Alexandrov, D. V.

Galenko, P. K.

Titova, E. A.

Institutional repository of Ural Federal University named after the first President of Russia B.N.Yeltsin

1Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distributionof this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Published under licence by IOP Publishing Ltd1234567890STPM2017  IOP PublishingIOP Conf. Series: Materials Science and Engineering 192 (2017) 012002 doi:10.1088/1757-899X/192/1/012002A relaxation time of secondary dendritic branches totheir steady-state growthE A Titova1, D V Alexandrov1 and P K Galenko21 Department of Theoretical and Mathematical Physics, Laboratory of Multi-ScaleMathematical Modeling, Ural Federal University, Ekaterinburg 620000, Russian Federation2 Friedrich-Schiller-Universita¨t-Jena, Physikalisch-Astronomische Fakulta¨t, 07743 Jena,GermanyE-mail: ekatitova@mail.ruAbstract. In the present article, we use the selection theory to estimate the non-stationaritytime of evolving dendrites to their steady-state growth.1. IntroductionThe microstructure predictions in solidification processes have the deep scientific and practicalroots (see [1–5] and references therein). A development of experimental methods in solidificationprocessing made possible to reach a much broader range of measurable crystal growth velocities,temperature gradients and cooling rates [6]. For instance, splats and films can be quenched withthe cooling rate of the order of 107 K/s, the temperature gradients can have the order of 108 K/sin laser annealing of sample surfaces, and containerless methods of droplets processing providethe deep undercoolings having the values of 200-400 K prior to the primary crystallization. Alarge driving force for transformation, arising in such methods, leads to fast solidification froma liquid metastable state as well as the high-speed solid-state transformations of metastablecrystalline phases [7]. For instance, the experimentally measured solidification velocity has theorder of 10−1−102 m/s in droplets processed by the electromagnetic levitation facility [7,8]. Assuch, the total duration of primary solidification in small droplets is rather short and estimatedas [8]: 10−5 − 10−3 s.For the analytical calculations of rapid solidification regimes and theoretical estimations ofmicrostructural parameters of dendritic and eutectic crystals, different models of non-equilibriumcrystallization are used, as a rule, formally developed for the steady state scenario [9–11].However, the steady state approximation for conditions of rapid solidification of small samples(films, splats, droplets) is questionable and such approach is often criticized. A very commonview is that a steady state scenario may be expected to exist but not in rapidly solidifying smallsamples [12]: the stationary regime of solidification can be reached at long times that is possiblein usual experimental circumstances (directional solidification) or well-known classic technologies(continuous casting and low intensive brazing). In other words, during the short periods oftime, the steady state is not achieved and, consequently, the quasi-stationary approximationin modeling of rapid solidification does fail. The detailed calculations of solidification regimeslead, however, to the remarkable behavior of the transient time between the non-stationary andstationary regimes of solidification: the non-stationary time sharply depends on the interface21234567890STPM2017  IOP PublishingIOP Conf. Series: Materials Science and Engineering 192 (2017) 012002 doi:10.1088/1757-899X/192/1/012002Table 1. Material parameters for succinonitrile (SCN) and nickel (Ni)Parameter SCN NiMelting temperature TA (K) 331.233 [27] 1728 [28]Latent heat L (J/(m3) 4.781 · 107 [27] 2.67 · 106 [30]Heat capacity cp (J/(m3K)) 2.08 · 106 [27] 6.39 · 103 [30]Solid-liquid interface energy σ (J/m2) 8.9 · 10−3 [27] 0.275 [30]Capillary length d0 (m) 2.821 · 10−9 [27] 4.92 · 10−10 [29]Surface energy stiffness αd (–) 0.0825 [27] 0.27 [30]Thermal diffusivity DT (m2/s) 1.13 · 10−7 [27] 1.2 · 10−5 [30]Selection constant σ0 (–) 0.1 [22] 0.191 [29]Relaxation time for ∂φ/∂t (∗) τφ (s) 2.5 · 10−7 5.5 · 10−7Maximum speed of φ-field(∗) VφB (m/s) 11.5 25.25Diffusion coefficient of φ(∗) νφ (m2/s) 5.5 · 10−5 3.5 · 10−4(∗)The numerical value of this parameter is chosen in the present work.velocity and undercooling in solidifying samples [13]. In particular, for the planar solid-liquidinterface, it has been shown that the non-stationary time rapidly decreases as the crystal growthvelocity increases taking the values of the order of milli- or even micro-seconds with the velocityof the order of centimeters per second [13].Moreover, using the theoretical model developed in [13], it is straightforward to show thatthe non-stationary time sharply increases with the decrease of crystal growth velocity takingthe values of the order of seconds or even minutes for the interface velocity of the order ofmicrometer per second. Such remarkable dependence should, obviously, exist not only for theplanar interface but also for many other crystal growth shapes. As one of the main crystalgrowth forms is the dendritic crystal shape (for slow and rapid regimes of rapid solidification[1–3]), we provide analytical treatments for obtaining the non-stationary time which could bequantitatively estimated in comparison with the time for primary solidification of experimentallyinvestigated samples. Our studies are limited to two dimensional dendritic growth. Many ofproperties of metallic samples (for instance, the time for homogenization) strictly depend on theinterdendritic spaces between secondary branches of dendrite. Applicability of the steady-statemodels to the growth of secondary branches is an actual task. Therefore, the present analysisgives some estimations of a non-stationary period of growing secondary branches in the slowand rapid solidification conditions.2. Relaxation time to a steady-state regimeThe relaxation time τT characterizes how fast the dendrite tip velocity,v(t) ∝ Av(t) exp(−Amt/τT ) (1)relaxes to the steady-state regime of dendritic growth [14]. Here Av(t) is the time-dependentamplitude of relaxation, Am is the factor which depends on the model parameters, t is thetime and the relaxation time τT represents a relaxation parameter to the steady state regimeof thermal dendrite. Let us consider the symmetric model of dendritic growth in the positive zdirection with a constant velocity v. Let ζ = ζ(x, t) and ρ designate the instantaneous coordinateof the crystallization front and the radius of curvature of the dendrite tip, respectively. It isconvenient to measure lengths in units of ρ and times in units of ρ/v. In the case of zero surface31234567890STPM2017  IOP PublishingIOP Conf. Series: Materials Science and Engineering 192 (2017) 012002 doi:10.1088/1757-899X/192/1/012002tension, the dendrite shape represents the Ivantsov parabola [14–17]ζiv = −x22.In the case of nonzero surface tension, this solution should be modified by a shape correctionζ0(x). The steady-state solution in the form of ζiv + ζ0(x) exists only if the so-called selectionparameterσ∗ = d0ρPg=2d0DTρ2v=d0v2DTP 2g(2)takes a special value determined from the microscopic solvability theory [18–25]. HerePg =ρv2DT(3)is the growth Pe´clet number, d0 is the capillary length, and DT is the thermal diffusivity.0.0 0.10.00.20.40.60.81.0SCNnon-stationary time, (s)Peclet number, Pg0.00 0.02 0.040.0000.0010.002NiPeclet number, Pgnon-stationary time,(s)Figure 1. The thermal dendrite growth from the undercooled nickel and SCN melts. Therelaxation time τT is a function of the growth Pe´clet number Pg = ρv/2DT . Calculations aremade using Eqs. (9) and (10).It is important to know what is the real relaxation time when the parabolic dendrite shapecan be considered as steady-state. In order to answer this question let us introduce a small time-dependent correction ζ1(x, t) to the steady-state solution and present the secondary branchesζ(x, t) as:ζ(x, t) = ζiv + ζ0(x) + ζ1(x, t). (4)The time-dependent addition ζ1(x, t) has been found in [26] on the basis of WKB theory. Theresult isζ1(x, t) = f(2z)1/8σ∗1/4Im{exp[(2z)1/4c3√σ∗(1− iz − t√2z)3/2]}, (5)where c =√3/2, z = x2/2, and f is a constant of unity order. Rising the complex number1− i(z− t)/√2z to the power 3/2, separating the real and imaginary parts of expression (5) andintroducing the dimensional variablesτ =ρtv, ξ = xρ, (6)41234567890STPM2017  IOP PublishingIOP Conf. Series: Materials Science and Engineering 192 (2017) 012002 doi:10.1088/1757-899X/192/1/0120020.0 0.5 1.00.06.0x10-6 non-stationary time(s) Ni SCNGrowth velocity, V (m/s)Figure 2. Growth of the thermal dendrite from the undercooled nickel and SCN melts. Therelaxation time τT is a function of the growth velocity. Calculations are made by Eqs. (3) and(10).one can represent the final result asζ1(x, t) = A exp−(βδ)4/3 + ((βδ)2/32− ττT)23/4 , (7)where the following designations are introducedτT =ρv(βδ)2/3, β =√ξc3√ρσ∗, α =ξ2ρ− vτξ, δ = | cos(ϕ/2)|,A = f(ξσ∗ρ)1/4sin [β sin(ϕ/2)] , ϕ = arctan(α2 − 31− 3α2). (8)To obtain a dependence of the thermal relaxation time τT of secondary dendritic branches on thegrowth Pe´clet number Pg let us express the dendrite tip velocity v from the solvability conditionderived in Refs. [21–23] for arbitrary Pe´clet numbersv(Pg) =2DTα7/4d σ0P2gd0(1 + a1√αdPg)2, (9)where αd is the surface energy stiffness, σ0 is the selection constant and a1 =√8σ0/7(3/56)3/8.Taking into account expression (8) and considering a vicinity of dendritic tip (ξ ∼ ρ) one cangetτT ≈ 2DTPgc2δ2/3v2(Pg)(d0v(Pg)2DTP 2g)1/3, (10)with the following estimated constants: α ≈ 1/2 and δ ≈ 0.738. These estimations weremade for the thermal dendrite with the following values of the parameters: the relaxation timeτT ≈ 10−10...10−7(s), the scaled coordinate ξ ≈ ρ ≈ 10−6 (m) and the growth velocity v ≤ 10−251234567890STPM2017  IOP PublishingIOP Conf. Series: Materials Science and Engineering 192 (2017) 012002 doi:10.1088/1757-899X/192/1/012002(m/s). Then, using equation (8), one gets α ≈ 0.5− vτ/ρ with vτ/ρ ≤ 10−3  1/2 and obviousestimation for the parameter δ. Expression (10) determines the thermal relaxation time as afunction of the growth Pe´clet number Pg. Together with the dendrite tip velocity v given bythe selection criterion (9) this makes possible to quantitatively estimate the relaxation time tothe steady-state regime of dendrite growth.3. ConclusionsIn summary, the growth of secondary dendritic branches represents a complex phenomenon withmany effects and parameters for a wide range of undercoolings. Therefore, we plan to advancethe present formalism for estimation of non-stationary time to include the kinetic effects [23],non-equilibrium effects (such as solute trapping, solute drag, trapping of disorder [16]) and threedimensional dendritic structure.4. References[1] Trivedi R and Kurz W 1994 Int. Mater. Rev. 39 49–74[2] Herlach D M 1994 Mater. Sci. Eng. R 12 177–272[3] Glicksman M E, Koss M B and Winsa E A 1994 Phys. Rev. Lett. 73 573–576[4] Alexandrova I V, Alexandrov D V, Aseev D L and Bulitcheva S V 2009 Acta Physica Polonica A 115 791–794[5] Herlach D 2015 Crystals 5 355–375[6] Herlach D M, Galenko P and Holland-Moritz D 2007 Metastable solids from undercooled melts (Amsterdam:Elsevier)[7] Hartmann H, Galenko P K, Holland-Moritz D, Kolbe M, Herlach D M and Shuleshova O 2008 Journal ofApplied Physics 103 073509[8] Eckler K 1992 (Ph.D. thesis), (Ruhr-Universita¨t)[9] Galenko P K and Danilov D A 1997 Phys. Lett. A 235 271–280[10] Galenko P K and Herlach D M 2006 Phys. Rev. Lett. 96 150602[11] Gao J, Han M, Kao A, Pericleous K, Alexandrov D V and Galenko P K 2016 Acta Mater. 103 184–191[12] Das A K 1998 Defect and Diffusion Forum 162 97–118[13] Galenko P K and Danilov D A 2000 J. Cryst. Growth 216 512–526[14] Langer J S 1980 Rev. Mod. Phys. 52 1–28[15] Pecle´ P and Pomeau Y 1986 Stud. Appl. Math. 74 245–258[16] Galenko P K, Danilov D A and Alexandrov D V 2015 Int. J. Heat Mass Trans. 89 1054–1060[17] Alexandrov D V and Galenko P K 2017 Physica A 469 420–428[18] Pecle´ P and Bensimon D 1987 Nucl. Phys. B 2 259–270[19] Alexandrov D V and Pinigin D A 2013 Technical Physics 58 309–315[20] Alexandrov D V and Galenko P K 2013 J. Phys. A: Math. Theor. 46 195101[21] Alexandrov D V and Galenko P K 2013 Phys. Rev. E 87 062403[22] Alexandrov D V and Galenko P K 2014 Phys. Usp. 57 771–786[23] Alexandrov D V and Galenko P K 2015 Phys. Chem. Chem. Phys. 17 19149–19161[24] Alexandrov D V, Danilov D A and Galenko P K 2016 Int. J. Heat Mass Trans. 101 789–799[25] Galenko P K, Danilov D A, Reuther K, Alexandrov D V, Rettenmayr M and Herlach D M 2017 J. Cryst.Growth 457 349–355[26] Barber M N, Barbieri A and Langer J S 1987 Phys. Rev. A 36 3340–3349[27] Koss M B, LaCombe J C, Tennenhouse L A, Glicksman M E and Winsa E A 1999 Metall. Mater. Trans. A30 3177–3190[28] Hultgren R, Desai P D, Hawkins D T, Gleiser M, Kelley K K and Wagman D D 1973 Selected values of thethermodynamic properties of the elements (Ohio: ASM International, Metals Park)[29] Gao J, Kao A, Bojarevics V, Pericleous K, Galenko P K and Alexandrov D V 2016 J. Cryst. Growthhttp://dx.doi.org/10.1016/j.jcrysgro.2016.11.069[30] Galenko P K, Phanikumar G, Funke O, Chernova L, Reutzel S, Kolbe M and Herlach D M 2007 MaterialsScience and Engineering A 448–451 649–653AcknowledgmentsThis work was supported by the Russian Science Foundation (grant no. 16-11-10095).

A relaxation time of secondary dendritic branches to their steady-state growth

https://elar.urfu.ru/bitstream/10995/72904/1/10.1088-1757-899X-192-1-012002.pdf

A relaxation time of secondary dendritic branches to their steady-state growth

Abstract

Similar works

Full text

Available Versions

Institutional repository of Ural Federal University named after the first President of Russia B.N.Yeltsin