Effects of Atomic Mixing in Inertial Confinement Fusion by Multifluid Interpenetration Mix Model

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1 Commun. Theor. Phys. (Beijing, China) 52 (2009) pp c Chinese Physical Society and IOP Publishing Ltd Vol. 52, No. 6, December 15, 2009 Effects of Atomic Mixing in Inertial Confinement Fusion by Multifluid Interpenetration Mix Model GU Jian-Fa and YE Wen-Hua Institute of Applied Physics and Computational Mathematics, Beijing , China (Received February 4, 2009; Revised July 10, 2009) Abstract The effects of atomic-level mixing are systemically investigated in a multifluid interpenetration mix model, and results are compared with the single-fluid model s simulations and experimental data. It is shown that increasing the model free parameter α, shock Mach number, and the initial density discontinuity makes the mix length and fraction of mixing particle increase, resulting in the lower shock temperatures compared with the results of single-fluid model without mixing. Recent high-compressibility direct-drive spherical implosions on OMEGA are simulated by the interpenetration mix model. The calculations with atomic mixing between fuel and shell match quite well with the observations. Without considering any mixing, the calculated neutron yields and ion temperatures are overpredicted; while inclusion of the interpenetration mix model with the adjustable parameter α could fit the simulated neutron yields and ion temperatures well with experimental data. PACS numbers: z, wj, Ne, Ex Key words: multifluid interpenetration mix model, atomic mixing, inertial confinement fusion 1 Introduction Fluid mixing is an important issue in inertial confinement fusion (ICF). In the process of ICF capsule implosion, some cold shell material (such as plastic or glass) mixes with the hot fuel (usually deuterium and tritium) due to instability growth, and this may quench the nuclear burn and degrade the performance of capsule. [1] ICF experiments have found that the measured yields of gas filled capsules degrade with increasing convergence ratio, [2 4] and the shell-fuel mixing is widely accepted as the dominated mechanism for the degradation. So, it is important to control and mitigate mixing for achieving ignition. Meanwhile, hydrodynamic simulations without any mixing overpredict the neutron yield compared with experiments, [5] then various theoretical models have been proposed to describe fluid mixing, such as Hybrid model [6] and Youngs model, [7 8] in which the Rayleigh Taylor (RT) instability growth seeded by nonuniformity makes chunks of shell and fuel into bubbles and spikes. The mixing above is on large-scales, but direct experimental evidence for atomic-level mixing has been obtained from deuterated plastic (CD) and CD offset capsule implosions on OMEGA laser system. [9 11] It is proved that the small-scale mixing (i.e. atomic mixing) is an important component of mixing, in which fluid species are homogenously mixed (sharing the same volume) and individual atoms can directly react with each other in the mixing zone; while for the chunk mixing, chunks occupy separate volumes and reactions are prohibited between chunks. A multifluid interpenetration mix model was proposed by Scannapieco and Cheng, [12 13] which is used to describe atomic mixing. Unlike other models which have more than one undetermined parameters and are lack of rigorous derivation, this model is derived rigorously from the collisional Boltzmann equation in a self-consistent manner. [14] It has only one free parameter α, which is included in the effective collisional frequency to describe binary collisions at an atomic level, and its value can be determined by experimental data. Several authors have used the mix model to research indirectly and directly driven implosions and many useful results were obtained. [15 17] However, the investigations of atomic mixing effects on multifluid system still lack. Moreover, the single-fluid model generally used in the existing hydrodynamic codes is not adequate to describe complex phenomena of the multiscale and multifluid ICF system. Consequently it is necessary to investigate the multifluid physical model with mixing. In this paper, we carry out those researches systemically, and the influences of the model parameters, the shock Mach number and the initial density discontinuity on the mixing properties of shock problem are also analyzed in detail. Besides, recent high-compressibility direct-drive spherical implosions on OMEGA with a lower laser intensity W/cm 2[18] are simulated by the multifluid interpenetration mix model, and the results are compared with experimental data and the clean simulations without mixing. The rest of this paper is organized as follows. In Sec. 2, the models and the basic formulae are given. In Sec. 3 we describe the results. The conclusions are presented in Sec Multifluid Interpenetration Mix Model In the interpenetration mix model, the dynamical equations of species s in a multifluid system are written Supported by the National Basic Research Program of China under Grant No.2007CB815100, and the National Natural Science Foundation of China under Grant Nos and

2 No. 6 Effects of Atomic Mixing in Inertial Confinement Fusion by Multifluid Interpenetration Mix Model 1103 as follows: [12] t ρs + (vj x ρs ) + ( Uj s j x ρs ) = Scoll s, (1) j t (ρs Uj s ) + (vi x ρs Uj s ) + (Pij s i x + Rs ij ) i ρs ρ (Pij + R x ij) + ρ s Ui s vj i x i = (A s j ) coll, (2) t es + (vj x es ) + ( Uj s j x es ) j + x j (Q s Tj + P s ji U s i ) + (P s ij + R s ij) x i v j = ρs ρ Us j x i (P ij + R ij) + E s coll. (3) Summing Eqs. (1) (3) over all species, we get the dynamical equations of the bulk fluid: t ρ + (vj ρ ) = 0, (4) x j ρ ( t v j + vi vj x i t e + (vj x e ) + Q Tj j x j + s ) + x i (P ij + R ij) = 0, (5) x j [ U s j (e s + P s ) + Π s ji U s i ] + (Pij + Rij) vj = 0, (6) x i where v and Uj s denote the mean-mass weighted bulk flow velocity and the peculiar velocity of specie s. The terms on the right side of Eqs. (1) (3) are the collision terms, which represent the time rate of change of mass, momentum, and energy due to collisions of particles with one another. The model assumes that binary collisions provide the main contribution in the collisional process, and multiple-particle interaction can be thought of as the sum of simultaneous binary collisions. Therefore, the mixing at an atomic level can be researched by the mix model. In the collision terms, a phenomenological collisional frequency is included as νeff ss = /ρ )(C (ρs s /λ ss ), in which the enhanced mean free path of specie s is given by λ ss = λ c + α ss Uj s Us j dt, (7) where the first term λ c represents the normal plasma collisional mean free path, [19] and the second term is the mean free path enhanced due to the difference of the peculiar velocities between species. α ss is the only free parameter and can be determined by experimental data. Other notations in the above formulae are same as those used in Ref. [12]. Together with the equations of state (EOS) of fluid species, Eqs. (1) (6) can be solved self-consistently and we could, in principle, uniquely determine the dynamics of the multifluid system. We have successfully completed a one-dimensional hydrodynamics code for the multifluid interpenetration mix model in the Cartesian and spherical coordinates. To describe mixing at the atomic level, a high resolution Eulerian Flux Corrected Transport (FCT) algorithm with sixth-order phase accuracy is used in the code. [20] Meanwhile, we introduce the adaptive mesh refinement by tracing the ablation front and the interface between distinct fluids. Moreover, the code includes a series of subroutines such as the electron flux-limited thermal conduction, laser inverse-bremsstrahlung absorption, and electron bremsstrahlung. [21] Using a series of test problems, the results calculated by the mix model s code are in quite good agreement with the exact solutions. 3 Simulation Results 3.1 Shock Problem To research the mixing effects of the multifluid interpenetration model, a shock problem is researched in an ideal gas system. A strong shock wave passes from specie 1 (ρ 1 = 1.0, v 1 = 0, and T 1 = 100 K) with the inflow boundary condition to the other specie 2 (ρ 2 = 0.01, v 2 = 0, and T 2 = 10 4 K) with the solid wall boundary condition. Since α is the only free parameter of the mix model, we change the parameter values to qualitatively investigate mixing effects on hydrodynamics properties of the multifluid system. Figure 1 depicts spatial distribution of the bulk mass density, pressure, mean-mass weighted bulk flow velocity, and temperature with the shock mach number Ma = 30. The results with the mix model are compared with those without any mixing. It is seen that inclusion of the mix model makes the propagation of shock wave faster than the result without mixing. Increasing the parameter α, the shock wave becomes even faster. Meanwhile, mixing with larger α depresses the shock temperature, and the length of mixing zone increases when the larger values of α are taken. By comparing the mean free path enhanced by the mix model with the normal plasma collision mean free path λ c, we found that the mean free path enhancement due to the turbulent eddies (i.e. the second term of Eq. (7)) is two orders larger than λ c in the mixing zone. This leads to the effective collisional frequency much smaller than the result without mixing. Consequently the interpenetration mixing appears quite easier among particles of distinct fluids; since the decrease of collisions frequency makes less kinetic energy of fluid convert into internal energy in the mix zone, the shock temperatures with atomic mixing are lower than simulations without any mixing. Using the same shock problem as Fig. 1, we calculate the mix quantities with different Mach numbers. Figure 2 shows the mix length and the fraction of particle number of specie 1 in the mix zone with α = We can see that the mix length and the fraction increase with the increasing Mach number. When the Mach number is small (Ma = 5 and 10), mixing effects are not remarkable, and results of the interpenetration mix model are similar to those without mixing. When larger Mach numbers are taken (Ma = 20 and 30), the mix quantities dramatically increase and mixing effects become quite significant compared with simulations without mixing.

3 1104 GU Jian-Fa and YE Wen-Hua Vol. 52 Fig. 1 The bulk mass density, pressure, mean-mass weighted bulk flow velocity, and temperature profiles at t=1ns for the shock problem with the mix model parameter 5α and 20α. The results are compared to that of singlefluid model without mixing. Fig. 2 The mix length and the fraction of mixing particle 1 in the mixing zone for the shock problem with Ma = 5, 10, 20, and 30. In Fig. 3, we plotted the mix length with the different initial densities of specie 2 in the same shock problem. The shock Mach number is 20, and the parameter α = One can see that the mix length calculated by the mix model increases when the initial density discontinuity increases. The reason is that the deceasing density of specie 2 produces the lower effective collisional frequency, consequently the appearance of interpenetration mixing becomes easier. Therefore, in a multifluid system with strong shock wave or large initial discontinuity, it is not adequate to simulate the physical properties only by the singlefluid model without mixing, and inclusion of the multifluid interpenetration mix model will give more reasonable results. Fig. 3 The mix length versus time for the shock problem with Ma = 20 and the initial density ρ 2 = 0.1, 0.01, and g/cm Simulation of OMEGA s Direct-Drive Implosion Smalyuk reported high-compressibility direct-drive spherical implosion experiments, which were carried out on the 60-beam OMEGA laser system (λ = 351 nm) with

4 No. 6 Effects of Atomic Mixing in Inertial Confinement Fusion by Multifluid Interpenetration Mix Model 1105 a 3 ns square pulse shape. [18] The total energy is 12.5 kj and the laser intensity W/cm 2. The capsules have an outer radius of 430 µm, 24 µm, and 35 µm CH shell filled with 15 atm of D 2 gas. The laser beams with and without 1 THz, two dimensional smoothing by spectral dispersion (2D SSD) were used in the implosions. The experiments provide a direct relation between modulation levels produced by RT instability and spherical target performance. We simulate the implosions by the multifluid interpenetration mix model, and the calculated results are compared with observed data. for thicker shell with SSD are more stable, the implosion velocities are smaller due to the heavier mass of shell, resulting in the less compressibility and less observed yields; while the performances of implosions without SSD are less sensitive to thickness of shell due to the stronger mixing effects. We can conclude that the multifluid interpenetration mix model successfully describes the yield degradation of high-compressibility implosions. Fig. 5 Same as Fig. 4 but for ion temperature as a function of shell thickness. Fig. 4 DD neutron yield as a function of target thickness. The observed results of the OMEGA spherical implosions are compared to the calculated results by the multifluid interpenetration mix model with α = 0.05, 0.07, and 0.1. The black stars denote unmixed calculations. Figure 4 plotted the DD neutron yield versus target thickness with the mix model parameter α = 0.05, 0.07, and 0.1. The results without mixing (clean calculation) are also represented. It is shown that the observed yields with SSD are higher than those without SSD. This means that implosions with SSD can mitigate the RT instability growth and decrease the mixing effects, resulting in the increasing yields. The clean calculated yields are about one order higher than the measured results. The interpenetration mix model can effectively decrease simulated yields close to experimental data. For 35-µm-thick shell, the calculation with α = 0.07 matches the observed yields without SSD, and α = 0.05 (0.1) describes too little (much) yield degradation. In 24-µm-thick shell, the observed yields with SSD are matched by the α = 0.07 calculation, and the result with α = 0.1 is in good agreement with the yields without SSD; while α = 0.05 could not provide enough yield degradation. As we know, the implosions with thick shell and with SSD are more stable than those with thin shell and without SSD, the ratios of observed yield and the yield calculated without mixing (YOC or yield over clean) for 35-µm-thick shell are higher than those of 24-µm-thick target. This means that mixing effects of thinner shell are greater than those of thicker shell. Therefore, the values of α fitted for thinner shell (α ) are larger than the values fitted for thicker shell (α ). Although the implosions Figure 5 shows ion temperatures of 24- and 35-µmthick capsules. The observed temperatures with SSD are higher than those without SSD, and the temperatures are not sensitive to thickness of shell. The calculated temperatures without any mixing are much higher than the observed values, and inclusion of the interpenetration mix model could effectively decrease the calculated temperature to fit for the observed data. The ion temperatures with atomic mixing match quite well with the values without SSD, and are lower than those with SSD. Meanwhile, the temperatures obtained from the mix model are not sensitive to the model parameter α and shell thickness. 4 Conclusions In this paper we have systemically investigated the effects of atomic mixing by the multifluid interpenetration mix model, and the results are compared with those of singlefluid model without any mixing and the experimental data. In the shock problem, increasing the free model parameter α could make the shock wave propagate faster and shock temperature lower; that is because the mean free path of particles in the mix zone increases as the larger value of α is taken, and the lower binary collisional frequency results in the greater interpenetration mixing and less conversion of kinetic energy into internal energy. Moreover, higher shock Mach numbers and lower initial densities ρ 2 increase the mix length and fraction of mixing particle, indicating the greater mixing effects. Recent high-compressibility direct-drive spherical implosions on OMEGA are simulated by the multifluid interpenetration mix model. The clean calculation without any mixing overestimates neutron yield and ion temperature, and the degradation of observed results could be

5 1106 GU Jian-Fa and YE Wen-Hua Vol. 52 explained by mixing between fuel and shell using the interpenetration mix model. The calculated DD neutron yield with α = matches the observed results for 35-µm-thick shell; while the measured yields for 24- µm-thick shell are in good agreement with the calculated results with α = This indicates that thinner shells have greater mixing effects. Besides, the calculated ion temperatures are not sensitive to the shell thickness and the parameter α. Further studies on atomic mixing of direct- and indirect-drive implosions will be carried out by the interpenetration mix model. Acknowledgments We would like to thank Xiao-Jin Yu, Jun-Feng Wu, Chang-Li Liu, Heng Yong, Yao-Ming Gao, and Wen-Bing Pei for help discussions. References [1] J.D. Lindl, Phys. Plasmas 2 (1995) [2] T.R. Boehly, D.L. Brown, R.S. Craxton, et al., Opt. Commun. 133 (1997) 495. [3] G.R. Bennett, J.M. Wallace. T.J. Murphy, et al., Phys. Plasmas 7 (2000) [4] P. Amendt, R.E. Turner, and O.L. Landen, Phys. Rev. Lett. 89 (2002) [5] P.B. Radha, J. Delettrez, R. Epstein, et al., Phys. Plasmas 9 (2002) [6] C.W. Cranfill, A New Multifluid Turbulent-Mix-Model, Los Alamos National Laboratory, Report LA-UR (1992). [7] D.L. Youngs, Physica D 133 (1984) 84. [8] D.L. Youngs, Physica D 37 (1989) 270. [9] C.K. Li, F.H. Seguin, J.A. Frenje, et al., Phys. Rev. Lett. 89 (2002) [10] J.R. Rygg, J.A. Frenje, C.K. Li, et al., Phys. Plasmas 14 (2007) [11] J.R. Rygg, J.A. Frenje, C.K. Li, et al., Phys. Rev. Lett. 98 (2007) [12] A.J. Scannapieco and B.L. Cheng, Phys. Lett. A 299 (2002) 49. [13] B.L. Cheng and A.J. Scannapieco, Phys. Rev. E 72 (2005) [14] E.H. Holt and R.E. Haskell, Foundations of Plasma Dynamics, Macmillan, New York (1965). [15] D.C. Wilson, A.J. Scannapieco, C.W. Cranfill, et al., Phys. Plasmas 10 (2003) [16] D.C. Wilson, C.W. Cranfill, C.R. Christensen, et al., Phys. Plasmas 11 (2004) [17] C.R. Christensen, D.C. Wilson, C.W. Barnes, et al., Phys. Plasmas 11 (2004) [18] V.A. Smalyuk, R. Betti, J.A. Delettrez, et al., Phys. Plasmas 14 (2007) [19] S.I. Braginskii, Reviews of Plasma Physics, Vol. 1, Consultants Bureau, New York (1965). [20] J.P. Boris, A.M. Landsberg, E.S. Oran, et al., LCPFCT- Flux Corrected Transport Algorihm for Solving Generalized Continuity Equations, Naval Research Laboratory, Washington DC (1993). [21] J. Zhang, T.Q. Chang, et al., Fundaments of the Target Physics for Laser Fusion, National Defense Industry Press, Beijing (2004).