radius unit. The ef-fective average charge Q e is defined as Q e = √ Γ e ak B T / e . Weuse the RDFs of QMD’s as the input of EOCP model, theresults agree well with those extracted from long time MDsimulations, especially when tempeature is low. However, thepredictions overestimate ionic diffusions compared with mod-ified QMD results. This is because of the insensitivity of staticproperties as showed in Fig 1. The good reproduction of QMDsimulations for the EOCP model reflects the importance ofacquirng precise effective interactions. While, the differencebetween the modified QMD results and EOCP predictions re-minds us again to pay attention to the instantaneous dynamiccollisions which is lost in those static potentials. VI. CONCLUSION
We have performed QMD, OFMD, and (C)EFF simula-tions to determine the RDFs and the ionic self-diffusion co-efficients of warm dense hydrogen at the densities of 5g/cm and 10g/cm and temperatures between 50kK to 300kK. Theresults from (C)EFF-MD method are carefully compared with0the results from QMD/OFMD methods based on the BO ap-proximation. In EFF method, the static properties are in-sensitive to electron-ion collisions, however, the diffusion ofions decreases significantly with the increase of electron-ioncollisions. The ionic diffusion coefficients calculated from(C)EFF agree well with the QLMD results, but largely dif-fer from QMD or OFMD simulations, revealing key role ofelectron-ion collisions in warm dense hydrogen. Most im-portantly, we proposed a new analytical model which intro-duce the electron-ion collisions induced friction (CIF) effects,constructing a formula to calculate self-diffusion coefficientswithout doing non-adiabatic simulations. The CIF model hasbeen verified to be valid over a wider range of temperature,density and materials. However, since the CIF model is de-rived from the fitting of simulation results, whether it can beapplied for more complex elements should be verified fur-ther. We also show the results from analytical models of OCP,YOCP and EOCP. The YOCP model shows good agreementwith QMD results at the density of 5g/cm , while overesti-mates electronic screening at higher density. Dynamic colli-sions suppress the motion of ions, partly offsetting the elec-tronic screening, make the prediction of OCP model close tothe modified QMD results. However, the effect of screen-ing and dynamic collisions can not be distinguished in OCPmodel. Based on the static information, EOCP model repro-duces QMD simulations better, but worse for modified QMDresults, suggesting that non-adiabatic dynamic collisions af-fect significantly on the tranport properties of WDM. VII. ACKNOWLEDGMENTS
The authors thank Dr. Zhiguo Li for his helpful discussion.This work was supported by the Science Challenge Project un-der Grant No. TZ2016001, the National Key R&D Program ofChina under Grant No. 2017YFA0403200, the National Natu-ral Science Foundation of China under Grant Nos. 11774429and 11874424, the NSAF under Grant No. U1830206. Allcalculations were carried out at the Research Center of Super-computing Application at NUDT.
VIII. DATA AVAILABLE
The data that support the findings of this study are availablewithin the article. V. E. Fortov,
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