A preliminary assessment of the sensitivity of uniaxially-driven fusion targets to flux-limited thermal conduction modeling
D. A. Chapman, J. D. Pecover, N. Chaturvedi, N. Niasse, M. P. Read, D. H. Vassilev, J. P. Chittenden, N. Hawker, N. Joiner
AA preliminary assessment of the sensitivity of uniaxially-driven fusiontargets to flux-limited thermal conduction modeling
D. A. Chapman, a) J. D. Pecover, N. Chaturvedi,
1, 2
N. Niasse, M. P. Read, D. H. Vassilev, J. P.Chittenden, N. Hawker, and N. Joiner First Light Fusion Ltd., Unit 9/10 Oxford Industrial Park, Mead Road, Yarnton, Kidlington OX5 1QU, United Kingdom Centre for Inertial Fusion Studies, Blackett Laboratory, Imperial College, London SW7 2AZ, United Kingdom (Dated: February 19, 2021)
The role of flux-limited thermal conduction on the fusion performance of the uniaxially-driven targets studied by Der-entowicz et al. is explored as part of a wider effort to understand and quantify uncertainties in ICF systems sharingsimilarities with First Light Fusion’s projectile-driven concept. We examine the role of uncertainties in plasma micro-physics and different choices for the numerical implementation of the conduction operator on simple metrics encapsu-lating the target performance. The results indicate that choices which affect the description of ionic heat flow betweenthe heated fusion fuel and the gold anvil used to contain it are the most important. The electronic contribution is foundto be robustly described by local diffusion. The sensitivities found suggest a prevalent role for quasi-nonlocal ionictransport, especially in the treatment of conduction across material interfaces with strong gradients in temperature andconductivity. We note that none of the simulations produce neutron yields which substantiate those reported in Ref. ,leaving open future studies aimed at more fully understanding this class of ICF systems. I. INTRODUCTION
First Light Fusion (FLF) is investigating a novel approachto controlled inertial confinement fusion (ICF) using efficient,low-cost, projectile-based driver technology. A key character-istic of the concept is that the implosion process initially pro-ceeds with a single, strong shock directed along a single axis,although the dynamics later-in-time can have convergent, non-planar aspects. As with all routes to ICF, a crucial element oftarget design and optimization is the validation of the numer-ical tools, and the multitude of options which underpin them.Sensitivity studies seek to assess the impact of factors suchas code configuration choices, numerical methods and the ba-sic properties of materials and are crucial for enhancing theconfidence in simulation-led predictions. Such studies are be-coming commonplace in the ICF community .For the planar driver geometry of interest there are veryfew experiments accessing the fusion regime against whichcode benchmarking can be undertaken. One notable class ofexperiments where the fuel is collapsed directly by a singleplanar shock are are those of Derentowicz et al. , which fea-ture a fuel-filled conical cavity in a metal anvil driven by auniform, planar Mach wave produced by the implosion of aconical liner. Several variations of this experiment have beenperformed that consider different driver technologies, suchas electrical discharge explosion , direct laser ablation andrelativistic electron beams . The reported output of these tar-gets of 10 − neutrons is of substantial utility from theperspective of validating both integrated simulations and di-agnostic calibration. Of these, the original concept providesthe simplest and therefore most valuable validation case.As for an ICF experiment, the modeling is challenging:compressible hydrodynamics, thermal conduction, viscousdrag, radiation generation, transport and absorption and de-tailed thermophysical material properties across a wide range a) Electronic mail: dave.chapman@firstlightfusion.com of density-temperature space must all be considered. Initialexaminations have shown that the most important process fordetermining fuel energetics is thermal conduction, which actsto homogenize strongly-localised heating resulting from re-verberating shock waves.For plasmas featuring gentle thermal gradients the rate ofconduction loss from a small volume of heated fuel is dic-tated principally by the thermal conductivities of the elec-trons and ions; κ e and κ i , respectively. In low-density, high-temperature, hydrogen-like systems the electronic contribu-tion dominates due to their higher mobility, with ionic con-duction only competing when T i (cid:38) T e . Since shock-heatingcouples most efficiently to the ions, nonequilibrium condi-tions where both electronic and ionic conduction are impor-tant are readily attainable. Additionally, electron-ion tem-perature relaxation directly affects not only the plasma reac-tivity but also influences the balance between electronic andionic conduction loss. Although the theoretical descriptionsof these properties are well-known under the near-ideal con-ditions of the fuel, the solid target components are predicted tobe driven into warm dense matter (WDM) states, with close-to-solid densities and temperatures of a few eV. Under suchextreme conditions even basic thermophysical properties canbe substantially uncertain .The system studied in Ref. uses a planar shock in cop-per to drive the collapse of a deuterium-filled conical cav-ity with a full internal angle at the tip of 60 ◦ . Between thecopper and the fuel is a polythene layer, referred to in thiswork as the ‘coverslip’. The material in which the cavity ismade is gold and is referred to here as the ‘anvil’. At thestagnation point of the system, where the converging flowinto the tip of the cavity is reflected from the anvil, a tran-sient nonequilibrium state is produced which sets up a thermalwave across the interface between the fuel and interior anvilwall. The resulting heat flow is driven by both electronic andionic conduction and occurs in the presence of sharp densityand temperature gradients between regions with very differ-ent conductivities and heat capacities. Accurately capturing a r X i v : . [ phy s i c s . p l a s m - ph ] F e b the heat flux under such conditions is in general a problemwhich can only be fully understood using state-of-the-art ki-netic codes (see, e.g. Refs. ) and is one of the many chal-lenges of present concern in the ICF community . Reducedkinetic models, such as the Schurtz-Nicolaï-Busquet (SNB)model are available which reproduce many of the fea-tures expected from heat flow in the kinetic regime , al-though recent experiments have demonstrated even largerheat flow suppression than predicted.Commonly-used simple models, such as local flux-limitedthermal conduction , offer a computationally tractablemethod used in many ICF simulation codes . Although con-ceptually straightforward, the precise manner in which theyare implemented is subject to uncertainty due to the largenumber of free parameters involved and adequate evidence-based values are seldom available. Of particular importanceare: the choices of the magnitude of the flux-limited heatflow, i.e. the values of the flux limiter coefficients; the man-ner in which the transition between purely diffusive and flux-limited conduction is undertaken; and how the relevant quan-tities are to be utilized within the numerical framework of thehost code, e.g. in the spatial discretization scheme.In this work we report on the initial results of an ongoingsimulations-based study investigating the influence of uncer-tainties in FLF’s predictive modeling capability using the ex-perimental results from Ref. as a benchmark. The scopeof this paper is focused on the impact of flux-limited ther-mal conduction on an idealized model of the target, as this iscurrently believed to be the largest effect on the fusion out-put. With suitably well-converged simulations we establisha reference case against which the impact of configurationchanges can be quantifiably measured using a few simple met-rics. The basic form of the conduction operator is described,with emphasis given to the main sources of uncertainty aris-ing from free choices in its implementation. The changes withthe strongest influence on the evolution of the target dynamicsand fuel energetics are found to be the accuracy of the electronthermal conductivity of the fuel, the magnitude of the maxi-mum ionic heat flux and the treatment of interface conduc-tion, both of which can only be fully understood with moreadvanced tools. Since the thermophysical properties of thefuel under the states produced are thought to be well-known,we conclude that an improved understanding of ionic conduc-tion in the kinetic regime is therefore crucial if we are to beable to use this experimental platform as a means to validatesimulations of relevance to FLF’s mission. II. SIMULATION SETUPA. Idealized target model
The present work is undertaken using Hytrac; one of thetwo in-house radiation-hydrodynamics codes developed byFLF. The code is based on the front tracking approach andalso implements cell-based adaptive mesh refinement, simi-lar to, e.g., RAGE . The multi-physics model includes two-temperature conduction, viscous fluxes and radiation trans- port via the P / -AFL method . The code further supportseither analytic or tabulated models for material equation ofstate (EoS), plasma microphysics and fusion reactivity.The work presented in Ref. shows that the drive into thefuel is well-approximated with a single, uniform shock intothe plastic coverslip. The pressure in the copper is inferredfrom shock velocity measurements to be 46 Mbar, which leadsto a shock in coverslip with a pressure of 13 Mbar. The latteris modeled as PMMA as a surrogate for polythene due to diffi-culties matching the principal Hugoniot using the FEOS EoSmodel . Since the densities of the two materials are suffi-ciently similar this compromise is believed to be reasonable.The target mesh geometry used for our simulations isshown in Fig. 1a. The number of AMR refinement levels is4, giving a minimum cell size of 1 . µ m on the Eulerian grid(EGrid). The inset shows how the cavity tip has had a radiusof curvature of 10 µ m added to account for expected manufac-turing tolerances. The shocked copper region, denoted as the‘driver’, at the top of the domain is initialised from the thermo-dynamic state resulting from shocking initially ambient cop-per to the experimentally inferred pressure along the principalHugoniot. All other materials are initialised at a pressure of1 bar and their appropriate ambient mass densities to produceinitially static interfaces.In these simulations the viscous flux and radiation transportoperators are not used. In the case of viscous effects this isbecause the predominant influence is on the damping of theincident shock and not the energetics of the cavity collapse,from which the bulk of the yield is expected. Full radiation-hydrodynamics modelling suggests the internal cavity dynam-ics may be influenced by bremsstrahlung emission from thefuel to a small degree through radiative ablation of the wall,although the conduction losses are vastly more important tothe energetics. Disabling these operators also makes tractablea far larger study focused on the physics of conduction, whichis our principal focus here. More details of the cavity collapsedynamics and the impact of the full range of multi-physicsoptions will be discussed in future work. B. Assessment of convergence
Assessment of the impact of changes to the setup of themulti-physics model must be done on the basis that any differ-ences observed can confidently be attributed to the predictedinfluence of the options, rather than due to lack of simula-tion convergence. We pursue convergence through variationof the EGrid base resolution at a constant refinement leveland comparing the difference between key metrics of progres-sively higher-resolution runs.The metrics used for this purpose are the burn-weighted av-erage (BWA) histories of the ion and electron temperaturesand mass density (cid:104) X (cid:105) b ( t ) = (cid:90) V d r ∂ Y n ∂ V ∂ t X ( r , t ) (cid:90) V d r ∂ Y n ∂ V ∂ t . (1) (b) h T i i b [ e V ] h T e i b [ e V ] h ρ i b [ g c m − ]
10 12 14 16 18 20 22 24 26 28 30 t [ns] Base res. = 10Base res. = 15Base res. = 20Base res. = 25 Base res. = 30Base res. = 35Base res. = 40 (c) C o n v e r g e n ce p a r a m e t e r , ε t o t ∆ min [ µ m] Base res. = 10Base res. = 15Base res. = 20Base res. = 25Base res. = 30Base res. = 35Reference case: Base res. = 40
Figure 1. (a): Simulation geometry and EGrid zoning for the idealized model, featuring a shocked Cu driver, PMMA coverslip, deuterium fueland gold anvil. The left-hand part of the plot shows a logarithmic map of mass density whilst the right-hand side shows the material ‘color’indicator. The tip of the conical cavity is rounded with a radius of curvature of 10 µ m, as shown in detail in the inset. (b): Time histories ofthe BWA ion (upper panel) and electron (middle panel) temperatures and mass density (lower panel). The regions with the gray backgroundindicate the time window over which simulation convergence is assessed. (c): Convergence metric ε tot as a function of minimum cell size. Here, X stands for the quantity of interest, ∂ Y n / ∂ V ∂ t repre-sents the total neutron yield emission per unit volume per unittime and V denotes the volume occupied by the fusion fuel.The convergence parameter ε with respect to variable X for asimulation with minimum cell size ∆ min relative to a suitablereference case is then defined as‘ ε X = ∑ i max i min (cid:0) (cid:104) X (cid:105) b ( t i ; ∆ min ) −(cid:104) X (cid:105) b (cid:0) t i ; ∆ refmin (cid:1)(cid:1) i max − i min / . (2)Here, the index i runs between times bounded by the onset( t i min ) and secession ( t i max ) of neutron production. These are respectively defined as the earliest and latest times where theneutron production rate exceeds 10% of its peak value. Theconvergence metric is then defined as the quadratic averageover all variables, i.e. ε tot = (cid:0) ∑ X ε X / ∑ X (cid:1) / . For pure deu-terium fuel at the modest temperatures predicted one may ap-proximately write ∂ Y n ∂ V ∂ t ≈ n (cid:104) σ v (cid:105) d ( D , He ) n . (3)In Eq. (3) the thermal reactivity (cid:104) σ v (cid:105) is given by fits due toBosch and Hale .A strong convergence trend can be seen in both the tim-ing and shape of the peaks in (cid:104) T i (cid:105) b and (cid:104) T e (cid:105) b with increasingbase resolution (Fig. 1b), indicating that the hydrodynamics ofthe cavity implosion are properly captured in our simulations.This is further exemplified in Fig. 1c. The two main spikes inthe BWA ion temperature correspond to the reflection of theincident shock from the wall and tip of the cavity. The subse-quent reverberations in the collapsing cavity produce a long-lived plasma in which conduction losses are roughly balancedby the heating due to compression. The electrons do not showthe same strongly peaked structure since they are treated as in-compressible and are therefore isentropically heated throughthe shock reflection. Nonequilibrium states with T i / T e > T ∼ −
180 eV, which persists for around 10 ns. Therole of ion conduction in dissipating the energy imparted tothe fuel at the tip of the cavity is therefore likely to be im-portant, as is the rate of temperature equilibration. Overall,the strength of the observed trends indicates that our resultsat a base resolution of 40 cells (smallest cell size of 1 . µ m)provides a suitable reference case for quantifying the impactof changes due to conduction physics, without conflation withissues related to lack of convergence. III. CONDUCTION MODELING IN ICF SIMULATIONSA. Flux-limited thermal conduction
It is well-known that the modeling of losses due to ther-mal conduction constitute a crucial aspect of the power bal-ance in prospective ICF systems . A commonly-used imple-mentation strategy for the conduction operator in radiation-hydrodynamics codes follows the local flux-limited diffusionapproach to Fourier’s law Q Four. = − κ ∇ T , (4)the magnitude of which is restricted to a fraction, α , of thefree-streaming limit Q max = α n v th k B T ; v th = ( k B T / m ) / , (5)in order to prevent unphysically large fluxes from arising inregions with steep temperature gradients. Such conditionsarise routinely near the ablation front of laser-irradiated solids,e.g. in short-pulse laser-driven targets or the heating ofhohlraum walls at NIF . In the present study sharp temper-ature gradients are expected to occur in proximity to strongshocks and also at interfaces between materials with very dif-ferent heat capacities.The flux-limited model represents a phenomenological cor-rection to purely diffusive (local) energy transport and doesnot capture the preheating effect arising from long-range (nearcollisionless) propagation of the high-energy tail of the parti-cle distribution. The impact of nonlocal transport is now ofgrowing concern in the ICF community and has driventhe development of numerous high-fidelity kinetic simula-tion tools and accurate reduced models . Unfortunately, these more accurate capabilities are seldom practical for im-plementation in fully-integrated simulations and the flux-limited approach remains in common usage. Even for thissimplest model, however, there are a number of free param-eters to set and numerical choices to be made; independentlyfor the electrons and ions. B. Flux limiter coefficients
For the electrons, the value of α e is typically tuned to matchexperimental data or designed to match results from high-fidelity simulations, such as kinetic codes, with α e ≈ . . Despite being informed by numericalresults a broad range of values may be needed to match dif-ferent experimental data sets, with values up to α e = .
15 (theso-called ‘high-flux’ model ) being required. On the otherhand, recent work by Meezan et al. has shown that a flux-limited approach with a lower value of α e = .
03 can quali-tatively explain results from several independent diagnosticssimultaneously. Thus, a wide range of values for this parame-ter can be substantiated for sensitivity studies.The ion heat flow is often either not flux-limited or its con-tribution is neglected entirely on the basis that electronic heatflow is usually dominant. The value of α i is generally takento be much larger than α e , although there is presently little-to-no consensus on a canonical value. Indeed, ionic VFPsimulations strongly discredit the idea that a single value of α i can universally describe ionic heat flow, especially in con-verging shocks . The default value used in this work is takento be 0 . and DEIRA . Weconsider the value of α i to be one of the most uncertain aspectsof our modeling. C. Interpolation functions
Another important free choice in the implementation offlux-limited conduction is the manner in which the capping ofthe Fourier heat flow to the maximum value given by Eq. (5)is achieved. A particularly prevalent choice in the literature isto simply take the minimum value Q eff = min ( Q Four. , Q max ) , (6)which leaves the Fourier heat flow unmodified until the tran-sition point. The impact of such a hard switch is discussedby Meezan et al. , wherein the propagation of a nonlinearthermal wave is studied for α e = . − .
25. The sharp frontfound for smaller flux-limiter coefficients results directly fromthe discontinuous profile in the heat flux due to Eq. (6), whichis unlikely to be representative of the true solution. Indeed,our simple conduction verification problems have shown thatthis approach can lead to spurious break-up of the wave frontand often shows undesirable numerical noise (see the relatedcurve in Fig. 2a).A smooth transition can of course be achieved in numerousways, the most popular of which is to consider a harmonicinterpolation Q eff = ( / Q Four. + / Q max ) − . (7)Application of Eq. (7) reduces the effective heat flux beforethe transition point (where Q Four. = Q max ), which removes theblunted shape of the thermal wave profile but also undesirablyretards the propagation of the front (Fig. 2a). Less well-knownsmooth schemes that do not strongly alter the wave speed inthe weakly flux-limited regime include the ‘Larson’ model Q eff = (cid:0) / Q + / Q (cid:1) − / , (8)and the ‘exponential’ model Q eff = Q max [ − exp ( − Q Four . / Q max )] . (9)Both of these originate in treatments of flux-limited radiationdiffusion and show similar behavior in how they approach (a) T e m p e r a t u r e , T [ e V ] Position, x [ µ m] Arithmetic : ExponentialArithmetic : HarmonicArithmetic : Min-cappedArithmetic : LarsonContraharmonic : ExponentialHarmonic : ExponentialInitial discontinuity position ( x = 5 µ m) T = 1 keV T = 293 K (b) κ e ff / κ m a x κ tab /κ max Min-cappedLarsonExponentialHarmonic (c) − f j + / / f j − − f j +1 /f j ArithmeticHarmonicContraharm.
Figure 2. (a): Spatial temperature profile for a simple nonlinear1D thermal wave propagation test in an ideal gas with Spitzer-likethermal conductivity , comparing different options for the interpo-lation functions (b) and for evaluating quantities on the cell faces (c).The harmonic mean case (dashed pink curve) moves just over 2 . µ mfrom the initial temperature discontinuity ( x = µ m) and maintainsa temperature of over 250eV behind the front. (b): Interpolationfunctions f interp governing the transition between κ tab and κ max . (c):Interpolation functions for evaluating quantities on cell faces. Thevalues at cell face j + / j . Q max (Fig. 2b) and accordingly lead to similar temperatureprofiles for nonlinear thermal waves (Fig. 2a).The electron and ion heat flux calculations are forced tofollow Fourier’s law (4) in Hytrac, subsequently introducing alimited form of thermal conductivity κ max = Q max / | ∇ T | . (10)All the heat fluxes in the foregoing expressions (6)–(9) cantherefore be replaced with thermal conductivities, such that Q eff = − κ eff ∇ T by definition and κ eff = f intrp ( κ tab , κ max ) , (11)where κ tab refers to the tabulated thermal conductivity at agiven set of thermodynamic conditions and f interp representsone of the interpolation functions Eqs. (6)-(9). Use of Eq. (10)crucially allows the diffusion time step ∆ t dif = ( ∆ x ) ˜ C V / ρκ in the Runge-Kutta-Legendre-2 explicit super time-steppingintegration scheme to be self-consistent with the heat flowin the flux-limited regime, thereby greatly reducing run time. D. Evaluation of cell-face quantities
As with all codes based on the finite volume method, theheat fluxes between EGrid cells must be calculated on the cellfaces and are evaluated via ∂ E ∂ t (cid:12)(cid:12)(cid:12)(cid:12) cond = − ∇ · Q eff V → = V ∑ { faces } ( S · Q eff ) face , (12)in which V is the cell volume, S is the cell face area vector andthe summation extends over all cell faces. Thus, the effectivethermal conductivity (11) must be evaluated on the cell faces.Following Eq. (10) the temperature gradient is approximatedwith a linear finite difference based on cell-centred values,e.g. | ∇ T | j + / ≈ (cid:12)(cid:12) T j + − T j (cid:12)(cid:12) / δ , where δ is the displacementbetween cell centres.Assuming that the thermodynamic fields within the systemare smooth and continuous then at sufficient resolution a linearinterpolation is the most natural choice to evaluate κ tab and Q max on the cell faces. For the conduction stencil used inHytrac this amounts to the arithmetic mean (exemplified hereusing the tabulated thermal conductivity) κ j + / = (cid:16) κ j + + κ j tab (cid:17) , (13)in which the superscripts refer to cell center (integer) and face(half-integer) indices. Weighting the cell-centred quantitiesusing their nonlinear scaling on thermodynamic variables, e.g.by T / / Z ∗ i for the electron thermal conductivity in the idealplasma regime , gives a better representation of their varia-tion over the EGrid in comparison to Eq. (13). For the generalcase one can simply self-weight the quantities, resulting in acontraharmonic mean κ j + / = ( κ j + ) + ( κ j tab ) κ j + + κ j tab . (14)For ideal states the T / scaling of the conductivity means thatthe heat flux resulting from Eq. (14) is driven by the hotterof the cells in thermal contact, i,e. the properties of the celldownstream of the temperature gradient have no bearing onthe flux it may accept.An alternative interpolation method used in thermal trans-fer as a phenomenological description of heat flow at high-thermal impedance junctions is to take a harmonic mean κ j + / = κ j + κ κ j + + κ . (15)This model enforces continuous heat flux and temperature be-tween cells and a discontinuity of the temperature gradientand conductivity and can be thought of as complimentaryto the arithmetic mean (Fig. 2c). For the case of a solidinsulator-conductor interface Eq. (15) approximately capturesthe contact resistance resulting from microscopic voids andis therefore unlikely to give good results if one of the materialsis a gaseous plasma.For the propagation of nonlinear thermal waves between re-gions having conductivities differing by orders of magnitudeEq. (15) fails to recover analytic results. For example, well-established test cases with zero conductivity in the un-heated part of the domain fail entirely since the resulting heatflux at the cell boundary representing the front is exactly zero.Similar results are shown in Fig. 2a, where the harmonic meancurve travels only 2 . µ m from the an initial temperature dis-continuity at x = µ m ( T = T =
293 K) com-pared to the 45 − µ m predicted using other model combi-nations. For cases where a large gradient in conductivity doesnot exist results using Eq. (15) perform similarly to thoseusing Eqs. (13) and (14).Despite these concerns, we are motivated to include the har-monic mean option in our study due to its inclusion in nu-merical solutions to the heat equation and other simulationcodes used for HEDP research . Furthermore, the choice ofhow to evaluate the maximum heat flux Q max in Eq. (10) atcell faces is also arbitrary and, in principle, independent tothe choice applied for κ tab . In this case the same rationale be-hind the arithmetic and contraharmonic mean interpolationsapplies, although there is no clear analogue to the reasoningbehind use of the harmonic mean approach. We consider thelatter as a means of examining flux restriction across materialinterfaces under conditions where the heat flow is dominatedby the free-streaming contribution. E. Conduction across a material interface
With the foregoing discussion in mind one of the most im-portant uncertainties in our modeling is the treatment of heatflow through material interfaces; especially for the deuterium-gold boundary, where the heated fuel is typically several or-ders of magnitude more conductive and the heat flow couldalso be flux-limited. If the interface is highly conductive thenthe internal energy will quickly leak into the high-heat capac-ity anvil and quench the yield. The same concern also holds for the later stages, when the fuel is being compressed withinthe collapsing cavity. On the other hand, if the interface isstrongly insulating, which would result from use of Eq. (15),then the fuel ions will tend to stay hot during the stagnationphase. Moreover, the evolution of the state is coupled to elec-tronic conduction through equilibration, the rate of which isalso influenced by the fact that the initial nonequilibrium pro-duced by the shock is enhanced when ion conduction is sup-pressed.Using Spitzer’s well-known expressions for the thermalconductivities we may estimate the conditions for which theconduction across an interface between fully-ionized deu-terium (D) and some high-Z material (Z) is driven by the con-ditions on one side only. Solving for the states that lead to a10-fold disparity, i.e. κ D e , i ≥ κ Z e , i , gives T e D (cid:38) . T e Z Z ∗ Z ( electrons ) , (16) T i D (cid:38) . T i Z ( Z ∗ Z ) / (cid:18) A D A Z (cid:19) / ( ions ) , (17)in which A denotes the atomic mass and Z ∗ Z the effective ioncharge for transport processes in the high-Z material.Taking Z ∗ Au ∼ Z ∗ PMMA = T e D (cid:38) . T e Au and T e D (cid:38) . T e PMMA . The equivalent criteria for the ions are T i D (cid:38) . T i Au and T i D (cid:38) . T i PMMA . These conditions arewell-fulfilled for both the D-Au and D-PMMA interfaces inour simulations throughout the period with the highest reac-tivity. A similar argument using flux-limited forms leads tocriteria which are also satisfied at all times of interest. UsingEqs. (13)-(14) the properties of the fuel will dominate whereasusing Eq. (15) the heat flow will be dictated by the propertiesof the high-Z material. Thus, we expect a strong correlationbetween target performance and the choice of cell face inter-polation scheme. Naturally, these conclusions hold only underthe restriction of the flux-limited model and may not properlycapture the true nature of interfacial transport.
F. Uncertainties in conduction microphysics
Aside from the uncertainties in the implementation of theconduction operator itself one of the largest sources of un-certainty is the accuracy of the tabulated microphysics mod-els which underpin the energetics. In these simulations weuse the well-known Lee-More and Stanton-Murillo mod-els for the conductivities of the electrons and ions, respec-tively. The Lee-More model is corrected to include electron-electron scattering following the approach of Apfelbaum .For the electron-ion energy exchange rate we use the f -sumrule approach , which been shown to perform well com-pared to molecular dynamics simulations . Higher-orderconsiderations such as the coupled mode effect can safelybe ignored since the nonequilibrium states produced in oursimulations almost always have T i > T e . All the microphysics Figure 3. Cell trajectory plots on the ρ − T i plane for the fuel (left), coverslip (middle) and anvil (right) materials. Contours for Γ ii = Γ e = D e = W = { . , . , . } (orange) are shown. For the initially solid materials (PMMA and gold) the Maxwellconstruction region, where the accuracy of the FEOS model is highly uncertain, is denoted by the horizontal isobars; There are no states in thisregion at any time in the simulation. The time evolution of the simulation is denoted by the color scale (blue - early time and dark red - latetime). The initial conditions of the deuterium fuel are shifted from true STP conditions for P = have been found to make negligible difference to the BWA profiles or finalneutron yield in these simulations, as expected for fuel initially in the gaseous phase. models are driven by the ionization predicted by FEOS forconsistency with the EoS .Whilst each of these models include corrections for electrondegeneracy and strong ion coupling they are still not expectedto be very accurate under conditions in warm dense matter(WDM) regime. Of particular concern is the electron thermalconductivity as this is strongly influenced by many-body ef-fects such as screening, structural effects and the ionizationequilibrium of high-Z systems .A simple but informative a priori indicator for model accu-racy is encapsulated by the WDM parameter W ( ρ , T e ) = S ( Θ ) S ( Γ ee ) , S ( x ) = / ( x + / x ) , (18)which provides a simple measure of the nonideality of thermo-dynamic states from the perspective of theoretical modeling.In Eq. (18), the usual definitions of the degeneracy parameter, Θ , and electron coupling parameter, Γ ee , are used . Cell tra-jectory plots extracted from the reference simulation (Fig. 3)show that the deuterium fuel is mostly ensconced within theideal plasma regime, where D e (cid:28) Γ ee , ii (cid:28)
1, such that W (cid:28)
1. Although at late time, where the neutron emissionrate is highest, the degree of nonideality in the fuel is seento steadily increase. Significantly larger uncertainties can beexpected for the properties of the coverslip and anvil materi-als as they produce states with nonideal electrons, W ∼
1, andstrongly coupled ions Γ ii (cid:38) s ,covering two orders of magnitude; s = { . , . , , } . Thesemay be applied uniformly across phase space, referred to asa ‘blanket approach’, or conversely by a ‘targeted approach’which utilizes the WDM parameter Eq. (18) to focus their ap-plication toward regions with high uncertainty. The effectivescaling parameter for a particular point in ρ − T e space is then s (cid:48) ( ρ , T e ) = +( s − ) W ( ρ , T e ) . (19) The targeted form of the scaling factor crucially ensures thatthe tabulated microphysics are not unreasonably distorted un-der conditions where uncertainty in their forms is believed tobe negligible, e.g. in the high-temperature, low-density limitwhere the well-established Spitzer-type models apply. G. Kinetic (nonlocal) effects
A simple estimate of the degree of locality with regard tocollision-driven transport processes is the Knudsen number N K j = λ j L . (20)Here the collisional mean free path λ j of species j = e , i canbe estimated from simple expressions throughout the simu-lation domain as a function of local thermodynamic variables.For the scale length, L , at a particular location in the system,we take the harmonic mean of inverse logarithmic derivatives L ≈ N X (cid:32) ∑ X | ∇ X | X (cid:33) − , (21)where X = { T i , T e , ρ , Z ∗ i } represents the set of variables whichmost strongly influence the collisional mean free path and N X = .For the times relevant to the fusion output from our simu-lations N K e (cid:28)
1, such that electronic conduction is likely tobe well-described by purely local diffusion (4). On the otherhand, we find N K i ∼ . − Figure 4. Heat maps of ion temperature (left-hand side) and ion Knudsen number (right-hand side) computed using Eqs. (20) and (21) forthe reference simulation discussed in Section II B for different times throughout the cavity implosion: Just prior to stagnation of the incidentshock at the tip of the cavity (left); After stagnation, where the reflection from the tip drives up into the collapsing cavity (middle); During thequasi-isothermal compression phase (right). The Knudsen number is initially of order unity close to the hottest region but quickly falls to alevel at which local ion conduction is appropriate. In these plots the bright green contour denotes the Lagrangian grid representing the materialinterface. nonlocal ionic transport into the anvil material at the pointof shock stagnation where the highest ion temperatures occur(Fig. 4). This nonlocal layer dissipates quickly after shockreflects from the cavity tip.Since the conditions in the fuel throughout the periodswhere nonlocal effects may be significant are conducive to alarge contribution from the ions we expect this to have a stronginfluence on the fuel energetics and, thus, the fusion perfor-mance. Moreover, the reduction of the thermal reactivity is likely to be substantial for the initial shock stagnation event.This is presently being investigated and will reported in a fu-ture publication.
IV. RESULTS AND DISCUSSION
To assess the impact of the configuration changes relativeto the reference simulation (Tab. I) we track three simple met-rics which are sensitive to both the dynamics of the incidentshock and the fuel energetics throughout the period of neutronemission:1. Time of first reflection of the incident shock from theaxis of the simulation, t axis
2. Maximum BWA ion temperature during the stagnationphase, T max .3. Mean value of BWA ion temperature during the com-pression phase, T av .These are presented as as percentage changes relative to thecorresponding values obtained from the reference simulation,i.e. δ i = × (cid:0) M i / M ref i − (cid:1) , where M i stands for the relevantmetric ( i = { , , } ). We originally considered several othermetrics such as the full-width-half-maximum duration of the Variable ValueElectron flux limiter interp. ExponentialIon flux limiter interp. ExponentialElectron flux limiter coef. 0.05Ion flux limiter coef. 0.5Conductivity cell face interp. Arithmetic meanMax heat flux cell face interp. Arithmetic meanTable I. Configuration options of Hytrac’s flux-limited conductionoperator used in the reference simulation. stagnation phase and the standard deviation of the BWA iontemperature in the compression phase; both of these provedunreliable in producing stable trends over the wide range ofperturbations to the reference configuration. Most impor-tantly, we have chosen not to track the total neutron yield re-sulting from each configuration change. This is principallybecause it scales so nonlinearly with ion temperature that anytransient artefacts resulting from small pockets of fuel caughtin the complex structures formed lead to spurious results thatare both essentially impossible to correct for and distort other-wise meaningful and insightful trends. This is an unfortunatebut inescapable consequence of the very low yields ( Y n (cid:38) s = { . , . , , } havebeen applied (individually) using the ‘blanket’ and ‘targeted’manner as described in Section III F. To simplify the analysiswe consider only order-one perturbations, i.e. only a single -40-2002040 δ [ % ] Electron thermal conductivity
Metric 1 (blanket)Metric 2 (blanket)Metric 3 (blanket) -10010 δ [ % ] -10010 δ [ % ] − Effective scaling factor, s Ion thermal conductivity
Metric 1 (targeted)Metric 2 (targeted)Metric 3 (targeted) − Effective scaling factor, s Electron-ion energy exchange rate − Effective scaling factor, s P MM A c o v e r s li p G o l d a n v il D e u t e r i u m f u e l Figure 5. Effect of scaling factors applied on the conduction microphysics models featured the reference simulation; electron thermalconductivity (left-hand column), ion thermal conductivity (middle column) and electron-ion energy exchange rate (right-hand column). Theseare varied in the deuterium fuel (top row), gold anvil (middle row) and PMMA coverslip (bottom row). The three metrics described in the textare plotted in each panel as a percentage change relative to the reference case: Metric 1 (blue markers/line); Metric 2 (orange markers/line);Metric 3 (green marker/line). The horizontal shaded band in each panel between δ = ±
10% represents the maximum impact relative to thereference simulation that we (arbitrarily) deem significant. In each panel the blanket (filled marker connected by solid lines) and targeted (openmarkers connected by dashed lines) approaches to applying the scaling factors are used. aspect of the system is perturbed at a time. This has the dis-advantage that interactions between uncertainties are ignored,which is surely important in a two-temperature, conduction-dominated plasma. On the other hand, establishing the ba-sic trends is of prime importance and lays the foundations formore complex, coupled sensitivity studies in future.Strong trends are found with respect to all the fuel micro-physics using the blanket approach to applying the scaling fac-tors (filled markers connected by solid lines in Fig. 5), partic-ularly in the maximum BWA ion temperature during stagna-tion. This verifies the assertion that transport and equilibrationplay a crucial role in the energetics of the fuel at stagnation.However, the strongest correlation is between the electronicconductivity of the fuel and the average temperature in thecompression phase, with substantially less sensitivity to theionic conductivity and equilibration rate. Electron conduc-tion is therefore the most important loss mechanism duringthe period over which the majority of the neutron output isexpected. This can be readily understood since the plasma isessentially in thermal equilibrium during this phase, such thatthe relative importance of conduction is dictated by the ratio κ e / κ i ∼ ( m i / m e ) / (cid:29)
1. Weaker correlations are evident forthe properties of the anvil and coverslip. The lack of any clearinfluence of any of the scaling factors on the arrival time ofthe first shock on the simulation axis (Metric 1) can be takento mean that the early time dynamics of this target are mostlyhydrodynamic in nature. In all cases the sensitivity of the metrics to any of the scal-ing factors drops when the targeted approach (19) is used(open markers connected by dashed lines in Fig. 5). In thismore realistic case none of the perturbations lead to a changebeyond ± δ [ % ] Contra. Harmonic
Conductivity cell face interpolation method
Metric 1Metric 2Metric 3 -15-10-5051015 δ [ % ] Harmonic Larson Min-cap.
Electron flux limiter interpolation method -20020406080100 δ [ % ] Electron flux limiter coefficient, α e Contra. Harmonic
Max heat flux cell face interpolation method
Harmonic Larson Min-cap.
Ion flux limiter interpolation method
Ion flux limiter coefficient, α i Figure 6. Same as in Fig. 5, but considering the influence of the choice of interpolation scheme for evaluating the tabulated conductivities(left) and max heat fluxes (right) on cell faces using cell-centered quantities (top row), the electron (left) and ion (right) flux limiter coefficients(middle row) and the interpolation method which applies the limited form of the electron (left) and ion (right) heat flux (bottom row). h T i i b [ e V ]
11 12 13 14 15 16 17
Reference caseHarmonic cell face conductivityHarmonic cell face max heat fluxIon flux limiter coefficient = 0.05 Y n [ n e u tr . ]
12 14 16 18 20 22 24 26 28 30 t [ns] Figure 7. Comparison of the BWA ion temperature (top panel)and cumulative yield (bottom panel) from the reference case to thosefrom simulations with the three most significant changes to the stag-nation temperature. Note the change in the temporal scale on the x -axis between the top and bottom panels. mean approximation. This has not been repeated here sincethe indications of the test cases discussed in Sec. III D arethat using Eq. (15) will not properly capture nonlinear ther- mal wave propagation through large conductivity gradients.Nevertheless, it is still interesting to note its wider impacts onthe target performance and in the absence of direct verifica-tion or validation evidence it cannot presently be completelyruled as a modeling option.As shown in the top row of Fig. 6, where the interpolationof cell-centered quantities to the cell faces is performed usingcontraharmonic and harmonic averages, Eqs. (14) and (15),the impact of using the harmonic mean in particular is indeedsignificant. For the harmonic mean case the insulation of thefuel resulting from the dominance of the non-conductive anvilin setting the conduction flux increases the peak stagnationtemperature by a factor of two to three ( T max (cid:38) − . ∼
30% higher ion temperatures achieved when only the max-imum heat flux is constructed on the cell face using Eq. (15).The corresponding amplification in the yield is much larger,being roughy ∼ α i = α e = .
05, which amountsto a ten-fold reduction in the maximum ion heat flux from thereference simulations, again leads to a peak BWA ion temper-ature nearly twice as hot as the reference case during stagna-tion and similarly increased neutron total yield from stagna-tion, whilst the other metrics are essentially unaffected, but issubstantially reduced relative to the purely diffusive expres-sion (4). This is consistent with the findings from the scalingfactor study where the role of ion conduction during the com-pression phase was shown to be negligible. The fact that thereis very little change to any of the metrics if the fraction ofthe maximum ion heat flux is allowed to increase arbitrarily,i.e. for α i ≥
1, again suggests that the ionic heat flow is notstrongly flux-limited. This is further backed up by the ob-servation that switching to the softer harmonic interpolationmethod (7) acts to slightly increase the stagnation temperatureby cutting the effective heat flux, whereas the min-capped ap-proach (6) slightly enhances it. The ratio Q eff / Q max is thenclose to unity (see Fig. 2b) since in the strongly flux-limitedregime one would expect a much larger effect, especially fromusing harmonic interpolation. In contrast, the evidence doesnot suggest that electron heat flow is flux-limited as no sig-nificant sensitivities are observed for any of the configurationchanges considered, in agreement with the expectations notedin Sec. III G.Despite the indication being that the ionic heat flux is quasi-local it is important to note that only separate perturbations tothe operator configuration have been examined, meaning thatthe potential for large changes to the performance caused bycoupling of multiple configuration changes must be assessedbefore stronger conclusions can be drawn. Taking into ac-count the observed strong sensitivities to the treatment of con-duction at the interface and the lack of any compelling evi-dence for the reasonableness of a single, universal value of α i (cid:29) .
05, as well as the fact that high ion Knudsen num-bers of order 0 . − . Nevertheless, we notethat none of the changes considered here result in a total yield close to the lowest reported yields of Derentowicz et al., withthe highest total yield being roughly fives smaller. The workrequired to bring together all these threads in a properly cou-pled manner is presently ongoing, as are efforts to developour understanding of the possible role and impact of kineticion transport on these targets and the influence of the otherradiation-hydrodynamic phenomena left out of this work. V. CONCLUSIONS
We have undertaken an initial examination of the influenceof flux-limited thermal conduction modeling on the robust-ness and performance of the uniaxially-driven conical fusiontargets used in the experiments of Derentowicz et al. . Thiskind of target is of interest to FLF as it is one of the few ex-amples of fusion targets in the literature with observable neu-tron yield, where the initial collapse is driven by a single pla-nar shock produced by conditions within reach of our currentexperimental facilities. The attention given to thermal con-duction in this paper is justified since other phenomena suchas radiation transport and viscous effects are not expected toplay as important a role in determining the energetics of thefusion fuel.The uncertainties which principally affect conduction lossare associated with the related plasma microphysics and thedegree to which the transport is nonlocal. To assess the im-pact of uncertainties in the microphysics we have undertakenscaling factor studies in which the tabulated models are scaledboth uniformly, i.e. a blanket approach, and with a more re-alistic method using the WDM parameter to target regionsof density-temperature space with larger theoretical uncer-tainty. The reference case was produced using an idealizedmodel that was rigorously converged on the basis of root-mean-square differences between time-dependent profiles ofburn-weighted average (BWA) plasma parameters from suc-cessively higher resolution simulations. Simple metrics whichcapture the dynamics of the initial stages of the cavity col-lapse and the energetics of the fusion fuel over the time ofneutron emission were defined, from which direct comparisoncan be made between different simulations and the referencecase. Uniform application of the scaling factors showed thatelectron conduction in the fuel during the compression phasemost strongly affects the target performance. With the morerealistic targeted approach we found very little influence onany of the metrics. This is due to the fact that it is the proper-ties of the fuel that determine its energetics in the compressionphase, which remains strongly ideal, and therefore minimallyuncertain, throughout the simulations. We found that the roleof the nonideal states expected to be produced in the plasticcoverslip and gold anvil is minimised due to the numericaltreatment of the heat flux at material interfaces.Considering alternative interpolation schemes for evaluat-ing cell-centered quantities at cell faces leads to significantchanges to the predicted conditions and also in the charac-ter of the neutron emission. In particular, we found that thelargest influence on target performance comes from changingthe cell-face averaging from the arithmetic mean used in the2reference case to a harmonic mean. Although the latter is notthought to be an especially physical choice as it was shownto be incapable of propagating nonlinear thermal waves be-tween regions of substantially different conductivities, it can-not be explicitly ruled out without direct validation. With thischange the suppression of conduction losses substantially im-pacts all of the metrics, crucially changing the fusion out-put from one dominated by heating due to quasi-isentropiccompression during the cavity collapse to one in which a sin-gle neutron flash is produced by the reflection of the incidentshock from the cavity tip during stagnation. A similar changewas found to result from restricting the ionic heat flux in thereference case from the stagnated fuel to the anvil through areduced flux limiter (set equal to the widely used canonicalvalue used in flux-limited electron conduction ). Based onthe results we conclude that the ionic heat flow is likely to bemoderately flux-limited, i.e. nonlocal kinetic effects may beimportant, whereas the electronic heat flow is evidently morediffusive.If the ionic conduction is indeed minimally restricted, eitherbecause Hytrac’s treatment of the heat flux across material in-terfaces is reasonable or because a small value of the ionicflux limiter coefficient is unreasonable, then our referencecase simulations should be relatively robust. Consequently,the question of why the predicted total yield of these tar-gets is several orders of magnitude less than those reported inRef. remains an open one. Moreover, we noted that none ofthe singular changes to the reference case considered resultedin neutron yield amplified by more than an order of magni-tude. More extensive sensitivity studies incorporating multi-ple, simultaneous, interacting configuration changes as well asmore elements of Hytrac’s multi-physics model, e.g. radiationtransport and viscous effects, are needed to develop a clearerpicture.Finally, we note that the findings related to the potentiallyimportant role of ionic conduction are interesting as they sug-gest that targets similar to those studied here may provide anexperimental platform for accessing states of matter in thequasi-local regime, in which energy transport is close to thetransition between local diffusion and the kinetic transportregime. With suitable diagnostics to provide time-resolvedhistories of neutron production and electron and ion tempera-tures from the stagnated plasma, it may be possible to experi-mentally distinguish between predictions based on simple iontransport models (such as flux-limited conduction) and state-of-the-art kinetic models, whilst minimizing uncertainty as-sociated with electron transport and equilibration. Such datamay prove to be invaluable for benchmarking integrated sim-ulation codes for ICF. DATA AVAILABILITY
The data that support the findings of this study are availablefrom the corresponding author upon reasonable request.
ACKNOWLEDGEMENTS
The authors gratefully acknowledge R. King and J. Gar-diner for providing HPC support and A. Venskus and J. Her-ring for invaluable contributions to the robustness and stabil-ity of Hytrac. We also recognize insightful conversations withA. Crilly and G. Kagan (Imperial College London) regardingequation of state and ionic transport modeling.
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