THE ONE-COMPONENT AND BINARY METALLIC
MELTS' CRYSTALLIZATION MECHANISMS WITHIN THE
SMALL AND FINITE SUPERCOOLINGS OF THE MELTCRYSTAL INTERFACE
P.D.Sarkisov, Yu.A.Baikov, V.P.Meshalkin
Russian University of chemical technology n.a. Mendeleev D.I.,
Moscow, Russia
Introduction
One of the most important points of modern materials' science is
a crystalline phase formation when the metallic melts crystallizing.
Important problem among ones connected with the crystallization
processes is a study of the one-component and binary metallic melt
crystallisation fluctuation mechanisms acting within some small and
finite supercoolings' region at the melt-crystal interface (see [l]).
The grounds of given fluctuation mechanisms have been stated
in the [2-6] works. In given publications there have been described three
different models of so-called spontaneous thermal equilibrium-like
fluctuations of the solid state particle concentrations. Similar
concentration fluctuations act usually in the monoatomic thickness
layers of some diphase transition zone which separates two bulk phasesthe melt and crystal- and lead to a normal (linear) crystal growth kinetics
within the small (less 0,1K) and finite (less 1K) supercoolings' region
T = T0 – T(T = Tl-T) where T-current temperature, T0, Tl are onecomponent melting temperature and the liquidus one for single or binary
systems, respectively.
1.The micro- and macrosystem fluctuation mechanisms leading
to normal (linear) crystallization kinetics in the case of one-component
metallic melt.
When studying some metallic one-component and binary melts'
crystallization it's very important to take into account its specific
character-so-called diphase transition zone's presence. This zone is some
melt-crystal interface, which consists of a limited number of
monoatomic thickness layers in which the metallic particles are partly in
solid or liquid states.
172
Let Ci=
mi
(i=1,…,n) be the solid state particle concentration in
N
the i-layer of diphase transition zone (n is a full number or separate
monoatomic thickness layers of given zone) where mi is the solid state
particles number in the i-layer, N is so-called macroscopic parameter,
i.e. a full number of particles to be either in solid or liquid states. In such
a case in the concentration image above-mentioned diphase zone can be
presented as shown in fig. 1,2,3 depending on various fluctuation
mechanisms in question or composition of systems considered. In each
of 1,2,3 figures the hatched columns identified by 1,…,i-1,i,i-1,…,n
indices are the solid state particle concentrations Ci  1 in a proper layer
of diphase zone. To the left from the n-length diphase zone is pure
crystalline phase (C1=1) and to the right is the pure melt (C1=0). In
each monolayer characterized by the a-parameter (mean interatomic
distance) at a discrete time ti=t0+iτ(t0 is initial time of crystallization, τ is
mean period of one spontaneous fluctuation) is some stochastic move of
the solid state particle concentration Cj(t) in an j-layer (j  i) from initial
Cj(t0) to final Cj(t0+iτ) values. Here we keep in mind: 0  Cj(t0),
Cj(t0+iτ)  1. During the τ-period only one spontaneous fluctuation is in
one of the n-monolayers of diphase transition zone. Let us consider all
three worked out spontaneous fluctuation mechanisms.
Fig.1. A physical scheme of diphase
transitional zone consisting of “n” the
monoatomic thickness layers (in the
concentration image). To the left of it a
homogeneous crystalline phase is, to the
right the melt is situated. In each the i
monolayer (i=1,…,n) the hatched columns
are the solid state particle concentrations
Ci, the unhatched columns are the liquid
state particle concentrations. Given scheme
illustrates a model of spontaneous
fluctuations of the «unlimited» spectrum
solid state particle concentrations.
A). Spontaneous fluctuations with an "unlimited" spectrum in
changes in the concentration of crystalline particles for one-component
173
metallic micro- and macrosystems.
Physical scheme of diphase transition zone resulted in by given
mechanism is presented in fig.1. Specific character of fluctuation
mechanism considered is: as a result of a spontaneous fluctuation the
solid state particle concentration can be varied from zero (the pure melt)
to 1 (pure crystalline phase) in each monolayer of diphase zone. A
spontaneous fluctuation period τ(T) depends on a crystallizing onecomponent or binary metallic system temperature.
In the [3,7-9] works, given mechanism has been developed for
one-component metallic microsystems where in all possible details
statistical thermodynamics and so-called normal kinetics (V=K*ΔT) of
metallic crystal growth have been considered. Here V is a mean
crystallization rate of diphase transition zone, K -is so-called kinetic
coefficient determined by crystallization parameters.
Fig.2. Physical scheme in
concentration Image of diphase
transition zone comprising "n"
sheets of single atom thickness.
This scheme describes the model
of spontaneous fluctuations of
limited spectrum of solid state
particle concentration changes.
As shown in above-quoted works given kinetic coefficient can
be
K
а q
( N, )
 kTT0
written as follows:
( N, ) 
А
А 
1  exp(  ) r  
 
4 
174
(1)
A=N(r+Θ), q -is latent melting heat .per atom,  
2q
is so-called
3kT
"roughness" parameter of the melt-crystal interface, k is the Boltzmann
constant, r=-4ln2. Further, given fluctuation mechanism has been
developed for one-component metallic macrosystems when the
macroscopic parameter tends to infinity ( N   ). As a result there has
been got a working formula to be used for metallic crystal normal
kinetics within the small supercoolings’ region associated with the meltcrystal system: V*= lim V  K*ΔT where K* is so-called effective
N 
kinetic coefficient which can be written as follows:
К
а
3q
 (8 ln 2)kTT0
(2)
Fig.3. Physical scheme of diphase
transition zone between an A+B melt
and crystalline bulk phase A. The
diphase region consists of "n"
sandwiched
monoatomic
layers
labelled as I=1,2,…,n where hatched
layers designate the concentration
profile of solid A and B species and the
unhatched ones figure the share of
liquid A and B species.
When from micro- to macrosystems transiting ( N   ) given
mechanism of spontaneous fluctuation in each monolayer of diphase
zone degenerates into only one spontaneous fluctuation realization as a
result of which the solid state particle concentra-tion in each monolayer
is of about 1/2 -value [10,11]. As seen from the (1), (2) formulas in the
case of given sponta-neous fluctuation mechanism the mean
crystallization rate of the one-component melts within the small
supercoolings’ region (ΔT/T0<<1) does not depend on diphase transition
zone size in the case of micro- and macrosystems. The K(N)-kinetic
coefficient dependence on the macroscopic parameter is just only for
175
metallic microsystems. This is a peculiarity of fluctuation mechanism in
question.
B). Ordered, cut-off spontaneous fluctuations' mechanism in the
case of micro- and macrosystems.
The previous spontaneous fluctuation mechanism has not led to
real configuration of diphase transition zone which as experiments show
is of step-like form, i.e. when increasing the i-layer number the solid
state particle concentration decreases gradually from crystalline phase to
the pure melt. That's why the ordered, cut-off spontaneous fluctuations'
mechanism in the case of one-component metallic microsystems [12]
and proper macrosystems crystallizing from the melts within the small
supercoolings' region has been worked out. In given mechanism the
spontaneous fluctuations are ordered: first fluctuation is in the i=1-layer
at t1=t0+τ-discrete time etc. the last fluctuation is in the i=n layer_at
tn=t0+nτ discrete point with compulsory conditions Хi1i1  Хii  Хi1i1
where Хi 1i1  Сi1(ti1) , Хii  Сi (t i ) , Хi 1i1  Сi1(ti1) are the mean
solid state particle concentrations in the i-1, i, i+1 layers of diphase zone
at the ti-1, ti, ti+1, -discrete times, respectively. In addition, within the
small supercoolings’ region and for very developed diphase transition
zones (

 1 )(see [2-4,12]) given mechanism leads to normal
r
crystallization law in the case of one-component metallic microsystems:
V=
а q
Φ(n,N,Θ)=ΔT=KΔT
 kTT0
 ( n , N , ) 
n 

 n i 1  exp(  i )
4 
n  i 1

1
n
(3)
1
r   
2
i 1
ni=N Xi1i r  Xi1i where Xi1i  Ci1 (t i ) a mean solid state particle
concentration in the (i-1) layer at ti=t0+iτ discrete time. In the case of
one-component metallic macrosystems within the small supercoolings'
region this mechanism leads to next normal crystallization kinetics with
the following effective kinetic coefficient:
a  n 1  q

  8 ln 2  kTT0
V*= lim V  K * T, K*  
N
176
(4)
As seen from the (3) and (4) formulas the fluctuation mechanism
in question used for one-component metallic micro- and macrosystems
leads to the K(n) and K*(n) dependences on diphase zone's size (the nparameter). For metallic macrosystems ( N   ) given mechanism
results in only single spontaneous fluctuation in each layer of diphase
transition zone and as a consequence the solid state particle
concentration is equal to Ci (t i )  Xii  (1/ 2)i (i=1,…,n) within the small
supercoolings' region. As a consequence of given spontaneous
fluctuations’ mechanism diphase transition zone's size is limited equally
for micro- and macrosystems by next condition: n<<1+1,4lnN.
C). Spontaneous fluctuations with a limited spectrum of changes
in the concentration of crystalline particles in the transition region.
Specific character of above-pointed mechanism is: as a consequence of spontaneous fluctuation, for example, in the i-layer
(i=1,…,n) at the tK=t0+kτ discrete time (k=1,...,n; i  k) the solid state
particle concentration in given layer is less compared with one in the (i1)layer at previous discrete time tK-1=t0+(k-1)τ and more in comparison
with one in the (i+1) layer at the same time tK-1, i.e. next conditions (see
[3]) are:
Xi+1k-1<Xik< Xi-1k-1 where Xik=Ci(tk); Xi+1k-1=Ci+1(tk-1);
Xi-1k-1=Ci-1(tk-1).
Compulsory condition, which conditions the spontaneous
fluctuation mechanism action can be formulated in the same manner:
n<<1+1,4lnN. Given mechanism of spontaneous fluctuations results in
the step-like configuration of the melt-crystal interface (see fig.2), its
form is determined by the Xii  (1 / 2)i parameter set. Within the small
supercoolings' region of the melt-crystal interface given mechanism as
two previous ones for very developed diphase zones leads to the same
normal crystallization law in the case of microsystems V=KΔT and
proper metallic macro-ones: V*= lim V  K * T . Here the K -kinetic
  

N
and K* -effective kinetic coefficients are defined by the same (3),(4) formulas which have been received in the case of previous fluctuation
mechanism. As seen from the K - and K* -formulas in the case of
fluctuation mechanism in question diphase transition zone's length has
an influence on the crystallization kinetics (the more diphase zone's
177
length the more its crystallization rate).
In the 1A- and 2A-tables a comparison of theoretical and
experimental values of mean crystallization rates due to spontaneous
fluctuations with a limited spectrum of changes in the concentration of
crystalline particles for particular one-component systems are presented
within some small supercoolings' region (ΔT<1K).
Moreover, diphase transition zone sizes in the case of the Ga,
Hg, Cd, In metallic melts have been characterised by n=7 value (table
1A).In the case of the alkaline K and Rb melts which according to
experimental data have more developed diphase transition zone the n=30
parameter has been accepted (see table 2 A).
Table 1A.
A comparison of theoretical and experimental data on kinetics of crystal
growth for Ga, Hg, Cd, In-metals within the small supercoolings' region.
The relative errors of experimental data
(Kexp / Kexp)100%  (Vexp / Vexp)100% (for the here presented, onecomponent metallic systems are: 1,9%; 1,1%; 1,8%; 9,1%, respectively).
Diphase zone’s size is characterized by n=7 - parameter.
Metal
Ga
Hg
Cd
In
T
0,1
0,3
0,5
0,05
0,1
0,5
0,02
0,1
0,3
0,5
0,05
0,1
0,2
V*, (m/sec)
-3
5,810
1,710-2
2,910-2
7,910-3
1,610-2
7,910-2
1,310-3
6,510-3
1,910-2
3,210-2
3,710-3
7,210-3
1,410-2
178
Vexp, (m/sec)
5,310-3
1,610-2
2,610-2
4,610-3
9,210-3
4,610-2
1,110-3
5,510-3
1,610-2
2,710-2
5,510-3
1,110-2
2,210-2
Table 1B
A comparison of theoretical and experimental data concerning the mean
crystallization rates of some binary alloys within a small supercoolings'
region. Diphase transitional zone’s size is characterized by n=7
parameter. The relative errors of experimental data
(Kexp / Kexp)100%  (Vexp / Vexp)100% for the here presented binary
systems are 11,8%; 1,0%; 1,2%; 1,5%; 2,6%, respectively.
Binary system
In + 9 at.% Bi
In + 4,6 at.% Pb
In + 7,1 at.% Pb
In + 9,5 at.% Pb
Ga + 8 at.% Sn
T
0,1
0,3
0,5
0,1
0,3
0,1
0,34
0,1
0,8
0,5
1,0
V*, (m/sec)
1,910-4
5,710-4
9,510-4
2,410-3
7,210-3
2,510-3
8,510-3
2,410-3
1,910-2
7,510-3
1,510-2
Vexp, (m/sec)
1,710-4
5,110-4
8,510-4
1,010-2
3,010-2
2,610-3
9,010-3
1,310-3
1,110-2
1,910-2
3,810-2
Table 2A
A comparison of theoretical and experimental data on kinetics of crystal
growth for alkaline K, Rb-metals within the small supercoolings' region.
The relative errors of experimental data
(K exp / K exp )100%  (Vexp / Vexp )100% for the here presented alkaline
metals are: 2,4%; 2,0%, respectively. Diphase zone's size is
characterized by n=30 parameter.
Metal
K
Rb
T
0,01
0,05
0,08
0,26
0,28
0,36
V*, (m/sec)
6,810-3
3,410-2
5,410-2
2,010-2
2,210-2
2,710-2
179
Vexp, (m/sec)
4,110-3
2,010-2
3,310-2
7,510-3
9,610-3
1,410-2
Table 2B
A comparison of theoretical and experimental data of the mean
crystallization rates within the small supercoolings’ region for very
dilute binary solid solutions with well-developed diphase transition zone
characterized by the n=30 parameter. The relative errors of experimental
data (K exp / K exp )100%  (Vexp / Vexp )100% for the presented binary
systems are 1,0%; 2,6%, respectively.
T
0,1
0,3
0,5
1,0
Binary system
In + 4,6 at.% Pb
Ga + 8 at.% Sn
V*, (m/sec)
1,310-2
3,810-2
4,010-2
8,010-2
Vexp, (m/sec)
1,010-2
3,010-2
1,910-2
3,810-2
2.Fluctuation crystallization mechanism for micro- and
macrosystems in the case of binary metallic melts.
To describe properly the binary metallic melts crystallization
kinetics within the small and finite supercoolings’ region and compare
with available experimental data above-stated spontaneous fluctuation
mechanism with a limited spectrum of changes in the concentration of
crystalline particles has been used (see [5,6,14-16]). Physical scheme of
diphase zone in the concentration image for the binary metallic melts is
shown in fig.3, A step-like configuration of the melt-crystal interface is a
con-sequence of given accepted fluctuation mechanism. For the binary
metallic micro- and macrosystems the n<<1+1,4lnN condition remains
in force at any rate. As shown in the [5,6] works within the small
supercoolings' region in the case of the binary melt -crystal
microsystems given fluctuation mechanism for the very dilute binary
solid solutions leads to the same normal crystallization kinetics:
V
a q AA
(n, N,  AA )
 kTTl
(n, N,  AA ) 
1
n

(5)
n  i1

180

 n i   1 i1
 r     AA  r (  iA ) 2

 4   2 
 n i 1  exp  

1
n i  N 
2
i 1
1
r  
2
i 1

AA
 r (  iA ) 2

where a is the mean interatomic distance of crystal in question,
 AA  2q AA / 3kT ,
 
 iA  k iA
1
,
kiA  C1i A / Ci2A
-the A-component
distribution coefficient depending on the i-layer number: C1i A , Ci2A -the
A-component concentrations in the i-layer of diphase zone in a
crystalline and liquid states, respectively. qAA -the A-component latent
melting heat per 1 atom; ΔT=Tl-T, Tl -the liquidus temperature of binary
system considered, T -current temperature of the binary melt-crystal
interface. The (5) formula is just for the binary metallic microsystem
when the condition of very developed diphase zone
 AA
<<1 is
r
fulfilled. In the [6,16]-works it has been shown that given fluctuation
mechanism acting in the diphase zone layers within the small and finite
supercoolings’ region and for very developed transition zones leads in
the case of macrosystems to the normal kinetics: V*= lim V  K * T
N


 n i 1

a 
1
 q AA
where K*=


 kTT
j

1
n r i 2 j1  1 
l
j 2 

1    (k A
)


2
 

As seen from the (5)-formula fluctuation mechanism in question
in the case of binary microsystems determines normal crystallization
kinetics by means of the macroscopic parameter "N" and diphase
transition zone length "n". In the case of binary macroscopic systems
effective kinetic coefficient depends only on the n-parameter.
In the tables 1B and 2B some comparison of theoretical and
experimental values of mean crystallization rates are presented within
the small and finite supercoolings' region for several' very dilute binary
solid solutions. The mean periods of spontaneous fluctuations with
limited spectrum of changes in the concentration of crystalline particles
for all the In-binary alloys have been estimated due to the [l7-19] -works
and were equal to τ=10-11 sec. and for the Ga+8 at.% Sn-alloy the same
was equal to τ=10-10 sec.
181
References
1. Borisov V.T. Theory of diphase zone of metallic ingot.-Moscow,
Metallurgiya, 1987,p.223.
2. Zelenev Yu.V. ,Baikov Yu.A., Molotkov A.P.On the theory of normal
growth of crystals in binary systems.-Kristall und Technik, 14,
N4,1978,pp.389-398.
3. Baikov Yu.A.,Zelenev Yu.V.,Haubenreisser W. A fluctuation theory of
normal crystal growth for metallic one-component systems.Phys.stat.sol.(a) 55.N1 ,1979,pp.123-135.
4. Chistyakov Yu.D.,Baikov Yu.A.,Schneider H.G., Ruth V.- Fluctuation
mechanism of normal crystal growth during solidification of
metals.-Acta Metallurgica, 29,N2, 1981, pp.415-423.
5. Chistyakov Yu.D., Baikov Yu.A., Schneider H.G., Ruth V.Fluctuation mechanism of normal crystal growth during
solidification of binary metallic melts.-Crystal Research and
Technology ,18, N6,1983,pp.711-723.
6. Chistyakov Yu.D.,Baikov Yu.A. A fluctuation theory of normal crystal
growth for the very dilute binary metallic solid solutions'
crystallization in the case of macroscopic systems.- Crystal
Research and Technology 20, N10,1985,pp. 1301 -1307.
7. Baikov Yu.A.,Meshalkin V.P. Statistical thermodynamics of di-phase
transitional zone for the one-component metallic melts.- Neorgan.
Materialy 35, N9,1999,pp.106l-1064.
8. Baikov Yu.A.,. Meshalkin V.P. Statistical thermodynamics of the twophase transition region for one-component metallic melts.-Inorganic
Materials,35, N9,1999, pp.899-901.
9. Chistyakov Yu.D., Baikov Yu.A. ,Akulionok M.V. Fluctuation theory
of the one-component and binary metallic melts’ crystallization in
the case of micro- and macrosystems.-Crystal Research and
Technology 23,N5,1988,pp.609-619.
10. Chistyakov Yu.D., Baikov Yu.A., Akulionok M.V. A fluctuation
mechanism of crystallization processes for the one-component
metallic melts.-Crystal Research and Technology 23, N3, 1988,pp.
299-305.
11. Chistyakov Yu.D., Baikov Yu.A.A fluctuation theory of normal
crystal growth in the case of one-component metallic
macrosystems.-Crystal
Research
and
Technology
20,N9,1985,pp.1143-1148.
182
12. Chistyakov Yu.D.,Baikov Yu.A.,Manec V.N.Eine Untersuchung des
Kristallwachstums mechanismus bei geringen Unterkühlungen im
Falle metallischer Systeme.-Vereinigung für Kristallographie.
Mitteilungen 2,13 Jahrgang,Berlin, 1978,s.3-18.
13. Borisov V.T.,Matveev Yu.E. The Ga-alloys crystallization at
significant growth rates.- Kristallographiya 16,N2,1971, pp.249253.
14 .Baikov Yu.A,,Zelenev Yu.V.,Haubenreisser W. Thermodynamics of a
two-phase transitional region between an A+B melt and a
crystalline bulk phase A. - Phys.stat.sol.(a) 59,N1,1980,pp.K81-K86.
15. Baikov Yu.A.,Zelenev Yu.V.,Haubenreisser W.Kinetic coefficient of a
two-phase transitional region between an A+B melt and a
crystalline bulk phase.-Phys,stst. sol,(a) 59,N2,1980,pp.K155 K158.
16. Baikov Yu.A.,Zelenev Yu.V. Kinetic coefficient of the normal
crystallization law for macrosystems in the case of dilute binary
solid solution.-Izv. AN SSSR, Neorgan. Mater. 22,N10,
1986,pp.1656-1660.
17. Swalin R.A. On the theory of self-diffusion in liquid metals.- Acta
Metallurgica 7,N11,1959,pp. 736-740.
18. Swalin R,A. Concerning the mechanism of diffusion in liquid.- Acta
Metallurgica 9,N4,1961,p.379.
19. Leak V., Swalin R.A. Diffusion of silver and tin in liquid silver.Trans. Met. Soc. AIME, 230,1964,pp. 426-432.
183