Nonlocal Thermal Transport Research Articles

Nonlocal effects on the shock-augmented ignition scheme of laser inertial fusion

Shock-augmented ignition (SAI) has recently been proposed as a viable approach to optimize the shock ignition scheme of laser fusion, which may substantially reduce laser power and intensity requirements and, as a result, greatly inhibit the growth of laser-plasma instabilities. However, the nonlocal thermal transport effect, which is believed to play a significant role in shock ignition, has not been considered in evaluation of the SAI scheme yet. Here, by self-consistently including modeling of the space-time-dependent nonlocal thermal transport into the radiation hydrodynamic simulations, we revisit the whole implosion and ignition dynamics of SAI numerically and theoretically. We find that, due to the nonlocal effect, on the one hand, the time-delay window between the igniter spike and the compression pulse in the drive laser for achieving high-gain fusion is much broadened from about 0.45 to 0.7 ns, relaxing the difficulty of laser pulse shaping; on the other hand, both the coast time and the fusion burn duration are significantly reduced by, respectively, 0.22 ns and 47 ps, in favor of increasing the hot-spot pressure around the stagnation stage. Published by the American Physical Society 2024

Space-time dependent non-local thermal transport effects on laser ablation dynamics in inertial confinement fusion

In inertial confinement fusion (ICF), electron thermal transport plays a key role in laser ablation and the subsequent implosion processes, which always exhibits intractable non-local effects. Simple modifications of the local Spitzer–Härm model with either an artificially-assumed constant flux limiter or a purely time-dependent one are applied to explain some experimental data, but fail to simultaneously reproduce the space-time evolution of the whole laser ablation process. Here, by carrying out a series of one-dimensional and two-dimensional radiation hydrodynamic simulations where the space-time-dependent non-local thermal transport model proposed by Schurt, Nicolaï and Busquet (the SNB model) are self-consistently included, we systematically study the non-local effects on the whole laser ablation dynamics including those occurring at the critical surface, the conduction zone and the ablation front. Different from those obtained previously, our results show that due to the non-local heat flow redistribution and redirection, at the critical surface the thermal flux is more inhibited, in the conduction zone the lateral thermal transport is suppressed, and ahead of the ablation front the plasma is preheated. When combined together they eventually result in significant improvement of the laser absorption efficiency, extension of the conduction zone, increase of both the mass ablation rate and shock velocity. Furthermore, the dependence of these laser ablation dynamics on different drive laser intensities is investigated, which provides beneficial enlightenments on potential laser pulse shaping and/or ignition scheme optimization in ICF.

Nonlocal thermal transport in magnetized plasma along different directions

Nonlocal thermal transport in magnetized plasmas is studied theoretically and numerically with the Vlasov–Fokker–Planck (VFP) model, in which the magnetic field has nonzero components both perpendicular to and along the temperature gradient. Nonlocal heat transport is found in both the longitudinal and transverse directions, provided the temperature gradients are sufficiently large. The magnetic field tends to reduce the nonlocality of the thermal transport in the direction perpendicular to the magnetic field, i.e., the difference between the heat fluxes predicted by the Braginskii theory and the VFP simulation decreases with increasing magnetic field strength. When the initial temperature gradient is steep, the nonlocal heat flux depends not only on the present temperature profile, but also on its time history. Moreover, the contribution of high-order terms in the spherical harmonic expansion of the electron distribution function becomes important for a magnetized plasma, in particular for thermal transport in the direction perpendicular to the temperature gradient.

Non-local linear-response functions for thermal transport computed with equilibrium molecular-dynamics simulation

We establish an approach to compute linear-response functions to elucidate heat waves and non-local thermal transport. The theory is able to describe the response of a system to external heat sources that are nonuniform in space and time. The response functions are computed using equilibrium molecular-dynamics simulations of an Ar crystal modeled using the standard Lennard–Jones potential. It is shown that for low temperatures and short length scales, transport can be partially or even completely ballistic, with the response primarily limited by the group velocity of lattice waves. By contrast, at longer length scales and higher temperatures, the response functions correspond more closely to diffusive transport characteristic of Fourier’s law. It is also shown how the effective thermal conductivity can be determined in a partially ballistic regime. The results demonstrate the known reduction in the effective thermal conductivity observed when system dimensions are smaller than the mean-free path for lattice waves. Finally, we show how the determination of relevant response functions can be used to model heating of a crystal without requiring additional atomic-scale simulations. Differences between computed results and predictions from Fourier’s law represent wave-like, partially ballistic transport.

Beyond phonon hydrodynamics: Nonlocal phonon heat transport from spatial fractional-order Boltzmann transport equation

Spatially convoluting formulations have been used to describe nonlocal thermal transport, yet there is no related investigation at the microscopic level such as the Boltzmann transport theory. The spatial fractional-order Boltzmann transport equations (BTEs) are first applied to the description of nonlocal phonon heat transport. Constitutive and continuity equations are derived, and two anomalous behaviors are thereafter observed in one-dimensional steady-state heat conduction: one is the power-law length-dependence of the effective thermal conductivity, κeff∝Lβ with L as the system length, and the other is the nonlinear temperature profile, Tx−Tx=0∼x1+η. A connection between the length-dependence and nonlinearity exponents is established, namely, β=−η. Furthermore, we show that the order of these BTEs should be restricted by the ballistic limit. In minimizing problems, the nonlocal models in this work give rise to different results from the case of Fourier heat conduction, namely that the optimized temperature gradient is not uniform.

Space-time dependent thermal conductivity in nonlocal thermal transport

Nonlocal thermal transport is generally described by the Peierls-Boltzmann transport equation (PBE). However, solving the PBE for a general space-time dependent problem remains a challenging task due to the high dimensionality of the integro-differential equation. In this work, we present a direct solution to the space-time dependent PBE with a linearized collision matrix using an eigendecomposition method. We show that there exists a generalized Fourier type relation that links heat flux to the local temperature, and this constitutive relation defines a thermal conductivity that depends on both time and space. Combining this approach with ab initio calculations of phonon properties, we demonstrate that the space-time dependent thermal conductivity gives rise to an oscillatory response in temperature in a transient grating geometry in high thermal conductivity materials. The present solution method allows us to extend the reach of our computational capability for heat conduction to space-time dependent nondiffusive transport regimes. This capability will not only enable a more accurate interpretation of thermal measurements that observe nonlocal thermal transport but also enhance our physical understanding of nonlocal thermal transport in high thermal conductivity materials that are promising candidates for nanoscale thermal management applications.

Incorporating nonlocal parallel thermal transport in 1D ITER SOL modelling

Accurate modelling of the thermal transport in the ‘scrape-off-layer’ (SOL) is of great importance for assessing the divertor exhaust power handling in future high-power tokamak devices. In conditions of low collisionality and/or steep temperature gradients that will be characteristic of such devices, classical local diffusive transport theory breaks down, and the thermal transport becomes nonlocal, depending on conditions in distant regions of the plasma. An advanced nonlocal thermal transport model is implemented into a 1D SOL code ‘SD1D’ to create ‘SD1D-nonlocal’, for the study of nonlocal transport in tokamak SOL plasmas. The code is applied to study typical ITER steady-state conditions, to assess the relevance of nonlocality for ITER operating scenarios. Results suggest that nonlocal effects will be present in the ITER SOL, with strong sensitivity in simulation outputs observed for small changes in upstream density conditions, and drastically different temperature profiles predicted using local/nonlocal transport models in some cases. Global flux limiters are shown to be inadequate to capture the spatially and temporally changing SOL conditions. Introducing impurity seeding, under conditions where detached divertor operation is achieved using the flux-limited Spitzer-Härm models used in standard SOL codes, simulations using the nonlocal thermal transport model under equivalent conditions were found to not reach detachment. An analysis of the connection between SOL collisionality and nonlocality suggests that nonlocal effects will be significant for future devices such as DEMO as well. The results motivate further work using nonlocal transport models to study disruption events and low collisionality regimes for ITER, to further improve accuracy of the nonlocal models employed in comparison to kinetic codes, and to identify more appropriate boundary conditions for a nonlocal SOL model.

Direct-drive double-shell implosion: A platform for burning-plasma physics studies.

Double-shell ignition designs have been studied with the indirect-drive inertial confinement fusion (ICF) scheme in both simulations and experiments in which the inner-shell kinetic energy was limited to ∼10-15kJ, even driven by megajoule-class lasers such as the National Ignition Facility. Since direct-drive ICF can couple more energy to the imploding shells, we have performed a detailed study on direct-drive double-shell (D^{3}S) implosions with state-of-the-art physics models implemented in radiation-hydrodynamic codes (lilac and draco), including nonlocal thermal transport, cross-beam energy transfer (CBET), and first-principles-based material properties. To mitigate classical unstable interfaces, we have proposed the use of a tungsten-beryllium-mixed inner shell with gradient-density layers that can be made by magnetron sputtering. In our D^{3}S designs, a 70-μm-thick beryllium outer shell is driven symmetrically by a high-adiabat (α≥10), 1.9-MJ laser pulse to a peak velocity of ∼240 km/s. Upon spherical impact, the outer shell transfers ∼30-40 kJ of kinetic energy to the inner shell filled with deuterium-tritium gas or liquid, giving neutron-yield energies of ∼6 MJ in one-dimensional simulations. Two-dimensional high-mode draco simulations indicated that such high-adiabat D^{3}S implosions are not susceptible to laser imprint, but the long-wavelength perturbations from the laser port configuration along with CBET can be detrimental to the target performance. Nevertheless, neutron yields of ∼0.3-1.0-MJ energies can still be obtained from our high-mode draco simulations. The robust α-particle bootstrap is readily reached, which could provide a viable platform for burning-plasma physics studies. Once CBET mitigation and/or more laser energy becomes available, we anticipate that break-even or moderate energy gain might be feasible with the proposed D^{3}S scheme.

Influence of atomic kinetics on inverse bremsstrahlung heating and nonlocal thermal transport.

This paper describes a computational model that self-consistently combines physics of kinetic electrons and atomic processes in a single framework. The formulation consists of a kinetic Vlasov-Boltzmann-Fokker-Planck equation for free electrons and a non-Maxwellian collisional-radiative model for atomic state populations. We utilize this model to examine the influence of atomic kinetics on inverse bremsstrahlung (IB) heating and nonlocal thermal transport. We show that atomic kinetics affects nonlinear IB absorption rates by further modifying the electron distribution in addition to laser heating. We also show that accurate modeling of nonlocal heat flow requires a self-consistent treatment of atomic kinetics, because the effective thermal conductivity strongly depends on the ionization balance of the plasma.

Measuring heat flux from collective Thomson scattering with non-Maxwellian distribution functions

Heat flux was measured in coronal plasmas using collective Thomson scattering from electron-plasma waves. A laser-produced plasma from a planar aluminum target created a temperature gradient along the target normal. Thomson scattering probed electron-plasma waves in the direction of the temperature gradient with phase velocities relevant to heat flux. The heat-flux measurements were reduced from classical values inferred from the measured plasma conditions in regions with large temperature gradients and agreed with classical values for weak gradients. In regions where classical theory was invalid, the heat flux was determined by reproducing the measured Thomson-scattering spectra using electron distribution functions consistent with nonlocal thermal transport. Full-scale hydrodynamic simulations using both flux-limited thermal transport (FLASH) and the multigroup nonlocal Schurtz, Nicolaï, and Busquet models underestimated the heat flux at all locations.

Optimization of laser-driven cylindrical implosions on the OMEGA laser

Laser-driven cylindrical implosions were conducted on the OMEGA laser as part of the laser-driven mini-MagLIF (Magnetized Liner Inertial Fusion) Campaign. Gated x-ray images were analyzed to infer shell trajectories and study the energy coupling in these implosions. Two-dimensional and three-dimensional HYDRA simulations were performed and post-processed to produce synthetic x-ray self-emission images for comparison. An analysis technique, which could be applied to both experimental and simulated x-ray images, was developed to characterize the shape and uniformity of the implosion. The analysis leads to a measurement of the average implosion velocity and axial implosion length, which can then be used to optimize the beam pointing and energy balance for future experiments. Discrepancies between simulation results and experiments allude to important physical processes that are not accounted for in the simulations. In 2-D simulations, the laser beam's azimuthal angle of incidence is not included because the ϕ-direction is not simulated, and thus, energy absorption is over-predicted. The 3-D simulation results are more consistent with the experiments, but the simulations do not include the calculation of cross-beam energy transfer or non-local thermal transport, which affects the energy coupled to the implosion. By appropriately adjusting the simulated energy balance and flux limit, the simulations can accurately model the experiments, which have achieved uniform implosions over a 700-μm-long region at velocities of approximately 200 km/s.

Mitigating laser-imprint effects in direct-drive inertial confinement fusion implosions with an above-critical-density foam layer

Low-density foams of low-/mid-Z materials have been previously proposed to mitigate laser imprint for direct-drive inertial confinement fusion (ICF). For foam densities above the critical density of the drive laser, the mechanism of laser-imprint mitigation relies on the reduced growth rate of Rayleigh–Taylor instability because of the increased ablation velocity and density scale length at the ablation surface. Experimental demonstration of this concept has been limited so far to planar-target geometry. The impact of foams on spherical implosions has not yet been explored in experiments. To examine the viability of using an above-critical-density foam layer to mitigate laser-imprint effects in direct-drive ICF implosions on OMEGA, we have performed a series of 2-D DRACO simulations with state-of-the-art physics models, including nonlocal thermal transport, cross-beam energy transfer, and first-principles equation-of-state tables. The simulation results indicate that a 40-μm-thick CH or SiO2 foam layer with a density of ρ = 40 mg/cm3 added to a D2-filled polystyrene (CH) capsule can significantly improve the moderate-adiabat (α ≈ 3) implosion performance. In comparison to the standard CH target implosion, an increase in neutron yield by a factor of 4 to 8 and the recovery of 1-D compression ρR are predicted by DRACO simulations for a foam-target surface roughness of σrms ≤ 0.5 μm. These encouraging results could readily facilitate experimental demonstrations of laser-imprint mitigation with an above-critical-density foam layer.

Overview of recent HL-2A experiments

Since the last Fusion Energy Conference, significant progress has been made in the following areas. The first high coupling efficiency low-hybrid current drive (LHCD) with a passive–active multi-junction (PAM) antenna was successfully demonstrated in the H-mode on the HL-2A tokamak. Double critical impurity gradients of electromagnetic turbulence were observed in H-mode plasmas. Various ELM mitigation techniques have been investigated, including supersonic molecular beam injection (SMBI), impurity seeding, resonant magnetic perturbation (RMP) and low-hybrid wave (LHW). The ion internal transport barrier was observed in neutral beam injection (NBI) heated plasmas. Neoclassical tearing modes (NTMs) driven by the transient perturbation of local electron temperature during non-local thermal transport events have been observed, and a new type of non-local transport triggered by the ion fishbone was found. A long-lasting runaway electron plateau was achieved after argon injection and the runaway current was successfully suppressed by SMBI. It was found that low-n Alfvénic ion temperature gradient (AITG) modes can be destabilized in ohmic plasmas, even with weak magnetic shear and low-pressure gradients. For the first time, the synchronization of geodesic acoustic mode (GAM) and magnetic fluctuations was observed in edge plasmas, revealing frequency entrainment and phase lock. The spatiotemporal features of zonal flows were also studied using multi-channel correlation Doppler reflectometers.

Evidence of enhanced self-organized criticality (SOC) dynamics during the radially non-local transient transport in the HL-2A tokamak

Self-organized criticality (SOC) dynamics have been investigated in non-local transport regimes induced by the supersonic molecular beam injection in the HL-2A tokamak. Experimental evidence shows that the SOC or avalanche behaviors, such as the Hurst parameter, self-similarity and large-scale radial correlations in turbulence, are remarkably enhanced during the prompt non-local phase, together with an increase of inward propagation of turbulent events. These results highlight the importance of SOC paradigm during the transient non-local thermal transport in magnetically confined plasmas.

Measurements of the Conduction-Zone Length and Mass Ablation Rate in Cryogenic Direct-Drive Implosions on OMEGA.

Measurements of the conduction-zone length (110±20 μm at t=2.8 ns), the averaged mass ablation rate of the deuterated plastic (7.95±0.3 μg/ns), shell trajectory, and laser absorption are made in direct-drive cryogenic implosions and are used to quantify the electron thermal transport through the conduction zone. Hydrodynamic simulations that use nonlocal thermal transport and cross-beam energy transfer models reproduce these experimental observables. Hydrodynamic simulations that use a time-dependent flux-limited model reproduce the measured shell trajectory and the laser absorption but underestimate the mass ablation rate by ∼10% and the length of the conduction zone by nearly a factor of 2.

Optothermal Modeling of Plasmonic Nanofocusing Structures With Nonlocal Optical Response and Ballistic Heat Transport

Nanoscale optical energy focusing using plasmonic structures is crucial for many applications, such as imaging and lithography. Thermal management for these nanostructures is of great importance to maintain their reliabilities but has not been investigated extensively yet, especially when the strong nonlocalities present in the nanostructures. Here, we report a multiphysics model to study the coupled optical and thermal responses of plasmonic nanofocusing structures. We applied the hydrodynamic Drude model to describe the nonlocality in the optical response and derived ballistic–diffusive equations for both electrons and phonons to model the nonlocal thermal transport. Strong nonlocal optothermal responses were observed.

Influence of spatial geometrical curvature on nonlocal electron heat transport in expanding plasmas

The electron thermal transport in fluid theory would be inaccurate when the collisionality is not enough, and the Fokker-Planck (FP) simulations are usually employed to resolve the inadequacies. In this paper, the one-dimensional Fokker-Planck code is extended to handle the cylindrical and spherical geometries in which the electron distribution functions are solved in the reference frame of the ion fluid. The FP code is validated in the fluid limit by comparing with fluid (MULTI) simulations. Then, the expansions of plasmas in different spatial geometries are simulated with the FP and fluid codes. As the main characters of nonlocal transport, the electron thermal transport inhibition and preheating are investigated in expanding plasmas. The spherical nonlocal theory can give the thermal transport inhibition and preheating phenomenon, which is exploited to fit the heat flux with variation of fitting parameter . The spherical nonlocal theory will reproduce Spizer-Hrm expression as = 0. Then we analyze the heat flux after the plasma expanding 200 ps with a uniform initial temperature T = 100 eV and density ne= 1 1021 /cm3. By comparing the heat flux computed by spherical nonlocal thermal transport theory and FP simulation, it is found that (n-1)/r term in Eq. (3a) cannot be neglected when the radius is small and the geometrical curvature effect will decrease the nonlocality of transport in outer expanding plasmas. The geometrical curvature effect leads to a smaller thermal transport inhibition and preheating in the expanding plasmas as comparing the spherical case with the planar one. The expansions of plasmas in different spatial geometries are also simulated with the FP and fluid codes under the initial conditions which are similar to the inertial confinement fusion. The same influence of geometrical curvature on nonlocal electron thermal transport are also obtained.

Nonlocal thermal transport across embedded few-layer graphene sheets

Thermal transport across the interfaces between few-layer graphene sheets and soft materials exhibits intriguing anomalies when interpreted using the classical Kapitza model, e.g. the conductance of the same interface differs greatly for different modes of interfacial thermal transport. Using atomistic simulations, we show that such thermal transport follows a nonlocal flux-temperature drop constitutive law and is characterized jointly by a quasi-local conductance and a nonlocal conductance instead of the classical Kapitza conductance. The nonlocal model enables rationalization of many anomalies of the thermal transport across embedded few-layer graphene sheets and should be used in studies of interfacial thermal transport involving few-layer graphene sheets or other ultra-thin layered materials.

Modeling high-compression, direct-drive, ICF experiments

The success of direct-drive-ignition target designs depends on two issues: the ability to maintain the main fuel entropy at a low level and the control of the nonuniformity growth during the implosion. Modelling the ICF experiments requires an accurate account for all sources of shell heating, including the shock heating, radiation and suprathermal electrons preheat, and small-scale perturbation growth. To increase calculation accuracy, a new heat-transport model has been developed and implemented in the 1-D hydrocode LILAC. This model includes both the effect of the resonance absorption and the nonlocal thermal transport. The OMEGA experiments designed with the help of the new model have achieved high-areal-density (ρR > 200 mg/cm2) fuel assembly in the low-adiabat cryogenic shell implosions.

Early stage of implosion in inertial confinement fusion: Shock timing and perturbation evolution

Excessive increase in the shell entropy and degradation from spherical symmetry in inertial confinement fusion implosions limit shell compression and could impede ignition. The entropy is controlled by accurately timing shock waves launched into the shell at an early stage of an implosion. The seeding of the Rayleigh-Taylor instability, the main source of the asymmetry growth, is also set at early times during the shock transit across the shell. In this paper we model the shock timing and early perturbation growth of directly driven targets measured on the OMEGA laser system [T. R. Boehly et al., Opt. Commun. 133, 495 (1997)]. By analyzing the distortion evolution, it is shown that one of the main parameters characterizing the growth is the size of the conduction zone Dc, defined as a distance between the ablation front and the laser deposition region. Modes with kDc>1 are stable and experience oscillatory behavior [V. N. Goncharov, Phys. Rev. Lett. 82, 2091 (1999)]. The model shows that the main stabilizing mechanism is the dynamic overpressure due to modulations in the blow-off velocity inside the conduction zone. The long wavelengths with kDc<1 experience growth because of coupled Richtmyer-Meshkov-like and Landau-Darrieus instabilities [L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, New York, 1982)]. To match the simulation results with both the shock timing and perturbation growth measurements a new nonlocal thermal transport model is developed and used in hydrocodes.