Effect of the misorientation angle and anisotropy strength on the initial planar instability dynamics during solidification in a molten pool

International Journal of Heat and Mass Transfer 130 (2019) 204–214

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Effect of the misorientation angle and anisotropy strength on the initial planar instability dynamics during solidification in a molten pool Fengyi Yu a, Yanzhou Ji b, Yanhong Wei a,⇑, Long-Qing Chen b a b

College of Material Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA

a r t i c l e

i n f o

Article history: Received 18 October 2017 Received in revised form 29 March 2018 Accepted 29 March 2018

Keywords: Anisotropy strength Preferred crystalline orientation Morphology instability Transient analytic model Phase-field method

a b s t r a c t The initial planar instability will appear with the solute accumulation ahead of the solid/liquid (S/L) interface during solidification in a molten pool. The instability process is dominated by the misorientation angle and the surface tension anisotropy strength, where the misorientation angle is the angle between the preferred crystalline orientation of base metal and the thermal gradient direction in front of the S/L interface. In this study, their effects on the initial planar instability during gas tungsten arc welding of an Al-alloy are investigated using a modified analytic model and a quantitative phase-field model, respectively. Specifically, we apply the uniform fluctuation spectrum assumption, Ax(0) = kBTM/{c0[1
15c4cos(4h0)]x2}, to represent the influence of thermal noise on S/L interface evolution. The incubation time, average wavelength and detailed interface morphology of the initial planar instability are investigated with varying surface tension anisotropies (determined by anisotropy strength c4 and misorientation angle h0). The results indicate that -c4cos(4h0) is a reasonable indicator for the effect of surface tension anisotropy on the initial planar instability. Moreover, rather than influencing solute diffusion, the surface tension anisotropy just affects the planar interface stability during the solidification. Finally, the experimental observations with the same welding parameters was carried out, which are in general agreement with the simulated results. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction The mechanical properties of fusion weld joints are directly dependent on the solidification structures in the fusion zone. Solidification involves complicated interplays between interface dynamics and transfer processes of heat and mass, which could result in complex solid/liquid (S/L) interface morphology [1]. Understanding solidification dynamics during welding could provide useful guidance for optimizing the welding process by obtaining desired microstructure morphology [2]. However, due to the opaqueness of metal, high-temperature and instantaneity of welding process, experimental characterization methods could hardly obtain the detailed dynamic evolution information of solidification structures in a molten pool [3], which limits current understanding of solidification behavior during welding. Fortunately, numerical simulations can efficiently predict the S/L interface dynamic evolution during general solidification [4,5], which sheds lights on the related investigations for welding [6]. Nevertheless, since the welding process includes sophisticated ⇑ Corresponding author. E-mail address: [email protected] (Y. Wei). https://doi.org/10.1016/j.ijheatmasstransfer.2018.03.106 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.

interaction between the electric arc and welded material [7], integrating the whole welding process into one comprehensive model is fundamentally challenging and numerically expensive. As a compromise, most existing numerical investigations typically focus on a certain stage of welding, ranging from the planar growth stage, the competitive growth stage to the quasi-steady growth stage [8,9]. This simplification, although efficient, neglects the fact that solidification is a history-dependent process, in which the planar interface instability in the early stage can be deterministic for cellular/dendritic array, tilted cellular/dendritic array or seaweed patterns in the late stages [10–12]. Therefore, the initial planar instability is a key phenomenon during solidification which does not only deserve thorough investigations, but its effect should also be effectively considered in the subsequent solidification stages in the molten pool. Initial planar instability has been investigated during the past few decades by theoretical calculations, simulations and experiments. The instability of planar interface was first analyzed by Tiller et al. [13] based on the redistribution of solute process in directional solidification. Then Mullins and Sekerka (MS) [14] analyzed the initial planar instability dynamics by using the limitation of infinitesimal amplitude of S/L interface deformation, which can

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predict the critical velocity of instability. Warren and Langer (WL) [15] generalized the MS theory to non-steady-state dynamics for predicting the incubation time and average wavelength of interface instability, whose results show good agreement with the experimental observations of real time synchrotron X-ray radiography [16]. But the detailed S/L interface structures during initial planar instability could not be easily predicted. To directly predict the S/ L interface morphology, Wang et al. [17] developed an analytic model combining the Fourier synthesis method and the timedependent linear stability analysis. The model is verified by both phase-field simulations and experimental observations under steady-state conditions of directional solidification [18]. As for welding, the pulling speed and thermal gradient vary considerably across the molten pool due to the movement of heat source, which cannot be regarded as steady-state conditions [19]. Dong et al. [20] modified the steady-state model to a non-steady-state model, formulating the time-dependent equations for the pulling speed and thermal gradient. The model was extended to transient conditions of gas tungsten arc welding (GTAW) for predicting the onset of initial planar instability [21]. Although prominent achievements have been made in initial planar instability predictions, most of such investigations are limited to single crystal with the preferred crystalline orientation parallel to the thermal gradient direction. In reality, due to the movement of heat source and the nonplanar shape of the molten pool, there will be a misorientation angle between the preferred crystalline orientation of base metal and thermal gradient direction in front of S/L interface during welding [22]. In addition, investigations on S/L interface patterns also reveal that the surface tension anisotropy plays an important role in the morphology evolution of solidification structures [23,24]. The surface tension anisotropy is determined by anisotropy strength and misorientation angle, where the anisotropy strength reflects the stiffness degree of surface tension. Therefore, it is essential to investigate the effect of surface tension anisotropy on the initial planar instability during solidification process in a molten pool. The phase-field method is a suitable choice to investigate the S/L interface morphology evolution due to the mesoscale nature of solidification microstructures [25]. It has been applied to study the solidification dynamics for more than two decades and a lot of achievements have been made [26,27]. By introducing a phenomenological ‘‘antitrapping current” term [28], the phase-field method can predict the cell/dendrite arm spacing [29], dendrite morphology [30,31], equiaxial growth [32,33] and sidebranching [34] quantitatively. The simulation results are consistent with the experimental observations in both general solidification and welding process. The simulation results ensure that the phasefield model could be applied to validate the analytic investigation

about the effect of surface tension anisotropy on the initial planar instability during welding process. In this study, we develop a transient analytic model to describe the evolution dynamics of the initial planar instability in a GTAW molten pool of an Al-6wt.%Cu alloy, considering the effect of surface tension anisotropy. The transient analytic model predictions are performed using the equilibrium fluctuation spectrum and the uniform fluctuation spectrum, respectively. Meanwhile, the phasefield equations with surface tension anisotropy are also formulated to reproduce the corresponding initial planar instability processes under transient conditions, which can be used for verifying the accuracy of the analytic model. Then the incubation time and average wavelength of initial planar instability under different anisotropy strengths and misorientation angles are investigated in detail to reveal the effect of surface tension anisotropy on initial planar instability dynamics in the molten pool. 2. Modeling and experiment 2.1. Analytic model Based on the WL theory [15], Wang’s Model [17] and isotropic non-steady-state model [20], a modified analytic model, incorporating the surface tension anisotropy effect, was presented to predict the onset of initial planar instability under transient conditions. The sketch of the coordinate system during the initial transient solidification stage is shown in Fig. 1. As shown in Fig. 1, z axis is the pulling direction of S/L interface. According to so-called ‘‘frozen temperature approximation”, the temperature field has relationship of T(z, t) = TM + G(t)z, where TM is the melting temperature of the pure solvent, G(t) is the thermal gradient along the z axis. A typical representation of surface tension anisotropy with fourfold symmetry is c = c0[1 + c4cos4(h + h0)], where c0 is the isotropic part of the surface tension, c4 is the anisotropy strength, h is the angle between the normal vector of interface and a fixed direction (here the thermal gradient direction), and h0 is the misorientation angle. Based on the local equilibrium assumption and the one-sided model of solidification, the free boundary conditions, with the consideration of surface tension anisotropy, could be expressed as:

cS ¼ kcL

ð1Þ @cL @z

ð2Þ

T I ¼ T M þ mcL
jC½1
15c4 4ðh þ h0 Þ

ð3Þ

V tip ðcL
cS Þ ¼
D

where cL and cS are concentration at solid side and liquid side of the S/L interface respectively, k is the partition coefficient, Vtip is the instantaneous interface velocity, D is the diffusion coefficient of solute in the liquid phase, TI is the temperature at the S/L interface, m is the slope of liquidus line, j is the S/L interface curvature; and C = c0TM/L is the Gibbs-Thomson coefficient, L is the latent heat. The term [1
15c4cos4(h + h0)] describes the surface tension stiffness from the expression (c + c00 ) [35]. In the linear stability analysis, the value of h is infinitesimal, hence the anisotropy term of the capillary effect can be simplified as 1
15c4cos4(h + h0)  1
15c4cos (4h0) [36]. During the initial transient stage, the time-dependent concentration profile can be approximated by a function of the form [15]:


c0 ðz; tÞ ¼ c1 þ ½c0 ðz0 ; tÞ
c1  exp

Fig. 1. The sketch of the coordinate system during the initial planar instability.


2ðz
z0 Þ l

ð4Þ

where c1 is the solute concentration in the far-away field, c0(z0, t) = G(t)z0/m, z0 is the S/L interface position, l is the time-dependent

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solute length. The values of z0 and l can be calculated from two coupled differential equations [15]:

@z0 2Dðz0
z1 Þ
V P ðtÞ ¼ lð1
kÞz0 @t

ð5Þ

@l 4Dðz1
kz0 Þ l @z0 ¼
@t lð1
kÞz0 z0
z1 @t

ð6Þ

where z1 is the steady-state position of the planar interface with the relation of z1 = mc1/G(t). With the definition of instantaneous interface velocity Vtip(t) = VP + oz/ot, the concentration field c0(z0, t) at S/L interface can be expressed as a function of the form:

c0 ðz0 ; tÞ ¼

2Dc1 2D þ V tip ð1
kÞl

ð7Þ

Based on the time-dependent linear stability analysis and assumption of an infinitesimal sinusoidal perturbation with spacing frequency x, the increase rate of perturbation amplitude can be given by Eq. (8):

rx ðtÞ ¼ ½dAx ðtÞ=dt=Ax ðtÞ

ð8Þ

where Ax(t) is the amplitude, which can be assumed to be comparable to thermal capillary length d0 = C/|m|(1
k)c0l before amplification, i.e., the S/L interface perturbation satisfies Ax(t) << l(t). Based on the time-dependent interface position and solute concentration (Eqs. (5) and (6)), the linear stability analysis of accelerating planar interface yields the dispersion relation of the perturbation under transient conditions, as shown in Eq. (9) [18], where the surface tension anisotropy is considered.

  2ðz0
z1 Þ C½1
15c4 cosð4h0 Þx2 ¼ qx 1 þ þ l GðtÞ   V I 2ðz0
z1 Þ V tip ðtÞ rx ðtÞ 1 C½1
15c4 cosð4h0 Þx2 þ þ þ þ l D V tip ðtÞ z0 D GðtÞ ð9Þ 2

2 1/2

where qx = Vtip(t)/(2D) + {x + [Vtip(t)/(2D)] } is the characteristic length of solute concentration fluctuation attenuation near the S/L interface along the z direction. The solution of Eq. (8) is


Z Ax ðtÞ ¼ Ax ð0Þ exp



t

t0

rx ðtÞdt

ð10Þ

where t0 is the critical time when rx changes from negative to positive. The value of Ax(0) can be calculated based on the equilibrium fluctuation spectrum [17] or the uniform fluctuation spectrum [37]. Then the position of S/L interface could be presented as Fourier series [17]:

zðx; tÞ
z0 ðtÞ ¼

X

Ax ðtÞ cos ½xx þ ux ðtÞ

ð11Þ

where z0(t) is the basic S/L interface position, x is the space frequency, ux(t) is the phase which is a stochastic variable with an average distribution within [0, 2p] before t0 and ux(t) = ux(t0-) after t0, respectively. 2.2. Phase-field model The quantitative phase-field model used in this paper was developed by Echebarria et al. [38] for directional solidification and modified by Zheng et al. [39] to transient conditions of welding. To investigate the planar interface instability in the initial transient stage, the transient conditions of welding process must be considered. Fig. 2(a) shows the sketch of GTAW molten pool in the longitudinal section, where the blue region reflects the domain used for numerical analysis. The shape of the molten pool can be regarded as the combination of two half ellipsoids, where al and bl are the depth and rear length of the pool, respectively. At the initial transient stage, the crystal growth orientation was assumed to be always perpendicular to the fusion line, as shown in the yellow area of Fig. 2(b). Therefore, the time-dependent pulling speed VP and thermal gradient G can be expressed in terms of the temperature difference and welding speed [39]:

al V 2 t V p ðtÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2 4 V 2 t2 a2l
bl þ bl

ð12Þ

TP
TL GðtÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 V t þ a2l 1
Vb2 t

ð13Þ

l

where V is the welding speed, TP and TL are the highest temperature of molten pool center and liquidus temperature, respectively. The values of al, bl and TP could be obtained by the finite element simulation, then the evolutions of VP and G with time can be obtained as input for the phase-field simulation. The detailed derivations and validations of the thin-interface phase-field model can be found in the reference [38]. Here we just present the equations describing the evolution of the phase-field field and the solute concentration field. In the formulation, a scalar variable /(r, t) is introduced to determine the phase at the fixed point and time, which takes the value / = 1 (/ =
1) in the solid (liquid) phase and varies smoothly across the diffuse interface. The solute concentration c(r, t) is characterized by a generalized supersaturation field U(r, t), which is given by U = (2kc/c1)/ [1 + k
(1
k)/]
1/(1
k). In terms of the field /, c and U, the

Fig. 2. The sketch of GTAW molten pool.

F. Yu et al. / International Journal of Heat and Mass Transfer 130 (2019) 204–214

governing equations of the quantitative phase-field are defined by Eqs. (14) and (15):

"

_ 2

s0 aðn Þ 1
ð1
kÞ

z


Rt 0

# 0 V p ðt 0 Þdt @/ @t lT "

z
! _ 2! ¼ W 2 r ½aðn Þ r / þ /
/3
kgð/Þ U þ

Rt

V ðt0 Þdt 0 p lT

0

#

ð14Þ

acteristic parameters of planar solidification process in the initial transient stage. For a small time interval Dt, it is easy to show analytically that [15]

l



1=2 8DDt 3

ð16Þ

pffiffiffiffiffiffiffi V P ðtÞ 2D ðDtÞ3=2 z0 ¼ z1
V P ðtÞDt þ pffiffiffi 3jz1 jð1
kÞ

1þk 1
k @U
/ 2 2 @t

! ! _ ! @/ r / 1 @/ ¼ r Dqð/Þ r U þ aðn ÞW ½1 þ ð1
kÞU  þ ½1 þ ð1
kÞU  @t j! 2 @t r /j !

ð15Þ 2 2

where the function g(/) = (1-/ ) is the double-well function which ensures g(
1) = g(1) = 0, the function q(/) = (1
/)/2 dictates that solute diffusivity is 0 in the solid, D in the liquid, and varies across the interface. a(n)  a(h + h0) = 1 + c4cos4(h + h0) imposes a fourfold anisotropy for a 2-D system. lT(t) = |m|c1(1
k)/k/G(t) is the thermal length, and m is the liquidus slope. W and s0 are the interface width and relaxation time, which are the length scale and time scale, respectively. Neglecting the kinetic effect, the phase-field variables can be linked to the physical quantities by W = d0k/a1 and s0 = a2kW2/D, respectively, where a1 = 0.8839 and a2 = 0.6267 are the numerical constants, and k is the coupling constant. 2.3. Experiment design and computational procedures The experimental observations for the same set of welding parameters were carried out to verify the simulated results. The welded material of base metal was Al-Cu alloy 2219 and the dimensions of plates were 160 mm  100 mm3 mm. The GTAW was the adopted welding method without filler metal, with current of 130 A, voltage of 15.6 V and welding speed of 3.5 mm/s. The metallograph at the longitudinal section of the welded joint was observed to represent the solidification structures near the fusion line by an optical microscope. The program codes ware written by FORTRAN and executed on the platform of Institute for Cyber Science Advanced Cyber Infrastructure (ICS-ACI) in the Pennsylvania State University in this study. In the simulation experiments, the welded material alloy 2219 was assumed to be a dilute binary Al-6.0 wt%Cu alloy. The physical parameters of the material from literature [40] are summarized in Table 1. As for the surface tension anisotropy, although previous experimental results give different values for a specific material [41], the property of surface tension anisotropy should be almost constant. In this paper, to investigate the general behavior about the effect of surface tension anisotropy, the values of which are intentionally varied as 0.01, 0.02 and 0.04, respectively. Meanwhile, the misorientation angles were defined to be 0°, 15°, 30° and 45° due to the fourfold symmetry. By solving Eqs. (5–7) in the transient analytic model, we can predict the instantaneous interface velocity, solute length, and solute concentration at the S/L interface, which are the most char-

Table 1 Physical properties of Al-Cu alloy and related parameters. Symbol

Value

Unit

Liquidus temperature, TL Alloy composition, c1 Liquid diffusion coefficient, D Liquidus slope, m Anisotropy of surface energy, e0 Equilibrium partition coefficient, k Gibbs-Thomson coefficient, C

922.9 6.0 3.0  10
9
2.6 0.01, 0.02, 0.04 0.14 2.4  10
7

K wt.% m2/s K/wt.%

Km

207

ð17Þ

which can be regarded as the initial conditions when solving Eqs. (5–7) via finite difference method (FDM). The solutions of Eqs. (10) and (11) could predict the S/L interface evolution and the detailed interface morphologies in the linear growth regime of solidification during welding. In the phase-field simulations, the most important calculation parameter is the interface width W. With the decrease of W, the precision of simulation results increases while the computational cost increases dramatically [42]. According to reference [38], W only needs to be one order of magnitude smaller than a characteristic length scale of the microstructures. The experimental observations show that the characteristic length of solidification structures during the initial stage is about 5–10 lm, so W was chosen to be 0.10 lm in this simulation. The Eqs. (14) and (15) were solved by explicit FDM with a fixed grid size dx = 0.8 W. The Neumann boundary conditions with zero-flux were used for both the phase-field and solute field. A time step size was chosen below the threshold of numerical instability for the diffusion equation in a 2-D system, i.e., Dt < (Dx)2/(4D). The initial phase-field variable p was set as /(z, 0) = tanh(z/ 2) along the normal of the planar interface, and the initial supersaturation field U was set to a steadystate diffusion profile along the normal of the interface. Moreover, to consider the infinitesimal perturbation of thermal noise on the S/L interface, a fluctuating current JU induced concentration fluctuation was included in the governing equations. The components are random variables obeying a Gaussian distribution with variance [21]. 0 ~ n ~0 0 ~ ~0 hJ m U ðr; t ÞJ U ðr ; t Þi ¼ 2Dqð/ÞF U dmn dðr
r Þdðt
t Þ

ð18Þ

The fluctuating current explicitly depends on the phase-field via the solute diffusivity Dq(/). The magnitude FU = FU0[1 + (1
k)U] could be defined by the relation of [43]:

hðdUÞ2 i ¼

hðdcÞ2 i ðDc0 Þ2



FU DV

ð19Þ

where the constant noise magnitude FU0 = kv0/(1
k)2/NA/c0 is the value of FU for a reference planar interface at temperature T0(U = 0). Dc0 = c1(1/k-1) is the concentration jump across the S/L interface, v0 is molar volume of the solute atoms, and NA is the Avogadro constant. 3. Results and discussion 3.1. Dynamic evolution of the S/L interface Since the temperature decreases in front of the S/L interface as time goes on, solidification takes place to maintain the local thermodynamic equilibrium. At the initial growth stage, the interface keeps planar and advances slowly towards the liquid region. With the accumulation of solutes ahead of the S/L interface, the planar interface loses its stability, i.e., the initial planar instability appears. The following dynamic evolution information of the interface instability is from the simulated results with anisotropy strength 0.01 and misorientation angle 0°. As the most characteristic parameters of solidification in the initial transient stage, the

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F. Yu et al. / International Journal of Heat and Mass Transfer 130 (2019) 204–214

Fig. 3. The evolution with time of interface velocity (a) and the interface solute concentration (b) obtained by the analytic model and phase-field simulation (c4 = 0.01, h0 = 0°).

interface velocity and the solute concentration in front of S/L interface obtained by the transient analytic model and phase-field model are presented in Fig. 3. As shown in Fig. 3, the analytic model and phase-field model show good agreement with each other, indicating that the increase rate of interface velocity remains nearly stable while that of the interface solute concentration increases with time. The simulated results are different from those obtained under steady-state conditions, where both increase rates decrease with time [44]. The discrepancy between the transient conditions and steady-state conditions reveals that the latter is not accurate enough for transient conditions during welding even for the initial stage. Because the pulling speed and thermal gradient change dramatically with time in the molten pool, as shown in Fig. 4. The solutions of Eqs. (8–10) could predict the dynamic evolution of S/L interface perturbed by fluctuation. The following simulated results are from the transient analytic model with equilibrium fluctuation spectrum assumption, i.e., the initial interfacial fluctuation Ax(0) is equal to the capillary length d0, as shown in Fig. 5. The evolutions of amplitude spectrum increase rate from Eq. (8) are shown in Fig. 5(a). At the initial solidification stage, the increase rate is lower than zero for any spacing frequency before

Fig. 4. Time-dependent solidification parameters in the molten pool (calculated by Eqs. (12) and (13)).

the critical time tc. When the solidification reaches the critical time of marginal stability, i.e., the perturbation increase rate is positive for some spacing frequencies, the initial planar instability happens. The evolution of the amplitude spectrum of the disturbed S/L interface after tc is shown in Fig. 5(b), where the amplitude spectrum enlarges with time in a particular frequency range of perturbations, which could indicate the instability of planar interface. Then the morphology evolution can be predicted directly after taking Fourier transformation, as shown in Fig. 5(c). It is important to point out that at the critical time tc, the perturbations at the S/L interface are still infinitesimal, which cannot be observed at the microscale. Specifically, although the perturbation increase rate rx turns into positive at t = 0.163 s around the peak x = 0.6 (Fig. 5(a)), the S/L interface is still relatively planar at that time, while the microscale visible interface perturbation appears at t = 0.244 s (Fig. 5(c)). Based on the above results, we can know that even if an external perturbation is imposed on the S/L interface, the perturbation will decay or even disappear due to the negative linear growth coefficient (Fig. 5(a)), which is consistent with literature [45]. Fig. 5(b) illustrates that the particular frequencies where increase rate is positive determine the range of the perturbation wavelengths resulting in the initial planar instability. The differences between the interface structures in Fig. 5(c) indicate that it will take some time for the bulges of the fluctuation to grow until they become microscopically observable. Moreover, the spacing of the visible interface perturbation is not constant but distributes irregularly within a finite range, which is consistent with the structures observed by experiments [45]. As a validation, the phase-field simulation was carried out to reproduce the initial planar instability under the transient conditions of GTAW. The S/L interface morphologies at different times are shown in Fig. 6, with the anisotropy strength 0.01 and misorientation angle 0°. The incubation time and average wavelength of initial planar instability from the phase-field simulation are 0.317 s and 5.27 lm, respectively. The values from phase-field simulations are different from those obtained by the previous analytic simulations: 0.244 s and 6.38 lm. The differences may result from the different selections of the disturbance amplitude. So the initial interfacial fluctuation was taken as the uniform fluctuation spectrum [43]:

Ax ð0Þ ¼

kB T M

c0 ½1
15c4 cosð4h0 Þx2

ð20Þ

F. Yu et al. / International Journal of Heat and Mass Transfer 130 (2019) 204–214

209

Fig. 5. Dynamic evolution of the initial planar instability predicted by the transient analytic model during welding: (a) The evolution of amplitude spectrum increase rate, (b) The evolution of the amplitude spectrum, (c) The S/L interface structures at different time. (c4 = 0.01, h0 = 0°).

Fig. 6. The morphology evolution of S/L interface from phase-field simulations (c4 = 0.01, h0 = 0°).

which is different from the previous selection: Ax(0) = d0, i.e., equilibrium fluctuation spectrum. The S/L interface morphologies of initial planar instability from the different calculation methods are shown in Fig. 7. The incubation time and average wavelength from the uniform fluctuation spectrum assumption are 0.316 s and 5.38 lm, which are closer to the values from the phase-field simulations (0.317 s and 5.27 lm) than those from the equilibrium fluctuation spectrum (0.244 s and 6.38 lm).

The comparisons between the different analytic models and the phase-field model reveal that the uniform fluctuation spectrum is more reasonable than the equilibrium fluctuation spectrum to represent the effect of thermal noise on the S/L interface evolution. Moreover, Eq. (20) in the uniform fluctuation spectrum assumption can also reflect the effect of surface tension anisotropy on the initial interfacial fluctuation. That is to say, the effect of surface tension anisotropy on both the frequency and amplitude of perturbation could be considered in the transient analytic model via the

Fig. 7. The morphology of S/L interface: (a) analytic model with equilibrium fluctuation spectrum (t = 0.244 s); (b) analytic model with uniform fluctuation spectrum (t = 0.316 s); (c) phase-field simulation (t = 0.317 s).

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Fig. 8. The evolution of incubation time (a) and average wavelength (b) with misorientation angle under different anisotropy strengths obtained by analytic model.

uniform fluctuation spectrum, which has its physical footstones [26]. 3.2. The effect of surface tension anisotropy In this section, the dynamic evolution of initial planar instability with different anisotropy strengths and misorientation angles were simulated to investigate the effect of surface tension anisotropy. Due to the sudden change of the transient behavior of the S/L interface in a molten pool, it is difficult to directly evaluate the instantaneous critical interface velocity of planar instability by experimental observations or numerical simulations. First of all, the criteria to reflect the effect of surface tension anisotropy should to be confirmed. Because the destabilizing effects of solute diffusion can shorten the stage of planar interface, while the stabilizing effects of surface tension and temperature field could extend the stage of planar interface. In addition to instantaneous interface velocity, the dynamic evolution of the interface instability can also reflect the effect of surface tension anisotropy. That is to say, the incubation time, the average wavelength and the detailed S/L interface morphology could also reveal the effects of surface tension anisotropy on the initial planar instability. The three variables will be discussed in detail. The evolutions of incubation times and average wavelengths with varying misorientation angles under different anisotropy strengths from the transient analytic model are presented in Fig. 8.

Fig. 8 shows that with a fixed c4, both the incubation time and average wavelength increase with the increase of h0. For example, the incubation times for c4 = 0.02, h0 = 0° and c4 = 0.02, h0 = 45° are 0.309 s and 0.337 s, respectively, and the wavelengthes are 4.77 lm and 5.98 lm, respectively. Moreover, the increase rates are greater if the crystal has larger c4, which is represented by the different slopes of the lines in Fig. 8. As a validation, the interface structures at different times of planar instability from phase-field simulations under various values of h0 with c4 = 0.02 are shown in Fig. 9. As h0 increases from 0 to 45°, the planar interface will last a longer time and the initial wavelength becomes larger. That is to say, the phase-field simulations agree well with the analytic model results. On the other hand, for a given h0, the effect of c4 on initial planar instability is not changing monotonously. The results of linear stability analysis reveal that the surface tension anisotropy destabilizes the S/L interface when cos(4h0) > 0 while stabilizes the interface when cos(4h0) < 0, i.e., the critical angle of surface tension anisotropy effect is hc = 22.5°. The incubation time and average wavelength are negatively related to c4 when h0 < hc, while these two variables are positively related to c4 when h0 > hc. For example, according to Fig. 8(a), the incubation times for c4 = 0.01, h0 = 0° and c4 = 0.04, h0 = 0° are 0.316 s and 0.294 s, respectively; while the incubation times for c4 = 0.01, h0 = 45° and c4 = 0.04, h0 = 45° are 0.332 s and 0.348 s, respectively.

Fig. 9. Dynamic evolutions of planar interface instability with different misorientation angles from phase-field simulations (c4 = 0.02, the time interval between two lines is 0.015 s).

F. Yu et al. / International Journal of Heat and Mass Transfer 130 (2019) 204–214

To verify the conclusion, the interface morphology evolution of initial planar instability under different values of c4 with h0 = 0° and h0 = 45° were obtained by the phase-field simulations respectively, as shown in Fig. 10. The different interface morphologies in Fig. 10(a) reveal that the planar interface becomes unstable at an earlier time and the average wavelength is smaller with a larger c4 when h0 = 0° (h0 < hc). Nevertheless, Fig. 10(b) shows that the surface tension anisotropy has a reverse behavior with h0 = 45° (h0 > hc): the planar interface becomes unstable later and the initial wavelength becomes larger as c4 increases. These results from the phase-field simulations are consistent with those from the transient analytic model. Based on the above analysis, the conclusion could be obtained: the effect of surface tension anisotropy on the initial planar instability is dependent on the multiplication of c4 and cos(4h0), i.e.,
c4cos(4h0). Then the variations of incubation time and initial average wavelength with respect to
c4cos(4h0) were obtained from both the transient analytic model and phase-field simulations, as presented in Fig. 11.

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The results of both methods confirm that either the incubation time or initial average wavelength increases with the increase of
c4cos(4h0). This phenomenon results from the stabilizing effect of surface tension anisotropy on the planar interface evolution. The surface tension stiffness in Eq. (3) c0[1
15c4cos4(h + h0)] can be simplified as c0[1
15c4cos(4h0)] in the initial transient stage. That is to say, the stiffer the S/L interface is, the longer time period for planar interface and the larger initial average wavelength can be obtained. The S/L interface morphologies with different surface tension anisotropies from phase-field simulations are shown in Fig. 12, with the calculation domain width of 200 lm at the same time. The distinct morphology structures in Fig. 12 illustrate that the surface tension anisotropy does obviously influence the S/L interface evolution. To get a more thorough understanding about the effect of surface tension anisotropy on S/L interface evolution, the timedependent characteristic parameters, the interface velocity and the solute concentration in front of S/L interface, need to be inves-

Fig. 10. Dynamic evolutions of planar interface instability with different anisotropy strengths from phase-field simulations: (a) h0 = 0°; (b) h0 = 45° (the time interval between two lines is 0.015 s).

Fig. 11. The incubation time (a) and average wavelength (b) with
c4cos(4h0) obtained by the analytic model and phase-field simulation, respectively.

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Fig. 12. The S/L interface morphologies with different surface tension anisotropies obtained by phase-field simulations (t = 0.317 s).

tigated. Here we focus on the evolution of solute concentration due to the difficulty of obtaining distinguished instantaneous critical interface velocity of planar instability. According to Eq. (3), the transient analytic model indicates that the surface tension anisotropy does not influence the solute diffusion process, but affects the planar interface stability by changing the boundary condition of the free boundary problem. To validate the conclusion, the evolutions of solute concentration with time for different values of
c4cos(4h0) are simulated using the phase-field model, as presented in Fig. 13. As shown in Fig. 13, the curves completely overlap with each other before the critical time of initial planar instability, which means the surface tension anisotropy indeed does not influence the solute diffusion process. After the critical time, the initial planar instability happens with the appearance of cellular crystals. The small-spacing cellular crystal has large interface curvature that may promote the diffusion of the rejected solute from the tip position to both sides of the interface. So, the sooner the initial planar instability happens, the larger the solute concentration gradients are. In turn, the local increase in the solute concentration gradient will decrease the critical wavelength; the sooner the initial planar instability happens, the smaller the average wavelengths are. From the above discussions, the surface tension anisotropy and the solute diffusion process can be treated as two independent factors for the initial planar instability. However, after the interface instability, the interface dynamics depend on the coupling of these two factors.

3.3. Experiment results The evolution of solidification structures could be obtained from the optical metallograph of the welded joint with the given welding parameters, as shown in Fig. 14. The red dotted line is

the fusion line, and the whole area is located at the bottom of the molten pool. In the initial growth stage, the S/L interface remains planar and advances slowly towards the liquid region, i.e., the planar growth stage. Compared with the whole solidification stage, the crystal growth distance is very narrow in initial stage, which cannot be clearly shown in the metallograph. There are some short cellulars near the fusion line, which have been eliminated from the final array because of the competitive growth. Since the figure is one of the intersecting surface of the weld, unbroken dendrite morphologies can hardly be preserved on this intersecting surface. Fortunately, the average wavelength could be obtained by counting the number of crystals near the fusion line, including the eliminated cellulars and the survived dendrites. The average wavelength from the experimental result is about 7.87 lm, which is a little larger than that from the simulation results (4.0–7.0 lm). The discrepancy may result from the following reasons: (1) the simulations are in 2-D system, while the experimental results are in 3-D system; (2) the above analyzes are based on the fact that the solute diffusion is used as the driving force. However, other transfer approaches (for example, convection) also have significant effects on the morphology evolution of S/L interface, which are difficult to be considered in the analytic and phase-field investigations; (3) there are some critical parameters that cannot be determined accurately, which will lead to the deviation of the simulation results. Normally, a complete quantitative agreement of the transient analytic model and phase-field results with experimental observations is hard to obtain. Nevertheless, the initial average wavelengths from the simulation results and experimental observations are in general good agreement, indicating the validation of the current analytic model and phase-field model. 4. Conclusions The effect of surface tension anisotropy (defined by misorientation angle and anisotropy strength) on the initial planar instability during GTAW of an Al-alloy is investigated by both the transient analytic model and the quantitative phase-field model. The simulation results indicates that the surface tension anisotropy indeed significantly influence the S/L interface evolution. The following conclusions can be drawn based on the simulated results:

Fig. 13. The evolution with time of interface solute with different
c4cos(4h0) from phase-field simulations.

(1) In the transient analytic model, the uniform fluctuation spectrum, with consideration of the surface tension anisotropy, is more reasonable than the equilibrium fluctuation spectrum to represent the effect of thermal noise on the S/L interface evolution. (2) The effect of surface tension anisotropy on the initial planar instability dynamics is only dependent on the
c4cos(4h0). Both the incubation time and average wavelength increase as the
c4cos(4h0) increases. Specifically, with a fixed c4, the critical time of planar instability comes later with a larger initial wavelength as h0 increases. For a given h0, the

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Fig. 14. The microstructure near the fusion line.

effect of c4 on the incubation time and initial wavelength is related to cos(4h0), i.e., the critical misorientation angle is hc = 22.5°. (3) The surface tension anisotropy does not influence the solute diffusion process, but the stability of S/L interface during the solidification. Moreover, the surface tension anisotropy and solute diffusion can be treated as two independent factors for the initial planar instability. It should be pointed out that the S/L interface normal changes directions all the time during welding. In this paper, the interface morphology was assumed to be perfect plat form and the interface

normal parallel to the temperature gradient direction. The deflection angle should be considered at the late stage. Conflicts of interest The authors declare that there is no conflicts of interest. Acknowledgements The study is supported by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD) and the

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financial support of the project from the Fundamental Research Funds for the Central Universities NP2016204. The authors acknowledge Xiaoxing Cheng in Pennsylvania State University for the help in the development of phase-field codes. The author Yu was supported by the China Scholarship Council as a visiting graduate student at Pennsylvania State University.

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