Borisov, V T

Umantsev, Alexander

Vinogradov, V V

English

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·tal results of measurements of the refractive indices of NLC will be given in a further article. The authors thank A. G. Petrov and E. M. Aver'yanov for fruitful discussions. 1w. 11. de Jeu, Physical Properties of LitJuid Crystalline Materials, Gordon and Hr.,ach, New York- London- Paris (1980). ~!!.E. Ncugebau.,r, Can. J. Phys. 32, l (1954). 3 D. A. llunmur and R. W. Munn, Ch.,m. Phys. 7G, 249 (1~1~3). 4 ~1. r. Vuks, Opt. Spektrosl;. 20, 644 (19GG). sw. 11. de jeu and P. llerdewijk, J, Chern. Phys. Gt:, 109 (1\J78). 6!. ll. thrahim aud w. Haase, ~1olcc. eryst, Liq. Cryst.~, 18>1 (1,)01). 7E. ~~. Aver'yanov, and V. F. Shabanov, Kristallografiya 23, :120 (1973) [Sov. Phys. crysrallogr. 23, 177 (1978)]. 'A. llerzhanskii and A, Petrov, Dokl. llolg. Akad. Nauk, 5(;9 (H171). 9A. Hauger, G. Pels!, C. Selbmann, et al., J.lokc. Cryst. Liquid Cryst • .£!, 97 (1\)1\3). 10H. Lorentz, Theory of Electrons [Russian translation}, Gostekhizdat, Mos-cow- Lenin/;'rad (1 \l34). llyu. L. Klimontovich, Kinetic Theory of Electromagnetic Processes [in "/) Russian], Nauka, :>.1oscow (19i:l0), p. 1!ill. 'l> 12Yu. L. Klimontovich, Statistical Physics [in Russian], Nauka, :\1nsco1v (19H2). 13 D. Bedeaux and P. ~Iazur, Phvslca 67, 23 (1973). 14 E. M. Zemskov, V. N. Semin~gov, and A. L. Yamnov, Fiz. Zhidk, Sostoyaniya, No. 10, 95 (1982). 15G. R. Luckhurst and G. W. Gray (eds .), Molecular Physics of Liquid Crys-tals, Academic Press, New York (1979). 16ll, W. van dcr Mcer and G. Vertogcn, J. Phys. Colloq. (Paris)~, CJ, 2~:! ( 11l7~J). 17H. A. Stuart, lvlolekulstruktur, 3rd Ed., Springer, flcrlin (1%7), p. 41G, 1'E. M. Avcr'yunov anJ V. 1'. Shabanov, Kristallografiya 2G, 174 (UJ\::1) [Sov. Phys. Crystallogr. 2G, 96 (1981)]. 19L. D, Landau anu E. M. Li fshots, Electrodynamics of Continl!ous ~lcU1a [in Russian1 Fizmatgiz, Moscow ( 1957), p. 88. Translated by S. G. Kirsch Mathematical model of growth of dendrites in a supercooled melt A. R. Umantsev, V. V. Vinogradov, and V. T. Borisov Institute of Metallurgy and Physics of Metals, and Central Scientific-Research Institute of Ferrous Metallurgy (Submitted February IS, 1984} Kristallografiya 30, 455-460 (May-June 1985) By means of a mathematical model, the authors analyze the formation and growth of dendrites in a supercooled melt. They find how the parameters of dendrite growth depend on the supercooling. They compare the theoretical and experimental results. INTRODUCTION As a rule, the growth of crystals from supercooled melts of pure substances is accompanied by the formation of dendrites. Much experimental work has been done on this problem.1- 3 However, owing to the mathematical dif-ficulties, theoretical work has been limited to the descrip-Uon of acicular, unbranched crystals; following Ivantsov ,4 authors usually assume that the tip of the dendrite is para-bolic. s-7 This hypothesis has been verified for slightly supercooled systems.2•8 However, for relatively great degrees of supercooling it is not justified. The problem thus arises of determining the shape and describing the growth kinetics of dendrites under such degrees of super.-cooling. This problem can be solved by means of a mathematical model. Smith9 has attempted to do this, but obtained no quan-titative results. Below we suggest a method and a direct mathematical model of the gro, .. th kinetics and shape for-mation of a dendrite in a supercooled melt of a pure sub-stance. DESCRIPTION OF MODEL We shall make a model for the two-dimensional prob-lem. Let us introduce a system of Cartesian coordinates (X, Z). In the region of space 0 ::::; x ::::; X, 0 ::::; z ::::; Z with unit length in the third dimension, filled with a super-cooled solidifying melt, and for the time variable t > 0, we introduce the difference network Wx.z ,t == {xi = i.ix, i = 0, 1, ••• , I; zj jAz, j = 0, 1, .•• , J, tk = kAt, k == 0, 1, ••• } with steps Ax = X/ I, Az = Z/ J in space and At in time. The spatial network divides the region into cells with di-mension AxAz in the XZ plane. Let the numbers of the cells be those of their top right-hand corners in the net-work. The state of a cell with number (i, j) at time tk will be characterizedby an average temperature Tijk and a fraction of solid phase gijk equal to the ratio of the volume of substance crystallizing in the cell to the vol-ume of the cell. For cells filled with solid phase, gijk 1; for cells filled with melt, gijk = 0; and in solidifying cells, i.e., those which at a given moment contain the phase boundary, 0 < gijk < 1. In time At the crystaJlization process leads to a change in the fraction of solid phase in a solidifying cell by an amount Ag1l and to the separation of an amount LAxAzAgij)( of heat, where L is the latent heat of crys-tallization of unit volume. The equation of heat baJance for such a cell is (1) where C and X are the specific heat per unit volume and the thermal conductivity of the medium, which are as-sumed to be equal for the liquid and solid phases. For the nonsolidifying cells the heat balance obeys the same 262 Sov. Phys. Crystallogr. 30(3), May-June 1985 0038-5638/8 5/03 0262-04 $03.90 © 1986 American Institute of Physics 262 -----------------------------------------• equation, but with l:.gijk 0. Suppose that part of the boundary, of length l ijk, adjoining the (i, j) -th solidifying cell, advances at a rate Vn, measured along the normal to the phase boundary. Then the change in the function P:i{ in this cell in time L:.t can be represented in the form ' V M l • t;g,, = n AXAZ ij • (2) If the crystal grows by a normal mechanism,10 then the rate of advance of the front is related as follows to its temperature T1: 1' .. =!1 ( TE-T,)' (3) ~ind!t. where 1.1 is the ePitieal growth coefficient, and TE is the temperature of equilibrium between the solid and liquid phases, related to the curvature of the phase boundary surface K by the Gibbs -Thompson equation: (4) where ri is the surface energy of the phase boundary and Tm is the melting point. We shall assume that the solid phase can grow only on previously formed solid phase and that nucleation of crystals ahead of the front is not allowed. To realize this condition, in the model we introduce the type of the cell Wijk characterizing the ability of the cell (i, j) to solidify. We put w~. "'1 for solidifying cells (0 < gijk < 1), i.e., for those ce\ls through which the crystallization front passes at a given moment. We also put Wijk 1 for a purely liquid cell (gijk = 0) if it has a common side with at least one completely solidified cell. The remain-ing purely liquid and purely solid cells are assigned to the type wijk = o. Let us introduce dimensionless variables: where T0 is the initial temperature of the supercooled melt, P 0 and T 0 are the characteristic length and time of the problem, related to the characteristic velocity v0 by the expressions where a = A/ c is the thermal diffusivity of the medium •. For metals, Po ..... IQ-5 cm,T 0 -1o-9 sec, and v0 .... 104 em/ sec. ~-/ FIG,l. Dependence of B on outflow -~ / q, Curve: analytical solution from 1/ Ref. 11; points: results from mode I. 263 Sov. Phys. Crystallogr. 30(3), May·June 1985 FIG. 2. Shape of growing dendrite at successive moments of time. 1) Seed at start; 2) acicular crystal: 3) shape of tip, not varying with time; 4) den-drite with side branches. Suppose ax = L:.z. Then in the new variables, sys-tem of equations (1)-(4) can be written in the form T~+'= (1-4v) Tt/+v[r,:+,+r,:_,+T,!,1+T:_,;] +t..g,;\ (5) (6) (7) where v Cll:!..t/t:..x2 , 'Y =v0L:.t/t:..x, A= /H1Tm/aL, L:.8 = C(Tm T0VL is the relative supercooling of the melt, and ~ 1t l ijk/ t:..x is the relative length of the front in the cell; ~{ is the curvature of the interphase surface in cell (i, J), expressed as a multiple of p 0 -1. At the starting time, in the supercooled melt a crystal with the same temperature is placed. It occupies the cell with numbers j = 0, i and on its planar boundary is placed a disturbance in the form of one solidifying cell in the middle (g01/2,1 1). At the boundary cells of the re-gion with i = 0, i I, and also j = 0, boundary conditions of the second kind are postulated, corresponding to the absence of heat fll1X and symmetry of the crystal; at the boundary j J conditions of the first kind are postulated, Tijk 0, corresponding to supercooling of the liquid to a temperature T0 < T m· System of equations (5)-(7) with these boundary con-ditions is a closed system and completely represents the growth kinetics of the new phase and the redistribution of heat in the system. The results of solution of this problem for various physical systems are similar, with similarity coefficients Po, v o, and v0• The physical parameters which determine the evolution of the system are the relative supercooling AO and the coefficient A (7) characterizing the role of the T 0.5 U.~'ILL O.l5 o.o; . 115 I UJ 48 I u -zu t1 z FIG. :1. _Temperature distribution along axis of dendrite under supercooling t:iJ ~ 0,;-,, Inset: dependence of Gl on t:JF). Umantsev et al. 263 l.mS 5 Z.5 0 zt-~. 1L::_ 0 0.5 1.0 48 l J tndz FIG. 4. Loganthm of volume of upper part of demlnte, lnS vs logarithm of its length, ln e:.z. lnse t: $hape factor n of dendrite vs supercooling .::lP. surface energy of the botmdary in this process, together with v and "I which are parameters of the numerical scheme. In the model the initial seed gave rise to a dendrite which grew in the direction of the cooler botmdary z = z. As the crystal approached this boundary the whole region was periodically lowered, corresponding to a transition to a system of coordinates associated with the tip of the dendrite. The dimension of the region along the x axis was chosen to be large enough that the influence of the boundaries had practically no effect on the growth of the crystal. In the calculations the function ~ ijk was re-placed by its mean value equal to unity. The growth rate v of the dendrite was determined as the ratio of the length of the cell to the time in which it was traversed by the tip. The curvature of the surface of the dendrite Kijk at the k-th moment of time in the neighborhood of the cell with number (i, j) was determined as follows. It is lmown that the curvature of a curve represented by the equation X= X(Z) is K( d':ddz' d [ dxldz ] z)= [1+(dx/dz)']"' = dz l't+(dxldz)' . The gradient dx/dz of the tangent to the curve bounding the crystal can be found if at the k-th moment of time 0.5 (8) FIG. 5. Growth rare v and curvature K of dendrite rip vs supercooling t>®. o.z 264 Sov. Phys. Crystallogr. 30(3), May-June 1985 v Q.¥ o.z FIG. 6. Influence of surface energy on growth rate and curvature of tip of dendrite, 1) A 0 or 0,01; 2) 0,1. we know the field {gijkh At the tip the gradient must be put equal to zero, Then by means of Eq. (8) we can find the curvature of the part ofthe curve in the neighborhood of the cell (i, j). A program in FORTRAN-IV was written to solve system (5)-(7) numerically; the algorithm was as follows. At the initial moment of time k 0 we assigned the initial state of the system. For the change from the k-th to the (k + 1)-th time layer we successively examined all the interior cells of the region (the model showed that the results of the calculations do not depend on the order in which the cells are examined). At first from the field {g1jk} for each cell we determined its type Wijk· Then from Eq. (7) we found the change of the fraction of the solid phase ~ijk• a;fter which from (5) and (6) we deter-mined the temperature Tijk-t-1 and the fraction of solid phase g1jk+1 in the cell in the next time layer. At the end of the calculations, in accordanceowith the boundary conditions we restored the values gijk-tl and Tijk+i in the cells with numbers i "' 0, i = I, j = 0, j = J. During the calculation we periodically computed the rate of gro\\th of the tip of the dendrite and the curvature of its surface. In order to choose the optimal values of the param-eters v and 'Y and for an indirect check on the operation of the proposed algorithm, we used the latter to solve two test problems. In the first we examined the advance of a planar crystallization front in a supercooled liquid. As the second we chose stefan's problem of solidification with a continuou~ly acting outflow along the axis perpen-dicular to the pVme (X, Z), which at t > 0 removes a quantity q of heat per unit time. This problem has an exact solution11: The new phase solidifies in the form of a cylinder at a rate proportional to t-112• Figure 1 plots the proportionality coefficient {j vs the outflow q, obtained in Ref. 11, and also as a result of our model for v 0.08 and "I 0.4. The discrepancies between the numerical results for both test problems and the corre-sponding analytical solutions were less than the error Umantsev et al. l'lc;. 7. Depen~encc of growth rare of acicular JenJrites on supercooling .:, n. 1) Experimental1; 2) theoretical6 ; points, results of moJel. in the numerical solutions. In modeling the main problem we chose the same values for 11 and 'Y. HESULTS FROM MODEL The model showed that in a supercooled liquid the plane boundary ofthe phases if? tinstgbl~, i.e., a siight ·· clis6irbance to the front (Fig. 2, curve 1) ultimately grows into a needle directed along the Z axis (2), and its shape is independent of the form of the initial disturbance. At some time~ after the start of the process, a quasista-tionary state is set up in the system: The tip of the dendrite moves at a constant rate, and a region behind it acquires a constant shape (3). The existence of such a state has been observed in experiments performed direct-ly on transparent organic substances. In the model this result was also obtained for A = 0, i.e., without taking account of the surface energy of the crystal. This proves that the kinetic process of attachment of particles at the phase boundary (3) has a stabilizing action on the growth rate and on the form of the tip of the dendrite. This is becaus~ owing to the conditions at the advancing bound-ary (3) an increase in the velocity of the needle leads to a reduction in the temperature of its surface, and this in turn leads to a reduction in the intensity of the heat flux from the needle. In the later stages of growth the needle is converted to a dendrite with side branches (Figs. 2 -3!Eid 4). Thus it appears that'the shape of the dendrite is stable not only relative to short-wave perturbations of its surface, due to the network character of the solution (the wavelength being of the order of the network dimen-sion), but also relative to long-wave perturbations of the side structure type. Figure 3 shows a typical temperature distribution along the axis of the dendrite, obtained for the case t:.O 0.5. The position of the tip of the dendrite corresponds to the point z = 0; for z < 0 we get a crystal in which the temperature is almost uniform; and when z > 0 we get a liquid phase in which there is a large temperature gra-dient near the tip of the dendrite. The inset to Fig. 3 plots the temperature gradient G l, averaged over the cell, in the liquid phase at the tip of the dendrite vs the. relative supercooling ,t,.(J of the melt. 265 Sov. Phys. Crystallogr. 30(3). May-June 1985 For numerical analysis the shape of the dendrite surface was determined by the volume S of its upper part, cut off by the plane z = z0 , as a function of the distance from this plane to the tip of the dendrite. A typical result fort:. 0 = 0.5 is shown in Fig. 4. It was found that the sur-face of the needle, except for a small neighborhood near the tip, can be represented by the equation x = Y.zt/n with n < 2. This conflicts with the usual theoretical assump-tion that the surface of the dendrite is parabolic. The model (inset to Fig. 4) showed that as the supercooling decreases, the index n tends toward two, i.e., the shape of the surface approaches the parabolic. The points in Fig. 5 show the dependence of the growth rate v and the curvature of the dendrite tip, K, averaged over the cell, on the supercooling, !:::. 0. In this figure the continuous lines correspond to the theory in Ref. 7 with A = 0, in which it was assumed that the den-drite has a parabolic shape. Comparing the results in Fig. 5, we see that parabolic dendrites grow more slowly than dendrites obtained in the model, which have no lim-itation on the shapes. For an analysis of the influence of the surface energy on the dendrite growth process we made a model for vari-ous values of the surface energy: A = 0 and A = 0.01 (typical metals), curve 1; and A= 0.1 (organic compounds), curves 2 in Fig. 6. The results showed that in the range of supercooling .:lO ~ 0.4 the surface energy has prac-tically no influence on the growth rate or the curvature of the tips of metallic dendrites. We also made a model of the growth of dendrites having the shape of figures of rotation (acicular). Figure 7 shows the dependence of the growth rate of such den-drites on the degree of supercooling. In the range of supercooling 0.2 ~ .:lO ~ 0.5 this dependence can be ap-proximated by the function v = 0.48 (.:lO )1•7, in agreement with the experimental results on the growth of dendrites of silver, obtained in Ref. 3, with a kinetic coefficient J.L = 124 em/ sec· oc. The solid curve in this figure plots results on the growth of dendrites of phosphorus,1 with J.L = 17.7 em/ sec · K, and the dashed curve plots the theo-retical results in Ref. 6. 1M. E. Glicksman anJR. J. Schaefer, J. Cryst. Growth_!, 297 (1967). 2s-c. Huang anJ :--.1. E. Glicksman, Acta :\1etallurgica 29, 701 (1981). 3T. Suzuki, s. ToyoJa, T. llmcda, andY. Kimura, J, cryst. Growth~, 123 (1977). 4 G. P, Ivantsov, Dokl. Ak?J. Nauk SSSR 58, 567 !1947). 5 D, E. Temkin, Dokl. AkaJ. Nauk SSSR 1:!2, 13) 7 (1%0) (Sov, Ph)'S, Dokl. 5, 609 (1960)]. "R. TrivcJy, Acta 1-.!etallurgica 18, 287 (1970). 7R. TriveJy, Scr, 1-.letall. :1, G1:J(i%9). 'J. S. Langer an\l...!L..M(iliZr-Krumbhaar, Acta 1-.lctal!urgica 2:1, 1-!5 (1!J~1). 9]. ll. Smith, J. Comput. Phys. :J:l, 112 (1\J81). -10V. T. lloriso\', Dokl. AkaJ. Naul< SSSR 151, 1:111 ( HJG:l) (Sov. Pltys. JETP 8, H:l4 (19G4) ]. 11G. Karelou anJ D. jaeger, Tl1ermal Conductivity of Solids [Russian trans-larwn]. Nau ka, ~\los cow ( !9G4 ). Translated by S. G. Kirsch Umantsev et al. 265 

Mathematical model of growth of dendrites in a supercooled melt

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Mathematical model of growth of dendrites in a supercooled melt

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