PHYSICOCHEMICAL ANALYSIS AND MODELING OF THE PRIMARY
CRYSTALLIZATION PROCESSES OF A METAL DURING WELDING
V. Mazurovsky, M. Zinigrad, L. Leontev, and V. Lisin
ABSTRACT. The processes occurring during the primary crystallization of the
multicomponent molten metal in a weld pool with an unrestricted alloying range are
examined. A brief physicochemical analysis of these processes is presented, and a new theory
of carbide formation in a high-alloy iron-carbon weld deposit is proposed on its basis. A
phenomenological model of the primary crystallization of the molten metal in a weld pool is
developed. The model takes into account the involvement of some d alloying elements and
carbon in carbide formation during primary crystallization and permits prediction of the
chemical composition of the matrix of the weld deposit as well as the quantitative and
qualitative composition of the strengthening phases (primary carbides).
Fusion welding is a very complicated process, primarily because of the formation of
the weld pool, which is a multicomponent, multiphase system with a nonuniform temperature
field and complex mass- and heat-transfer processes. The mass-transfer processes occurring
on the metal–slag and metal–gas boundaries determine the chemical composition of the weld
metal and, consequently, largely determine its mechanical properties. However, the shaping
of the properties of a weld, especially a welded joint, is determined not just by the chemical
composition of the metal. The nature of the crystallization of the weld has a great influence
on its properties. For many alloy steels and alloys, this stage in the formation of welded joints
determines their mechanical properties [1, 2, 4]. The chemical composition of the weld metal
is determined by the original chemical composition of the welding material and the base
metal and by the nature of the physicochemical interactions between the molten metal and the
slag. The chemical composition of a weld deposit can be predicted by a procedure based on a
kinetic analysis of the simultaneous diffusion-controlled reactions that occur between the
molten metal and the slag . This procedure takes into account the mutual influence of the
reactions and the diffusion of all the reactants in the metal and the slag. The oxidation of the
elements in the molten metal can be represented by the reaction
[ Ei ] (FeO) ( Ein O m ) Fe, (1)
where Ei denotes the elements dissolved in the molten metal (Mn, SI, W, Mo, V, etc.), and
EinOm denotes the oxides in the molten slag.
A calculation of the rates of reactions of type (1) for each element does not present
any difficulties. However, a separate analysis of each reaction does not correspond to the real
processes occurring in the weld pool. The mutual influence of both the components of the
interacting molten phases and the heterogeneous reactions taking place in these complex
systems must be considered. Within the approach developed, the rates V Ei of the reactions of
type (1) for all the metal components with consideration of their mutual influence are defined
by the expression
x m K im
Ein O m
VEi , (2)
K Ein O m
VEli Ei VElinOm
where x is the ratio between the concentration of iron oxide in the slag and the concentration
of iron in the molten metal on the boundary between the interacting phases, VEli and VElinOm are
the limiting diffusion fluxes of the components of the molten phases, [Ei] and (EinOm) are the
initial concentrations (wt. %) of the elements and oxides in the molten phases, respectively,
Ki is the equilibrium constant of reaction (1) for the i-th component of the molten metal, and
n and m are stoichiometric coefficients.
Taking into account that the VE are the rates of passage of the i-th element from the
molten metal into the molten slag or vice versa, we can use them to calculate the
corresponding concentrations of the elements in the metallic and slag phases as functions of
time. As a result, for the current and final compositions of the molten metal, in accordance
with  we have:
Vd [ Ei ]d Vbm [ Ei ]bm 100 M Ei Ap VEi d
[ Ei ] 0
where Ap is the interfacial interaction area, Vcryst is the rate of crystallization of the weld pool,
M Ei is the molar mass of the i-th element, Vd and Vbm are the rates of supply of the electrode
and base metals to the weld pool, and [Ei]d and [Ei]bm are the concentrations of the i-th
element in the electrode and base metals (wt. %).
Thus, the proposed method can be used to find the chemical composition of the
molten metal in the weld pool, i.e., of the metal in a welded joint. This chemical composition
is the starting point for determining the quantitative and qualitative composition of the phases
of the weld deposit.
Physicochemical aspects of primary crystallization and carbide
The subsequent transformations of the molten metal in the weld pool are associated
with the primary and secondary crystallization processes, i.e., the phase transformations in
the multicomponent alloy. Let us use the chemical composition of the liquid molten metal in
the weld pool as a starting point for examining the primary crystallization process. As we
know from the theory of welding processes , crystallization of the weld pool proceeds
under highly nonequilibrium conditions in the absence of convective stirring of the metal in
the "tail" of the weld pool, i.e., at the crystallization front. Therefore, the process of
distributing the components between the liquid and solid phases is controlled only by
diffusion. Another important factor that determines the distribution of the components is the
concentration buildup occurring at the crystallization front. These factors produce
concentration-induced supercooling, which, together, with thermal supercooling, is
responsible for the cyclic character of weld pool crystallization and the chemical
nonuniformity of the crystallized weld metal. At any moment during crystallization of the
weld pool, the amount of the i-th component that has passed from the liquid phase into the
solid phase can be defined as [1, 4]:
Ei( s ) Ei0 [1 (1 K eff ) exp( )] , (4)
where Ei(s) is the concentration of the i-th component in the solid phase at the crystallization
time t, Ei0 is the initial mean concentration of the i-th component in the molten phase, Keff is
the effective distribution coefficient, Lt is the distance from the crystallization starting point
(the length of the crystallite at the crystallization time t), Vcryst is the crystallization rate, and
Di(l ) is the diffusion coefficient of the i-th component in the molten phase.
After determining the concentration of the i-th component in the solid phase at the
crystallization time t, we still cannot determine its distribution between the austenite and the
strengthening phases that form during crystallization. In most cases, we have one solid
solution in the crystallized weld metal, that is, austenite phase, and several phases of primary
carbides. We need to know the distribution of the i-th component between the solid solution
and these phases. The factors that influence carbide formation can be divided into two
groups: physicochemical factors, which determine the nature of the carbide-formation
process, and technological factors, which, in a final analysis, influence the carbide-formation
process by altering parameters of the former group. Let us examine these groups of factors in
somewhat greater detail. First of all, we must take into account the concentration of each
carbide-forming element and its chemical affinity to carbon. The latter is determined by the
change in the Gibbs free energy G 0 upon formation of the particular carbide under
standard conditions. The value of G 0 is related to the equilibrium constant K p of the
corresponding carbide-formation reaction by the familiar relation
G 0 RT ln K p (5)
This value of G 0 corresponds to the equilibrium concentration of the carbide formed, i.e.,
its limiting concentration. Since the formation of different carbides of the same element is
possible, the order of their formation and the ratio between the concentrations of the different
carbides in the equilibrium state is also determined by the affinity of the carbide-forming
element to carbon. Of course, the values of G 0 and, accordingly, of K p are temperature-
dependent, as is illustrated by the presence of several stable phases on the binary phase
diagram of the carbide-forming-element–carbon system.
However, in fast welding process, during which melting, the chemical reaction
between the phases, and crystallization take place with high rates, an equilibrium state is
generally not attained. Therefore, we should take into account kinetic factors, among which
we should include the rate constant K C of the purely chemical step of the heterogeneous
carbide-formation reaction, as well as the diffusion coefficients of carbon ( DC ) and the
carbide-forming element ( DEi ). The corresponding step of the reaction can be represented in
the simplified form
xEi yC ( Ei ) x C y (6)
It should be recalled that the rate of a reaction is determined not only by the kinetic constants
K C , DC , and DEi , but also by how far the state of the system is from equilibrium and, when
liquid phases are involved, by the hydrodynamic conditions of the process.
The distance from equilibrium is determined by the ratio between the actual
concentrations of the reactants in the system under investigation and the equilibrium constant
K p . The kinetic and thermodynamic constants cited depend greatly on the temperature,
while the hydrodynamic parameters are determined by the thermal and geometric conditions
of the processes in the weld pool. Therefore, the second group of factors that influence
carbide formation, viz., the technological factors, takes on special significance. The most
important factors in this group are the parameters of the welding process (the welding
current, the voltage, the electrode feed rate, the welding speed, the dimensions and shape of
the electrode, polarity, etc.), the dimensions of the parts being welded, and the thermal
regime of the welding and crystallization processes.
To obtain a more complete picture of the mechanism of carbide formation, let us
analyze the electronic structure of d metals and carbides.
The band structure of transition metals has several special features that are related to
the presence of an incompletely filled d subshell [7-10]. When these metals form a crystal
lattice, the outer s electrons are completely delocalized. In contrast, the wave functions of the
d electrons remain concentrated in the cores of the atoms. However, the crystal field lowers
the potential barrier in the interstitial regions of the lattice, providing for the tunneling of
some of the d electrons through the potential barrier and the formation of narrow d bands.
The width of the d bands formed is of great significance because these bands determine the
nature of the binding of electrons and the binding energy in d metals. As we know, the band
width depends on the value of the exchange integral Ae according to the expression [13, 14]
Ed 12 Ae . (7)
The exchange integral increases with increasing values of the principal quantum number n.
Thus, the higher is the number of the d sublevel in the expression nd, the higher is the value
of the exchange integral and the broader is the d band. If the number of d electrons does not
exceed five, electrons fill only the lower part of the band, providing a gain in binding energy.
This gain is higher, the broader is the band. If the number of d electrons exceeds five,
electrons begin to fill the upper part of the band, and the energy gain drops. Thus, in the case
of d metals, the binding energy increases with increasing period (with increasing atomic
radius) as long as the number of d electrons does not exceed five. Therefore, the d metals
with the largest atomic radii and three or four electrons in the d sublevel form the strongest
bonds. Taking into account that only some of the d electrons participate in the formation of
the d bands (in metallic bonding), we must assume that there is covalent bonding in transition
metals, which further increases the binding energy. This accounts, for example, for the high
melting points of tantalum and tungsten. When carbon is inserted into the crystal lattice of a
transition metal, the formation of carbides of two types is possible (in accordance with Hagg's
rule [7, 8, 10]). If the ratio between the atomic radius of carbon and the atomic radius of the
metal is less than 0.59, so-called interstitial phases having the lattice of the metal solvent
form. Otherwise, carbides of complex composition with a crystal lattice differing from the
lattice of the metal solvent form. These carbides are characteristic of chromium, manganese,
and iron, which have five or more d electrons. They have lower melting points and hardness
than do carbides of the first group. Carbon atoms occupy tetrahedral or octahedral cavities,
depending on the type of crystal lattice. When a carbon atom enters the crystal field of the
lattice of a d metal, it experiences the coordinating action of an atom of the d metal. This
effect is stronger, the higher is the number of the d sublevel of the metal and the smaller is the
number of electrons filling the d sublevel. A covalent bond is formed between the carbon
atom and the metal atom by a donor–acceptor mechanism. The carbon atom also acts on the
coordinating atom and removes the degeneracy in the d sublevel, splitting it into two new
levels eg and t2g [8, 9]. This accelerates the chemical reaction and provides an additional gain
in binding energy. The smaller is the occupancy of the degenerate d sublevel (before it is
split), the greater is the gain in binding energy [13, 14]. On this basis, it may be asserted that
the amount of carbon that is used to form the carbide of the i-th metal is proportional to the
atomic radius of the metal (Ri) and is inversely proportional to the number of electrons in the
d sublevel of the metal. We introduce the concept of the absolute carbide-forming ability of
the i-th d metal and represent it as
i . (8)
It follows from an analysis of (8) that the carbide-forming ability increases along the
series consisting of Fe, Mn, Cr, Mo, W, Nb, V, Ta, Ti, Zr, and Hf, in good agreement with
the results in [7-10]. At the same time, this series specifies the thermal stability of carbides in
As we have already noted, the distribution of impurities (alloying elements and
carbon) between the liquid and solid phases is given by (4). We next need to elucidate the
distribution of carbon and alloying elements between these solid solutions. We shall assume
that carbide-forming elements tend to bind all the carbon and carbides in accordance with
their carbide-forming ability (see the carbide-formation series). The amount of carbon bound
by any carbide-forming element will correspond to the stoichiometry of the compound
(MexCy), and the total amount of carbon participating in carbide formation should not exceed
the concentration of carbon determined at a specific crystallization time from Eq. (4). It is
reasonable to assume that only the portion of the alloying elements and carbon that cannot be
dissolved in austenite at the applicable temperature is used in carbide formation. That is, the
quantity of an alloying element that can be bound in a carbide by the undissolved carbon will
be used in carbide formation. The distribution of carbon between the carbide phases in an
alloy will be proportional to the relative carbide-forming ability of the respective transition
and its atomic concentration in the alloy ai (l is the number of alloying
elements in the alloy).
We note that all the parameters of the crystallization process are determined from
known relations [1-4] for a thermal cycle during welding. For the most part, the types of
welding that are employed in practice are based on local concentrated heating of portions of
the articles being welded to their melting points or yield temperatures. The structural and
phase transformations and the mechanical, technological, and service properties of welded
joints depend on the degree of heating of the metal and the nature of the distribution of heat
and strains in the article. The appearance of weld stresses also depends on the heating and
cooling cycle of the article being welded. In addition, the intensity of the thermal processes
predetermines such important parameters of the welding process as the throughput and the
technical and economic efficiency. Thus, practically all processes that occur in metals during
welding depend on the thermal cycles during welding and can be determined to a
considerable extent by their parameters.
Phenomenological model of the primary nonequilibrium crystallization
of the weld pool and formation of the weld metal (weld metal deposit).
Prediction of the quantitative and qualitative composition of
Now that the basic principles governing carbide formation in alloys have been
formulated, let us turn to the primary crystallization of the weld metal. Determining the
distribution of alloying elements between the solid solutions (austenite and primary carbides)
is a fairly complex problem. Based on the foregoing physiochemical analysis of the primary
crystallization process and the principles governing carbide formation, we can make several
Primary crystallization results in the formation of a supersaturated solid solution that
is specified by expressions (3) and (4) at each moment in time t during the process.
That is, we have a crystallite (its length is Lt = Vcrystt, where t is the current primary
crystallization time and Vcryst is the crystallization rate of the weld pool) of variable
At the time t, at which a portion of the supersaturated solid solution (a crystallite of
length Lt) has formed, it undergoes a separation of phases into austenite and carbide
phases according to a diffusionless mechanism.
All alloying transition elements except for chromium, manganese, and iron, which are
assumed to be completely dissolved in the austenite phase, participate in carbide
formation at this stage.
Carbon is distributed between the austenite and the carbide phases in accordance with
the principles governing carbide formation.
The heat of crystallization of any alloy component in the liquidus–solidus temperature
range is a constant.
The molten metal in the weld pool and the supersaturated solid solution formed are
ideal solutions with no mutual solubility.
Based on the principles governing carbide formation in alloys, we shall assume that
carbide-forming elements tend to bind all the carbon in carbides in accordance with their
carbide-forming ability (see the carbide-formation series). The probability of carbide
formation decreases from hafnium to zirconium, titanium, etc. The amount of carbon bound
by any carbide-forming element corresponds to the stoichiometry of the compound (MexCy)
and can be determined from the following expression:
ECci ) Ei( c )
where x and y are stoichiometric coefficients, AC and Ai are the atomic weights of carbon and
the i-th carbide-forming element, and Ei(c ) is the concentration of the i-th carbide-forming
element in the carbide phase. For primary carbides, the value of x is always equal to 1, and y
takes values from 0.4 to 1.0, depending on the homogeneity region of the respective carbide.
Of course, the total concentration of carbon involved in carbide formation should not exceed
the carbon concentration determined for the crystallization time from (4). Moreover, it is
known [7, 10] that not all the carbon and carbide-forming elements are consumed in carbide
formation and that a portion of these elements remains dissolved in the austenite phase. It has
been established [7-10] that all the carbide-forming elements apart from chromium,
manganese, and iron (which are completely dissolved in austenite because their atomic radii
are very close) participate in carbide formation during primary crystallization and form
primary carbides (of the type MeC1–y) with different homogeneity regions. As was suggested
above, only the portion of the alloying elements and carbon that cannot be dissolved in
austenite at the respective temperature will be used for carbide formation, and precisely as
much of the alloying elements as can be bound in carbides by the carbon that is not dissolved
in austenite will be used for carbide formation:
ECc ) t k ECs ) t k EC )t k
( ( (lim
where E Cc ) t k is the concentration of carbon that is not dissolved in austenite, E Cs ) t k is the
carbon concentration given by (4) at the crystallization time tk, and EC )tk is the solubility
limit of carbon in austenite at the respective crystallization temperature at the time tk.
According to the principles governing carbide formation, the distribution of carbon between
the carbide phases and the alloy is proportional to the relative carbide-forming ability of the
respective transition element l
and its concentration (at. %) in the alloy ai. The
corresponding proportionality factor is:
i n i . (11)
Then the concentration of the i-th carbide-forming element dissolved in austenite at the time
tkt can be defined as (wt. %)
Ei( c )tk i ECc )tk
Therefore, the concentration of the i-th carbide-forming element dissolved in austenite
at the time tk equals (wt. %):
Ei( a ) t k Ei( s ) t k Ei( c ) t k , (13)
where Ei( s )tk is the concentration of the carbide-forming element at the crystallization time tk
given by (4).
The concentration of carbides formed at the time tk (wt. %) is the sum of the carbon
concentration and the total concentration of the carbide-forming elements involved in carbide
Qktk ECc ) tk Ei( c ) tk
In the next time interval (tk+1), the compositions of the austenite and carbide phases
will be different. The calculation is repeated z times (z = trc/tkt, where trc is the cooling time,
which is determined by the parameters of the thermal-straining cycle during welding and
depends on the process scheme adopted). The total carbide concentration (wt. %) at the end
of primary crystallization (trc) is defined as
Q I Qkk .
Then the austenite concentration (wt. %) is
S ( a) 100% Q I . (16)
The mean concentrations (wt. %) of carbon and the alloying elements in the austenite
phase are defined, respectively, as
( k 1
100% ; Ei( a ) k 1
zS ( a ) zS ( a )
Thus, at the end of primary crystallization, we know the mean chemical composition
of the austenite phase, as well as the quantitative and qualitative composition of the carbide
phases in different zones of the welded joint. Equations (4) and (8)-(17) comprise a
phenomenological model of the primary nonequilibrium crystallization of the weld pool and
the formation of the weld metal (the weld deposit). In summation, at the end of primary
crystallization, we have a weld deposit of complex phase and structural composition that
consists of primary carbides, on the one hand, and of austenite (the weld deposit matrix), on
the other hand. Equations (4) and (8)-(17) can be used to calculate the concentrations of
carbon and alloying elements in the matrix, the amounts of carbon and alloying components
that are used to form carbides in the weld deposit during primary crystallization, and, as a
result, the concentration of the carbides themselves. The qualitative composition of the
carbide phases can be determined in parallel.
Practical application of the model developed
In the new approach [15-18] that we have proposed for designing modern welding
materials, modeling of the physicochemical processes taking place during welding is the
basis of a new methodology. In particular, modeling of the primary crystallization of the weld
metal (which was described in this paper) together with the mathematical model of the
industrial welding process  is the basis for calculating the composition (electrode formula)
of the welding material being designed. The results of designing welding materials for
various purposes (hardfacing with the deposition of layers having special properties, welding
of special steels and alloys) were presented in [19-22]. All the welding materials designed
have high welding-technological properties and ensure the formation of a weld deposit with a
structure and properties close to the calculated. This indicates the adequate level of
equivalence of the model developed.
1. A physicochemical analysis of the primary crystallization of the weld metal and the
accompanying formation of primary carbides has been presented. Principles governing
carbide formation in iron-carbon alloys during nonequilibrium crystallization have been
2. The principles for developing a formalized description of the nonequilibrium
crystallization of the weld pool on the basis of mathematical modeling have been
analyzed. It has been shown that the construction of such a model requires consideration
of numerous parameters of simultaneously occurring physicochemical processes.
3. A phenomenological model of the nonequilibrium primary crystallization of the weld
pool has been developed, and mathematical relations that can be used to calculate the
chemical composition of the weld deposit matrix and the qualitative and qualitative
composition of the strengthening phases of the weld deposit have been derived on its
4. The model developed has been implemented within a new approach to developing
welding materials for the purpose of calculating their electrode formula. The structure and
properties of the metal deposit provided by these materials are close to the calculated
structure and properties, indicating the adequate level of equivalence of the model
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