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7

Solute trapping and diffusionless solidification in a binary system

Peter Galenko∗

German Aerospace Center, Institute of Materials Physics in Space, Cologne 51170, Germany

(Dated: October 30, 2018)

Numerous experimental data on the rapid solidification of binary systems exhibit the formationof metastable solid phases with the initial (nominal) chemical composition. This fact is explainedby complete solute trapping leading to diffusionless (chemically partitionless) solidification ata finite growth velocity of crystals. Special attention is paid to developing a model of rapidsolidification which describes a transition from chemically partitioned to diffusionless growth ofcrystals. Analytical treatments lead to the condition for complete solute trapping which directlyfollows from the analysis of the solute diffusion around the solid-liquid interface and atomicattachment and detachment at the interface. The resulting equations for the flux balance atthe interface take into account two kinetic parameters: diffusion speed VDI on the interface anddiffusion speed VD in bulk phases. The model describes experimental data on nonequilibrium solutepartitioning in solidification of Si-As alloys [M.J. Aziz et al., J. Cryst. Growth 148, 172 (1995);Acta Mater. 48, 4797 (2000)] for the whole range of solidification velocity investigated.

PACS numbers: 81.10.Aj; 05.70.Fh; 05.70.Ln; 81.30.Fb

I. INTRODUCTION

The concept of “solute trapping” has been introducedto define the processes of solute redistribution at the in-terface which are accompanied by (i) the increasing ofthe chemical potential [1] and (ii) the deviation of thepartition coefficient for solute distribution towards unityfrom its equilibrium value (independently of the sign ofthe chemical potential) [2].

In experimental investigations of rapid solidification, acomplete solute trapping leading to diffusionless (chem-ically partitionless) solidification was first observed byOlsen and Hultgren and Duwez et al. in experiments onrapid solidification [3]. They showed that rapidly solid-ifying alloy systems lead to the originating of supersat-urated solid solution with the initial (nominal) chemicalcomposition of the alloy. Later on, crystal microstruc-tures with the initial chemical composition were foundby Biloni and Chalmers in rapidly solidified pre-dendriticand dendritic patterns [4].

Backer and Cahn [1] have shown that with the finitesolidification velocity in a Cd-Zn system the coefficient ofthe Cd distribution becomes equal to the unit that char-acterizes diffusionless solidification. This fact has beenconfirmed in many binary systems by Miroshnichenko[5]. He investigated dendritic crystal microstructure af-ter quenching from the liquid state by splat quench-ing and melt spinning methods. The results of Mirosh-nichenko’s microstructural analysis show that at a cool-ing rate greater than some critical value (depending on analloy and experimental method this value is in the range105−106 K/s) a core of main stems of dendrites has initial(nominal) chemical composition of the alloy. A criticalvalue for undercooling in the transition to purely ther-

∗e-mail: Peter.Galenko@dlr.de

mally controlled growth with a homogeneous distributionof chemical composition in Ni-B solidifying samples pro-cessed by an electromagnetic levitation facility has beenobtained by Eckler et al. [6]. Finally, it is necessaryto note that many eutectic systems undergo chemicallypartitionless solidification with an initial composition [5]that can be explained by the transition to diffusionlesssolidification [7].

As a consequence, experimental investigations [1, 3,4, 5, 6] show that with increasing driving force of so-lidification solute traps are much more pronounced bysolidifying microstructure. At a finite value of the crit-ical governing parameters (undercooling, cooling rate ortemperature gradient) complete solute trapping occurs.Because the finite value of the governing parameter de-fines the concrete solidification velocity, complete solutetrapping and diffusionless solidification begin to proceedwith a fixed critical growth velocity of crystals.

The main purpose of the present paper is to describe amodel for solute trapping and the transition from chem-ically partitioned to diffusionless solidification in a bi-nary system. Using the local nonequilibrium approach torapid solidification, an analysis of diffusion mass trans-port in bulk phases together with conditions of atomicattachment and detachment on the solid-liquid interfaceis given.

The paper is organized as follows. In Sec. II, previousinvestigations of solute trapping are shortly reviewed. InSec. III, an analysis of solute diffusion leading to pro-nounced solute trapping and complete solute trapping isgiven. The nonequilibrium solute partitioning functionfor atoms on the interface is derived in Sec. IV. A com-parison with previous models and experimental data onsolidification of binary systems is presented in Sec. V.Finally, in Sec. VI conclusions of the work are summa-rized.

2

II. PREVIOUS INVESTIGATIONS

For the simplest case of an atomic system, let us con-sider an isobaric and isothermal binary system (the pres-sure P and temperature T are constant) with concentra-tion XA and XB of atoms A and B, respectively. In thisarticle, we denote X as the concentration of the atoms ofB sort. For a brief overview, we summarize the equilib-rium and nonequilibrium solute distribution on the solid-liquid interface.

A. Equilibrium

In equilibrium, the concentration of atoms X at thephase interface is not equal from both sides of the in-terface due to the different solubility of atoms in phases.During the equilibrium coexistence of phases (gas-solid,liquid-solid, gas-liquid) the atoms are distributed alongthe interface in consistency with the diagram of a phasestate. A difference in atomic concentration in phasesat the interface can be characterized by the equilibriumcoefficient ke of the atomic distribution between phases.For equilibrium coexistence of phases (e.g., between crys-tal and melt, vapor and crystal, crystal and liquid), thecoefficient ke can be expressed in the general form [8]

ke(XL, XS , T ) =Xe

S

XeL

≡ exp

(

−∆µ′

RT

)

. (1)

In Eq. (1), XeL and Xe

S are the mole fractions of the Bcomponent in the liquid phase (L) or crystal (S), respec-tively, R is the gas constant, and ∆µ′ is the difference inchemical potentials described by

∆µ′ = ∆µ′

B −∆µ′

A, (2)

with

∆µ′

B = µ′

BS − µ′

BL, ∆µ′

A = µ′

AS − µ′

AL, (3)

where ∆µ′

A and ∆µ′

B are the driving forces for redistri-bution of atoms A and B, respectively, which are definedby redistribution potentials µ′

A and µ′

B for phases L andS. The differences ∆µ′

A and ∆µ′

B , Eq. (3), define thesign of ∆µ′ in Eq. (2). For instance, if ∆µ′ is negative(∆µ′

A > ∆µ′

B), one has ke < 1 - the case of smaller solu-bility of atoms B in the phase S in comparison with theirsolubility in the phase L.As a general characteristic of phase equilibria in binary

systems, expression (1), together with Eqs. (2) and (3),is usually considered as a measure of the driving forcefor atomic redistribution at the phase interface. It canalso be considered as one of the main parameters for theconstruction of the diagrams of a phase state.

B. Nonequilibrium

Expressions (1)-(3) assume local equilibrium at the in-terface, which is a useful approximation for many systems

transforming at small interface velocities. At a large driv-ing force for the interface advancing and with increasingof the interface velocity, the local equilibrium is not main-tained [1]. Therefore, the condition for local interfacialequilibrium was relaxed by taking into account a kineticinterface undercooling and deviations from chemical equi-librium at the alloy’s solidification front [8, 10].A number of models [2, 9, 10, 11, 12, 13] have been pro-

posed to account for solute trapping and related phenom-ena observed during rapid phase transformations. One ofthe well-established boundary conditions for solute redis-tribution can be taken from the continuous growth model(CGM) applied to solute trapping by Aziz and Kaplan[2, 13, 14]. The CGM assumes alloy solidification at a“rough interface”; i.e, all interface sites are potential sitesfor crystallization events. With a high solidification rate,the atom can be trapped on a high-energy site of thecrystal lattice. This leads to a local nonequilibrium onthe interface and to the formation of metastable solids(see examples in Ref. [15]). As a result, the solute parti-tioning function at the solid-liquid interface is describedby [2, 13]

k(V ) =ke + V/VDI

1 + V/VDI, (4)

where VDI is the speed of diffusion at the interface andke is the value of the equilibrium partition coefficientgiven by Eq. (1), i.e., with the negligible interface mo-tion, V → 0. Equation (4) evaluates the ratio XS/XL

at the interface for dilute solutions of B (“solute”) in A(“solvent”).The interfacial diffusion speed VDI is the kinetic pa-

rameter describing the deviation from chemical equilib-rium at the interface. It has been defined as the ratiobetween the diffusion coefficient DI at the interface andthe characteristic distance λ for the diffusion jump [2, 13]:VDI = DI/λ. The distance λ is assumed to be equal tothe width of the solid-liquid interface (few interatomicdistances) and the diffusion jumps are taken along thedirection of growth. Therefore, this definition for VDI

is corrected by results of molecular dynamic simulations[16]. They include diffusion in all spatial directions; i.e.,the diffusion speed is VDI = 6DI/λ, where the factorof 6 accounts for the possibility of jumps along the six(±x, y, z) Cartesian axes.Outcomes following from the solute partitioning func-

tion (4) were compared in the modeling of solute trap-ping using numerical computations based on the phase-field theory of alloys solidification. Wheeler et al. [17]naturally included an energy penalty for high composi-tion gradients in the liquid that supresses the partition-ing of solute at a rapidly moving interface and leads tosolute trapping. They also showed that the construc-tion of common tangents to the curves of free energy (inthe spirit of Baker and Cahn [18]) has to be defined fornonequilibrium concentrations which already depend onthe solidification velocity. In order to eliminate or re-duce the solute trapping effect by the diffuse interface at

3

small growth velocity, Karma and co-workers proposedan ad hoc suitable antitrapping condition to the diffu-sion flux [19]. These works [17, 19] showed that whenthe solute trapping effect comes to modeling alloy so-lidification with both phase and concentration fields, acrucial issue arises concerning the relative magnitudes ofthe gradients of the two fields within the solidificationfront as well as the relative thickness of the concentra-tion jump interface. Additionally, Conti [20] investigatedthe usual one-dimensional (1D) formulation of the phasefield model without the concentration gradient correc-tions of Wheeler et al. [17]. He resolved the governingequations numerically for the interface temperature andthe solute concentration field as a function of the growthvelocity. The partition coefficient k(V ) is monotonicallyincreasing towards unity at large growth rates followingthe predictions of the continuous growth model (4). How-ever, in contrast to the results of natural experiments[1, 3, 4, 5, 6], numeric predictions [17, 20] were not ableto reach the complete chemically partitionless (diffusion-less) solidification at a finite solidification velocity.One of the deficiencies of the function (4) is the diffi-

culty to describe complete solute trapping at the finitesolidification velocity: Equation (4) predicts k → 1 onlywith V → ∞. Contrary to this prediction, a transitionto partitionless solidification occurs at a finite solidifica-tion velocity as it has been shown in numerous experi-ments [1, 3, 4, 5, 6]. Molecular dynamic simulations alsoshow that the transition to complete solute trapping isobserved at a finite interface velocity in rapid solidifica-tion of a binary system [16]. Therefore, as an extensionof Eq. (4), a generalized function for solute partitioningin the case of local nonequilibrium solute diffusion withinthe approximation of a dilute system has been introducedby Sobolev [21]. This yields

k(V ) =(1− V 2/V 2

D)ke + V/VDI

1− V 2/V 2

D + V/VDI, V < VD,

k(V ) = 1, V ≥ VD. (5)

The diffusion speed VD introduced in Eq. (5) is the char-acteristic bulk speed. It is defined as a maximum speedfor the solute diffusion propagation or as a speed for thefront of solute diffusion profile. In particular, the speedVD is obtained by the speed of propagation of the planeharmonic wave away from the solid-liquid interface (seethe Appendix in Ref. [22]). As the velocity V of theinterface is comparable by magnitude to the speed VD,the high frequency limit takes place: ωτD >> 1, whereω is the real cyclic frequency of the plane harmonic waveand τD is the time for relaxation of the diffusion flux toits steady state. In this case, VD has to be consideredfinite and it is defined as VD = (D/τD)1/2, where D isthe diffusion coefficient in bulk liquid.In the local equilibrium limit, i.e., when the bulk diffu-

sive speed is infinite, VD → ∞, expression (5) reduces tothe function k(V ) which takes into account the deviationfrom local equilibrium at the interface only as described

by Eq. (4). The function (5) includes the deviation fromlocal equilibrium at the interface (introducing interfacialdiffusion speed VDI) and in the bulk liquid (introducingdiffusive speed VD in the bulk liquid). As Eq. (5) shows,complete solute trapping k(V ) = 1 proceeds at V = VD.This result has been introduced by Sobolev from a postu-lation about the zero value for the diffusion coefficient atV ≥ VD. The next section further details that the con-dition for complete solute trapping follows directly fromthe analysis of solute diffusion flux.

III. DIFFUSION MASS TRANSPORT AND

SOLUTE TRAPPING

In 1D solidification along the z-axis, the mass balanceis given by

∂X

∂t= −

∂J

∂z, (6)

where t is the time and J is the diffusion flux. To considersolute trapping in 1D local nonequilibrium solidification,we take one of the results from a model of rapid phasetransitions [23]. Using this model, the evolution equationfor diffusion flux J along the z-axis is described by

J = M

[

∂

∂z

(

∂s

∂X+ ε2x

∂2X

∂z2

)

− αj∂J

∂t

]

, (7)

where s is the entropy density, εx the factor proportionalto the correlation length, and M is the diffusion mobilityof atoms. The latter is defined by

M = TD, D = D(∂(∆µ)/∂X)−1, (8)

where D is the diffusion coefficient and ∆µ is the differ-ence of chemical potentials between solvent and solute.From known thermodynamic expressions [24] one can ac-cept that

∂s

∂X= −

∆µ

T, αj =

τDTD

∂(∆µ)

∂X=

τD

TD, (9)

where τD is the time for diffusion flux relaxation to itssteady state. Then, omitting the term responsible foratomic correlation, i.e., assuming that εx = 0, one canget from Eq. (7) the expression

J = M

[

1

T

∂(∆µ)

∂z−

τD

TD

∂J

∂t

]

. (10)

Using Eq. (8), the evolution equation (10) results asfollows

J = −D∂(∆µ)

∂z− τD

∂J

∂t. (11)

We find the solution for the diffusion flux J which hassignificance in the analysis of solute trapping. Using the

4

expression for the diffusion speed, VD = (D/τD)1/2, fromEqs. (6) and (11) one gets

τD∂2J

∂t2+

∂J

∂t= D

∂2J

∂z2. (12)

Equation (12) is a partial differential equation of hyper-bolic type. It describes the flux J in the so-called “hy-perbolic evolution”, which proceeds with a sharp front ofthe profile for the solute transport. It occurs due to boththe diffusive and propagative nature of the transport inthe high frequency limit ωτD >> 1 with V ∼ VD.For a steady-state regime of interfacial motion Eq. (12)

takes the form

D

(

1−V 2

V 2

D

)

d2J

dz2+ V

dJ

dz= 0, (13)

which is true in a reference frame moving at constantvelocity V with the interface z = 0. A general solutionof Eq. (13) is

J(z) = c1 + c2 exp

(

−V z

D(1− V 2/V 2

D)

)

. (14)

To define a particular solution one can assume the fol-lowing boundary conditions: the balance on the interfaceis J(z = 0) = V (X∗

L − X∗

S), and the flux is limited bythe expression J(z → ∞) = 0 far from the interface withz → ∞. The latter condition gives c1 = 0 for any veloc-ity V and one gets c2 = 0 for V ≥ VD. Also, from theinterfacial balance with z = 0 one gets c2 = V (X∗

L−X∗

S)for V < VD. As a result, solution (14) transforms intothe particular solution

J(z)

= V (X∗

L −X∗

S) exp

(

−V z

D(1− V 2/V 2

D)

)

, V < VD,

J(z) = 0, V ≥ VD, (15)

where X∗

L and X∗

S are the liquid concentration and solidconcentration, respectively, on the interface.Solution (15) gives the condition for complete solute

trapping with finite velocity V ≥ VD. This is expressedby the expression for the solute partitioning function

k(V ) = X∗

S/X∗

L 6= 1, V < VD,

k(V ) = 1, X∗

S = X∗

L, V ≥ VD. (16)

The latter condition in Eq. (16) defines the equality ofthe concentrations in the phases and leads to completesolute trapping.To obtain an explicit form for the solute partitioning

function (16), we analyze the balance of diffusion fluxeson the interface. Taking again the steady state regimeof solidification constant velocity V , from the system (6)and (11) one can obtain the equations

dJ

dz= V

dX

dz, J = −D

d(∆µ)

dz+ τDV

dJ

dz, (17)

from which we get the single equation for the diffusionflux J . This yields

J = −D

(

∂(∆µ)

∂X

)

−1d(∆µ)

dz+D

V 2

V 2

D

dX

dz. (18)

The above-defined thermodynamic parameter D and thediffusion speed VD in bulk have been taken into account.Defining the gradient of the difference of the chemicalpotentials as d(∆µ)/dz = [∂(∆µ)/∂X ]dX/dz, Eq. (18)gives

J = −D

(

∂(∆µ)

∂X

)

−1 [

1−V 2

V 2

D

]

d(∆µ)

dz. (19)

This equation is a general expression for the steady dif-fusion flux into the liquid from the interface. Withinthe local equilibrium limit VD → ∞ one can obtain theknown Fickian approximation which has been used pre-viously for analysis of solute trapping [25, 26].Analytical solutions [27] for solidification under local

nonequilibrium diffusion show that the concentration inboth phases becomes equal to the initial (nominal) con-centration and the diffusion flux is absent for V ≥ VD. Itis also given by Eq. (15). Therefore, in addition to Eq.(19), one can finally obtain

J = −D

(

∂(∆µ)

∂X

)

−1 [

1−V 2

V 2

D

]

d(∆µ)

dz, V < VD,

J = 0, V ≥ VD. (20)

At the phase interface one assumes in Eq. (20), first,that the term D((∂∆µ)/∂X)−1 is proportional to con-centration such that

D

(

∂(∆µ)

∂X

)

−1

=DIX

∗

L

RT. (21)

Second, the chemical inhomogeneity (solutal segregation)exists due to the jump of the chemical potential ∆µwhich has the interfacial gradient −d(∆µ)/dz ∼= ∆µ/W0

at a small distance W0 of the order of a few inter-atomic distances. Third, in an approximation of ideal(or even real) solutions, one can assume for the interfa-cial difference of chemical potentials ∆µ = ∆µL(X) −∆µS(X) ∼= RT (lnXe

L − lnX∗

L) − RT (lnXeS − lnX∗

S) =RT ln(X∗

S/X∗

L) − RT ln(XeS/X

eL) = RT [lnk(V ) − ln ke],

where XeL and Xe

S are equilibrium concentrations on theinterface from the liquid phase and solid phase, respec-tively, and k(V ) is the ratio of concentrations on the in-terface defined by Eq. (16). Taking into account theselast evaluations, one can get for the chemical potentialgradient the expression

−d(∆µ)

dz∼=

RT

W0

lnk(V )

ke. (22)

Integration of the balance (17) on the interface givesthe flux

J = V (X∗

L −X∗

S). (23)

5

Substituting Eqs. (21) and (22) into the expression fordiffusion flux Eq. (20) with using the balance (23) givesthe following expression for the solute partitioning func-tion:

[

1−V 2

V 2

D

]

lnk(V )

ke=

V

VDI[1− k(V )], V < VD,

k(V ) ≡ X∗

S/X∗

L = 1, V ≥ VD, (24)

where VDI = DI/W0 is the speed for solute diffusion onthe interface.Equation (24) gives the evaluation of solute trapping

effect through the solute partitioning function k(V ) de-rived initially from the analysis of the evolution equation(7) for the diffusion flux J . This equation takes into ac-count finite diffusion speeds on the interface and in bulkliquid. The introduction of these two speeds is a conse-quence of the local nonequilibrium both on the interfaceand in bulk liquid. As Eq. (24) shows, the complete so-lute trapping k(V ) = 1 proceeds at V = VD. Equation(24) transforms into a previously known expression forthe function k(V ) derived in Refs. [25, 26] with relaxinglocal equilibrium on the interface and using local equi-librium in bulk liquid (VD → ∞) for the diluted binarysystem (X∗

L << 1).

IV. SOLUTE PARTITIONING FUNCTION

We use a model of diffusion in which particles moveby diffusion jumps in random time between two phases(states). This model was called the “two-level model ofdiffusion” and it was introduced in the context of variousapplication, e.g., in chromatography [28] or for a longi-tudinal solute dispersion in a tube with flowing water(Taylor’s dispersion) [29].Let Pi(t, z) be the probability density of a particle po-

sition in the phase i = L or in the phase i = S at themoment t. Then local conservation of the probabilitydensity in a point with coordinate z belonging to thephase i is defined by

∂Pi

∂t= −

∂Ji∂z

. (25)

If the interface moves with a velocity comparable to thesolute diffusion speed VD in bulk phases, then the fluxJi(t, z) of the density probability depends on the pre-history of the diffusion process. The flux, therefore, isdefined by

Ji(t, z) = −

∫ t

−∞

Di(t− t∗)∂Pi(t

∗, z)

∂zdt∗. (26)

The relaxation function Di(t− t∗) can be chosen in theform Di(t− t∗) = Di(0) exp[−(t− t∗)/τD] of exponentialdecay. In such a case, Eq. (26) is reduced to the Maxwell-Kattaneo equation

τD∂Ji∂t

+ Ji +Di(0)∂Pi

∂z= 0. (27)

It is accepted in Eq. (27) thatDi(0) is the diffusion coeffi-cient at the final moment of relaxation prehistory so thatDi(0) = D. System (25) and (27) gives a single equationof a hyperbolic type for the density of probability

τD∂2Pi

∂t2+

∂Pi

∂t= Di

∂2Pi

∂z2, (28)

or for the flux,

τD∂2Ji∂t2

+∂Ji∂t

= Di∂2Ji∂z2

. (29)

As was shown in Ref. [30], the density of probabilitydescribed by Eq. (28) gives a positive entropy productionfor the particle exchange between two levels (between twosubsystems or phases).Integration of Eq. (28) by an infinitesimal layer in-

cluding an interface leads to the balance

[

Di∂Pi

∂z+ τD

∂(V Pi)

∂t+ Ji

]∣

∣

∣

∣

L

S

= 0. (30)

In the steady-state regime one can get the following equa-tion for the i-th phase

Di∂Pi

∂z+ τD

∂(V Pi)

∂t+ Ji = Di

dPi

dz− τDV 2

dPi

dz+ Ji

= Di

(

1−V 2

V 2

Di

)

dPi

dz+ Ji, (31)

which is true in a reference frame moving with constantvelocity V and placed on the interface where the balance(25) is described as dJi/dz = V dPi/dz. Using Eq. (31),the balance (30) is

JL − JS

= −

[

DL

(

1−V 2

V 2

DL

)

dPL

dz−DS

(

1−V 2

V 2

DS

)

dPS

dz

]

.

(32)

In Eq. (32) we introduce the speeds VDL and VDS

of interfacial solute diffusion from the liquid and solidphases, respectively. They are defined by

VDL = DL/lD = νLlD, VDS = DS/lD = νSlD, (33)

where DL and DS are the diffusion coefficients in thephases, lD scales for diffusion within which the diffusionjumps occur in phases (or on the interface), and νL andνS are the frequencies of diffusion jumps in phases (oron the interface). From the theory of the transitive state[31] one can define the frequencies of atomic jumps as

νL = ν0 exp

(

−QD

RT

)

,

νS = ν0 exp

(

−QD +∆µ′

RT

)

, (34)

6

where ν0 is the attempt frequency of atomic jumps of theorder of the vibrational frequency [8, 32], QD the acti-vation barrier for atomic diffusion through the interface,and ∆µ′ is the difference of chemical potentials definedby Eqs. (2) and (3). Obviously, interfacial equilibriumexists for

νS/νL = exp[−∆µ′/(RT )] ≡ ke. (35)

From the interfacial balance (32) there follows

JL − JS = −

(

1−V 2

V 2

DL

)

d

dz[DLPL −DS(V )PS ] , (36)

where DS(V ) is the function of the interfacial velocity Vdefined by

DS(V ) = DS1− V 2/V 2

DS

1− V 2/V 2

DL

=

0, VDS << VDL,

DS , VDS ≈ VDL,

DS(1− V 2/V 2

DL)−1, VDS → ∞.

(37)

The function (37) describes the following cases: (a)VDS → 0, negligible diffusion in solid (DS = 0) in com-parison with the diffusion in liquid; (b) VDS ≈ VDL, ap-proximate equality for diffusion speeds in the liquid andsolid around the interface; and (c) VDS → ∞, conditionof local equilibrium in the diffusion field of the solid (thatoccurs with high frequency jumps of atoms in solid).From now on, the above case (b) for approximate

equality of diffusion speeds in phases around the inter-face is taken. First, we use the finite difference −dx = lDin the balance (32). Second, we take into account thatthe factor (1− V 2/V 2

DL) is related to the bulk diffusion.Finally, using the definition (33), the balance (32) is de-scribed by

JL − JS =

(

1−V 2

V 2

D

)

[VDLPL − VDSPS ], (38)

where VD is the solute diffusion speed in bulk liquidaround the interface. Using Eqs. (33)-(35), this balancecan be rewritten as

JL − JS = VDL

(

1−V 2

V 2

D

)

[PL − kePS ]. (39)

For the concentrated binary system the probabilitiesPL and PS in Eq. (39) are directly proportional to theatomic concentrations in phases. This leads to

PL = XS(1−XL)/Ω, PS = XL(1−XS)/Ω, (40)

where Ω is the atomic volume. Therefore, Eq. (39) canbe rewritten as

JL − JS =

(

1−V 2

V 2

D

)

×[XS(1 −XL)− keXL(1 −XS)]VDL

Ω. (41)

We further use the already obtained result (20) accordingto which the diffusion flux is absent at V ≥ VD. Then,the difference (41) of fluxes on the interface takes theform

JL − JS =

(

1−V 2

V 2

D

)

×[XS(1−XL)− keXL(1−XS)]VDL

Ω, V < VD,

JL = JS , V ≥ VD. (42)

The net flux (42) must be equal to the diffusion flux

JD = (XL −XS)V

Ω. (43)

From the equality of Eqs. (42) and (43) one gets

(XL −XS)V

VDI=

(

1−V 2

V 2

D

)

×[XS(1−XL)− keXL(1−XS)], V < VD,

XL = XS , V ≥ VD, (44)

in which VDI = VDL is the diffusion speed on the inter-face from the liquid phase. Equation (44) can be eas-ily resolved regarding the function k(V ) = XS/XL ofnonequilibrium solute partitioning. This yields

k(V,X∗

L)

=(1− V 2/V 2

D)ke + V/VDI

(1− V 2/V 2

D)[1− (1− ke)X∗

L] + V/VDI, V < VD,

k(V,X∗

L = X0) = 1, V ≥ VD, (45)

where X∗

L is the solute concentration in the liquid at theinterface.

V. DISCUSSION AND COMPARISON WITH

EXPERIMENTAL DATA

Expression (45) gives the general functional depen-dence of solute partitioning at the phase interface forconcentrated binary systems, and with application torapid solidification, it exhibits the two known limits. Inthe first limit, when solidification proceeds with localnonequilibrium at the interface only, i.e., with VD → ∞,Eq. (45) leads to the solute partitioning function of Azizand Kaplan [2]. In the second limit, as the concentrationX∗

L of the second dissolved component becomes small,i.e., the term (1− ke)X

∗

L might be negligible in compar-ison with the unity, Eq. (45) transforms into Eq. (5) assuggested by Sobolev [21].

7

FIG. 1: Predictions of the model given by Eq. (47). Con-stants of the binary system are equilibrium partition coeffi-cient ke = 0.1, bulk diffusion speed VD = 25 (m/s), and in-terface diffusion speed VDI = 20 (m/s). The curves present:diluted system, (1−ke)X0 << 1 (dotted line); slightly concen-trated system, X0 = 0.10 (dashed line); concentrated systemX0 = 0.25, (dash-dotted line); and equiconcentrated system,X0 = 0.50 (solid line).

From the analytical solution of the problem of rapidsolidification under the steady-state regime [27], the con-centration at the planar interface is given by

X∗

L =X0

k(V ), V < VD,

X∗

L = X0, V ≥ VD, (46)

where X0 is the nominal (initial) concentration of thesolute in the system. In accordance with the solutionsobtained in Refs. [27], a source of concentration pertur-bations, i.e., the solid-liquid interface, moving at a ve-locity V equal to or higher than the maximum speed VD

of these perturbations, cannot change the concentrationor create the concentration profile ahead of itself. Asa result for the interface, one obtains in Eq. (46) thatXL = XS = X0 with V ≥ VD. Then the substitution ofEq. (46) into Eq. (45) leads to the following expressionfor nonequilibrium solute partitioning function:

k(V,X0)

=(1− V 2/V 2

D)[ke + (1− ke)X0] + V/VDI

1− V 2/V 2

D + V/VDI, V < VD,

k(V,X0) = 1, V ≥ VD. (47)

Figure 1 demonstrates the behavior of solute partition-ing, Eq. (47), as a function of the interface velocity at

FIG. 2: Nonequilibrium solute partitioning function k(V,X0)given by the various models. Constants of the binary systemare nominal concentration of a solute X0 = 0.05 mole frac-tion, equilibrium partition coefficient ke = 0.22, bulk diffusionspeed VD = 19 (m/s), and interface diffusion speed VDI = 16(m/s). The dotted line is given by the model of Aziz [13] forthe diluted system (1− ke)X0 << 1, the dashed line is givenby the model of Aziz and Kaplan [2], the dash-dotted line isgiven by the model of Sobolev [21] for diluted system, and thesolid line is predicted by the present model given by Eq. (47).

various nominal solute concentrations. As the systemdeviates from a diluted one, the trapping of a solute be-comes much more pronounced. Also, Eq. (47) shows that,independently from the solute concentration within thesystem, the complete solute trapping k(V,X0) = 1 pro-ceeds when the interface velocity becomes equal to orgreater than the diffusion speed, i.e., with V ≥ VD. Thecondition of equality of concentrations in the liquid andsolid [see Eqs. (44) and (45)] means that the lines of thenonequilibrium kinetic liquidus and solidus in the kineticphase diagram are merging. It can also be considered asthe characteristics of diffusionless processes.As a general outcome, Eq. (47) includes the follow-

ing important cases for nonequilibrium phase transfor-mations: (i) the dilute limit described by Aziz’s model[13], Eq. (4),

(1− ke)X0 << 1, and VD → ∞,

(ii) the dilute limit described by Sobolev’s solute parti-tioning function, Eq. (5),

(1− ke)X0 << 1, and with the finite VD,

(iii) the concentrated system described by Aziz and Ka-plan’s model, Ref. [2],

VD → ∞, for arbitrary concentration X0.

8

FIG. 3: Solute partitioning versus interface velocity for exper-imental data [33, 34] on solidification of Si-As alloys. Curves1′ and 2′ are given by Eq. (47) with VD → ∞ for 4.5 at.%and 9.0 at.% of As in Si, respectively (that gives the model ofAzis and Kaplan [2]). They describe experiment at small andmoderate solidification velocities. Curves 1 and 2 are given by(47) with the finite speed VD for 4.5 at.% and 9.0 at.% of Asin Si, respectively. These show ability to describe experimentin a whole region of investigated solidification velocities forboth alloys. Data for calculations are given in Table I.

In comparison with the present model’s prediction de-scribed by Eq. (47) these limits are plotted in Fig. 2.

Figure 3 exhibits theoretical predictions for solute par-titioning in comparison with experimental data on thesolidification of Si-As alloys. Introducing the deviationfrom equilibrium at both the interface and bulk liquid al-lows one to describe the whole set of experimental data.Particularly, the complete solute trapping is predicted byEq. (47) for Si-4.5 at.%As with VD = 2.5 m/s and forSi-9.0 at.%As with VD = 2.1 m/s (Table I). This pro-vides a much better agreement with experiments thanthat shown by the Aziz-Kaplan model.

As can be seen in Fig. 3, predictions of the model ofAzis and Kaplan [Eq. (47) with VD → ∞] disagree withexperimental data in the region 1.7 < V (m/s)< 2.2 ofsolidification velocities. One may note that at the samesolidification velocity, i.e., below about V = 2 (m/s), the”interface temperature - velocity” relationship also ex-hibits a clear deviation from experimental data (see Fig.11 in Ref. [34]). One may also attribute this deviationto the increasing influence of local nonequilibrium solutediffusion around the interface and intensive solute trap-ping. Thermodynamic analysis and numeric evaluationsconfirm the idea about the pronounced influence of localequilibrium in bulk liquid on solute trapping and ”inter-face temperature - velocity” relationship at high solid-ification velocity [22, 35]. This example confirms thatlocal nonequilibrium in the solute diffusion field is re-sponsible for nonequilibrium effects appearing in rapid

solidification (such as solute trapping and solute drag)and essential influence on the interface response func-tions (temperature, concentration, velocity) [35]. Thus,the agreement between Eq. (47) and experimental datademonstrates the pronounced effect of deviation from lo-cal equilibrium in bulk liquid on solute trapping at highersolidification velocity.Summarizing the behavior for solute partitioning

shown in Figs. 1-3, one can conclude that during rapidsolidification the consequences of deviations from localchemical equilibrium are threefold. First, the partitioncoefficient becomes dependent on the growth velocity.Second, the liquidus and solidus lines approach eachother. For these two cases it can be enough to introduceinto the theory deviation from local equilibrium at theinterface only. Third, in the extreme case (if the solid-ification velocity is equal to or greater than the atomicdiffusive speed in bulk liquid) the partition coefficientk(V ) becomes unity and the liquidus and solidus linescoincide. This leads to a solid being far from chemicalequilibrium upon diffusionless solidification. Such threeconditions are of special importance in the preparationof metastable supersaturated solutions [15].

VI. CONCLUSIONS

Solute trapping in rapid solidification of a binary al-loy’s system has been considered. It has been shown thatthe condition for complete solute trapping leading to dif-fusionless solidification follows directly from the solutionfor the diffusion task. This task assumes both the low-frequency regime (purely diffusion) and high-frequencyregime (diffusion and propagative regime) of atomic mo-tion in a phenomenological statement.The two-level model has been used to define the solute

partitioning function. This model has been used previ-ously (e.g., in chromatography and for investigation oflongitudinal solute dispersion), and it has been formallyreduced to expressions for an extended version of the con-tinuous growth model. The extended version adopts twokinetic parameters: solute diffusion speed VDI on the in-terface and solute diffusion speed VD in bulk liquid.A condition of complete solute trapping at the finite so-

lidification velocity equal to the diffusion speed, V = VD,has been found. This fact is expressed by the general ex-pression (16) for the solute partitioning function. Thiscondition defines the equality of the concentration in thephases and describes complete solute trapping. Analysisleads to concrete forms for the solute partitioning func-tion. The first function is given by Eq. (24) and thesecond function for solute partitioning is described byEq. (45). Both these functions predict a sharp finishingof solute trapping and the onset of diffusionless crystalgrowth at the solidification velocity V equal to the solutediffusion speed VD in bulk liquid. A concrete expressionfor the liquid concentration X∗

L at the interface allows usto give predictions comparable with experimental data.

9

TABLE I: Interface diffusion speed VDI and bulk diffusion speed VD for binary systems used in the calculations of the partitioningfunction k(V,X0) at the solid-liquid planar interface. Equilibrium partition coefficient is taken as ke=0.3 from Refs. [33, 34].

Model Binary system VDI (m/s) VD (m/s) Reference

Aziz and Kaplan’s model, Ref. [2] Si - 4.5 at.% As 0.46 – [33]

0.37 – [34]

Aziz and Kaplan’s model, Ref. [2] Si - 9 at.% As 0.46 – [33]

0.37 – [34]

Sobolev’s solute partitioning function, Eq. (5) Si - 4.5 at.% As 0.75 2.7 [21]

and Si - 9 at.% As

Present model, Eq. (47) Si - 4.5 at.% As 0.8 2.5 current data

Si - 9 at.% As 0.8 2.1 [22]

The model predicts the complete behavior for the so-lute partitioning function dependent on the solidificationvelocity and alloy concentration. In comparison with theexperimental data of Aziz et al. on solidification of Si-As alloys [M.J. Aziz et al., J. Cryst. Growth 148, 172(1995); Acta Mater. 48, 4797 (2000)], the model wellpredicts deviation of the solute partitioning from equilib-rium and complete solute trapping (Fig. 3). The tran-sition from chemically partition growth to diffusionlessgrowth at V = VD occurs sharply. As has been shownfor dendritic growth [36] such a sharp transition leadsto an abrupt exchange of growth kinetics in consistency

with experimental data.

Acknowledgments

The author thanks Professor Dieter Herlach and Pro-fessor Dmitri Temkin for numerous useful discussions.This work was performed with support from the Ger-man Research Foundation (DFG - Deutsche Forschungs-gemeinschaft) under project No. HE 1601/13.

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Phys. Rev. E 47, 1893 (1993); W.J. Boettinger, A.A.Wheeler, B.T. Murray, and G.B. McFadden, Mater. Sci.Eng. A 178, 217 (1994).

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10

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7

Solute trapping and diffusionless solidification in a binary system

Peter Galenko∗

German Aerospace Center, Institute of Materials Physics in Space, Cologne 51170, Germany

(Dated: October 30, 2018)

Numerous experimental data on the rapid solidification of binary systems exhibit the formationof metastable solid phases with the initial (nominal) chemical composition. This fact is explainedby complete solute trapping leading to diffusionless (chemically partitionless) solidification ata finite growth velocity of crystals. Special attention is paid to developing a model of rapidsolidification which describes a transition from chemically partitioned to diffusionless growth ofcrystals. Analytical treatments lead to the condition for complete solute trapping which directlyfollows from the analysis of the solute diffusion around the solid-liquid interface and atomicattachment and detachment at the interface. The resulting equations for the flux balance atthe interface take into account two kinetic parameters: diffusion speed VDI on the interface anddiffusion speed VD in bulk phases. The model describes experimental data on nonequilibrium solutepartitioning in solidification of Si-As alloys [M.J. Aziz et al., J. Cryst. Growth 148, 172 (1995);Acta Mater. 48, 4797 (2000) 1995-2000] for the whole range of solidification velocity investigated.

PACS numbers: 05.70.Fh; 05.70.Ln; 66.30.Jt; 81.10.Aj; 81.30.Fb

I. INTRODUCTION

A concept of “solute trapping” has been introducedto define the processes of solute redistribution at the in-terface which are accompanied by (i) the increasing ofchemical potential [1], and (ii) the deviation of the parti-tion coefficient for solute distribution towards unity fromits equilibrium value (independently of the sign of thechemical potential) [2].

In experimental investigations of rapid solidification, acomplete solute trapping leading to diffusionless (chem-ically partitionless) solidification has been first observedby Hultgren et al. and Duwez et al. in experiments onrapid solidification [3]. They showed that rapidly solid-ifying alloy systems lead to the originating of supersat-urated solid solution with the initial (nominal) chemicalcomposition of the alloy. Later on, crystal microstruc-tures with the initial chemical composition were foundby Chalmers et al. in rapidly solidified pre-dendritic anddendritic patterns [4].

Backer and Cahn [1] have shown that with the finitesolidification velocity in a Cd-Zn system the coefficientof Cd distribution becomes equal to the unit that char-acterizes diffusionless solidification. This fact has beenconfirmed in many binary systems by Miroshnichenko[5]. He investigated dendritic crystal microstructure afterquenching from the liquid state by splat quenching andmelt spinning methods. Results of Miroshnichenko’s mi-crostructural analysis show that at a cooling rate greaterthan some critical value (depending on an alloy and ex-perimental method this value is in the range 105 − 106

K/s) a core of main stems of dendrites has initial (nom-inal) chemical composition of the alloy. A critical valuefor undercooling in the transition to purely thermally-

∗e-mail: Peter.Galenko@dlr.de

controlled growth with a homogeneous distribution ofchemical composition in Ni-B solidifying samples pro-cessed by electromagnetic levitation facility has been ob-tained by Eckler et al. [6]. Finally, it is necessary to notethat many eutectic systems undergo chemically partition-less solidification with the initial composition [5] that canbe explained by the transition to diffusionless solidifica-tion [7].

As a consequence, experimental investigations [1, 3, 4,5, 6] show that with increasing driving force of solidifi-cation solute traps much more pronounced by solidifyingmicrostructure. At the finite value of critical governingparameters (undercooling, cooling rate or temperaturegradient) complete solute trapping occurs. Because thefinite value of the governing parameter defines the con-crete solidification velocity, complete solute trapping anddiffusionless solidification begin to proceed with the fixedcritical growth velocity of crystals.

The main purpose of the present paper is to describe amodel for solute trapping and transition from chemicallypartitioned to diffusionless solidification in a binary sys-tem. Using the local non-equilibrium approach to rapidsolidification, the analysis of diffusion mass transport inbulk phases together with conditions of atomic attach-ment/detachment on the solid-liquid interface is given.

The paper is organized as follows. In Sec. II, previousinvestigations on solute trapping are shortly overviewed.In Sec. III, the analysis of solute diffusion leading to pro-nounced solute trapping and complete solute trapping isgiven. The non-equilibrium solute partitioning functionfor atoms on the interface is derived in Sec. IV. A com-parison with previous models and experimental data onsolidification of binary systems is presented in Sec. V.Finally, in Sec. VI conclusions of the work are summa-rized.

2

II. PREVIOUS INVESTIGATIONS

For the simplest case of atomic system, let us consideran isobaric and isothermal binary system (the pressureP and temperature T are constant) with concentrationXA and XB of atoms A and B, respectively. In this arti-cle, we denote X as the concentration of the atoms of Bsort. For brief overview, we summarize equilibrium andnon-equilibrium solute distribution on the solid-liquid in-terface.

A. Equilibrium

In equilibrium, the concentration of atoms X at thephase interface is not equal from both sides of the in-terface due to different solubility of atoms in phases.During the equilibrium co-existence of phases (gas-solid,liquid-solid, gas-liquid) the atoms are distributed alongthe interface in consistency with the diagram of a phasestate. A difference in atomic concentration in phases atthe interface can be characterized by the equilibrium co-efficient ke of atomic distribution between phases. Forequilibrium co-existence of phases (e.g., between crystaland melt, vapor and crystal, crystal and liquid), the co-efficient ke can be expressed in the following general form[8]

ke(XL, XS , T ) =Xe

S

XeL

≡ exp

(

−∆µ′

RT

)

. (1)

In Eq. (1), XeL and Xe

S are the mole fractions of the Bcomponent in the liquid phase (L), or crystal (S), respec-tively, R is the gas constant, and ∆µ′ is the difference inchemical potentials described by

∆µ′ = ∆µ′

B −∆µ′

A, (2)

with

∆µ′

B = µ′

BS − µ′

BL, ∆µ′

A = µ′

AS − µ′

AL, (3)

where ∆µ′

A and ∆µ′

B are the driving forces for redistri-bution of atoms A and B, respectively, which are definedby redistribution potentials µ′

A and µ′

B for phases L andS. The differences ∆µ′

A and ∆µ′

B , Eq. (3), define thesign of ∆µ′ in Eq. (2). For instance, if ∆µ′ is negative(∆µ′

A > ∆µ′

B), one has ke < 1 - the case of smaller solu-bility of atoms B in the phase S in comparison with theirsolubility in the phase L.

As a general characteristic of phase equilibria in binarysystems, expression (1), together with Eqs. (2) and (3),is usually considered as a measure of the driving forcefor atomic redistribution at the phase interface. It canalso be considered as one of the main parameters for con-struction of diagrams of a phase state.

B. Non-equilibrium

Expressions (1)-(3) assume local equilibrium at the in-terface, which is a useful approximation for many systemstransforming at small interface velocities. At a large driv-ing force for the interface advancing and with increasingof the interface velocity, the local equilibrium is not main-tained [1]. Therefore, the condition for local interfacialequilibrium was relaxed by taking into account a kineticinterface undercooling and deviations from chemical equi-librium at the alloy’s solidification front [8, 10].A number of models [2, 9, 10, 11, 12, 13] have been pro-

posed to account for solute trapping and related phenom-ena observed during rapid phase transformations. One ofthe well-established boundary conditions for solute redis-tribution can be taken from the continuous growth model(CGM) applied to solute trapping by Aziz and Kaplan[2, 13, 14]. The CGM assumes alloy solidification at a“rough interface”, i.e all interface sites are potential sitesfor crystallization events. With a high solidification rate,the atom can be trapped on a high energy site of thecrystal lattice. This leads to a local non-equilibrium onthe interface and to the formation of metastable solids(see examples in Ref. [15]). As a result, the solute parti-tioning function at the solid-liquid interface is describedby [2, 13]

k(V ) =ke + V/VDI

1 + V/VDI, (4)

where VDI is the speed of diffusion at the interface, and keis the value of the equilibrium partition coefficient givenby Eq. (1), i.e. with the negligible interface motion, V →0. Eq. (4) evaluates the ratio XS/XL at the interface fordilute solutions of B (“solute”) in A (“solvent”).The interfacial diffusion speed, VDI , is the kinetic pa-

rameter describing deviation from chemical equilibriumat the interface. It has been defined as a ratio betweenthe diffusion coefficient, DI , at the interface and thecharacteristic distance, λ, for the diffusion jump [2, 13]:VDI = DI/λ. The distance, λ, is assumed to be equalto the width of the solid-liquid interface (few interatomicdistances) and the diffusion jumps are taken along thedirection of growth. Therefore, this definition for VDI

is corrected by results of molecular dynamic simulations[16]. They include diffusion in all spatial directions, i.e.,the diffusion speed is VDI = 6DI/λ, where the factor6 accounts for the possibility of jumps along the six(±x, y, z) Cartesian axes.Outcomes following from the solute partitioning func-

tion (4) were compared in modeling of solute trapping us-ing numerical computations based on the phase-field the-ory of alloy’s solidification. Wheeler et al. [17] naturallyincluded an energy penalty for high composition gradi-ents in the liquid that supresses the partitioning of soluteat a rapidly moving interface and leads to solute trap-ping. They also showed that the construction of commontangent to curves of free energy (in a spirit of Baker and

3

Cahn [18]) has to be defined for non-equilibrium concen-trations which already depend on solidification velocity.In order to eliminate or reduce the solute trapping effectby the diffuse interface at small growth velocity, Karma etal. proposed an ad hoc suitable antitrapping condition tothe diffusion flux [19]. These works [17, 19] showed thatwhen the solute trapping effect comes to modeling alloysolidification with both phase and concentration fields,a crucial issue arises concerning the relative magnitudesof the gradients of the two fields within the solidificationfront as well as the relative thickness of the concentrationjump interface. Additionally, Conti [20] investigated 1Dusual formulation of the phase field model without con-centration gradient corrections of Wheeler et al. [17]. Heresolved the governing equations numerically for the in-terface temperature and the solute concentration field asa function of the growth velocity. The partition coeffi-cient k(V ) is monotonically increasing towards unity atlarge growth rates following the predictions of the con-tinuous growth model (4). However, in contrast to theresults of natural experiments [1, 3, 4, 5, 6], the numericpredictions [17, 20] were not able to reach the completechemically partitionless (diffusionless) solidification at afinite solidification velocity.One of the deficiencies of the function (4) is the diffi-

culty to describe complete solute trapping at the finitesolidification velocity: Eq. (4) predicts k → 1 only withV → ∞. Contrary to this prediction, a transition topartitionless solidification occurs at a finite solidificationvelocity as it has been shown in numerous experiments[1, 3, 4, 5, 6]. Molecular dynamic simulations also showthat the transition to complete solute trapping is ob-served at a finite interface velocity in rapid solidificationof a binary system [16]. Therefore, as an extension ofEq. (4), a generalized function for solute partitioning inthe case of local non-equilibrium solute diffusion withinthe approximation of a dilute system has been introducedby Sobolev [21]. This yields

k(V ) =(1− V 2/V 2

D)ke + V/VDI

1− V 2/V 2

D + V/VDI, V < VD,

k(V ) = 1, V ≥ VD. (5)

The diffusion speed, VD, introduced into Eq. (5) is thecharacteristic bulk speed. It is defined as a maximumspeed for solute diffusion propagation or as a speed forthe front of solute diffusion profile. In particular, thespeed, VD, is obtained by the speed of propagation of theplane harmonic wave away from the solid-liquid interface(see Appendix in Ref. [22]). As the velocity, V , of theinterface is comparable by magnitude to the speed, VD,the high frequency limit takes place: ωτD >> 1, whereω is the real cyclic frequency of the plane harmonic waveand τD is the time for relaxation of the diffusion flux toits steady state. In this case, VD has to be consideredfinite and it is defined as VD = (D/τD)1/2, where D isthe diffusion coefficient in bulk liquid.In the local equilibrium limit, i.e. when the bulk diffu-

sive speed is infinite, VD → ∞, expression (5) reduces tothe function k(V ) that takes into account the deviationfrom local equilibrium at the interface only as describedby Eq. (4). The function (5) includes the deviation fromlocal equilibrium at the interface (introducing interfacialdiffusion speed VDI) and in the bulk liquid (introduc-ing diffusive speed VD in the bulk liquid). As Eq. (5)shows, the complete solute trapping, k(V ) = 1, proceedsat V = VD. This result has been introduced by Sobolevfrom a postulation about the zero value for the diffusioncoefficient at V ≥ VD. The next section further detailsthat the condition for complete solute trapping followsdirectly from the analysis of solute diffusion flux.

III. DIFFUSION MASS TRANSPORT AND

SOLUTE TRAPPING

In 1D solidification along the z-axis, the mass balanceis given by

∂X

∂t= −

∂J

∂z, (6)

where t is the time and J is the diffusion flux. To considersolute trapping in 1D local non-equilibrium solidification,we take one of the results from a model of rapid phasetransitions [23]. Using this model, the evolution equationfor diffusion flux J along the z-axis is described by

J = M

[

∂

∂z

(

∂s

∂X+ ε2x

∂2X

∂z2

)

− αj∂J

∂t

]

, (7)

where s is the entropy density, εx the factor proportionalto the correlation length, and M is the diffusion mobilityof atoms. The latter is defined by

M = TD, D = D(∂(∆µ)/∂X)−1, (8)

where D is the diffusion coefficient and ∆µ is the differ-ence of chemical potentials between solvent and solute.From known thermodynamic expressions [24] one can ac-cept that

∂s

∂X= −

∆µ

T, αj =

τDTD

∂(∆µ)

∂X=

τD

TD, (9)

where τD is the time for diffusion flux relaxation to itssteady state. Then, omitting the term responsible foratomic correlation, i.e. assuming that εx = 0, one canget from Eq. (7) the following expression

J = M

[

1

T

∂(∆µ)

∂z−

τD

TD

∂J

∂t

]

. (10)

Using Eq. (8), the evolution equation (10) results asfollows

J = −D∂(∆µ)

∂z− τD

∂J

∂t. (11)

4

We find the solution for the diffusion flux J which hassignificance in analysis of solute trapping. Using the ex-pression for the diffusion speed, VD = (D/τD)1/2, fromequations (6) and (11) one gets

τD∂2J

∂t2+

∂J

∂t= D

∂2J

∂z2. (12)

Eq. (12) is a partial differential equation of a hyperbolictype. It describes the flux J in the so-called “hyperbolicevolution”, which proceeds with the sharp front of profilefor the solute transport. It occurs due to both diffusiveand propagative nature of the transport in the high fre-quency limit ωτD >> 1 with V ∼ VD.For a steady-state regime of interfacial motion Eq. (12)

takes the following form

D

(

1−V 2

V 2

D

)

d2J

dz2+ V

dJ

dz= 0, (13)

which is true in a reference frame moving at constantvelocity V with the interface z = 0. A general solutionof Eq. (13) is

J(z) = c1 + c2 exp

(

−V z

D(1− V 2/V 2

D)

)

. (14)

To define a particular solution one can assume the fol-lowing boundary conditions: balance on the interface isJ(z = 0) = V (X∗

L −X∗

S), and the flux is limited by theexpression J(z → ∞) = 0 far from the interface withz → ∞. The latter condition gives c1 = 0 for any veloc-ity V and one gets c2 = 0 for V ≥ VD. Also, from theinterfacial balance with z = 0 one gets c2 = V (X∗

L−X∗

S)for V < VD. As a result, solution (14) transforms intothe following particular solution

J(z)

= V (X∗

L −X∗

S) exp

(

−V z

D(1− V 2/V 2

D)

)

, V < VD,

J(z) = 0, V ≥ VD, (15)

where X∗

L and X∗

S are the liquid concentration and solidconcentration, respectively, on the interface.Solution (15) gives condition for complete solute trap-

ping with the finite velocity V ≥ VD. This is expressedby the expression for the solute partitioning function

k(V ) = X∗

S/X∗

L 6= 1, V < VD,

k(V ) = 1, X∗

S = X∗

L, V ≥ VD. (16)

The latter condition in Eq. (16) defines equality of con-centration in the phases and leads to complete solutetrapping.To obtain the explicit form for the solute partitioning

function (16), we analyze the balance of diffusion fluxeson the interface. Taking again the steady state regimeof solidification constant velocity, V , from the system (6)and (11) one can obtain the following equations

dJ

dz= V

dX

dz, J = −D

d(∆µ)

dz+ τDV

dJ

dz, (17)

from which we get the single equation for the diffusionflux J . This yields

J = −D

(

∂(∆µ)

∂X

)

−1d(∆µ)

dz+D

V 2

V 2

D

dX

dz. (18)

It has been taken into account the defined above thermo-dynamic parameterD and the diffusion speed VD in bulk.Defining the gradient of difference of chemical potentialsas d(∆µ)/dz = (∂(∆µ)/∂X)dX/dz, Eq. (18) gives

J = −D

(

∂(∆µ)

∂X

)

−1 [

1−V 2

V 2

D

]

d(∆µ)

dz. (19)

This equation is a general expression for the steady dif-fusion flux into the liquid from the interface. Within thelocal equilibrium limit VD → ∞ one can obtain knownFickian approximation that has been used previously foranalysis of solute trapping [25, 26].Analytical solutions [27] for solidification under local

non-equilibrium diffusion show that the concentration inboth phases becomes equal to the initial (nominal) con-centration and the diffusion flux is absent for V ≥ VD. Itis also given by Eq. (15). Therefore, in addition to Eq.(19), one can finally obtain:

J = −D

(

∂(∆µ)

∂X

)

−1 [

1−V 2

V 2

D

]

d(∆µ)

dz, V < VD;

J = 0, V ≥ VD. (20)

At the phase interface one assumes in Eq. (20), first,that the term D((∂∆µ)/∂X)−1 is proportional to con-centration such that

D

(

∂(∆µ)

∂X

)

−1

=DIX

∗

L

RT. (21)

Second, the chemical inhomogeneity (solutal segrega-tion) exists due to the jump of the chemical potential∆µ which has the interfacial gradient −d(∆µ)/dz ∼=∆µ/W0 at a small distance W0 of the order of few in-teratomic distances. Third, in approximation of ideal(or even real) solutions, one can assume for the interfa-cial difference of chemical potentials: ∆µ = ∆µL(X) −∆µS(X) ∼= RT (lnXe

L − lnX∗

L) − RT (lnXeS − lnX∗

S) =RT ln(X∗

S/X∗

L)−RT ln(XeS/X

eL) = RT (lnk(V )− ln ke),

where XeL and Xe

S are equilibrium concentrations on theinterface from the liquid phase and solid phase, respec-tively, and k(V ) is the ratio of concentrations on the in-terface defined by Eq. (16). Taking into account theselast evaluations, one can get for the chemical potentialgradient the following expression:

−d(∆µ)

dz∼=

RT

W0

lnk(V )

ke. (22)

Integration of the balance (17) on the interface givesthe flux

J = V (X∗

L −X∗

S). (23)

5

Substituting Eqs. (21) and (22) into expression for diffu-sion flux (20) with using balance (23) gives the followingexpression for the solute partitioning function

[

1−V 2

V 2

D

]

lnk(V )

ke=

V

VDI[1− k(V )], V < VD,

k(V ) ≡ X∗

S/X∗

L = 1, V ≥ VD, (24)

where VDI = DI/W0 is the speed for solute diffusion onthe interface.Eq. (24) gives the evaluation of solute trapping effect

through the solute partitioning function k(V ) derived ini-tially from the analysis of the evolution equation (7) forthe diffusion flux J . This equation takes into accountfinite diffusion speeds on the interface and in bulk liquid.Introduction these two speeds is a consequence of thelocal non-equilibrium both on the interface and in bulkliquid. As Eq. (24) shows, the complete solute trapping,k(V ) = 1, proceeds at V = VD. Eq. (24) transformsinto previously known expression for the function k(V )derived in Refs. [25, 26] with relaxing local equilibriumon the interface and using local equilibrium in bulk liquid(VD → ∞) for the diluted binary system (X∗

L << 1).

IV. SOLUTE PARTITIONING FUNCTION

We use a model of diffusion in which particles moveby diffusion jumps in random time between two phases(states). This model was called as “two level’s model ofdiffusion” and it was introduced in a context of variousapplication, e.g., in chromatography [28] or for a longi-tudinal solute dispersion in a tube with flowing water(Taylor’s dispersion) [29].Let Pi(t, z) is the probability density of a particle po-

sition in the phase i = L or in the phase i = S at themoment t. Then local conservation of the probabilitydensity in a point with coordinate z belonging to thephase i is defined by

∂Pi

∂t= −

∂Ji∂z

. (25)

If the interface moves with the velocity comparable withthe solute diffusion speed VD in bulk phases then theflux Ji(t, z) of density probability depends on prehistoryof the diffusion process. The flux, therefore, is defined by

Ji(t, z) = −

∫ t

−∞

Di(t− t∗)∂Pi(t

∗, z)

∂zdt∗. (26)

Relaxation function Di(t− t∗) can be chosen in a formDi(t − t∗) = Di(0) exp(−(t − t∗)/τD) of exponential de-cay. In such case, Eq. (26) is reduced to the Maxwell-Kattaneo equation

τD∂Ji∂t

+ Ji +Di(0)∂Pi

∂z= 0. (27)

It is accepted in Eq. (27) thatDi(0) is the diffusion coeffi-cient at the final moment of relaxation prehistory so thatDi(0) = D. System (25) and (27) gives a single equationof a hyperbolic type for the density of probability:

τD∂2Pi

∂t2+

∂Pi

∂t= Di

∂2Pi

∂z2, (28)

or for the flux

τD∂2Ji∂t2

+∂Ji∂t

= Di∂2Ji∂z2

. (29)

As it was shown in Ref. [30], the density of probabilitydescribed by Eq. (28) gives a positive entropy productionfor the particle exchange between two levels (between twosubsystems or phases).Integration of Eq. (28) by infinitesimal layer including

interface leads to the balance

[

Di∂Pi

∂z+ τD

∂(V Pi)

∂t+ Ji

]∣

∣

∣

∣

L

S

= 0. (30)

In the steady-state regime one can get the following equa-tion for the i-th phase

Di∂Pi

∂z+ τD

∂(V Pi)

∂t+ Ji = Di

dPi

dz− τDV 2

dPi

dz+ Ji

= Di

(

1−V 2

V 2

Di

)

dPi

dz+ Ji, (31)

which is true in a reference frame moving with the con-stant velocity V and placed on the interface where thebalance (25) is described as dJi/dz = V dPi/dz. UsingEq. (31), the balance (30) is

JL − JS

= −

[

DL

(

1−V 2

V 2

DL

)

dPL

dz−DS

(

1−V 2

V 2

DS

)

dPS

dz

]

.

(32)

In Eq. (32) we introduce the speeds VDL and VDS

of interfacial solute diffusion from the liquid and solidphases, respectively. They are defined by

VDL = DL/lD = νLlD, VDS = DS/lD = νSlD, (33)

where DL and DS are the diffusion coefficients in phases,lD scale for diffusion within which the diffusion jumpsoccur in phases (or on the interface), and νL and νS arefrequencies of diffusion hopes in phases (or on the inter-face). From the theory of the transitive state [31] onecan define frequencies of atomic jumps as

νL = ν0 exp

(

−QD

RT

)

,

νS = ν0 exp

(

−QD +∆µ′

RT

)

, (34)

6

where ν0 is the attempt frequency of atomic hopes of theorder of the vibrational frequency [8, 32], QD the acti-vation barrier for atomic diffusion through the interface,and ∆µ′ is the difference of chemical potentials definedby Eqs. (2) and (3). Obviously, interfacial equilibriumexists for

νS/νL = exp[−∆µ′/(RT )] ≡ ke. (35)

From the interfacial balance (32) there follows

JL − JS = −

(

1−V 2

V 2

DL

)

d

dz[DLPL −DS(V )PS ] , (36)

where DS(V ) is the function of the interfacial velocity Vdefined by

DS(V ) = DS1− V 2/V 2

DS

1− V 2/V 2

DL

=

0, VDS << VDL,

DS , VDS ≈ VDL,

DS(1− V 2/V 2

DL)−1, VDS → ∞.

(37)

The function (37) describes the following cases:(a) VDS → 0: negligible diffusion in solid (DS = 0) incomparison with the diffusion in liquid,(b) VDS ≈ VDL: approximate equality for diffusionspeeds in the liquid and solid around the interface,(c) VDS → ∞: condition of local equilibrium in the dif-fusion field of the solid (that occurs with high frequencyhopes of atoms in solid).From now on, the above case (b) for approximate

equality of diffusion speeds in phases around the inter-face is taken. First, we use the finite difference −dx = lDin the balance (32). Second, we take into account thatthe factor (1− V 2/V 2

DL) is related to the bulk diffusion.Finally, using the definition (33), the balance (32) is de-scribed by

JL − JS =

(

1−V 2

V 2

D

)

[VDLPL − VDSPS ], (38)

where VD is the solute diffusion speed in bulk liquidaround the interface. Using Eqs. (33), (34) and (35),this balance can be rewritten as

JL − JS = VDL

(

1−V 2

V 2

D

)

[PL − kePS ]. (39)

For the concentrated binary system the probabilitiesPL and PS in Eq. (39) are directly proportional to theatomic concentrations in phases. This leads to

PL = XS(1−XL)/Ω, PS = XL(1−XS)/Ω, (40)

where Ω is the atomic volume. Therefore, Eq. (39) canbe rewritten as

JL − JS =

(

1−V 2

V 2

D

)

×[XS(1 −XL)− keXL(1 −XS)]VDL

Ω. (41)

We further use the already obtained result (20) accordingto which the diffusion flux is absent at V ≥ VD. Then,the difference (41) of fluxes on the interface takes thefollowing form

JL − JS =

(

1−V 2

V 2

D

)

×[XS(1−XL)− keXL(1−XS)]VDL

Ω, V < VD,

JL = JS , V ≥ VD. (42)

The net flux (42) must be equal to the diffusion flux

JD = (XL −XS)V

Ω. (43)

From equality of Eqs. (42) and (43) one gets

(XL −XS)V

VDI=

(

1−V 2

V 2

D

)

×[XS(1−XL)− keXL(1−XS)], V < VD,

XL = XS , V ≥ VD, (44)

in which VDI = VDL is the diffusion speed on the inter-face from the liquid phase. Equation (44) can be easilyresolved regarding the function k(V ) = XS/XL of non-equilibrium solute partitioning. This yields

k(V,X∗

L)

=(1− V 2/V 2

D)ke + V/VDI

(1− V 2/V 2

D)[1− (1− ke)X∗

L] + V/VDI, V < VD,

k(V,X∗

L = X0) = 1, V ≥ VD, (45)

where X∗

L is the solute concentration in the liquid at theinterface.

V. DISCUSSION AND COMPARISON WITH

EXPERIMENTAL DATA

Expression (45) gives the general functional depen-dence of solute partitioning at the phase interface forconcentrated binary systems, and, with application torapid solidification, it exhibits the two known limits. Inthe first limit, when solidification proceeds with the localnon-equilibrium at the interface only, i.e. with VD → ∞,Eq. (45) leads to the solute partitioning function of Azizand Kaplan [2]. In the second limit, as the concentrationX∗

L of the second dissolved component becomes small,i.e., the term (1 − ke)X

∗

L might be negligible in com-parison with the unity, Eq. (45) transforms into Eq. (5)suggested by Sobolev [21].

7

FIG. 1: Predictions of the model given by Eq. (47). Con-stants of a binary system are: equilibrium partition coefficientis ke = 0.1, bulk diffusion speed is VD = 25 (m/s), and inter-face diffusion speed is VDI = 20 (m/s). The curves present:diluted system, (1 − ke)X0 << 1, (dotted); slightly concen-trated system, X0 = 0.10, (dashed); concentrated system,X0 = 0.25, (dashed-dotted); and equi-concentrated system,X0 = 0.50, (solid).

From the analytical solution of the problem of rapidsolidification under steady-state regime [27], the concen-tration at the planar interface is given by

X∗

L =X0

k(V ), V < VD,

X∗

L = X0, V ≥ VD, (46)

where X0 is the nominal (initial) concentration of thesolute in the system. In accordance with the solutionsobtained in Refs. [27], a source of concentration pertur-bations, i.e. the solid-liquid interface, moving at the ve-locity V equals to or higher than the maximum speed VD

of these perturbations, cannot change the concentrationnor create the concentration profile ahead of itself. Asa result for the interface, one obtains in Eq. (46) thatXL = XS = X0 with V ≥ VD. Then the substitution ofEq. (46) into Eq. (45) leads to the following expressionfor non-equilibrium solute partitioning function:

k(V,X0)

=(1− V 2/V 2

D)[ke + (1− ke)X0] + V/VDI

1− V 2/V 2

D + V/VDI, V < VD,

k(V,X0) = 1, V ≥ VD. (47)

Figure 1 demonstrates the behavior of solute partition-ing, Eq. (47), as a function of the interface velocity at

FIG. 2: Non-equilibrium solute partitioning function k(V,X0)given by the various models. Constants of a binary systemare: nominal concentration of a solute is X0 = 0.05 molefraction, equilibrium partition coefficient is ke = 0.22, bulkdiffusion speed is VD = 19 (m/s), and interface diffusion speedis VDI = 16 (m/s). The dotted line is given by the model ofAziz [13] for the diluted system, (1−ke)X0 << 1; the dashedline is given by the model of Aziz and Kaplan [2]; the dashed-dotted line is given by the model of Sobolev [21] for dilutedsystem; and the solid line is predicted by the present modelgiven by Eq. (47).

a various nominal solute concentration. As the systemdeviates from the diluted one, the trapping of a solutebecomes much more pronounced. Also, Eq. (47) showsthat, independently from the solute concentration withinthe system, the complete solute trapping, k(V,X0) = 1,proceeds when the interface velocity becomes equal to orgreater than the diffusion speed, i.e. with V ≥ VD. Thecondition of equality of concentrations in the liquid andsolid [see Eqs. (44) and (45)] means that the lines of thenon-equilibrium kinetic liquidus and solidus in the kineticphase diagram are merging. It can also be considered asthe characteristics of diffusionless process.As a general outcome, Eq. (47) includes the follow-

ing important cases for non-equilibrium phase transfor-mations: (i) dilute limit described by Aziz’s model [13],Eq. (4),

(1− ke)X0 << 1, and VD → ∞,

(ii) dilute limit described by the Sobolev’s solute parti-tioning function, Eq. (5),

(1− ke)X0 << 1, and with the finite VD,

(iii) the concentrated system described by the Aziz andKaplan’s model, Ref. [2],

VD → ∞, for arbitrary concentration X0.

8

FIG. 3: Solute partitioning versus interface velocity for exper-imental data [33, 34] on solidification of Si-As alloys. Curves1′ and 2′ are given by Eq. (47) with VD → ∞ for 4.5 at.%and 9.0 at.% of As in Si, respectively (that gives the model ofAzis and Kaplan [2]). They describe experiment at small andmoderate solidification velocities. Curves 1 and 2 are given by(47) with the finite speed VD for 4.5 at.% and 9.0 at.% of Asin Si, respectively. These show ability to describe experimentin a whole region of investigated solidification velocities forboth alloys. Data for calculations are given in Table I.

In comparison with the present model’s prediction de-scribed by Eq. (47) these limits are plotted in Fig. 2.

Figure 3 exhibits theoretical predictions for solute par-titioning in comparison with experimental data on solid-ification of Si-As alloys. Introducing the deviation fromequilibrium both at the interface and bulk liquid allowsone to describe the whole set of experimental data. Par-ticularly, the complete solute trapping is predicted byEq. (47) for Si-4.5 at.%As with VD = 2.5 m/s and forSi-9.0 at.%As with VD = 2.1 m/s (Table I). This pro-vides a much better agreement with experiments thanthat shown by Aziz and Kaplan model.

As it can be seen in Fig. 3, predictions of the model ofAzis and Kaplan [Eq. (47) with VD → ∞] disagree withexperimental data in the region 1.7 < V (m/s)< 2.2 ofsolidification velocities. One may note that at the samesolidification velocity, i.e. below about V = 2 (m/s), the”interface temperature - velocity” relationship also ex-hibits a clear deviation from experimental data (see Fig.11 in Ref. [34]). One may also attribute this deviationto the increasing influence of local non-equilibrium so-lute diffusion around the interface and intensive solutetrapping. Thermodynamic analysis and numeric evalua-tions confirm the idea about pronounced influence of localequilibrium in bulk liquid on solute trapping and ”inter-face temperature - velocity” relationship at high solidifi-cation velocity [22, 35]. This example confirms that localnon-equilibrium in solute diffusion field is responsible fornon-equilibrium effects appearing in rapid solidification

(such as solute trapping and solute drag) and essentialinfluence on the interface response functions (tempera-ture, concentration, velocity) [35]. Thus, the agreementbetween Eq. (47) and experimental data demonstratesthe pronounced effect of deviation from local equilibriumin bulk liquid on solute trapping at higher solidificationvelocity.Summarizing the behavior for solute partitioning

shown in Figs. 1-3, one can conclude that during rapidsolidification the consequences of deviations from localchemical equilibrium are threefold. First, the partitioncoefficient becomes dependent on the growth velocity.Second, the liquidus and solidus lines approach eachother. For these two cases can be enough to introduceinto the theory deviation from local equilibrium at theinterface only. Third, in the extreme case (if the solid-ification velocity is equal to or greater than the atomicdiffusive speed in bulk liquid) the partition coefficientk(V ) becomes unity and the liquidus and solidus linescoincide. This leads to a solid being far from chemicalequilibrium upon diffusionless solidification. Such threeconditions are of special importance in the preparationof metastable supersaturated solutions [15].

VI. CONCLUSIONS

Solute trapping in rapid solidification of a binary al-loy’s system has been considered. It has been shownthat the condition for the complete solute trapping lead-ing to the diffusionless solidification follows directly fromthe solution for the diffusion task. This task assumesboth low-frequency regime (purely diffusion) and high-frequency regime (diffusion and propagative regime) ofatomic motion in phenomenological statement.The two level model has been used to define the solute

partitioning function. This model has been used previ-ously (e.g., in chromatography and for investigation oflongitudinal solute dispersion) and it has been formallyreduced to expressions for extended version of the con-tinuous growth model. The extended version adopts twokinetic parameters: solute diffusion speed VDI on the in-terface and solute diffusion speed VD in bulk liquid.A condition of complete solute trapping at the finite so-

lidification velocity equal to the diffusion speed, V = VD,has been found. This fact is expressed by the general ex-pression (16) for solute partitioning function. This condi-tion defines equality of concentration in the phases anddescribes the complete solute trapping. Analysis leadsto concrete forms for the solute partitioning function.The first function is given by Eq. (24) and the secondfunction for solute partitioning is described by Eq. (45).These both functions predict the sharp finishing of solutetrapping and the onset of diffusionless crystal growth atthe solidification velocity V equal to the solute diffusionspeed VD in bulk liquid. The concrete expression forliquid concentration X∗

L at the interface allows to givepredictions comparable with experimental data.

9

TABLE I: Interface diffusion speed VDI and bulk diffusion speed VD for binary systems used in the calculations of the partitioningfunction k(V,X0) at the solid-liquid planar interface. Equilibrium partition coefficient is taken as ke=0.3 from Refs. [33, 34].

Model Binary System VDI (m/s) VD (m/s) Reference

Aziz and Kaplan’s model, Ref. [2] Si - 4.5 at.% As 0.46 – Ref.[33]

0.37 – Ref.[34]

Aziz and Kaplan’s model, Ref. [2] Si - 9 at.% As 0.46 – Ref.[33]

0.37 – Ref.[34]

Sobolev’s solute partitioning function, Eq. (5) Si - 4.5 at.% As 0.75 2.7 Ref.[21]

and Si - 9 at.% As

Present model, Eq. (47) Si - 4.5 at.% As 0.8 2.5 current data

Si - 9 at.% As 0.8 2.1 Ref.[22]

The model predicts the complete behavior for the so-lute partitioning function dependent on solidification ve-locity and alloy’s concentration. In comparison with ex-perimental data of Aziz et al. on solidification of Si-Asalloys, the model well predicts deviation of the solutepartitioning from equilibrium and the complete solutetrapping (Fig. 3). The transition from chemically par-tition growth to diffusionless growth at V = VD occurssharply. As it has been shown for dendritic growth [36]such sharp transition leads to the abrupt exchange ofgrowth kinetics in consistency with experimental data.

Acknowledgments

Author thanks Prof. Dieter Herlach and Prof. DmitriTemkin for numerous useful discussions. This workwas performed with support from the German ResearchFoundation (DFG - Deutsche Forschungsgemeinschaft)under the project No. HE 1601/13.

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