microsegregation in crystal growth from the melt: an analytical approach

8/17/2019 Microsegregation in crystal growth from the melt: an analytical approach

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Journal of Crystal Growth 13 1 (1993) 431—438 CRYSTALNorth-Holland GROWT H

Microsegregation in crystal growth from the melt:

an analytical approach

J.P. Garandet

DTA / CEREM/DEM/SMGP/LET, Commissariat a l’Energie Atomique, Centre d’Etudes Nucléaires de Grenoble, 85X,F-38041 Grenoble Cedex, France

Received 5 January 1993; manuscript received i n final form 22 March 1993

The influence of interface velocity variations on microsegregation is modelled analytically using a boundary layer formulationand a perturbation approach. The theoretical correlations between interface velocity and concentration fluctuations are seen to

compare well with existing numerical and experimental data, meaning that the solute boundary layer formalism i s well adapted to

the study of microsegregation. A possible field of application of this work is the derivation of the allowable pulling deviceinstabilities in a crystal growth furnace.

1. Introduction A correlation between interface and composi-tion fluctuations was obtained numerically by

Solute segregation at the microscopic level is Wilson [4,51in the early eighties for a sinusoidaloften a major problem for the crystal growth of growth rate in an idealized Czochralski configura-application oriented materials [1]. For instance, it tion. An interesting result was that microsegrega-

is well known that rotation of the sample in an tion scaled linearly with the amplitude of theasymmetrical thermal field, pulling device insta- velocity variations even up to the point of back -bilities or vibrations lead to time-dependent van- melting.ations of the interface velocity which in turn An hybrid analysis was carried out b y Van Run

often result in striations. [61and Favier and Wilson [71.Convection is ac-Another possible mechanism for microsegrega- counted for using a stagnant film model, and the

tion is related to unsteady convection in the melt, solution to the time dependent problem is then

It can act either directly on the solutal field, as in found numerically using either finite differencesthe case of g-jitter during solidification in micro- [6] or finite elements [71.Both approaches weregravity [2] or indirectly when the temperature seen to compare very well with Wilson’s data.fluctuations associated with the flow modify the A solution to the general problem, involvinggrowth velocity and thus the composition of the an analytical derivation of the correlations be-alloy. tween the interface velocity, temperature and

Kinetic effects on faceted interfaces are also to concentration variations is very difficult to find.

be considered. High lateral growth velocities at One of the reasons is that, without any simplify-macrosteps or local interface breakdown due to ing assumptions, the time dependent heat andan excessive undercooling can result in transient mass transfer equations are generally non-linear.solute incorporation [3], but a detailed study of As a consequence, the two outstanding effortsthese microscopic problems is outside the frame in the field [8,91 both relied on a perturbationof our present work. approach. Another common feature of these two

0022-0248/93/$06.OO © 1993 — Elsevier Science Publishers B.V. All rights reserved


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132 1 .1 ’. Gurandet / Microsegregation in crystal growth from the melt

models is that the effect of convection is ac- With the Z-axis oriented from the interface into

counted for via a stagnant film model; this ap- the liquid, W is necessarily negative.proach, as rightly recognized by the authors, is BPS obtained an analytical solution to the

only correct when convection dominates mass above set of equations and then proceeded totransfer, propose a simple stagnant film model that could

Our purpose in this paper is to extend the account for the macrosegregation phenomena.previous analytical results using a physically sound Indeed, even if we know that convection is pre-boundary layer approach. Our solution will thus sent, let us suppose that mass transport is purely

be valid all over the convecto-diffusive solute diffusive in a region of extent 6s~in front of the

transport range. An additional advantage of our interface, the bulk of the liquid being assumed to

analysis with respect to previous work is that the be homogeneous. Eqs. (1) and (3) become:algebra involved is much simpler, making it easier

d2C dCto use for the practician. D—~-+ V

A necessary prerequisite is to modify the stag- dZ~ ‘dZ = (4)nant film model first introduced b y Burton, Prim ~ = ~SF~ c = C~. (5)and Slichter (BPS, [101), which will be done in

section 2. We shall then focus on the limiting case BPS did not ascribe any physical meaning to 6~,of low frequency perturbations in section 3 before one of the main problems being that the solution

turning to the general problem in section 4. The of the stagnant film model led to a discontinuityresults are discussed and a comparison with the of the solute flux at Z = 6SF’ To quote their

words, “the somewhat arbitrary quantity 65F mayexisting numerical and experimental data will hepresented in section 5. be characterized b y defining it so that it yields the

same dependence of the composition upon thegrowth parameters that is given by the exact

2. The boundary layer concept solution”.

Afterwards however, the externally supplied

Let us briefly recall the basis of the stagnant parameter 65F was widely interpreted as the

film model; in their pioneering work, BPS consid- length scale of “diffusion dominated mass trans-ered a steady state, one-dimensional concentra- port”. Wilson [121showed that this idea was not

tion problem, the fluid velocity in their idealized correct, but a lot of misunderstanding and confu-

Czochralski configuration being given by the ap- sion can still be found in the literature.proximate analytical expression of Cochran [111 Our opinion is that the effect of diffusion and

In a frame moving with the interface, the govern- convection is to force all the significant composi-ing mass transfer equation, tion variations to take place in a region of extent

6 — the solutal boundary layer — in the vicinity of d2C dC the interface. Using Wilson’s [121definition,

D—+[V 1—W(Z)]—-—=0, (1)

dZ2 dZ C(0) C~

6= (6)was solved along with the boundary conditions: Z=0 (interface):

the physical meaning of 6 rests on stable physical

D —I =V 1(1-k)C(0), (2)(dC 1 grounds. One of the main interests of this bound-

— dZ )~ ary layer concept is that it is possible to account Z — s ~: C =C~. (3) very simply for the convection phenomena in the

melt. Indeed, let us consider the “diffusion con-V1, W, D, k and C~,respectively, stand for the trolled” mass transfer equation:

interface and convective velocities, solute diffu-d

2C dC sion and partition coefficients, and concentration D— ~- Vett~= 0. (7)

in the fluid phase far away from the interface. dZ2


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J.P. Garandet / Microsegregation in crystal growth from the melt 433

where a constant effective velocity fr~,yet to be A lot of misunderstanding could probably havedefined, has been substituted to the V

1 — W(Z) been avoided, at no additional cost in terms of term in eq. (1). The boundary conditions being mathematical complexity, if BPS had introduced

again eqs. (2) and (3), the solution is an effective velocity J”~instead of 65F~Of course,

for steady state problems, the difference is mere(1 — k)V

1 ~_

~ + i

1c~. intellectual satisfaction, but we shall now see that

C(Z) = —

(1 —

k)V 1 exp~ D ) j this boundary layer concept can be fruitfully used

(8) in the study of microsegregation.

So far we have merely changed the arbitrary

parameter necessary to match the exact solution 3. The low frequency limitof BPS at the interface, but a first advantage isthat there is no discontinuity of the concentration In this section, we shall assume that the fre-derivative in this formulation. Besides, from eqs. quency of the fluctuations is low enough for the

(6) and (8), we get ~ = D/6, meaning that the system to follow them. In other words, we sup-characteristic length in the exponential is — as it pose that the composition instantaneously reacts

should be — the boundary layer thickness. More- to a change of the growth conditions. If both

over, it is possible to show from scaling analysis solidification and convection velocities are un-arguments [13,141that the characteristic equation steady, the time-dependent mass transfer equa-governing the variations of 6 is tion in a one-dimensional frame moving with the

D/6 = V 1 — W(6). (9) interface at a rate V1(t) is

Eq. (9) was seen to compare very well with exist- aC a~C 3Cing numerical and experimental data [13—15]; D + [V1(t) — W(Z, t)] ~. (11)a t a z

2 a z D/6 is then expected to be close to V

1 — W(Z) at

the boundary layer scale 6. The substitution in Order of magnitude arguments indicate that if

eq. (7) i s thus justified on physical grounds. the concentration variations take place oven aA way to interpret eq. (9) is to say that, at the time scale T much larger than 6

2/D, aC/at will

boundary layer edge, a typical diffusive velocity be negligible with respect to DII2C/aZ2. In that D/6 balances the overall fluid motion V, — W(6) case, it is possible to write down a time depen-

towards the front. Moreover, depending on the dent version of eq. (9):relative magnitudes of V 1 and W(6), transport D/6 ( t) = ~, ( t) — W( 6(t), t), (12)

will be said to be diffusion controlled (‘~~>>

— W(6), J’~~= V1), or convection controlled (V 1 with W(6(t), t) standing for the effective convec- — W(6), fr~= — W(6)). It is then possible to tion velocity at the time dependent boundary

define an effective partition coefficient keff as layer scale 6(t) (see eq. (9)). Once 6(t) has been

kC(0) k derived, it is possible to use the convecto-diffu-(10) sive parameter ~i(t) = 6(t)V1(t)/D in the effec-keff = C~ = 1 — (1 — k ) ~‘ tive partition coefficient formula (eq. (10)). Let us

where the convecto-diffusive parameter ~ = now suppose that the convective contribution doesV16/D = V1/V~~~measures the respective contri- not depend explicitly on time, viz. W(6(t), t) =butions of convection and diffusion to mass trans- W(6(t)), and that the growth rate I/~(t)is givenport. It is interesting to note [13,141 that in the by:developed convective regime (~1~n 1 or 6 ~ V1(t) = ~7~(i +m(t)), (13)

D/V1), 665F~ Under this restriction, 65F is

indeed the characteristic length scale of the prob- m(t) being a modulation around the average value

1cm, but this result certainly does not hold when V~,assumed smaller than unity to avoid backmelt-

diffusion becomes important. ing. The time-averaged value of the boundary


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434 . 1 . P. Gurundet / Microsegregaoon in crystal growth from the ,m’lt

1 relative concentration variation velocity changes. It can also be considered as thelow frequency limit for the general problem we

k~O.Oi shall now address.0.8 ‘~-

0,6 k=O.1

4. The general problem

0,4 Our purpose here is to derive a correlation

- between interface velocity and concentration0,2 ~ k—O.5 variations. As was done previously in the analyti-

cal efforts already quoted [8,9], we shall rely on a

c — —— —~-~- perturbation approach. However, we shall not0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 I

dwell on the related temperature problem andthe interested reader is referred to the earlier

Fig. I. Relative microsegregaiion versus i in the low ‘re- papers for a derivation of the influence of ther-quency limit for various values of the segregation coefficient. mal oscillations on growth rate.

In refs. [8,91, it was assumed that the stagnant

film thickness did not depend on interface veloc-ity. We shall here drop this hypothesis, the key

layer thickness ~ can then he derived from eq. improvement being that our results arc expected

(9), i.e.. D/6 = V 1 W(6). If we now suppose the to hold whatever the convecto-diffusive state of

variations of the W(6(t)) term to be small. eqs. the melt, i.e. for i ranging from 0 to I. However,(12) and (13) yield the approach of refs. [8,91 is certainly correct in

the convective regime limit (6 ~< D/VI or i ~< 1 )

~(I) = + (14) and it will be used to check the validity of our1 +ni(t)~1 solution there.

— — — Due to our solutal boundary layer formulation.where i = V16/D. Using ~(t) thus defined as an the time-dependent version of eq. (7) will be theinput in the effective partition coefficient for- relevant mass transfer equation. The main van-

mula, we find that the concentration fluctuation ables of the problem (composition field, interfacefor lo w amplitude growth rate changes is and effective velocity) are expanded into Fourier

— k ) ~(1 — ~) series up to the order one; thus:(‘~(t) ~SS I ~(l —k)~ c’5sifl(t), (15) C(Z, r) =C°(Z) +C’(Z) exp(iwt), (16a)

where ~ is the average interface composition V~= V1°+ V1 t exp(iwt), (lOb)derived setting i=i in eq. (10). The relative V.~1=V.YF+ V, exp(iwt). (lbc)concentration variation is plotted in fig. 1 forvarious values of the partition coefficient k. An w being the pulsation of the perturbations, as-

expression similar to eq. (15) was earlier obtained sumed small with respect to the zeroth orderin the frame of the stagnant film model [11; as contributions. Since we focus only on the influ-expected, both approaches yield the same tran- ence of variations of the interface velocity, the

sient behaviour when convection dominates mass fluctuation of fr~reduces to V~l.transport (i ~< 1) , but significant differences are The zeroth order solution is simply eq. (8),observed when diffusion becomes important. setting fr~= D/6°; we shall assume that the

Eq. (15) is useful for prediction purposes, boundary layer thickness of the steady state prob-keeping in mind that it is only valid when the 1c m 5°is a given parameter. More interesting forboundary layer reacts instantaneously to interface our present purposes, the first order governing


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J.P. Garandet / Microse gre gation in crystal growth from the melt 435

equations in the bulk and at the interface are, interface velocity variations are proportional, theirrespectively, relative amplitude depending on three non-di-

mensional parameters:d

2C’ D dC’ dC° D dZ2 + ~T - iwCt = - V

1 t ~(17) ~°=5°V1

0/D, Fq =w(6°)2/D, k.

dC’ Eq. (21) may look a little unfriendly, but it is= (1 — k)(V 1

0C1 + v~tC0). (18) indeed quite simple with respect to the previouslydZ

published correlations (eqs. (13a), (14a) and (14b)An additional requirement is that the perturba- in ref. [8], eqs. (40), (41) and (42) in ref. [9]). Totion C’ vanishes far away from the interface: check the validity of our result, two limiting cases

Z—~, C1=0. (19) can be considered:. The low frequency limit, Fq — v 0 ; it is easy to

The solution to the homogeneous problem is show that eq. (21) reduces to eq. (15).simply: C~= A exp(m,Z), A being an integration • The convective regime, high frequency rangeconstant yet unknown and m, a complex parame- (~O~ 0 , Fq — s ~) where the previous solutionster: [8,9] are expected to be hold. Taking the limit

1 [

( 4iw(6°)2)i/2] Fq —* ~, eq. (21) becomes

1—i11+ 1+ I ~(1_k)~Fq~/2 (22)1260 D

~=Vi

Setting dC°/dZ= —B exp(—Z/6°), with B = which can be made to coincide with the publishedV~°C,°(l — k)/D, a particular solution of eq. results [1,9,16].

(17) is simply C,~= (iB/w) exp(—Z/6°). The

general solution is thus

iB ( Z 1 5. DiscussionC’ =A exp(m,Z) + —exp~ —

0)

Now that we have gained confidence in theWe now make use of eq. (18) to determine A , and validity of eq. (21), we can proceed to a system-we finally get for the value of C’ at the interface: atic presentation of the predicted correlations.

Shown in fig. 2 is the frequency dependence of V 1

1 (1 —k)C~

( iV,° ,~0)). the modulus of the relative concentration vania-

1— —(1 +mC:=_D+(lk)V0 0)6°

(20) —

The above expression can be transformed into ->

C V 1

1 — -1

-1.5C~ V,

1 — (i~i0/2Fq)[1 — (1 + 4iFq)iz”2] ~ -2 X ~[1+(l+4iFq)1/2] —(1—k)~°

—2.5

(21) — .3

—3 —2 —i 0 1 2 3 4

As noted in ref. [8], the occurrence of imaginary log (Fq~terms defines the phase difference between the Fig. 2. Norm of the relative concentration variation versusinterface concentration, and the driving term non-dimensional frequency for three values of .i°and k =

V

1 ’ exp(iwt). We find that the concentration and o.oi.


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436 . 1 . P . Garandet / Mic-ro.segregation in crystalgrowth from the roe/i

C ~ T

C ’

51

~ -~ r~C r)

U -.Cr

NCr~Cr- 6—

- oL~—I ~

3 2 1 0 1 2 3 4 4 5

log (-q)

Fig. 3. Interface velocity/concentration phase shifi sersus FigS. Norm of the relative concentration sariation versus the

non-dimensional frequency for the case Jo = ((.5 and — convecto-diffusive parameter tor three values ol Fq and k — ‘(1 .1)! . 0.))

tions for three values of the steady state con- Fq I ~ However, since microsegregation van-

vecto-diffusive parameter, i° 0.1, i°—0.3 and ishes both in the low (eq. (15)) and high (ccl. (22))= 0.9; in all cases, we took k = 0.01. Fq limits, a maximum in the medium frequency

The variations are monotonic, the expected range could have been expected.Fq I 2 power law being observed above a cutoff An alternative presentation of’ these results isfrequency Fq 1 . The associated phase shift of proposed in fig. 5, using i~as a variable. Forfig. 3 also exhibits a monotonic decrease; in the high values of Fq, the relative concentration flue-low frequency range, the concentration fluctua- tuations increase linearly with i°;it is only for Fq

tion i s able to follow the variation of the interface smaller than 0.03 that a maximum can he oh-velocity, but it lags ~/4 behind in the high fre- served. The influence of the partition coefficient

quency limit, has not been checked in detail, hut the corn-

A priori more surprising is the result obtained position variations always seem to increase withfor i°= 1 , i.e. for purely diffusive mass transport 1 — k H(see fig. 4). The relative composition variations Turning to a comparison with the numericalfirst increase with frequency, then pass through a data, the effect of fluctuating growth rate in a

maximum around Fq 0 .1 and finally decrease as Czochralski configuration was thoroughly studiedby Wilson [4,5.7] in the early eighties. A fullytime-dependent analysis was carried out. assurn-ing a variation following eq. (13). with n,( t) —

A sin(wt), where w stands for the angular veloc--

5 L- - ity of crystal rotation.

Shown in table I is a comparison between the

predictions of our eq. (21) and the numerical

Table I

(~onipartsonbetween our analytical predictions and Wilson’s

numerical results 1 7 1

-2 5 .~____________________________________ J° Fq k ( H / ( (‘iH ( H ’4 3 —2 1 0 1 2 3 4

leg. (21)) (numericalllog ~3r

Fig. 4. Norm of the relative concentration variation versus 1)33 6.09 001 1 ). 1 1 ) ((II

non-dimensional frequency for the case — I (purely diffu- ~ 2 54 ((.07 028 026

sive mass transport) and k ( (1)1. 1)78 2.54 1.25 0.0505 11.0515


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J.P . Garandet / Microsegregation in crystal growth from the melt 437

E

0 C)

51

c v _50 E

0

a 0 (b ) —.~. •

~ 3x10

19 • - 40 a,0 • l~~1~’~O • 30 —0 0

0 •••••a )o •~ ~ - 20

=0

Icycle 2 cycle 3 cycle — 102xtO’9 — I i I I 0 0

0 1 2 3 4 5 6 7 8 9 1 0 I I 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 c c

tb ) Length of crystal grown (pm1x280 (0 10

0

Fig. 6 . Quantitative growth rate and microsegregation analysis on a Ga doped Ge crystal (experimental result from ref. [17]).

results for the three cases presented in ref. [7]. with the predictions of eq. (21) is fair, the maxi-Even though we know that microsegregation re- mum discrepancy being roughly 50%.

mains proportional to A up to the point of back

melting [4], thus supporting the possibility of alinear approach even out of its range of validity Concluding remarks

stricto sensu, the agreement is well beyond what

could have been expected. Our purpose in this paper was to propose anAs for the experimental data, a very important analytical approach to the microsegregation prob-

paper was published by Witt et al. [17] in 1973. 1cm. Using a perturbation analysis, we were able

These authors were able to measure simultane- to derive a correlation between interface velocityously the dopant distribution (via spreading resis- and concentration fluctuations. The algebra in-

tance) and the microscopic interface velocity (via volved was seen to be quite simple with respect tothe distance between striations induced by cur- the previous theoretical treatments [8,9]. We thus

rent pulses sent at a given frequency) in think that our modelling can be used in practice,Czochralski grown Ga doped Ge. A typical out- for instance to assess the allowable pulling device

put of this procedure is plotted in fig. 6 . instabilities in a crystal growth furnace.Further work by the MIT team [18,19] on this We also demonstrated the possibility of ac-topic featured Sb doped Si alloys. All the results counting for the effect of convection in the melt

are summarized in table 2, only the experiments via a solute boundary layer formalism. As op-performed without backmelting being considered posed to the Burton, Prim and Slichter stagnanthere. Two sets of data were obtained from ref. film model [10], this approach stands on stable[18], as the two growth cycles were somewhat physical grounds. As the boundary layer concept

different (see fig. 2 in ref. [18]). The agreement was also successfully applied to radial problems[20,21], we think that it is a very powerful key tothe modelling of segregation phenomena in melt

Table 2

Comparison between our analytical predictions and the exper- crystal growth.imental results [17—19]

J° Fq k C~/C,° C~/C°(eq. ( 2 1 ) ) ( e x p e r i m e n t a l ) Acknowledgements

0.44 3.49 0.087 0.094 0.147 1 1 7 10.35 5 .6 1 0.023 0.058 0.08 1 1 8 1 Fruitful discussions on the topic with Drs. A.0.37 5.38 0.023 0.069 0.097 [181 Rouzaud, D. Camel and J.J. Favier are gratefully0.26 6.38 0.023 0.045 0.06 [191 acknowledged. The author would also like to


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438 . 1 . P. Garatu/et / Micro.segregation in crystal growtbt /rom the tile/i

thank his friends 0.. Z., J. and F . for their help in i5 Zeroth order convecto-diffusive parameter

improving the presentation of the manuscript. w Frequency of perturbation (s IThe present work was conducted within the frameof the GRAMME agreement between the CNES

and the CEA.References

I”~oiuenclature [I] JR. (arrutllers and A l ’ . Wilt. in: (rystal ( ~rowIli ,iiid(‘haracterizittion, lids. R. IJeda and i l ) . Mullin (North—

A Amplitude of growth rate modulation I lolland. Amsterdam. 197S).[2] .1.1.1). Alexander, I Ouazzani and I Rosenberiiet. .1 .

C Solute composition (mass or mole traction) (‘rysil Growth 97 (989) 285

~ Steady state interface composition (mass or [3] AN. Danilewskv nil K.W. I l e n z , J . (ryslal ( ~ussIbi 01

mole fraction) (1989) 571.

C ’~ ’ Zeroth order interface composition (mass [4] 1.0. Wilson. J. Crystal Growth 48 I 1980) 4.35or mole fraction) H [.0. Wilson. J. (‘rystal Growth 48 (1980) 451.

I ‘ . . . . [6] A.M.J.G. van Run. J . Crystal Growth 47 11979) 68ILC, Fluctuation of the interface composttion [7] JJ [~ and [.0 Wikon, . 1 Crystal (irowth 5$ I 1982)

(mass or mole fraction) 1(3.

I) Solute diffusion coefficient (rn s -- ) [8] DII. I l u r l c , F. Jake man and ER. l’ikc 3-stIll

Fq Non-dimensional frequency (irowih 3/4 (1968) 633.

k Solute partition coefficient [9] 1) . Thevcnard, A. Rouz~iud.J. (‘omera and .1.1. ~aviei. . 1 .(‘rvstal Growth 108 (I 991) 572.

m( I ) Modulation of growth rate [11)1 IA Burton. R (‘ Prim and W.P Sliehter, J. (hem PhysW Dimensional velocity component on Z axis 21 (1953) 987.

(m5_I) [I Ii W.G. (oehran. Proc. (‘ambridge Phil. Soc. 30 (1934) .365.

V~~-Effective diffusion velocity (rn ~ -- I) [12] L.C). Wilson, J. Crystal Growth 44 (197$) 247.

V Interface velocity (m ~ — - I [13] J.P. Garandet. T. Duffar and .1.1. Favicr.J.(rysI~l- . Growth 11(6 (1990) 437.

V, Average intirtact velocity (m s [14] 1 P Ci ir mdcl ii I sri nd D ( imcl in II indbook o hV ,

1 1 Zeroth order interface velocity (m ~ — I) Crystal Growth. Vol. 2, Ed. D.T.J. I lurle (North-I lolland,V, I Fluctuation of interface and effective veloc- Amsterdam. to he published).

ities (m ~ I) [IS] 3.1’. G:irandet. A. Rouzaud, T. Duffar and [1 Camel. .1 .

Z Space coordinate normal to interface (m) (‘rystal Growth 113 (1991) 587.I 6] D.T.J . I-lu ne and F. Jakcman, J . Crystal Growth S ( I 969)

227.S Solute boundary layer thickness (m) [17] A.F. Wilt, M. l.ichtcnsteigcr and Il.(. Gatoi,.I, [Ice-51 1 Zeroth order solute boundary layer thick- trochem. Soc. 1 211 (1973) 1119.

ness (m ) [IS] A. Murgai. I IC’. Gatos and A.F. Wilt. .1 . Eleetroclicm.Soc l~3(19761 114

~. Stagnant film thickness in BPS model (m) [19] A. Murgai, [C. Gatos and WA. Wesldor~~..I.Flee—I Convecto-diffusive parameter (0 ~ i ~ trochem Soc. 126 (1979) 2240.i Quasi-steady-state convecto-diffusive pa- [20] J.P. Garandei, J. Crystal Growth 114 (1991) 593.

rameter [21] JR Garandet, . 1 . Crystal Growth 125 (1992)112.

microsegregation in crystal growth from the melt: an analytical approach

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