Neon Cluster Formation and Phase Separation During White Dwarf Cooling
DDraft version October 2, 2020
Typeset using L A TEX twocolumn style in AASTeX63
Neon Cluster Formation and Phase Separation During White Dwarf Cooling
M. E. Caplan, C. J. Horowitz, and A. Cumming Illinois State University, Department of Physics, Normal, IL 61790 Center for Exploration of Energy and Matter and Department of Physics, Indiana University, Bloomington, IN 47405, USA Department of Physics and McGill Space Institute, McGill University, Montreal, QC H3A 2T8, Canada
Submitted to ApJLABSTRACTRecent observations of Galactic white dwarfs (WDs) with Gaia suggest there is a population ofmassive crystallizing WDs exhibiting anomalous cooling – the Q branch. While single-particle Nesedimentation has long been considered a possible heat source, recent work suggests that Ne mustseparate into clusters, enhancing diffusion, in order for sedimentation to provide heating on the observedtimescale. We show definitively that Ne cannot separate to form clusters in C/O WDs using moleculardynamics simulations, and we further present a general C/O/Ne phase diagram showing that strong Ne enrichment is not achievable for Ne abundance (cid:46) Ne cluster sedimentation and that Q branch WDs may have an unusualcomposition, possibly rich with heavier elements.
Keywords:
White dwarf stars (1799), Stellar interiors (1606), Degenerate matter (367), N-body simu-lations (1083) INTRODUCTIONThe Gaia space observatory has determined parallaxdistances to large numbers of Galactic stars (Babusiauxet al. 2018), which allow for unprecedented tests of whitedwarf (WD) models and evolution. Massive WDs havehigh central densities and strong gravities. Furthermore,some of them may have formed via mergers (Hollandset al. 2020) and they may have interesting compositions.Recently, Cheng et al. (2019) found that the popula-tion of massive WD known as the ‘Q branch’ appearto have an additional heat source that maintains a lu-minosity of order 10 − L (cid:12) for Gyrs. Latent heat fromcrystallization (Tremblay et al. 2019; Winget et al. 2009;Horowitz et al. 2010) and gravitational energy releasedfrom conventional Ne sedimentation (Bildsten & Hall2001a; Deloye & Bildsten 2002; Garc´ıa-Berro et al. 2008;Hughto et al. 2010b) do not appear to be large enoughto explain this luminosity (Camisassa et al. 2020; Cheng [email protected]@[email protected] et al. 2019). Heating from conventional electron captureand pycnonuclear (or density driven) fusion (Salpeter &van Horn 1969; Yakovlev et al. 2006; Horowitz et al.2008) reactions appear to need even higher densitiesand may depend too strongly on the density and ortemperature (Horowitz 2020). There are many worksdiscussing dark matter interactions in WD, see for ex-ample (Bertone & Fairbairn 2008; Graham et al. 2018;Hurst et al. 2015; Graham et al. 2015; Bramante 2015;Acevedo & Bramante 2019). However, WD may be toosmall to capture enough dark matter for its annihilationto produce the necessary heat (Horowitz 2020).The gravitational potential of a WD is large. There-fore, there is possibly enough energy available from sedi-mentation of neutron rich Ne to provide the necessaryheating (Bildsten & Hall 2001b). However the expecteddiffusion constant for Ne in a C/O mixture (Hughtoet al. 2010a) is too small to allow enough sedimenta-tion before the C/O mixture freezes. In addition, Nesedimentation is significantly slowed down by C/O crys-tallization. Therefore, even though sedimentation is alarge enough energy source, in practice, sedimentationis likely slow and most of this energy may remain un- a r X i v : . [ a s t r o - ph . S R ] S e p Caplan, Horowitz, & Cumming tapped by the time the star freezes (Deloye & Bildsten2002; Garc´ıa-Berro et al. 2008; Hughto et al. 2012, 2011).It is possible that the Ne sedimentation rate is en-hanced because of the formation of mesoscopic Neclusters. Recently, Blouin et al. (2020) speculate thatNe phase separation produces mesoscopic clusters thatcould enhance the heating from conventional Ne sed-imentation. Numerical calculations by Bauer et al.(2020) consider this possibility in more detail, showingthat clustering of N nuclei enhances the downward driftrate proportional to N / . This suggest that the appro-priate diffusion timescale is achievable with long livedclusters of only few hundred to a few thousand Neions, and is validated with MESA models.In this letter we directly address this possibility withmolecular dynamics (MD) simulations of Ne micro-crystals in C/O mixtures and new calculations of theC/O/Ne phase diagram. We find that our Ne micro-crystals are unstable in C/O liquid mixtures. There-fore enhanced heating from cluster formation or phaseseparation is unlikely to be important in conventionalC/O/Ne mixtures. MOLECULAR DYNAMICS SIMULATIONSIUMD is a CUDA-Fortran classical molecular dynam-ics code and has been extensively used to model astro-materials in WDs and neutron stars (Caplan & Horowitz2017). In our model nuclei are fully ionized and treatedas point particles of charge Z i which interact via ascreened two-body potential V ( r ij ) = Z i Z j e r ij exp( − r ij /λ ) . (1)with periodic separation r ij and screening length λ − =2 α / k F /π / using k F = (3 π n e ) / . The screeningassumes relativistic electrons, which is not the case inWDs. At 10 g/cm , k F ≈ . λ ; the true λ is 21% smaller. At 10 g/cm ,the true λ is 7% smaller. This is expected to have littleimpact on our MD (see Hughto et al. 2011; Hamaguchiet al. 1997). We use λ/a = 2 .
816 for our mixture (consis-tent with Hughto et al. 2010a) which forms a bcc latticeat low temperature (Vaulina et al. 2002).The mixture we consider has number abundances (cid:126)x =( x C , x O , x Ne ) ≈ (0 . , . , .
02) and is well motivatedastrophysically. The He(2 α, γ ) C and C( α, γ ) Oreactions set x C /x O ≈
1, with order 10% variation dueto reaction rates and WD mass (Lauffer et al. 2018).During helium burning the remaining CNO elements arethought to burn via N( α, γ ) F( β + ) O( α, γ ) Ne; as-suming solar metallicity, x Ne ≈ .
02 in the WD (Bild-sten & Hall 2001c).
Figure 1. (Color online) Initial conditions, Ne (red) isconcentrated in a bcc microcrystal in the center of the volumeand is surrounded by a C/O (white) liquid with trace Ne.The configuration is cubic, an orthographic projection is usedfor clarity.
The crystallization of a one-component plasma(OCP) depends on the dimensionless parameter Γ i = Z / i e /a e k B T with ion charge Z i , electron separation a e = (3 / πn e ), and thermal energy k B T . The OCPis solid (liquid) above (below) Γ crit ≈
175 (Potekhin& Chabrier 2000). Screening may raise Γ crit to about178 (Vaulina et al. 2002). Averaging over mixture com-ponents for the multi-component plasma (MCP) givesΓ
MCP = (cid:104) Z (cid:105) / e /a e k B T ; crystallization is more com-plicated and sensitive to the exact mixture. A binaryC/O mixture has Γ MCP ≈ Γ crit ≈ x Ne = 0 . (cid:46) Γ crit (cid:46)
300 (Hughto et al. 2011). If Ne clus-ters are unstable then we expect that the Γ crit we findwill be closer to 300 than 230, as the region of the phasediagram probed will be that of high Ne concentration atthe C/O liquid-Ne cluster surface.Our MD mixture contains 8000 C, 8000 O, and384 Ne ions ( N = 16384 = 2 , for GPU threading).Our initial configuration is shown in Fig. 1 and con-sists of a neon microcrystal embedded in a C/O liquidin a cubic volume with periodic boundaries. The neoncrystal is prepared by trimming ions from the edges andcorners of a cubic bcc crystal to produce a truncatedcuboctahedron containing 253 ions, 6 a on its longest di-ameter. The C/O fluid and the remaining Ne haverandom initial positions around the microcrystal; if the eon Clustering and Phase Separation in WDs Figure 2. (Color online) Intermediate (left) and final(right) configurations of simulations at Γ = 289 (top, solid)and Γ = 287 (bottom, liquid). Colors as in Fig. 1; for claritywe only show the cluster Ne in red.
Table 1.
Summary of MD runs.Γ (Γ C , Γ O , Γ Ne ) Outcome296 (222,359,520) C/O Crystallization around Ne291 (219,353,513) C/O Crystallization around Ne289 (217,351,509) C/O Crystallization around Ne287 (215,348,505) Ne cluster dissolves into C/O285 (214,346,501) Ne cluster dissolves into C/O244 (183,296,429) Ne cluster dissolves into C/O microcrystal is stable we expect it to grow by adsorptionof the remaining 131 Ne.We run isothermal simulations to resolve Γ crit andstudy the stability of the neon cluster at a rangeof temperatures. Constant temperature is approxi-mately achieved by rescaling the velocities to a Maxwell-Boltzmann distribution with the desired temperatureevery 100 timesteps. Our simulations therefore do notconserve energy; instead, in the long time limit our simu-lations trend toward equilibrium so E ( t ) allows us to re-solve melting or freezing ( e . g . heats of fusion). Our sim-ulations differ from past work ( e . g . Hughto et al. 2011)and our asymptotic states may not be true equilibrium( i . e . they may be superheated/cooled) because we onlyrun as long as needed to verify stability or instability ofthe neon cluster. In Tab. 1 we list isothermal simulations run usingthe initial configuration shown in Fig. 1. These simu-lations were run for between 10 and 10 MD timestepswith dt = 1 / ω p with ion plasma frequency ω p =(4 πe (cid:104) Z (cid:105) n/ (cid:104) M (cid:105) ) / . We clearly resolve a first ordertransition between 287 < Γ < ≥
289 the C/Oliquid is supercooled and quickly begins crystallizing,first nucleating around the Ne cluster (Fig. 2 top left)before growing to fill the volume (top right). At Γ ≤ Ne microcrystals are not expected inC/O WDs.Our Γ crit ≈
288 is high compared to past work forC/O/Ne mixtures, which find Γ crit ≈
230 (see Tab. 1in Hughto et al. 2012). Our system may be finding themelting point of a very Ne rich system. Furthermore,the simulation does not have enough time for diffusion tobring the composition of the solid phase into equilibriumwith the composition of the liquid phase. Both finite sizeand finite time effects may be important in comparingto an equilibrium phase diagram (computed below).We now consider what impurities might form stablemicrocrystals. Generally speaking, stronger separationis observed in mixtures with greater contrast in charge Z . While Na or an isotope of Mg may be presentin comparable abundances to Ne they may not havelarge enough Z to strongly separate. Simulations withMg in place of the Ne at Γ = 290 also show a slow subli-mation of the cluster into the liquid, though on slightlylonger timescales than the Ne. In simulations with Ferun at Γ = 240, 262, and 289 the microcrystal persistsafter a few times 10 timesteps. A configuration evolvedfor 4 . × timesteps at Γ = 289 is shown in Fig. 3.In all three simulations the microcrystal shows exchangewith the Fe in the background and evolution in morphol-ogy. Ions on raised facets seem more likely to escapeor migrate to adjacent faces to produce larger smoothsurfaces, possibly an octahedron (stability, growth, anddiffusion of various microcrystal morphologies may beof interest to future authors). Thus, while Ne doesnot form stable microcrystals, higher Z impurities areviable candidates for phase separation and clustering. Caplan, Horowitz, & Cumming
Figure 3. (Color online) Evolved Fe microcrystal at Γ =289 (Γ C = 187, Γ O = 301, Γ Fe = 2149). Some small amountof Fe initially in the microcrystal (orange) escapes into thebackground or rearranges at edges, while a similar amountof background Fe (yellow) is captured on the surface.3. C-O-NE PHASE DIAGRAMAlthough mixtures with x Ne ≈ .
02 cannot form sta-ble microcrystals, mixtures with larger Ne abundancescould. Therefore, we compute the ternary phase dia-gram to determine what Ne abundance may be requiredfor such strong phase separation. We use the code developed in Caplan et al. (2018) which implementsthe semi-analytic method of Medin & Cumming (2010).This method identifies pairs of points on the minimumfree energy surfaces which share a tangent plane ( i . e . thedouble tangent construction); these points correspond tocoexisting solid and liquid compositions. Compositionslying on the tangent line connecting them are thereforeunstable and phase separate. For a detailed discussionsee Medin & Cumming (2010); Caplan et al. (2018).In Fig. 4 we show C/O/Ne phase diagrams for three Γ.We report temperature in Γ C ∝ /T . Orange liquiduspoints are connected to corresponding blue solidus bygreen tie lines corresponding to the tangent in free en-ergy. Below (above) the orange (blue) curve is stableliquid (solid), while the green tie lines span the unstableregion.At the lowest temperature (Fig. 4a), it is possible toform solid particles that are substantially enriched inNe. The phase diagram shows solid-solid coexistencewith strongly Ne enriched mixtures, where both theNe-enriched and Ne-depleted crystals show x Ne > . The code is available at https://github.com/andrewcumming/phase diagram 3CP. Γ C = . N e O C (a) Γ C = . N e O C ( b ) Γ C = . N e O C ( c ) Figure 4. (Color online) C/O/Ne phase diagram. Theliquidus (orange) and solidus (blue) are connected by tie-lines (green) showing solid-liquid equilibrium. To find a givencomposition (cid:126)x = ( x C , x O , x Ne ) in the phase diagram lines ofconstant x i are projected from the slope of the tick mark onthe relevant axis. For example, pure Ne is found in the bot-tom left corner while two-component CO mixtures are foundalong the right axis, so our (cid:126)x = (0 . , . , .
02) mixture liesat a point near the middle of the right axis. see the emergence of stable liquid near the C/Ne axis(bottom), found in the white region under the liquidusnear x C ≈ .
75. Along the C/Ne axis we see solid- eon Clustering and Phase Separation in WDs Z Ne /Z C = 1 .
66 we resolve eutectic separation.At intermediate temperature (Fig. 4b) we reach Γ crit for our mixture, (cid:126)x = (0 . , . , . x Ne (cid:46) .
3. Observe that the coexistencelines are approximately parallel to the C/O axis, whichimplies they fall on lines of constant Ne. Therefore, theNe fraction in the solid and liquid are nearly equal whichsuggests that any C/O/Ne mixture with x Ne (cid:46) . . ≤ x Ne ≤ . Z /Z and Z /Z studied by Caplan et al. (2018). The central ‘wedge’between the break in tie lines contains unstable mix-tures which do not fall on a single coexistence line sothey cannot separate into a single solid and single liquidcomposition, but they can separate by forming appro-priate amounts of the two solids and one liquid at thecorners of this region.At high temperature (Fig. 4c) the solidus and liquiduscurves are continuous and the region for the eutecticseparation has closed. Though mixtures with x Ne ≈ .
30 now show weak enhancement in solidus Ne, thosewith x Ne (cid:46) .
30 are still largely consistent with two-component C/O separation without any Ne enrichmentor depletion.Even if there is factor of 2 variation in x O /x C it onlytranslates our mixture parallel to the O axis, which doesnot move our mixture into a region where it achievesstrong Ne purification at any Γ. We conclude that thebehavior of Ne microcrystals in MD is fully consistentwith the known C/O/Ne phase diagram, and that nomesoscopic effects exist that make Ne cluster forma-tion likely.As past MD has only studied phase coexistence up to20% Ne, we also perform a few MD simulations to val-idate the behavior around 30% Ne. Similar to Hughtoet al. 2012; Caplan et al. 2018, configurations are pre-pared by joining a cubic bcc crystal on one face witha cubic volume of liquid (as in Fig. 1 in Hughto et al.2012 and Fig. 1 in Caplan et al. 2018). Compositions forthe solid and liquid are chosen to approximately matchpredictions from the phase diagram at Γ C = 183, allow-ing us to verify the strong break in the center of Fig. 4b. Our first mixture has low Ne, (cid:126)x = (0 . , , , . (cid:126)x s ≈ (0 . , . , .
3) and a liq-uid (cid:126)x l ≈ (0 . , . , . (cid:126)x = (0 . , , , . (cid:126)x s ≈ (0 . , . , .
75) and (cid:126)x l ≈ (0 . , . , . timesteps with little ob-served evolution. Runs varying Γ C up and down respec-tively find quenching of diffusion in the liquid and melt-ing of the crystal, suggesting these temperature varia-tions have moved our mixtures into the region of sta-ble liquid and stable solid. Taken together, these sim-ulations have qualitative agreement with our phase di-agram, and we conclude that the separation behaviornear 30% Ne concentration is likely physical. Futurework may be interested in performing a more thoroughsurvey at high Ne concentration with MD, though thismay have limited astrophysical relevance. DISCUSSIONWe find that Ne microcrystals are always unstablein a C/O liquid. Either the temperature is high enoughthat the crystal melts and the Ne dissolves into the liq-uid, or the whole system including the C/O mixturefreezes. Note that even at temperatures below the melt-ing point of pure Ne, but above the C/O melting point,a large entropy of mixing causes the small concentra-tion of Ne to dissolve into the bulk liquid. The C/O/Nephase diagram suggests that very much more Ne is nec-essary before it phase separates. One needs not 2% but ≈
30% or more. As a result, a conventional C/O WDwith n Ne = 0 .
02 is not expected to form stable neonclusters with enhanced sedimentation. In summary, wefind that there are no conditions where a Ne-enrichedcluster is stable in a C/O WD, and therefore, enhanceddiffusion of Ne cannot explain the Q branch.What compositions could then explain the heatingthat Cheng et al . infer? As seen in our phase diagram,unless the C/O ratio or Ne abundance is tuned to ex-tremes we don’t expect strong Ne separation, so wesuggest that Q branch WDs may have an anomalouscomposition. For example, Camisassa et al. (2020) havesuggested that x Ne = 0 .
06 can provide heating on thedesired timescale considering only single-particle diffu-sive settling rather than clusters. Another possibility is ≈
1% abundance of another impurity, besides Ne, withan even larger charge Z which would allow it to phaseseparate even when Ne does not. This impurity wouldneed to be neutron rich ( Z/A < .
5) to be a sedimen-tary heat source and have an abundance of a percent ormore for there to be enough gravitational energy avail-able. Our MD with a high purity microcrystal showsthat Z = 11 Na and Z = 12 Mg should not strongly Caplan, Horowitz, & Cumming separate in a C/O mixture. Phase diagrams of C/O/Naand C/O/Mg mixtures produced using our semi-analyticmethod (omitted for length) are similar to the C/O/Nein that they do not separate to form a solid enrichedin the high Z impurity when it is only abundant at thepercent level, so isotopes such as Na or Mg are poorcandidates for clustering.Iron-group elements provide another possibility, as wereadily observe long lived Fe microcrystals in MD. AsΓ Fe ≈ C / O , Fe in C/O will phase separate and likelydoes not require fine tuning of the mixture. While sed-imentation of 0.1% Fe by mass may produce some no-table heating, if some astrophysical process enriches Qbranch WDs up to ≈
1% mass fraction then settlingout of Fe could provide heating for several Gyr as anotherwise conventional C/O WD cools. Thus, this workmotivates including Fe in WD cooling models. This willrequire new phase diagrams of Fe and a survey with MD of the clustering and the characteristic sizes of Feclusters, which will be the subject of future work.ACKNOWLEDGMENTSWe thank S. Cheng and E. Bauer for helpful discus-sions. CH’s research was supported in part by US De-partment of Energy Office of Science grants DE-FG02-87ER40365 and DE-SC0018083. The authors acknowl-edge the Indiana University Pervasive Technology Insti-tute for providing supercomputing and database, stor-age resources that have contributed to the research re-sults reported within this paper. This research was sup-ported in part by Lilly Endowment, Inc., through itssupport for the Indiana University Pervasive Technol-ogy Institute. AC is supported by an NSERC DiscoveryGrant, and is a member of the Centre de recherche enastrophysique du Qubec (CRAQ).REFERENCES
Acevedo, J. F., & Bramante, J. 2019, Phys. Rev. D, 100,043020, doi: 10.1103/PhysRevD.100.043020Babusiaux, C., van Leeuwen, F., Barstow, M. A., et al.2018, A&A, 616, A10, doi: 10.1051/0004-6361/201832843Bauer, E. B., Schwab, J., Bildsten, L., & Cheng, S. 2020,arXiv e-prints, arXiv:2009.04025.https://arxiv.org/abs/2009.04025Bertone, G., & Fairbairn, M. 2008, Phys. Rev. D, 77,043515, doi: 10.1103/PhysRevD.77.043515Bildsten, L., & Hall, D. M. 2001a, The AstrophysicalJournal, 549, L219, doi: 10.1086/319169—. 2001b, The Astrophysical Journal, 549, L219,doi: 10.1086/319169—. 2001c, The Astrophysical Journal Letters, 549, L219Blouin, S., Daligault, J., Saumon, D., B´edard, A., &Brassard, P. 2020, arXiv e-prints, arXiv:2007.13669.https://arxiv.org/abs/2007.13669Bramante, J. 2015, Phys. Rev. Lett., 115, 141301,doi: 10.1103/PhysRevLett.115.141301Camisassa, M. E., Althaus, L. G., Torres, S., et al. 2020,arXiv e-prints, arXiv:2008.03028.https://arxiv.org/abs/2008.03028Caplan, M. E., Cumming, A., Berry, D. K., Horowitz, C. J.,& Mckinven, R. 2018, The Astrophysical Journal, 860,148, doi: 10.3847/1538-4357/aac2d2Caplan, M. E., & Horowitz, C. J. 2017, Rev. Mod. Phys.,89, 041002, doi: 10.1103/RevModPhys.89.041002 Cheng, S., Cummings, J. D., & M´enard, B. 2019, TheAstrophysical Journal, 886, 100,doi: 10.3847/1538-4357/ab4989Deloye, C. J., & Bildsten, L. 2002, ApJ, 580, 1077,doi: 10.1086/343800Garc´ıa-Berro, E., Althaus, L. G., C´orsico, A. H., & Isern, J.2008, ApJ, 677, 473, doi: 10.1086/527536Graham, P. W., Janish, R., Narayan, V., Rajendran, S., &Riggins, P. 2018, Phys. Rev. D, 98, 115027,doi: 10.1103/PhysRevD.98.115027Graham, P. W., Rajendran, S., & Varela, J. 2015, Phys.Rev. D, 92, 063007, doi: 10.1103/PhysRevD.92.063007Hamaguchi, S., Farouki, R. T., & Dubin, D. H. E. 1997,Phys. Rev. E, 56, 4671, doi: 10.1103/PhysRevE.56.4671Hollands, M. A., Tremblay, P. E., G¨ansicke, B. T., et al.2020, Nature Astronomy, 4, 663,doi: 10.1038/s41550-020-1028-0Horowitz, C. J. 2020, Nuclear and dark matter heating inmassive white dwarf stars.https://arxiv.org/abs/2008.03291Horowitz, C. J., Dussan, H., & Berry, D. K. 2008, Phys.Rev. C, 77, 045807, doi: 10.1103/PhysRevC.77.045807Horowitz, C. J., Schneider, A. S., & Berry, D. K. 2010,Phys. Rev. Lett., 104, 231101,doi: 10.1103/PhysRevLett.104.231101Hughto, J., Horowitz, C. J., Schneider, A. S., et al. 2012,Phys. Rev. E, 86, 066413,doi: 10.1103/PhysRevE.86.066413 eon Clustering and Phase Separation in WDs7