1 Introduction

Solidification is known as a very complex process accompanied with complex heat and mass diffusions. During metallic solidification, latent heat is released during the liquid to solid transition, leading to the interface moving continuously. In metallic alloys, the solute diffusion is significantly affected by temperature, pressure, liquid flow, and these factors would lead to changes of microstructures [1,2,3,4,5]. It has been widely established that the microstructures formed during the solidification process have great influences on properties of materials. Thus, in casting industries as well as metallic material researches, multi-scale microstructure controlling has become the most important way to improve the properties of materials. Recent studies have found that the convection can obviously affect the microstructure development during solidifications [6,7,8,9,10]. Liquid metal flow during solidifications can significantly affect the morphology, solute segregation at solid–liquid interfaces, and temperature distributions during solidification, by changing the heat and solute diffusions during the liquid–solid phase transition. However, the mechanism of solidification with convection is still unclear since it is too hard to observe in situ. Meanwhile, numerical simulation is an effective method for studying heat and solute diffusions during solidification, which are believed to be the main factors determining microstructure formations in metal alloys [11,12,13,14,15,16,17]. Thus, in this paper, the phase field method is implemented to investigate the dendritic growth of the Ni–Cu alloy with liquid flow, with the aim to study the effect of liquid flow on the microstructure formation during solidifications. Particularly, the heat and solute diffusions during solidification are analyzed to specify their relations on the morphology development with the liquid flow.

2 Phase field model and numerical method

In accordance with [18], this phase-field model for simulating the effect of liquid flow on the microstructure formation of a Ni–Cu alloy during solidification is employed. The main equations are listed below.

Phase field governing equation (in 2D) is

$$\frac{{\partial \phi }}{{\partial t}}={M_\phi }{\bar {\varepsilon }^2}\left[ {\nabla \cdot \left( {{\eta ^2}\nabla \phi } \right) - \frac{\partial }{{\partial x}}\left( {\eta {{\eta ^{\prime}}_\beta }\frac{{\partial \phi }}{{\partial y}}} \right)+\frac{\partial }{{\partial y}}\left( {\eta {{\eta ^{\prime}}_\beta }\frac{{\partial \phi }}{{\partial x}}} \right)} \right] - {M_\phi }\left[ {\left( {1 - {x_{\text{B}}}} \right){H_{\text{A}}}+{x_{\text{B}}}{H_{\text{B}}}} \right],$$
(1)

where \({M_\phi }\) is the interphase driving force-related phase field parameters, which is determined by \({M_\phi }=\left( {1 - {x_{\text{B}}}} \right){M_{\text{A}}}+{x_{\text{B}}}{M_{\text{B}}}\) with \({M_{{\text{A}}/{\text{B}}}}=\frac{{{{\left( {T_{{\text{m}}}^{{{\text{A}}/{\text{B}}}}} \right)}^2}{\beta _{{\text{A}}/{\text{B}}}}}}{{6\sqrt 2 \left( {\Delta {H_{{\text{A}}/{\text{B}}}}} \right){\lambda _{{\text{A}}/{\text{B}}}}}}\). \(T_{{\text{m}}}^{{\text{A}}}\) and \(T_{{\text{m}}}^{{\text{B}}}\) are the melting point of pure A and pure B, respectively. \(\Delta {H_{\text{A}}}\) and \(\Delta {H_{\text{B}}}\) are the heat of fusion per volume. \(\beta =\arctan \left( {{{\left( {\partial \phi /\partial y} \right)} \mathord{\left/ {\vphantom {{\left( {\partial \phi /\partial y} \right)} {\left( {\partial \phi /\partial x} \right)}}} \right. \kern-0pt} {\left( {\partial \phi /\partial x} \right)}}} \right)\) gives an approximation of the angle between the interface normal and the orientation of the crystal lattice. \(\bar {\varepsilon }\) is the parameter of the surface energy \(\sigma\) and the interface thickness \(\lambda\); \(\eta\) is the parameter related to the interface anisotropy intensity, modulus and angle. \({H_{\text{A}}}\) and \({H_{\text{B}}}\) have the formula terms \({H_{{\text{A}}/{\text{B}}}}\left( {\phi ,T} \right)={W_{{\text{A}}/{\text{B}}}}g^{\prime}\left( \phi \right)+30g\left( \phi \right)\Delta {H_{{\text{A}}/{\text{B}}}}\left( {\frac{1}{T} - \frac{1}{{T_{{\text{m}}}^{{{\text{A}}/{\text{B}}}}}}} \right)\), where \({W_{\text{A}}}\) and \({W_{\text{B}}}\) are constants which can be calculated by \({W_{{\text{A}}/{\text{B}}}}=\frac{{3{\sigma _{{\text{A}}/{\text{B}}}}}}{{\sqrt 2 T_{{\text{m}}}^{{{\text{A}}/{\text{B}}}}{\lambda _{{\text{A}}/{\text{B}}}}}}\). \(g\left( \phi \right)={\phi ^2}{\left( {1 - \phi } \right)^2}\) is a double-well potential with minima at \(\phi =0,{\text{ }}1\).

Solute conservation equation is

$$\frac{{\partial {x_{\text{B}}}}}{{\partial t}}+\vec {V} \cdot \nabla {x_{\text{B}}}=\nabla \cdot \left\{ {D\left[ {\nabla {x_{\text{B}}}+\frac{{{V_{\text{m}}}}}{R}{x_{\text{B}}}\left( {1 - {x_{\text{B}}}} \right)\left( {{H_{\text{B}}}\left( {\phi ,T} \right) - {H_{\text{A}}}\left( {\phi ,T} \right)} \right)\nabla \phi } \right]+{{\vec {j}}_{{\text{at}}}}} \right\},$$
(2)

where xB is the mole fraction of solute B in solvent A and \(\vec {V}\) is the velocity vector. D is the diffusion coefficient and Vm is the mole volume. In binary alloys, \(D={D_{\text{S}}}+p\left( \phi \right)\left( {{D_{\text{L}}} - {D_{\text{S}}}} \right)\), where \({D_{\text{L}}}\) and \({D_{\text{S}}}\) are the diffusion rates in the liquid and solid phases, respectively, and \(p\left( \phi \right)={\phi ^3}\left( {10 - 15\phi +6{\phi ^2}} \right)\) is a smoothing function. \({\vec {j}_{{\text{at}}}}\) is the anti-trapping current to suppress the solute trapping due to the larger interface width used in simulations to get more quantitative predictions.

Temperature governing equation is

$${c_{\text{p}}}\left( {\frac{{\partial T}}{{\partial t}}+\vec {V} \cdot \nabla T} \right)+30g\left( \phi \right)\Delta \widetilde {H}\frac{{\partial \phi }}{{\partial t}}=\nabla \cdot K\nabla T.$$
(3)

In the formula, T is the temperature; \({C_{\text{p}}}\) is the specific heat capacity; \(\Delta \widetilde {H}=\left( {1 - {x_{\text{B}}}} \right)\Delta {H_{\text{A}}}+{x_{\text{B}}}\Delta {H_{\text{B}}}\); K is the thermal diffusion coefficient.

Continuity equation:

$$\nabla \cdot \left( {\phi \vec {V}} \right)=0.$$
(4)

Momentum equation:

$$\frac{{\partial \phi \vec {V}}}{{\partial t}}+\phi \left( {\vec {V} \cdot \nabla } \right)\vec {V}= - \frac{\phi }{\rho }\nabla P+v{\nabla ^2}\left( {\phi \vec {V}} \right) - \frac{{vh(1 - {\phi ^2})\phi \vec {V}}}{{{\lambda ^2}}},$$
(5)

where \(P\), \(\rho\), and \(\nu\)are the liquid flow velocity, pressure, density, and kinematic viscosity, respectively. \({\vec {M}_{\text{d}}}\) is the dissipative interfacial stress. A detail description for this phase-field model with convection can be found in Ref [18].

In the present work, a seed with an initial radius is assumed to nucleate in the domain at the beginning. The zero-Neumann boundary conditions are used in the calculation of phase field, solute field and temperature field. Equations (1–3) are discretized using a finite difference approach with a uniform spatial lattice. The total number of grids is 1200∆x × 1200∆y with ∆x = ∆y = 1.0 × 10−8 m, and the corresponding time step ∆t = 5.0 × 10−8 s. A circular solid seed is set at (600, 400) with a radius of 6∆x. The continuity Eq. (4) and Navier–Stokes Eq. (5) were solved with the semi-implicit method for pressure-linked equation (SIMPLE) scheme under zero-Neumann boundary conditions for left and right boundaries, and the initial flow direction is set to be from top to the bottom of simulation zone, with top boundary as the flow input boundary and the bottom boundary is set to be the flow output boundary. Based on the physical parameters of the Ni–Cu alloy, which can be calculated from parameters of pure Cu and Ni (as shown in Table 1) based on the simple linear mixing method, the Prandtl number of the melt is about 0.002 and the Schmidt number is about 6.0. This indicates the flow would not have significant influence on the temperature distribution, but the solute diffusion would be significantly changed by the liquid flow. Thus, with zero-Neumann boundary conditions, the effect of the initial liquid flow on the morphology evolution can be investigated with this simulation domain since the solute diffusion can also determine the phase transition during solidifications by changing the solute segregation near solid–liquid interfaces.

Table 1 Physical parameters used in the simulations
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3 Results and discussion

From the discussion in the previous studies [17,18,19,20,21,22,23], we can conclude that the liquid flow in the presence of solidifying interfaces can significantly change the morphology evolution by altering the solute and heat diffusions near solid–liquid interfaces, and the asymmetric development of dendrites is believed to be enhanced with the increasing intensity of the liquid flow. The morphology transition with increasing liquid flow is to be simulated, and the interface morphology is plotted in Fig. 1.

Fig. 1
figure 1

Dendritic morphology during dendritic growth process under liquid flow (0 ~ 0.4 m/s) at the same time step. a Solid–liquid interface morphology. The color loop curves from inner to outer represent solid–liquid interfaces with initial velocities of 0, 0.02, 0.06, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35 and 0.40 m/s. b Interface distances from the initial dendrite center detected along the colored straight lines in a, which have different included angles: 0.0°, 22.5°, 45.0°, 67.5°, 90.0°

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Without liquid flow, the solidifying morphology is the typical dendrite with weak side branches, and all side branches are oriented in the direction of primary dendrite arms due to the kinetic anisotropy. With relatively small initial liquid flow, the dendritic growth shows an asymmetric developing: in the upstream, the dendrite developing is enhanced and the secondary arms are all oriented in the direction of the initial flow field, while in the downstream primary dendrite arm and all secondary arms are suppressed. With increasing liquid flow intensity, asymmetric solidifying in upstream and downstream is significantly enhanced, and side branches in both downstream and upstream change in two different ways: side branches in the downstream intend to vanish while side branches in the upstream are combined into a continuous solid phase due to the different solidifying rate caused by the liquid flow. It is particularly important to note that though no side branches can be found when the initial liquid flow is larger than 0.15 m/s, the secondary dendrite arms in the upstream can be considered to combine into a continuous solid phase, while in the downstream the secondary dendrite arms are continuously suppressed till disappearing. With further increase of the initial liquid flow, the solidifying morphology begins to take on new characteristics: anisotropy of the solid phase growing is qualitatively changed, and growth superiority of the primary dendrite arms is unapparent with increasing liquid flow. A semi-circular solid is formed in the downstream region with a liquid flow of 0.3 m/s. There is an obvious dendrite to semi-circular morphology transition with increasing liquid flow. To characterize the effect of liquid flow on microstructure evolution, the distance of solid–liquid interface positions from the initial dendrite center is measured and plotted in Fig. 1b. Representative positions are selected from the intersections of the solid–liquid interface and curves of different included angle with respect to the rectangular coordinate axes, as labeled in Fig. 1a. It is obvious that the growth rate of the dendrite tip in the upstream is significantly higher than those of other regions. With increasing initial flow field, the growth rate difference of each region is gradually reduced, indicating that when the flow field is greater than a certain value, the dendrite intends to a spherical shape in the upstream region.

Figure 2 shows the concentration distribution maps at 2.0 ms for dendritic growth under different forced liquid flow, and the corresponding temperature distribution maps are plotted in Fig. 3. With induced liquid flow, the solute diffusion is changed asymmetrically in the whole region, leading to solute segregation discrepancy at interfaces in different regions. Liquid flow moves the solute from the upstream of the flow field to the downstream, and the vortex, as the result of interaction between growing dendrite and initial liquid flow, leads to solute enrichment so that enhanced solute segregation is formed at the interface in the downstream. This large difference of the solute distribution around the dendrite could be the reason for the asymmetrical morphology formation with induced liquid flow. The temperature distributions are mostly determined by the lantern heat since the high temperature is located at solid–liquid interfaces, since the influence of liquid flow on the temperature distribution could be negligible within the limited time frame with Prandtl number Pr = 0.002 in this study.

Fig. 2
figure 2

Concentration distribution in dendritic growth process at t = 2.0 ms. a v = 0 m/s; b v = 0.05 m/s; c v = 0.10 m/s; d v = 0.15 m/s; e v = 0.20 m/s; f v = 0.30 m/s

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Fig. 3
figure 3

Temperature distribution in dendritic growth process at t = 2.0 ms. a v = 0 m/s; b v = 0.05 m/s; c v = 0.10 m/s; d v = 0.15 m/s; e v = 0.20 m/s; f v = 0.30 m/s

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To further specify reason for the microstructure transition induced by the liquid flow, the concentration and temperature distribution profiles on the primary dendrite arm in the vertical direction (X = 600Δx, from the bottom to the top) with different initial liquid flow are plotted in Fig. 4. It is obvious that the solute partitions in the downstream regions are much higher than that in the upstream, while temperature in the upstream is higher than that in the downstream region as the result of large latent heat release from the liquid–solid phase transition. It has been well established that during solidification in metal alloys, the constitutional supercooling at the solid–liquid interface caused by the solute partition is the main factor determining the phase transitions. With increasing initial liquid flow, the solute partition difference induced by the liquid flow intends to be enlarged. Considering the dendrite morphology shown in Fig. 1, we can clearly conclude that with large initial liquid flow, the main reason resulting in the dendritic to semi-circular morphology transition in this study is the large solute distribution difference caused by the liquid flow, which has already been discussed in detail in previous studies on liquid flow on solidifications [16, 18].

Fig. 4
figure 4

Concentration (a) and temperature (b) distribution profiles of primary dendrite arm with different initial liquid forced flow

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In Fig. 5, the solid fraction of the simulation zone at the same time step (t = 2.0 ms) is measured and plotted as a function of initial liquid flow. With increasing initial liquid flow, the solid phase shows a non-linear increasing characteristic: rapidly increasing with increasing liquid flow and the increasing rate decreases with large liquid flow. In our simulations, the only heat input into the simulation zone is the latent heat release from the liquid–solid phase transition, thus with increasing solid phase rate, the temperature should be higher. However, as shown in Fig. 4b, the temperature with large initial liquid flow is lower along the primary dendrite arm. Comparing with the temperature distributions in Fig. 3, dendrite formation is enhanced in the upstream with higher temperature, which could further confirm that the morphology transition investigated in this study is attributed to the asymmetrical solute diffusion induced by the liquid flow.

Fig. 5
figure 5

Solid rate change with increasing initial liquid flow

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4 Concluding remarks

Liquid flow has a significant effect on the dendrite microstructure formation by changing the solute and heat diffusions during the solidification. In this paper, the dendritic growth and microstructure development with initial liquid flow were simulated by the 2-D phase field method to investigate the microstructure formation with initial liquid flow. The effect of liquid flow on heat transfer is neglected due to the limited size of simulation zone and parameters used in this study. The liquid flow during solidification can largely influence the solute diffusion during solidification, leading to solute segregation difference at solid–liquid interfaces in the downstream and upstream, which would result in the irregular dendrite development. A dendritic to semi-circular morphology transition can be observed with increasing initial liquid flow. By analyzing the concentration and temperature distributions in the simulation zone, the dendrite to semi-circular transition is believed to be attributed to the constitutional supercooling differences in the upstream and downstream sides induced by the liquid flow.