Plasma Modeling I: Modeling of plasmas. Annemie Bogaerts
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1 Plasma Modeling I: Modeling of plasmas Annemie Bogaerts Department of Chemistry, University of Antwerp (UA), Research group PLASMANT Belgium 2 th International School on Low Temperature Plasma Physics Bad Honnef, October 2, 216
2 Contents 1. General overview of plasma models - Explanation of different models - Advantages and disadvantages 2. Examples of some models for various applications + Typical results
3 1. General overview of plasma models
4 A. Analytical model Principle: Simple analytical formulas, valid for specific range of conditions e.g.: Condition for self-sustaining discharge: e - + A 2 e - + A L = 1 st Townsend coefficient (ionization) = 2 nd Townsend coefficient (sec. e - emission)
5 A. Analytical model (cont) Electron multiplication in plasma, by electron impact ionisation: dn Balance equation: e e dx N N N exp dx e - + A 2 e - + A + Solution: e e, - + At cathode: N e, N N exp L At anode: e e, L Hence: for 1 electron at cathode: exp(l) electrons at anode Hence formed in plasma: exp(l) 1 Number of electrons formed = number of ions formed Secondary electron emission: Number of sec.electrons formed by these ions: * (exp(l) 1) Condition self-sustaining discharge : * (exp(l) 1) = 1 L ln 1 1
6 A. Analytical model (cont) Advantage: Simple, fast Disadvantage: Approximation, limited validity
7 Principle: B. D chemical kinetics model Also called: Global model Rate equations (balance equations) for all species, based on production/loss by chemical reactions: ni Rprod irloss i t Rprod / Rloss k n n (or :k n n n, or :An ) 2 B B Electron impact reaction rates: From Boltzmann solver ( Look-up tables: k()) + Electron energy balance equation ( )
8 B. D chemical kinetics model (cont) Advantage: Simple, fast (detailed chemistry) Disadvantage: Approximation: No transport (assumes plasma = uniform) However: based on gas flow velocity: can deduce spatial behavior from temporal behavior (equivalence batch reactor plug flow reactor)
9 B. D chemical kinetics model (cont) Example: Plasma jet Plasma jet TU/e (2 slm Ar gas feed, 6.5 W, 5% relat. air humidity) W. Van Gaens and A. Bogaerts, J. Phys. D: Appl. Phys. 46, (213)
10 B. D chemical kinetics model (cont) Example: Plasma jet Plasma jet TU/e (2 slm Ar gas feed, 6.5 W, 5% relat. air humidity) O, O 3,O 2 (a), OH, H 2 O 2 NO, NO 2,HNO 2,HNO 3 O at longer distance O 3,O 2 (a) at longer distance OH dimerizes into H 2 O 2 (~ 1.2 cm) Cluster formation important in ion chemistry O 2 and NO 3 :relativelylow (ppb) levels W. Van Gaens and A. Bogaerts, J. Phys. D: Appl. Phys. 46, (213)
11 B. D chemical kinetics model (cont) Example: DBD reactor: filamentary character Reactor length + gas flow residence time (correlation spatial temporal behavior)
12 B. D chemical kinetics model (cont) Example: DBD reactor: filamentary character Nanosecond pulses Interpulse time ~µs 1 microdischarge pulse + afterglow -T e ~2-3eV (3 ns) - N e ~1 13 cm -3
13 B. D chemical kinetics model (cont) Example: DBD reactor: filamentary character 5 consecutive pulses: species pass through several filaments (as function of distance, or function of residence time)
14 Principle: C. Fluid model: 1D or 2D Moment equations of Boltzmann equation: Conservation of mass, momentum, energy Continuity equations (for all species i): n(z,r) i t j (z,r) Rprod (z,r) Rloss (z,r) i i i Momentum equation: often drift-diffusion approximation: Transport equations (drift-diffusion for charged species; diffusion for neutrals): j i(z,r) in i(z,r)e(z,r) D in i(z,r)
15 Advantage: Simple, fast (detailed chemistry) Coupling to Poisson equation for self-consistent E-field Disadvantage: C. Fluid model (cont.) Electron energy balance equation: n e jw e jw E ki ne Ng i t i 5 5 jw e ne E De ne e V n n n X X e EV Ions, neutrals: assumed T g Approximation (assumes species gain more or less same amount of energy from E-field as they lose by collisions
16 D. Boltzmann model Principle: Full solution of Boltzmann transport equation (terms with E-gain, E-loss) f j f j(r,v,t) : vrf j avf j C el(f j) C inel(f j) t Advantage: Accounts for non-equilibrium behavior Disadvantage: Mathematically complex Typically only used for electron behavior (e.g., 2-term approximation): e.g., BOLSIG+
17 E. Monte Carlo model Principle: Treats particles on lowest microscopic level For every particle during successive time-steps: * trajectory by Newton s laws: z z x y x y v z v v x y t t t qe 2m qe qe ax rad rad * collisions by random numbers t 2m 2m 2 cos( ) sin( ) t 2 v v z x qe m qe qe t cos( ) t m sin( ) 2 rad t v v t y v v z x y ax rad m
18 E. Monte Carlo model (cont) Probability of collision: Prob coll 1exp s(n coll (E)) Compare with RN ( - 1): If Prob < RN => no collision If Prob > RN => collision Kind of collision: partial collision prob. + compare with RN 1 Pexcit Pioniz Pelast
19 E. Monte Carlo model (cont) New energy + direction: scattering formulas + RN, e.g.: - after ionization: E prim = (E E ion )*RN ; E sec = E E ion E prim or: RN E prim ion diff E d, (, ) ion ( E ) - after excitation: E = E - E excit - scattering angles (, ): 2RN cos 1 ; 2 RN 18 (1RN) - new angles (, ): sin( )cos( ) sin( ) sin( ) cos( ) cos( )cos( ) sin( ) sin( )cos( ) sin( )cos( ) cos( ) sin( ) cos( ) sin( ) sin( ) sin( ) sin( ) x sin( ) cos( ) cos( )
20 E. Monte Carlo model (cont) Advantage: Accurate (non-equilibrium behavior) + simple Can be applied for all species Disadvantage: Long calculation time + not self-consistent In practice: used for electrons (in hybrid model) Or for species in sheath (detailed behavior, e.g., EDF)
21 F. Particle-in-cell / Monte Carlo model Principle: Similar to MC model (Newtons laws, random numbers) But at every time-step: calculation of electric field from positions of charged particles (Poisson equation) Advantage: Accurate + self-consistent Disadvantage: Even longer calculation time
22 Principle of PIC-MC (cycle): Real plasma particles: replaced by superparticles 1 superparticle = W real particles (W = weight, e.g. 2x1 7 ) Integration of equation of motion, moving particles F i v i x i Particle loss/gain at the boundaries (emission, absorption) Δt elec = 1 ps Δt ion = 1Δt elec Weighting (E,B) j F i (interpolate field to particles) Δt MC Collision? Yes Postcoll. velocities v i v i Integration of Poisson s equations on the grid () i (E) i Weighting (x,v) i () I (interpol. charges to grid) No
23 Which model should I use? Every model has advantages + disadvantages Solution: use combination of models!! -> hybrid model
24 G. Hybrid model Principle: Because some species = fluid-like, others = particle-like: Combination of above models, e.g.: Monte Carlo for fast (non-equilibrium) species Fluid model for slow (equilibrium) species + E-field (transfer from fast to slow group: if (E k + E p ) < threshold) Advantage: Combines the advantages + avoids the disadvantages Accurate + self-consistent + reduced calculation time Disadvantage: /
25 H. Collisional-radiative model (for excited levels) Example: Ar: 65 Ar levels (individual or group of levels) (9s) (8s) (7s) (6s) (5s) E (ev) E (1 cm -1 ) j c = 3/2 j c = 1/2 s p d f, s p d f, (5p) Ar + (3p 5 ) 2 P 3/ (3d) (9d) (9p) (8d) (8p) (7d) (6d) 35 (7p) (5d) (6p) (4d) (9s (8s ) (7s ) (6s ) (5s ) (5p ) 15 Ar + (3p 5 ) 2 P 1/2 48 (9d ) (9p ) (8d ) (8p ) (7d ) (6d ) 34 (7p ) 3 26 (5d ) 28 (6p ) (4d ) (3d ) 22 Energy level scheme for Ar* CR model (4p) (4p ) (4s) (4s ) (3p 6 ) 1
26 H. Collisional-radiative model (for excited levels) For each level: continuity equation (balance equation): n t Ar* D Ar* 1 r n r r r Ar* D Ar* Additional processes for 4s levels: 2 n z Ar* 2 Production + loss processes for each level: * Ar* - Ar* collisions -> (associative) ionization * Penning ionization of sputtered atoms * three-body collisions with Ar atoms * radiation trapping of resonant radiation R prod R * electron, Ar ion + Ar atom impact excitation, de-excitation, ionization * electron-ion three-body recombination, radiative recombination * radiative decay * Hornbeck-Molnar associative ionization (for Ar* levels with E > 14.7 ev) loss
27 I. Model for plasma-surface interactions: Molecular Dynamics simulations Newton s laws: Calc. of position + velocities of atoms, by interatomic interaction potentials F U a v r F / v r m a t v t 1 2 a t 2 Time-integrated trajectory of impacting C-atom on amorphous C substrate
28 I. Model for plasma-surface interactions: Molecular Dynamics simulations (cont) Interaction potential: empirical potential (parameters) Potential energy surface for impacting H-atom on diamond {111} surface
29 I. Model for plasma-surface interactions: Molecular Dynamics simulations (cont) E.g.: C-films: Brenner interaction potential Binding energy: sum of binding energies between atoms i and j : U V R( rij ) bijva( rij system ) i ji Repulsive (V R ) and attractive (V A ) terms: V V R A ( r ij ( r ij ) ) f f ij ij ( r ij ( r ij ) A ) B ij ij exp( r ij exp( r ij ij ij ) ) Many-body chemistry: binding order function b ij b ij = f(local binding topology)
30 I. Model for plasma-surface interactions: Molecular Dynamics simulations (cont) Semi-infinite surface: periodic ±{x,y} boundaries Time scale ~ ns (t ~.1-1 fs) Length scale ~ nm Initialisation of the simulation: Define substrate Particle impacts» Film growth: sequentially» Reaction mechanisms» Random orientation + {x,y} position above surface Bottom layers fixed Heat dissipation by heat bath
31 I. Model for plasma-surface interactions: Molecular Dynamics simulations (cont) Advantage: Very accurate + deterministic (self-consistent) Disadvantage: Long calculation time
32 Illustration of MD simulations for film deposition (diamond like carbon) 5-fold coordinated C-atom 4-fold coordinated C-atom 3-fold coordinated C-atom 2-fold coordinated C-atom 1-fold coordinated C-atom H-atom
33 Result: Calculated microscopic structure of the film 5-fold coordinated C-atom 4-fold coordinated C-atom 3-fold coordinated C-atom 2-fold coordinated C-atom 1-fold coordinated C-atom H-atom
34 Illustration of MD simulations for plasma catalysis CH 3 radicals on Ni catalyst surface
35 Illustration of MD simulations for carbon nanotube growth Mechanism of cap formation of single walled carbon nanotube on Ni nanoparticle
36 Illustration of MD simulations for plasma medicine Breaking of peptidoglycan (~ bacterial cell wall damage) upon impact of O radicals
37 2. Examples of models for various applications A. D chemical kinetics model: for DBD in CH 4 /CO 2 Greenhouse gas conversion (plasma chemistry) B. PIC-MC model: for magnetron discharge C. MC model for electrons: for magnetron discharge D. Hybrid model: for glow discharge with sputtering E. Hybrid model: for ICP etch reactor
38 A. D chemical kinetics model for DBD in CH 4 /CO 2 Greenhouse gas conversion Global warming ~ greenhouse gases conversion into value-added chemicals However: greenhouse gases = inert Classical processes: high T, p DBD reactor = very useful: T e >> T gas Enables reactions that would thermodynamically not occur at low T gas Electrode Dielectric barrier plasma Electrode Gas/vapor D model (DBD in CH 4 /CO 2 : greenhouse gas conversion)
39 Molecules Charged species Radicals CH 4 C 2 H 6, C 2 H 4, C 2 H 2, C 3 H 8, C 3 H 6, C 4 H 2 H 2 Plasma chemistry 75 species : CH 5+, CH 4+, CH 3+, CH 2+, CH +, C + C 2 H 6+, C 2 H 5+, C 2 H 4+, C 2 H 3+, C 2 H 2+, C 2 H +, C 2 + H 3+, H 2+, H +, H CH 3, CH 2, CH, C C 2 H 5, C 2 H 3, C 2 H, C 2, C 3 H 7, C 3 H 5 H O 3, O 2 CO 2, CO H 2 O, H 2 O 2 CH 2 O, CH 3 OH, C 2 H 5 OH, CH 3 CHO, CH 2 CO, CH 3 OOH, C 2 H 5 OOH O 4+, O 2+, O +, O 4, O 3, O 2, O CO 2+, CO + H 3 O +, H 2 O +, OH +, OH Electrons O OH, HO 2 CHO, CH 2 OH, CH 3 O, C 2 H 5 O, C 2 HO, CH 3 CO, CH 2 CHO, CH 3 O 2, C 2 H 5 O 2 D model (DBD in CH 4 /CO 2 : greenhouse gas conversion)
40 Plasma chemistry 188 reactions taken into account: 165 Electron-neutral collisions (ionization, excitation, dissociation, ) e.g.: e - + CH 4 e - + CH 3 + H 5 Electron-ion recombination reactions e.g.: e - + O 2+ O + O 873 Ion/neutral chemical reactions e.g.: CH 3 + OH (+M) CH 3 OH (+M) CH 4+ + CH 4 CH 5+ + CH 3 D model (DBD in CH 4 /CO 2 : greenhouse gas conversion)
41 Calculated conversion, yields CO 2 +CH 4 conversion + E efficiency, as f(sei): (exp: Tu et al., Univ. Liverpool) Trade off: conversion + E efficiency Also calculated: Selectivities of formed products: At 5/5 CO 2 /CH 4 : H 2 : 55% CO: 48% C 2,C 3 : 6%, 3% Formaldehyde: 13% Methanol: 3% Future: plasma catalysis D model (DBD in CH 4 /CO 2 : greenhouse gas conversion)
42 Calculated conversion, yields Computational optimization study (75 simulations): (Effect of CO 2 /CH 4 ratio, power, residence time, frequency) Max + min obtained total conversion + E efficiency: Best results: 84% conversion, 8.5% E efficiency Max. E efficiency: 15% (but low conversion ) DBD not competitive for industrial implementation Plasma catalysis to improve E efficiency + selectivity D model (DBD in CH 4 /CO 2 : greenhouse gas conversion)
43 Comparison CO 2 /CH 4 -CH 4 /O 2 Species densities: Different chemistry CO 2 /CH 4 :C x H y,ch 2 O, CH 3 CHO, H 2 /CO > 1 O 2 /CH 4 :H 2 O, CH 3 OH, CH 3 OOH, CO 2, H 2 /CO < 1 D model (DBD in CH 4 /CO 2 : greenhouse gas conversion)
44 Comparison CO 2 /CH 4 -CH 4 /O 2 Different reaction pathways: Explains different product formation CO 2 /CH 4 :C x H y,ch 2 O O 2 /CH 4 : H 2 O, CH 3 OH, CH 3 OOH, CO 2 D model (DBD in CH 4 /CO 2 : greenhouse gas conversion)
45 B. PIC-MC model: for magnetron discharge Reactor geometry + Magnetic field z Anode 24 Target (cathode) Rotation axis r Br max = 3 11 G Vext = -4 V p = 4-3 mtorr Argon PIC-MC model (magnetron discharge)
46 Calculated Electron and Ar + Ion Density Electrons: Ar + ions: Maximum ~ 1 17 m -3 at r = 1.85 cm (maximum B r ) PIC-MC model (magnetron discharge)
47 Fast Cu Atom Density and Thermal Cu atom density Fast Cu atoms: Thermal Cu atoms: Concentrated near cathode Max ~ 1 18 m -3 Broad distribution Max ~ 4x1 17 m -3 PIC-MC model (magnetron discharge)
48 Erosion profile: Calculated - Measured Normalised erosion profile Calculated Measured Good agreement r, [mm] PIC-MC model (magnetron discharge)
49 C. MC model for electrons: dual magnetron discharge Magnetic field Closed field Mirror field Single electron trajectory Closed field Mirror field MC model for electrons (dual magnetron discharge)
50 Calculated electron density ionization rate Closed field: Mirror field: MC model for electrons (dual magnetron discharge)
51 Comparison MC vs PIC-MC model MC = faster (minutes to 1 hour vs. several weeks for PIC) complex geometries possible But: not self-consistent: - Input B needed (e.g., from Gmsh ( & GetDP ( - Input E needed (e.g., from PIC-MC model) MC model for electrons (dual magnetron discharge)
52 D. Hybrid model for GD plasma with sputtering Combination of different models for various species Species: Model used: Ar atoms no model (assumed thermalized) or: heat conduction equation electrons Ar + ions Ar f fast atoms Ar* excited atoms Cu atoms Cu*, Cu +(*), Cu ++ Cu + ions MC for fast electrons fluid for slow electrons fluid (with electrons + Poisson) MC in sheath MC in sheath collisional-radiative model thermalization after sputtering: MC collisional-radiative model MC in sheath Hybrid model (GD)
53 Application: Analysis of solid materials: Sample to be analyzed = cathode of GD Sputtering Excitation, ionization in plasma OES, MS E.g.: Depth profiling: Concentr as f(depth) conc A B C Plasma A B C depth Hybrid model (GD)
54 Ar* CR model no e - MC model no n Ar,met n Cu arbitrary R Ar+, R e,slow Ar + /e - fluid model E ax and E rad d c (r), j Ar+,dc (r), j Ar+, (r) R e,ion,ar R i,ion,ar and R a,ion,ar convergence? convergence? yes final solution Ar + /Ar f MC model new R Ar+ and R e,slow Cu MC model F T Cu/Cu + CR model f Cu+ (,E) R ion,cu, j Cu+,dc (r) Cu + MC model yes Coupling of the models: Modeling network e - MC model: - R e,exc,ar, R e,exc,met, R e,ion,met : for Ar* m model - R e,ion,cu : for Cu model Ar + /Ar f MC model: - R i,exc,ar, R a,exc,ar : for Ar* m model - f Ar+ (,E), f Ar (,E) : for Cu model Ar + /e - fluid model: - n Ar+, n e,slow : for Ar* m model - n Ar+, V : for Cu model Ar* m model: - n Ar,met : for R e,exc,met and R e,ion,met - prod, loss: for n Ar+, n e,slow Cu model: - n Cu : for R e,ion,cu - n Cu+ : for E, V - asymm. CT : for n Ar+ - ioniz.terms: for n e,slow Hybrid model (GD)
55 Hybrid model (GD)
56 Input data for the modeling network Electrical data: Voltage, pressure, (gas temperature) Electrical current = calculated self-consistently Reactor geometry: (e.g., cylinder: length, diameter) Gas (mixture) Cross sections, rate coefficients, transport coefficients, All other quantities: calculated self-consistently Hybrid model (GD)
57 Typical calculation results General calculation results: * Electrical characteristics (current, voltage, pressure) * Electric field and potential distribution * Densities, fluxes, energies of the plasma species * Information about collisions in the plasma Results of importance for applications (sputtering, GD-OES, ): * Crater profiles, erosion rates at the cathode * Optical emission intensities * Effect of cell geometry, operating conditions Hybrid model (GD)
58 Calculated gas temperature: VG9 cell (1 V, 3 ma, 75 Pa): r (cm) (a) r (cm) (b) r (cm) Grimm-type cell (8 V, 5 ma, 4 Pa) + Corresponding Ar gas density: z (cm) z (cm) Hybrid model (GD)
59 Potential distribution: (VG9 cell 1 V, 75 Pa, 3 ma) CDS: ca. 2 mm long NG: major part (9 V) r (cm) Volt z (cm) Hybrid model (GD)
60 Densities: (VG9: 1 V, 75 Pa, 3 ma): Ar + ions ~ slow electrons: Fast electrons: cm -3 5E+11 4E cm E+7 3.E+7 2.5E+7 r (cm) E+11 2E+11 1E+11 5E+1 r (cm) E+7 1.5E+7 1.E+7 5.E+6 2.E E+1 1E+1 E E+6 5.E+5.E z (cm) z (cm) Hybrid model (GD)
61 Energies: Electron energy distribution (1 V, 75 Pa, 3 ma): Hybrid model (GD)
62 Energies: Argon ion energy distribution (1 V, 75 Pa, 3 ma): Calculated: Measured (MS): Hybrid model (GD)
63 Energies: Copper ion energy distribution (1 V, 75 Pa, 3 ma): Calculated: Measured (MS): Hybrid model (GD)
64 Information about sputtering at the cathode: Crater profile after 45 min. sputt. (VG9: 1 V, 75 Pa, 3 ma): Calculated: Crater edge effect due to anode front plate: Measured: r (cm) Volt z (cm) Hybrid model (GD)
65 Grimm cell Craters more flat Because equipot.lines // cathode depth (m) -2 Berekend Calculated 1 V (Volts) depth (m) r (mm) Gemeten Measured r (mm) r (mm) z (mm) Hybrid model (GD)
66 Optical emission 5.E+15 intensities: 5.E+15 4.E+15 4.E+15 3.E+15 3.E+15 2.E+15 2.E+15 1.E+15 5.E+14.E (nm) (nm) Intensity (a.u.) Intensity (a.u.) Ar(I) spectrum Calculated: Measured: Hybrid model (GD)
67 Optical emission intensities Int. (a.u.) Int. (a.u.) Int. (a.u.) Int. (a.u.) Calculated ArI (75.3 nm) ArI (811.5 nm) ArII (476.5 nm) CuI ( nm) z (cm) Int. (a.u.) Int. (a.u.) Int. (a.u.) Int. (a.u.) DC:.6 Torr 1.55 ma V Measured ArI (75.3 nm) ArI (811.5 nm) ArII (476.5 nm) CuI ( nm) 3.1 ma V 4.65 ma - 48 V 6.2 ma V 7.75 ma - 55 V z (cm) Hybrid model (GD)
68 E. Hybrid model for ICP etch reactor HPEM (M.Kushner, University of Michigan) Z(cm) Inlet Coil Quartz window Maxwell equations E, B fields Monte Carlo (electrons) n e, T e, EEDF, reaction rates 1 5 Silicon substrate Electrode R(cm) Monte Carlo (heavy particles) etch profile evolution Pump port Surface modeling surface reactions, Γ Ri Fluid (heavy particles) n i, Γ i, E s Monte Carlo (heavy particles) IEDF, IADF Hybrid model (ICP etch reactor)
69 Calculated species densities (Cl 2, e - ) Cl 2 : Maximum near inlet, then depletion (chemical reactions) Fairly uniform density profile Electrons: Maximum in center/near coil (ioniz.degree ~ 1-4 ) Hybrid model (ICP etch reactor)
70 Calculated fluxes + angul.distrib. to wafer Flux to substrate (cm -2 s -1 ) Cl 2 O 2 O Cl Position from center of the wafer (cm) Cl Ar Cl + O 2 + Ar + O + Angle distribution ( -1 ) Neutral angle distribution Ion angle distribution Angle from normal on substrate ( ) Ion fluxes ± 1x lower than neutral fluxes Ions: narrow angular distribution (directed by E-field) Neutrals: wide angular distribution Hybrid model (ICP etch reactor)
71 Calculated energy distributions to wafer Ions (a) Energy distribution (ev -1 ) Neutrals (b) Energy (ev) Ions: bimodal distribution (rf bias at substrate): Ions feel rf bias amplitude (time sheath < 1 rf cycle) Neutrals: Maxwellian distribution at low E Hybrid model (ICP etch reactor)
72 Input: Etch profile calculations - Fluxes, energy and angular distributions of bombarding species (from plasma model) - Surface reaction probabilities (etch, oxid, sputt, redepos) of the various species (Cl (+), Cl (+) 2, O (+), O (+) 2, Ar + ) on the various surface sites (Si, SiCl,SiCl 2, SiCl 3, SiO, SiO 2 ) Hybrid model (ICP etch reactor)
73 Summary Plasma modeling : Most appropriate model depends on application: - Fluid (1D or 2D) or (D) chemical kinetics modeling: * Detailed information on plasma chemistry * fast - Particle-in-cell Monte Carlo simulations: * Microscopic non-equilibrium behavior - Monte Carlo modeling: * Faster, but not self-consistent - Hybrid Monte Carlo - fluid simulations: * Combination: combines the advantages + eliminates the drawbacks of individual models * Extra information (etch profiles, OES,...) - Molecular dynamics simulations: Surface modeling
74
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