# Interface Response Functions for multicomponent alloy solidification- An
application to additive manufacturing

V S Hariharan<EMAIL_ADDRESS>B S Murty  Gandham Phanikumar

###### Abstract

The near-rapid solidification conditions during additive manufacturing can
lead to selection of non-equilibrium phases. Sharp interface models via
interface response functions have been used earlier to explain the
microstructure selection under such solidification conditions. However, most
of the sharp interface models assume linear superposition of contributions of
alloying elements without considering the non-linearity associated with the
phase diagram. In this report, both planar and dendritic Calphad coupled sharp
interface models have been implemented and used to explain the growth-
controlled phase selection observed at high solidification velocities relevant
to additive manufacturing. The implemented model predicted the growth-
controlled phase selection in multicomponent alloys, which the other models
with linear phase diagram could not. These models are calculated for steels
and a Nickel-based superalloy and the results are compared with experimental
observations.

###### keywords:

Additive Manufacturing, Calphad , Microstructure , Steels , Superalloys

††journal: Computational Materials Science

[inst1]organization=Department of Metallurgical and Materials Engineering,
Indian Institute of Technology Madras,city=Chennai, postcode=600036,
state=Tamil Nadu, country=India

[inst2]organization=Indian Institute of Technology Hyderabad,city=Kandi,
postcode=502284, state=Telangana, country=India

Calphad coupled dendritic growth model predicted the growth-controlled phase
selection occurring during additive manufacturing of steels, that is not
predicted by other growth models

Incorporating the effect of solute drag improved the predictions

Columnar to equiaxed transition (CET) calculation is coupled with KGT model
and benchmarked with experiments for Haynes 282 Ni-based superalloy

##  1 Introduction

Additive Manufacturing (AM) involves building three-dimensional parts layer-
by-layer based on a digital model. AM has proven its ability to produce
complex, near-net shape components making it appealing for biomedical and
aerospace applications, especially to manufacture high-value low-volume
components [1]. The near-rapid solidification condition and the thermal cycles
experienced during additive manufacturing can lead to non-equilibrium
microstructures [2]. Understanding the microstructure evolution during AM can
help in alloy design for AM as well as site-specific control of microstructure
within a component [3].

The solidification velocities in AM processes like electron powder bed fusion
can reach up to 10 cm/s [4]. Equilibrium thermodynamic calculations alone are
insufficient to understand the microstructure evolution during such rapid
solidification conditions. One must account for the deviation from equilibrium
at the solid-liquid interface. Analytical solidification models, also known as
sharp interface models have been developed to explain the solute segregation
behaviour and phase selection during processes such as welding and directional
solidification [5]. The sharp interface models assume that the solidification
occurs by an “interface reaction” at the solid-liquid interface and there is
long-range transport of atoms and heat across the interface. Interface
response functions appropriately describe the interface reaction. The
interface response functions (IRFs) can be used to predict the solid
composition and the interface velocity for a given composition and temperature
of the interface or vice-versa.

The concept of the IRF is developed using the principles of irreversible
thermodynamics [6]. IRFs have been used in the solidification and solid-state
transformations. We restrict ourselves to the context of solidification for
this study. As given by Hillert, at the interface, Gibbs energy dissipation
occurs due to crystallization and diffusion of solute species across the
interface (trans-interface diffusion) [7]. The relationship between the
driving forces and their corresponding fluxes at the interface dictates the
IRFs . One such relationship provided by Aziz and Kaplan in their continuous
growth model gives the relationship between partition coefficient and
solidification velocity [8]. They also show how the phase diagram varies with
solidification velocity (kinetic phase diagram) [8].

Aziz and Kaplan model considers only the planar interface. In order to arrive
at IRF for curved interfaces, one must know the conditions at which the planar
interface becomes unstable. Mullins and Sekerka derived the condition for
planar stability in the light of perturbation analysis for low Peclet numbers
[9]. Based on this work, Langer and Müller-Krumbharr proposed the Marginal
Stability criterion which assumes that the marginally stable wavelength gets
selected as the dendritic tip radius [10]. Trivedi and Kurz extended Mullins
and Sekerka’s analysis to rapid solidification conditions [11]. Building on
the above models, Kurz et al. have proposed the IRF for rapid solidification,
where one can obtain the interface composition and temperature as a function
of solidification velocity and thermal gradient [12]. This model is popularly
referred to as KGT model, and is widely used in the solidification community.

The KGT model has been used to explain the growth-controlled phase selection
in Fe-Cr-Ni alloys [13]. Fukumoto and Kurz observed that equilibrium delta
ferrite ($\delta$ BCC) is the primary solidification phase for low
solidification velocities and austenite ($\gamma$ FCC) is the primary
solidification phase for high solidification velocities [13]. This phase
selection was explained using interface temperature versus solidification
velocity plot obtained using KGT model. The phase with higher interface
temperature gets selected and since the interface temperature of $\gamma$ is
higher than $\delta$ at high solidification velocities, $\gamma$ is preferred
over $\delta$ [13]. Several peritectic systems of commercial interest such as
steels and Nd-Fe-B alloys show this kind of growth-controlled phase selection
[14]. Although KGT model was originally developed for binary alloys, it has
been used for multicomponent alloys by assuming a simple linear superposition
of individual solute contributions [15]. This approximation can lead to errors
in the predictions.

Ludwig solved this issue by extending Aziz and Kaplan’s continuous growth
model to multicomponent alloys by considering the thermodynamic effects
instead of linear superposition [16]. Recently, Du et al. have reformulated
Aziz and Kaplan’s model in terms of Gibbs energy instead of the original
chemical potentials and coupled with Calphad (Calculation of Phase Diagram)
databases [17]. The model was applied in the context of additive manufacturing
for a multicomponent aluminium alloy. The reformulated model by Du et al. and
Ludwig’s model are similar and yield same results. However, Ludwig’s model is
not widely applied due to its increased computational complexity, except for a
few studies [18]. These models are valid only for planar interface. Wang et
al. proposed a Calphad coupled model for dendritic growth that considers non-
Fickian diffusion in liquid, which will be relevant for rapid solidification
conditions experienced during additive manufacturing [19]. However, these
models are applied only for undercooling [19] and casting conditions [20].

In this article, we couple Ludwig’s model and Wang’s model for multicomponent
alloy solidification with Calphad databases and explain the growth-controlled
phase selection observed during additive manufacturing. The implemented model
is compared with the existing analytical solidification models that are based
on linear superposition, in terms of its predictive capability and the
relevant physics associated. As case studies, we take commercially relevant
two steel compositions (Fe-C-Mn-Al steel, H13 tool steel) processed under
welding [15] and additive manufacturing [21] conditions, for which in situ
synchrotron studies are available in the literature to determine the primary
solidification phase. In addition to phase selection, the prediction of
columnar to equiaxed transition also becomes important in the case of additive
manufacturing of Nickel-based superalloys [22]. In relation to the previous
models, we also calculate columnar to equiaxed transition map for Haynes 282
and the impact of model assumptions are discussed.

##  2 Models

The IRFs can vary depending on the model and its assumptions used. In this
report, we considered both planar and curved interface models that use linear
phase diagram and the ones that are calphad coupled. The models used in this
work are shown in Fig.1

Figure 1: Types of Interface Response Functions (IRFs) used in this work

These models assume one-dimensional and steady state growth conditions on the
heat and mass transfer scale that occurs during solidification. Regarding AM,
due to the metal-laser interaction, there is a spatial and temporal variation
of temperature within the melt pool. However, the models are applied on the
scale of solute diffusion length (Solute Diffusivity/solidification velocity
$\approx$ 10-9 m2s-1/ 10-2 m s-1 = 10-7 m), which is short in the scale of the
melt pool. Even though further research is needed to test this assumption, it
is plausible to assume that the analytical models can be used to predict
microstructure selection during AM.

###  2.1 Planar growth with linear phase diagram

The interface response (temperature) for a planar solid-liquid interface by
assuming linear superposition of the individual solute contributions is given
by Eqn.1.

$\displaystyle T_{P}$
$\displaystyle=T_{m}+\sum_{i}C^{i}_{o}\left(\frac{m^{i}_{v}}{k^{i}_{v}}\right)-\frac{v}{\mu_{k}}$
(1) $\displaystyle k^{i}_{v}$
$\displaystyle=\frac{k^{i}_{o}+\frac{v}{v_{D}}}{1+\frac{v}{v_{D}}}$ (2)
$\displaystyle m^{i}_{v}$
$\displaystyle=m^{i}_{o}\times\frac{1-k^{i}_{v}(1-\ln(k^{i}_{v}/k^{i}_{o}))}{1-k^{i}_{o}}$
(3)

In Eqn.1, $T_{P}$ is the temperature of the planar interface, $T_{m}$ is the
melting point of the pure metal, the summation over the alloying element ‘$i$’
allows for extension to multicomponent alloys, $C^{i}_{o}$ represents the
alloy composition, $m^{i}_{o}$ represents the equilibrium liquidus slope,
$k^{i}_{o}$ represents equilibrium partition coefficient, the letter ‘$v$’ in
the subscript represents that the terms are velocity-dependent. The velocity-
dependent partition coefficient and liquidus slope are given in Eqn.2 [8] and
Eqn.3 respectively. ‘$v$’ represents the solidification velocity, ‘$\mu_{k}$’
represents the kinetic coefficient, ’$v_{D}$’ is the diffusive speed of
elements in liquid, which is assumed to be the same as that of the solvent.
The velocity dependent partition coefficient is from Aziz and Kaplan model
[8]. The interface temperature can be obtained as shown in Eqn.1 only by
assuming that Henry’s law holds and the solution is dilute [23, 16]. Diffusion
in solid is assumed negligible.

###  2.2 Dendritic growth with linear phase diagram

The interface response of curved interface based on KGT model is given in Eqn.
4 [12]. Again, a linear superposition of individual solutes is assumed when
extended to multicomponent alloys [13, 15].

$\displaystyle
T_{D}=T_{L}+\sum_{i}(C^{i}_{L}m^{i}_{v}-C^{i}_{o}m^{i}_{o})-\frac{2\Gamma}{R}-\frac{v}{\mu_{k}}-\frac{GD_{i}}{v}$
(4) $\displaystyle
C^{i}_{L}=\frac{C^{i}_{o}}{1-\left((1-k^{i}_{v})Iv(Pe^{i})\right)}$ (5)
$\displaystyle
4\pi^{2}\Gamma\left(\frac{1}{R^{2}}\right)+(2\sum_{i}\left[m^{i}_{v}Pe^{i}(1-k^{i}_{v})C^{i}_{L}\xi^{i}_{C}\right])(\frac{1}{R})+G=0$
(6)
$\displaystyle\xi^{i}_{C}=1-\frac{2k^{i}_{v}}{2k^{i}_{v}-1+\sqrt{1+{\left(\frac{2\pi}{Pe^{i}}\right)}^{2}}}$
(7)

In Eqn.4, $T_{D}$ is the temperature of the curved interface, $T_{L}$ is the
equilibrium liquidus temperature of the alloy, the second term in right-hand
side of the equation represents solutal contribution, the third and the fourth
term represents the contribution due to curvature of dendritic tip and
attachment kinetic effects respectively. The last term accounts for the
cellular growth at low solidification velocities. $C^{i}_{L}$ is the liquid
composition at the interface corrected for the solidification velocity and
curvature at the tip. $\rm{Iv}(Pe)=Pe\times\exp(Pe)\times\rm{E1}(Pe)$ is the
Ivantsov solution, to describe the solute diffusion field ahead of the
dendrite , where E1 is the exponential integral and is calculated using
’scipy’ library [24]. Ivantsov solution is a function of the dimensionless
solutal Peclet number ($Pe^{i}=VR/2D^{i}$), which is the ratio of advective
solutal transport to diffusive solutal transport, at the scale of dendrite tip
radius ($R$). The expressions for velocity-dependent partition coefficient and
liquidus slope remain the same for both planar and curved interfaces. Equation
6 is solved iteratively for a given thermal gradient (G) and a range of
solidification velocities to obtain $R$ and then $T_{D}$ is subsequently
calculated. The bisection method was used to solve Eqn.6 [25].

###  2.3 Calphad coupled planar growth

Solidification occurs via the motion of the solid-liquid interface, for the
interface to move, there must be a driving force. This driving force is due to
the deviation from equilibrium at the interface. This deviation leads to a
redistribution of solute across the interface and migration of the interface.
Building on the above assumption and Aziz – Kaplan model, Ludwig [16] derived
the first response function given in Eqn.8.

$\displaystyle(C^{j}_{L}-C^{j}_{S})\left(\frac{v}{v_{D}}\right)=\sum_{i=1}^{n}(C^{i}_{L}C^{j}_{S}-\kappa^{i,j}C^{i}_{S}C^{j}_{L})$
(8)
$\displaystyle\kappa^{i,j}=\exp{\left(-\frac{\Delta\tilde{\mu^{j}}-\Delta\tilde{\mu^{i}}}{RT}\right)}$
(9)
$\displaystyle\Delta\tilde{\mu^{i}}=\Delta\mu^{i}-RT\ln(\frac{C^{i}_{S}}{C^{i}_{L}})$
(10) $\displaystyle\tilde{\mu^{i}}=\mu^{i}-RT\ln(C^{i})$ (11)

In Eqn.8, $C^{i}_{L}$ and $C^{i}_{S}$ are the composition (mole fraction of
‘$i$’th element) of liquid and solid at the interface. $\kappa^{i,j}$ are the
partitioning-parameters between the elements $i,j$, such that $i\neq j$. To
account for the local entropic effects, the difference in the redistribution
potentials ($\Delta\tilde{\mu^{i}}=\tilde{\mu^{i}_{S}}-\tilde{\mu^{i}_{L}}$)
are considered rather than the chemical potentials. The redistribution
potential of component ‘$i$’ is defined as the difference between the chemical
potential and ideal mixing entropy contribution.$n$-1 equations are to be
solved for a ‘$n$’ component system and the composition of the major element
(solvent) can be obtained from the constraint $\sum^{n}_{j=1}C^{j}_{S}=1$. The
second response function is based on chemical rate theory. It describes the
relationship between the driving force for interface motion and the velocity
of the interface as given in Eqn.12.

$v=v_{o}[1-\exp(\frac{\Delta G}{RT})]$ (12)

In Eqn.12, $v_{o}$ is the maximum velocity that occurs when there is infinite
driving force and is generally in the order of speed of sound in
liquid.$\Delta G$ is the driving force for solidification. The expression for
$\Delta G$ can vary depending on how the solute drag effect is treated.
$\Delta G_{DF}$ is the Gibbs energy difference between solid and liquid that
causes interface motion. The driving force is assumed to have two components,
one due to the crystallization of atoms and the solute redistribution across
the interface($\Delta G_{DF}=\Delta G_{C}+\Delta G_{D}$). The crystallization
and solute distribution energies are given below.

$\displaystyle\Delta G_{C}=\sum_{i=1}^{i=n}C^{i}_{L}\Delta\mu^{i}$ (13)
$\displaystyle\Delta G_{D}=\sum_{i=1}^{i=n}(C^{i}_{S}-C^{i}_{L})\Delta\mu^{i}$
(14)

If we assert that the driving force must account for all the solidifying
atoms, that is, neglecting solute drag, then $\Delta G_{nsd}=\Delta G_{DF}$
and can be written as in Eqn.15.

$\Delta G_{nsd}=\sum^{n}_{i=1}C^{i}_{S}.\Delta\mu^{i}$ (15)

On the other hand, if we assume that a part of the driving force is consumed
due to the solute diffusion at the interface, that is, including solute drag,
then $\Delta G_{sd}=\Delta G_{C}$ and can be written as in Eqn.16.

$\Delta G_{sd}=\sum^{n}_{i=1}C^{i}_{L}.\Delta\mu^{i}$ (16)

During calculation, if solute drag is included then $\Delta G=\Delta G_{sd}$
and if solute drag is neglected $\Delta G=\Delta G_{nsd}$. The implications of
accounting for the effect of solute drag during multicomponent alloy
solidification are investigated in the subsequent sections.

Figure 2: A flowchart to calculate interface temperature and composition using
Ludwig model. The orange colour indicates that it is Calphad coupled.

A flowchart representing the implementation of Ludwig’s model is shown in
Fig.2. Since chemical potentials are needed to solve the system of equations
(Eqn. 8 -16), TC-python® API of Thermo-Calc® software [26] is used along with
optimization routines in scipy library [24]. The calculation starts with a
very low velocity, so the solid’s composition and the interface’s temperature
are close to the equilibrium values obtained from Calphad databases. These
values are used as initial guess values for the ’fsolve’ function in ’scipy’.
While calculating chemical potentials, all the phases are rejected except the
phase for which the chemical potential is calculated. Once the solution is
converged, these values are stored in an array and used as guess values for
the next calculation with higher velocity.

###  2.4 Calphad coupled dendritic growth

Wang et al proposed a calphad coupled IRF for multicomponent alloy
solidification for dendritic growth [19]. This model (hereafter referred to as
Wang model) accounts for local non-equilibrium diffusion in liquid. This
results in improved predictions at high solidification velocities. Unlike
Ludwig model which requires fixing either solid or liquid composition at the
interface, Wang model allows the calculation of temperature, solid and liquid
compositions at the interface. The first response functions to calculate the
interface compositions are given below.

$\displaystyle\frac{v}{V_{m}}(C^{i}_{L}-C^{i}_{S})=M^{i}_{D}\left(\Delta\mu^{i}\psi^{i}-\frac{\sum_{j=1}^{n}M^{j}_{D}\Delta\mu^{j}\psi^{j}}{\sum_{k=1}^{n}M^{k}_{D}}\right)$
(17) $\displaystyle
M^{i}_{D}=\frac{v^{i}_{I}}{V_{m}}\left(\frac{\partial\mu_{i}^{L}}{\partial
C^{L}_{i}}\right)^{-1}$ (18)

where $V_{m}$ is the molar volume and is assumed to be same for all elements.
$\psi^{i}(=1-(v/v^{i}_{D})^{2})$ is the non-equilibrium diffusion factor,
$M^{i}_{D}$ is the mobility for trans-interface diffusion, $v^{i}_{I}$ is the
diffusive speed at the interface. The above set of equations are same for the
planar model proposed by Wang et al [18]. In case of dendrite growth, an
additional set of equations that accounts for the curvature must be
considered. Thus, Ivanstov solutions are used for the same.

$C^{i}_{L}=\begin{cases}\frac{C^{i}_{o}}{1-\left((1-k^{i}_{v})Iv(Pe^{i})\right)}&\text{if
}v<v^{i}_{D}\\\ C^{i}_{o}&v\geq v^{i}_{D}\end{cases}$ (19)

It can be seen from the above equations that at solidification velocities
higher than diffusive speed in bulk liquid, there is no solute partitioning,
and complete solute trapping occurs. The second IRF is given below.

$\displaystyle v=-M\Delta G$ (20) $\displaystyle M=\frac{v_{o}}{RT}$ (21)
$\displaystyle\Delta
G=\sum_{i=1}^{n}\left[C^{i}_{L}\Delta\mu^{i}-\frac{V_{m}}{2}\alpha^{i}_{L}(J^{i}_{L})^{2}\right]$
(22)

Here, in the driving force expression, first term on right-hand side is the
classical driving force with solute drag effect and the second term is due to
the local non-equilibrium diffusion in liquid. In the above equation, the
diffusion flux of component $i$ is given by
$J^{i}_{L}=(v/V_{m})(C^{i}_{L}-C^{i}_{S})$ and the non-equilibrium kinetic
coefficient ($\alpha^{i}_{L}$) is given as

$\alpha^{i}_{L}=\frac{V_{m}}{(v^{i}_{D})^{2}}\left(\frac{\mu^{i}_{L}-\mu^{i}_{S}}{C^{i}_{L}-C^{i}_{S}}\right)$
(23)

In this model, the velocity-driving force relation is assumed to be linear as
opposed to the exponential relation in the Ludwig model. This driving force
relation accounts for the non-Fickian solute diffusion at high solidification
velocities. Now, the dendritic tip radius can be given by stability criteria
as below.

$\displaystyle\omega=\frac{2\pi}{R}$ (24) $\displaystyle
S_{n}=-\Gamma\omega^{2}-(K_{S}G_{S}\xi_{S}+K_{L}G_{L}\xi_{L})-\frac{\left(\sum_{i=2}^{n}\frac{M^{i}_{L}\zeta^{i}}{N^{i}}\right)}{1+\sum_{j=2}^{n}\frac{M^{i}_{L}}{N^{i}}}$
(25)

According to Marginal stability criteria, the dendrite tip radius is assumed
equal to the marginally stable wavelength. In Eqn.25, $G_{S}$ and $G_{L}$ are
thermal gradients in solid and liquid, respectively and $K_{S}$ and $K_{L}$
are solid and liquid thermal conductivities, respectively. Other thermal
variables are given below.

$\displaystyle\xi_{S}=\frac{\omega_{S}+v/D^{T}_{S}}{K_{S}\omega_{S}+K_{L}\omega_{L}}$
(26)
$\displaystyle\xi_{L}=\frac{\omega_{L}-v/D^{T}_{L}}{K_{S}\omega_{S}+K_{L}\omega_{L}}$
(27)
$\displaystyle\omega_{L}=\frac{v}{2D^{T}_{L}}+\sqrt{\left(\frac{v}{2D^{T}_{L}}\right)^{2}+\omega^{2}}$
(28)
$\displaystyle\omega_{S}=-\frac{v}{2D^{T}_{S}}+\sqrt{\left(\frac{v}{2D^{T}_{S}}\right)^{2}+\omega^{2}}$
(29)

where $D^{T}_{L}$ and $D^{T}_{S}$ are solid and liquid thermal diffusivities
respectively. Other solutal parameters are given below.

$\displaystyle M^{i}_{L}=-\frac{\partial\Delta G}{\partial
C^{i}_{L}}/\left[\frac{\partial\Delta G}{\partial T}+R\frac{v}{v_{o}}\right]$
(31) $\displaystyle
N^{i}=\left[D^{i}\psi^{i}\omega^{i}_{C}+vC^{i}_{L}\frac{\partial
k_{v}^{i}}{\partial C^{i}_{L}}+v(k_{v}^{i}-1)\right]/vC^{i}_{L}\frac{\partial
k_{v}^{i}}{\partial T}$ (32)
$\displaystyle\zeta^{i}=D^{i}\psi^{i}G^{i}_{L}\left[\frac{v}{D^{i}\psi^{i}}-\omega_{C}^{i}\right]/vC^{i}_{L}\frac{\partial
k_{v}^{i}}{\partial T}$ (33) $\displaystyle
G^{i}_{L}=\frac{vC^{i}_{o}}{D^{i}\psi^{i}}(1-1/k^{i}_{v})$ (34)
$\displaystyle\omega^{i}_{C}=\frac{v}{2D^{i}\psi^{i}}+\sqrt{\left(\frac{v}{2D^{i}\psi^{i}}\right)^{2}+\frac{\omega^{2}}{\psi^{i}}}$
(35)

Eqn.31 represents the kinetic liquidus slope and Eqn. 34 represents the solute
gradient at the interface. In Eqn.34, $C^{i}_{o}$ is the overall alloy
composition. In this work, only the diagonal terms in the solute diffusion
matrix are considered, and the other terms are assumed to be zero. This
approximation enables faster computation and simplifies the equations [27].
The stability equations corresponding to the diffusion equation with non-zero
solute diffusion matrix can be referred to in the article by Wang et al [19].

Figure 3: A flowchart to calculate interface temperature and composition using
Wang model. The orange colour indicates that it is Calphad coupled.

In this calculation, the solid thermal gradients are assumed to be negligible
compared to thermal gradients in liquid. Both solid and liquid thermal
diffusivities are assumed to be the same. The diffusive speeds of all the
solutes in liquid are taken as equal to that of the solvent element. The
interface diffusive speeds of all elements are considered to be the same. When
the thermophysical properties of solid and liquid are assumed to be same, the
equations tends to be multicomponent extension of the model proposed by Li et
al. [28] and in the case of dilute binary alloys, the model reduces to that of
Galenko and Danilov [29]. A schematic representing the implementation of Wang
model is shown in Fig.3. The thermodynamic data is obtained via TC-Python.
Choose an initial velocity, and the initial guess values for radius, interface
temperature and compositions can be obtained by performing a calculation using
KGT model. The equations corresponding to the solid and the liquid
compositions can be solved using the ’fsolve’ function in ’scipy’. While
calculating the interface temperature using Eqn.20, the temperature calculated
corresponds to the planar interface and thus curvature contribution
($2\Gamma/R$) is added to account for the curved nature. The stability and
temperature equations are solved iteratively until the solution is reached. We
again use ’fsolve’ function of ’scipy’ to arrive at the solution. Calculating
kinetic liquidus slope (Eqn.31) values require computing the derivative of
driving force with respect to liquid composition. Since liquid composition is
a dependent variable, the driving force derivatives can not be calculated
directly. We follow the below numerical procedure for the same. The velocity
value is subject to a variation of $10^{-4}$ m/s above and below the current
velocity value. Using the new velocities, the compositions are obtained by
solving the first IRF (Eqn:17) and Ivantsov solutions (Eqn:19) . Now for the
new compositions, driving forces are calculated. The derivatives are now
calculated using the new compositions and their corresponding driving forces.

###  2.5 Columnar to equiaxed transition

Gaumann et al. have proposed a model for columnar to equiaxed transition based
on Hunt’s model [30]. This model implies that once the local undercooling
reaches a critical value, all equiaxed grains nucleate in front of the
columnar front, and if they are able to develop over a threshold volume
fraction, they obstruct the growing columnar front. Thus, the following
expression gives the thermal gradient at which a certain volume fraction of
equiaxed grains nucleate.

$G=\frac{1}{n^{\prime}+1}\sqrt[3]{\frac{-4\pi N_{o}}{3\ln(1-\phi)}}\Delta
T\left(1-\frac{\Delta T_{n}^{n^{\prime}+1}}{\Delta T^{n^{\prime}+1}}\right)$
(36)

where $n^{\prime}$ is a material-dependent parameter, $N_{o}$ is the
nucleation volume density, $\phi$ is the volume fraction of grains,$\Delta T$
is the local undercooling and $\Delta T_{n}$ is the critical undercooling
needed for the nucleation of equiaxed grains. Gaumann et al. neglected the
kinetic, curvature and thermal contributions of undercooling and accounted
only for constitutional undercooling. The constitutional undercooling was used
as a fitted function of solidification velocities obtained from experiments.
However, the kinetic and curvature effects are non-negligible under additive
manufacturing conditions. Thus, in this work, we couple KGT model with
Gaumann’s model and apply for a Ni-based superalloy for LPBF conditions. Based
on experimental studies, a fully equiaxed microstructure is obtained when
$\phi>0.49$ and a fully columnar microstructures are obtained when
$\phi<0.0066$.

The models described above require alloy properties like $T_{L}$, $m_{v}^{i}$,
$k_{v}^{i}$, which were calculated using Thermo-Calc® 2021B software along
with TCFE11 database and other parameters such as $\Gamma,\mu_{k},v_{o},D_{i}$
were obtained from literature. For comparison, the “Scheil with solute
trapping” model, which is based on Aziz and Kaplan’s model, implemented in
Thermo-Calc® is also used in the present study.

##  3 Results and Discussion

###  3.1 Case 1: Phase Selection in Fe-C-Mn-Al steel

Babu et al., have used in situ time-resolved X-ray diffraction technique to
investigate the primary solidification phase during slow and rapid cooling
conditions experienced during different arc welding parameters [15]. It has
been observed that non-equilibrium FCC ($\gamma$) is the primary
solidification phase during rapid cooling whereas; equilibrium BCC ($\delta$)
is the primary solidification phase during slow cooling.Since this phase
selection is observed under arc welding conditions, these calculations will be
helpful for application in wire arc additive manufacturing. Thermodynamic
equilibrium and IRF calculations are performed to understand and predict the
experimentally observed phase selection.

Figure 4: (a) Isopleth section showing a peritectic reaction (b)Phase fraction
versus temperature plot for Fe-C-Mn-Al steel

The composition of the Fe-C-Mn-Al steel used in the study is C-0.234, Mn-0.5,
Al-1.7 and Fe-remaining (in weight percent). The isopleth section of
equilibrium phase diagram is shown in Fig.4(a), indicates the presence of the
peritectic reaction (Liquid+BCC($\delta)\rightarrow\rm{FCC}(\gamma$)). The
phase fraction as a function of temperature for the composition under study
shown in Fig.4(b). From Fig.4, it is clear that the equilibrium primary
solidification phase is BCC ($\delta$). The parameters used for calculation of
IRFs is given in Table1. The value of interface diffusive speed is assumed to
be slightly less than the bulk diffusive speed [19] The thermal gradient was
considered 9$\times 10^{4}$ K/m [19].

Table 1: Parameters used to calculate IRFs for Fe-C-Mn-Al steel Parameter | BCC($\delta$) | FCC($\gamma$)
---|---|---
Liquidus temperature $T_{L}$ (K) | 1796.60 | 1780.98
Equilibrium partition coefficient of C $k^{C}_{o}$ | 0.2059 | 0.2716
Equilibrium partition coefficient of Mn $k^{Mn}_{o}$ | 0.7129 | 0.7065
Equilibrium partition coefficient of Al $k^{Al}_{o}$ | 1.098 | 0.9295
Equilibrium liquidus slope of C $m^{C}_{o}$ (K/wt.fraction) | -7015.3 | -6569.4
Equilibrium liquidus slope of Mn $m^{Mn}_{o}$(K/wt.fraction) | -467.9 | -496.7
Equilibrium liquidus slope of Al $m^{Al}_{o}$ (K/wt.fraction) | 195.2 | -588.7
Kinetic coefficient $\mu_{k}$ ($\rm{ms^{-1}K^{-1}}$) | 10.0 | 10.0
Solute diffusion coefficient $D^{i}$ ($\rm{m^{2}/s}$) | $5.0\times 10^{-9}$ | $5.0\times 10^{-9}$
Diffusive speed in liquid $v_{D}$ (m/s) | 1.0 | 1.0
max speed $v_{o}$ (m/s) | 300.0 | 300.0
Interface diffusion speed $v_{I}$ (m/s) | 0.9 | 0.9
Gibbs-Thomson coefficient $\Gamma$ (K/m) | $2.0\times 10^{-7}$ | $1.0\times 10^{-7}$
Solid and Liquid Thermal conductivity $K_{S},K_{L}$ ($\rm{Wm^{-1}K^{-1}}$) | 50 | 50
Solid and Liquid Thermal Diffusivity$D^{T}_{S},D^{T}_{L}$ ($\rm{m^{2}s^{-1}}$) | $5.0\times 10^{-5}$ | $5.0\times 10^{-5}$

The BCC($\delta$)-liquid and FCC($\gamma$)-liquid interface temperature as a
function of solidification velocity for all the above mentioned models are
shown in Fig.5. The planar solidification models with linear phase diagram
shows increase in the interface temperature with increasing solidification
velocity, for both BCC and FCC. However, the interface temperature is observed
to decrease for all other models. In the case of Ludwig’s model, BCC phase has
higher interface temperature than FCC for low velocities, whereas at high
velocities, FCC has higher interface temperature. The phase with higher
interface temperature will be the primary solidification phase [5]. Thus,
Ludwig’s model qualitatively predicts the phase selection that is
experimentally observed during welding. However, it overestimates the velocity
at which the transition occurs. Ludwig’s model shown in Fig.5(a) takes solute
drag into account. In the case of dendritic models, KGT model does not show
any phase selection whereas, Wang’s model shows FCC as the primary
solidification phase for velocities greater than 1 cm/s. Compared to Ludwig’s
model, Wang’s model predicts a lower velocity at which the transition occurs.

Figure 5: The vertical coloured areas indicate the experimental observations.
Plot of interface temperature for BCC($\delta$) and FCC($\gamma$)
solidification for (a) planar models (b)dendritic models

In both planar and curved interface models, the coupling of thermodynamic data
has improved the predictions. KGT model will not predict the phase transition
even if the partition coefficients and liquidus slopes are directly calculated
from calphad models for every iteration [15]. Mohan and Phanikumar showed that
the sum of the constitutional undercooling contribution of individual solutes
is less than the constitutional undercooling values calculated via Calphad
databases using the liquid composition [31]. The explicit addition of
individual undercoolings is valid only for dilute binary alloys but for non-
dilute alloys, the method is shown to be invalid by Ludwig [16]. Since both
Ludwig’s and Wang’s models account for the thermodynamic interactions among
the elements, the phase transition is being captured. As Wang’s model
considers additional physics, such as the curved nature of the interface and
non-Fickian diffusion, it can predict closer to experiments than the case of
Ludwig’s model.

Figure 6: (a)Interface temperature versus solidification velocity calculated
using Ludwig’s model with and without solute trapping for Fe-C-Mn-Al steel
(b)Magnified view showing the transition

The effect of solute drag on the interface temperature is shown in Fig.6.
Including the solute drag has improved the prediction and the transition
occurs at a lower velocity. As a part of the driving force is consumed due to
solute transfer across the interface, there is a decrease in driving force
available for interface motion, leading to decreased velocity. The temperature
versus fraction of solid plot for Fe-C-Mn-Al steel calculated using Thermo-
Calc®’s ’Scheil with solute trapping model’ is shown in Fig.7. Since it is
planar, we will compare the results of Ludwig’s model with Thermo-Calc’s
value.Note the temperature when fraction of solid is zero. When the
solidification velocity is increased, there is a decrease in the interface
temperature as expected. This observation matches qualitatively with our
calculation using Ludwig’s model, although there is a quantitative difference.
However, the primary solidification phase is BCC even at such high
solidification velocities. This is in contrast with the experimental data.
Thus, Ludwig’s model and Wang’s model are able to show the experimentally
observed phase selection but not other IRF models.

Figure 7: The temperature versus fraction of solid calculated using Thermo-
Calc ’Scheil with solute trapping model’ for two extreme solidification
velocities

###  3.2 Case 2: Phase Selection in H13 tool steel

Konig et al. have carried out in situ synchrotron-based high speed X ray
diffraction experiments in different LPBF conditions to understand the primary
solidification modes of H13 tool steels during LPBF [21]. BCC-$\delta$ ferrite
was the primary solidification phase when cooled with a cooling rate of $\sim
2\times 10^{4}$ K/s and FCC-$\gamma$ austenite was the primary solidification
phase when cooled with a higher cooling rate of $\sim 1\times 10^{6}$ K/s.
Thermodynamic and IRF calculations are performed to explain this phase
selection.

Figure 8: (a) Isopleth section showing a peritectic reaction (b)Phase fraction
versus temperature plot for H13 tool steel

The composition used in the study is C-0.35, Mo-2.24, Mn-0.45, Si-0.25, Fe-
Remaining (weight percent). The isopleth diagram shown in Fig.8(a) reveals the
presence of peritectic reaction. The phase fraction versus temperature plot in
Fig.8(b) corroborates that for the considered composition, BCC($\delta$) is
the primary solidification phase under equilibrium conditions. The parameters
used for IRF calculations are given in Table 2. The thermal gradient is taken
as 1$\times 10^{5}$ K/m, which is typical for LPBF.

Table 2: Parameters used to calculate IRFs for H13 tool steel Parameter | BCC($\delta$) | FCC($\gamma$)
---|---|---
Liquidus temperature $T_{L}$ (K) | 1771.89 | 1765.78
Equilibrium partition coefficient of C $k^{C}_{o}$ | 0.156 | 0.2943
Equilibrium partition coefficient of Mo $k^{Mo}_{o}$ | 0.723 | 0.6118
Equilibrium partition coefficient of Mn $k^{Mn}_{o}$ | 0.718 | 0.7564
Equilibrium partition coefficient of Si $k^{Si}_{o}$ | 0.672 | 0.6430
Equilibrium liquidus slope of C $m^{C}_{o}$ (K/wt.fraction) | -8580.037 | -6470.758
Equilibrium liquidus slope of Mo $m^{Mo}_{o}$(K/wt.fraction) | -261.256 | -382.766
Equilibrium liquidus slope of Mn $m^{Mn}_{o}$ (K/wt.fraction) | -521.308 | -411.965
Equilibrium liquidus slope of Si $m^{Si}_{o}$ (K/wt.fraction) | -1256.316 | -1315.602
Kinetic coefficient $\mu_{k}$ ($\rm{ms^{-1}K^{-1}}$) | 10.0 | 10.0
Solute diffusion coefficient $D^{i}$ ($\rm{m^{2}/s}$) | $5.0\times 10^{-9}$ | $5.0\times 10^{-9}$
Diffusive speed in liquid $v_{D}$ (m/s) | 1.0 | 1.0
max speed $v_{o}$ (m/s) | 300.0 | 300.0
Interface diffusion speed $v_{I}$ (m/s) | 0.9 | 0.9
Gibbs-Thomson coefficient $\Gamma$ (K/m) | $1.0\times 10^{-7}$ | $5.0\times 10^{-8}$
Solid and Liquid Thermal conductivity $K_{S},K_{L}$ ($\rm{Wm^{-1}K^{-1}}$) | 50 | 50
Solid and Liquid Thermal Diffusivity$D^{T}_{S},D^{T}_{L}$ ($\rm{m^{2}s^{-1}}$) | $5.0\times 10^{-5}$ | $5.0\times 10^{-5}$

The variation in the interface temperature with solidification velocity is
shown in Fig.9. For both BCC and FCC, the planar solidification models with
linear phase diagram predict a rise in interface temperature with increasing
solidification velocity. But, the interface temperature decreases for all
other models. This is similar to that observed in Case 1. Except planar model
with linear phase diagram, all other models predicted the phase selection and
the transition velocities predicted are very close to the experimentally
observed values.

Figure 9: The vertical coloured areas indicate the experimental observations.
Plot of interface temperature for BCC($\delta$) and FCC($\gamma$)
solidification for (a) planar models (b)dendritic models

Fig.10(a) depicts the influence of solute drag on interface temperature.
Adding solute drag improved the prediction, and the changeover occurred at a
lower velocity. The effect of solute drag is more drastic in this case than in
the previous one. A plausible explanation for such a pronounced effect might
be the increased concentration of solute elements like Mo and Si that
partition to liquid. When there is severe partitioning of elements into liquid
at the interface, the compositional difference between the solid and the
liquid at the interface increases leading to an increase in the Gibbs energy
dissipation due to the solute drag (in Eqn.14). This leads to decrease in
solidification velocity. A similar effect has been observed in FeCoNiCuSn high
entropy alloy when concentration of Sn was increased [32]. In Case 1, Al has a
partition coefficient close to 1.0, thus the interfacial composition
difference is minimal and the solute drag effect is not very predominant. The
temperature versus fraction of solid plot for H13 steel calculated using
Thermo-Calc®’s ’Scheil with solute trapping model’ is shown in Fig.10(b). The
initial temperature (when fraction of solid is zero) decreases with increase
in solidification velocity. This observation agrees with the calculations from
Ludwig’s model. However, it predicts $\delta$ as the primary solidification
phase in both extreme velocities, contradicting the experimental observation.

Figure 10: (a)Interface temperature versus solidification velocity calculated
using Ludwig’s model with and without solute trapping for H13 tool steel
(b)The temperature versus fraction of solid calculated using Thermo-Calc
’Scheil with solute trapping model’ for two extreme solidification velocities

From the above cases, it is clear that the thermodynamic coupling of IRFs
improved the predictions. Additionally, including the effect of solute drag
increased the accuracy. Instead of either including or excluding solute drag,
one can use a partial drag parameter to match with the experiments and such a
model is proposed by Hareland et al. [33]. In this work, non-diagonal terms in
diffusion matrix are neglected. Guillemot et al. have highlighted the
importance of non-diagonal terms in diffusion matrix and used Hunziker’s
algorithm to account for the cross-diffusion of solutes [34, 35]. From their
report, it is clear that even in a concentrated eight component solid
solution, neglecting the effect of cross-diffusion leads to a difference of 10
K for velocities close to 1 m/s (Fig. 7(b) in Ref. [34]). Thus, neglecting the
cross-diffusion effects seems to be a reasonable approximation. Sobolev et al.
recently highlighted the effect of cross-diffusion terms when hyperbolic type
(non-Fickian) of diffusion equation is considered for the re-solidification
occurring during femtosecond laser pulse melting where the solidification
velocities are much higher [36]. The deviations are expected to be much less
for the range of velocities considered in our study. Another approximation
used in both the dendritic models is that the marginally stable wavelength is
selected as the dendritic tip radius (Marginal Stability criterion). Such an
assumption was made based on experimental data and the physics behind such
selection is lagging in marginal stability criterion. Microsolvability
criterion fills that gap by taking anisotropic effects into account and has
been verified by phase-field simulations [37, 38]. In Marginal stability
criterion, the selection parameter takes the value of $(2\pi)^{-2}$, but in
Microsolvability theory, this parameter depends on the anisotropy. One can
calibrate the selection parameter using selected phase-field simulations [39],
but it is beyond the scope of the current work.

In spite of these limitations, the above-mentioned models can serve as
important tools to understand the kinetically controlled phase selections
occurring in metal additive manufacturing.

###  3.3 Case 3: Columnar to Equiaxed Transition in Haynes 282

Figure 11: (a)Variation in CET plot for Haynes 282 with varying $N_{o}$ (b)CET
plot corresponding to $N_{o}=1\times 10^{12}$ along with experimental data
from LPBF [40], Laser deposition [41],Laser welding [42],Gas tungsten arc
welding [43],Laser deposition of IN625 [44], Electron Powder Bed Fusion [45]

Haynes 282 is a $\gamma^{\prime}$ strengthened Ni-based superalloy designed
for high temperature strength and thermal stability without compromising
fabricability. Additive manufacturing of Haynes 282 is gaining attention as
the additively manufactured components have better mechanical properties than
the wrought alloys. Haynes 282 has been successfully fabricated using electron
powder bed fusion (EPBF) [46], LPBF [40], laser deposition [41] and wire arc
additive manufacturing [47]. Fernandez-Zelaia et al. have successfully
fabricated single crystalline Haynes 282 using EPBF [48]. Thus, understanding
and predicting CET becomes necessary. Here, CET plot is obtained by
iteratively solving KGT model for each solidification velocity. The parameters
used to calculate KGT model is given in Table 3. The critical undercooling for
nucleation ($\Delta T_{n}$) is taken as zero due to the epitaxial nature of
solidification commonly observed during welding and AM [30].

Table 3: Parameters used to calculate IRF (KGT model) for Haynes 282 Parameter | Value
---|---
Liquidus temperature $T_{L}$ (K) | 1641.99
Equilibrium partition coefficient of Cr $k^{Cr}_{o}$ | 1.005
Equilibrium partition coefficient of Co $k^{Co}_{o}$ | 1.144
Equilibrium partition coefficient of Mo $k^{Mo}_{o}$ | 0.788
Equilibrium partition coefficient of Ti $k^{Ti}_{o}$ | 0.429
Equilibrium partition coefficient of Al $k^{Al}_{o}$ | 1.002
Equilibrium liquidus slope of Cr $m^{Cr}_{o}$ (K/wt.fraction) | -315.687
Equilibrium liquidus slope of Co $m^{Co}_{o}$(K/wt.fraction) | 82.478
Equilibrium liquidus slope of Mo $m^{Mo}_{o}$ (K/wt.fraction) | -432.653
Equilibrium liquidus slope of Ti $m^{Ti}_{o}$ (K/wt.fraction) | -1786.874
Equilibrium liquidus slope of Al $m^{Al}_{o}$ (K/wt.fraction) | -778.161
Kinetic coefficient $\mu_{k}$ ($\rm{ms^{-1}K^{-1}}$) | 10.0
Solute diffusion coefficient $D^{Cr}$ ($\rm{m^{2}/s}$) | $4.0\times 10^{-9}$
Solute diffusion coefficient $D^{Co}$ ($\rm{m^{2}/s}$) | $2.8\times 10^{-9}$
Solute diffusion coefficient $D^{Mo}$ ($\rm{m^{2}/s}$) | $1.3\times 10^{-9}$
Solute diffusion coefficient $D^{Ti}$ ($\rm{m^{2}/s}$) | $4.7\times 10^{-9}$
Solute diffusion coefficient $D^{Al}$ ($\rm{m^{2}/s}$) | $3.8\times 10^{-9}$
Diffusive speed in liquid $v_{D}$ (m/s) | 10.0
max speed $v_{o}$ (m/s) | 300.0
Gibbs-Thomson coefficient $\Gamma$ (K/m) | $1.0\times 10^{-7}$
Material parameter $n^{\prime}$ | 2.0

The variation in CET plot for Haynes 282 with varying $N_{o}$ is shown in
Fig.11(a). Haines et al. studied the sensitivity of different variables on the
CET plot of IN 718 alloy and have demonstrated that the $N_{o}$ is the most
sensitive parameter that causes significant changes to the CET plot [25].
Based on the experimental evidence from different processing conditions, one
has to choose the $N_{o}$ and it has been treated as a fitting parameter in
several studies [30, 25]. The experimentally observed microstructures are
plotted along with the calculated CET plot for $N_{o}=1\times
10^{12}\rm{m^{-3}}$ in Fig.11(b). For the process conditions reported in the
literature, the thermal gradient and solidification velocities are estimated
using the Rosenthal solution [49]. Even though there are several
approximations like ignoring convection within the liquid pool and assuming
constant thermophysical properties, it gives an estimate of the processing
conditions for us to perform CET or IRF calculations. Due to the limited
availability of data related to Haynes 282, other related Ni-based superalloys
like IN 625 and IN 718 are used for comparison.

When single crystal or columnar microstructure are preferred, one has to
choose the process conditions, such that the gradients and velocities lie
above the columnar (blue) line. When equiaxed microstructures are preferred,
process conditions must be chosen such that they are in the region below the
equiaxed (red) line. In laser powder deposition, the solidification front at
the top of the melt pool experiences low thermal gradients and high
solidification velocities. Hence, equiaxed grains are commonly observed at the
top of the melt pool [22]. In EPBF, one can obtain either columnar or mixed
microstructure by adjusting the process conditions or scan patterns [50].
Whereas in LPBF, the thermal gradients are generally higher and difficult to
achieve equiaxed microstructure only by process parameter modification.
External particles or inoculants that increase the $N_{o}$ value are added to
induce heterogeneous nucleation and resulting in equiaxed microstructure [25,
51].

##  4 Conclusions

The study demonstrated the capability of Calphad coupled IRFs to predict the
phase selection during AM, that are otherwise difficult by models based on
linear superposition of phase diagram. The thermodynamic interactions among
the elements are ignored in models with linear superposition, which can become
non-negligible for multicomponent alloys and lead to erroneous results. The
methods described here allow the users to have a quick calculation for the
phase selection. These calculations can serve as preliminary results before
performing more sophisticated phase-field calculations. These models provide
qualitative trends and insights, thus serving as a guide for both process and
alloy design for additive manufacturing.

## CRediT authorship contribution statement

V S Hariharan: Conceptualization, Data Curation, Software, Writing, Formal
Analysis. B S Murty: Investigation, Reviewing and Editing, G Phanikumar:
Conceptualization, Formal Analysis, Funding, Reviewing and Editing.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgement

Hariharan thanks Dr. Dasari Mohan (IIT Bombay, India) for his technical
discussions and inputs which helped improve this work. Hariharan thanks Dr.
Kang Wang (University of Virginia, USA) for his suggestions in implementing
Wang model.

## Data Availability

https://github.com/ICME-India/solidification-IRF

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