Simulation of fluctuation effect on dendrite growth by phase field method

Simulation of fluctuation effect on dendrite growth by phase field method LIU Xin-mei(刘新妹), HAN Yan(韩 焱) Key Laboratory of Instrumentation Science and Measurement, Ministry of Education, North University of China, Taiyuan 030051, China Received 12 June 2008; accepted 5 September 2008 Abstract: The dendrite growth process was simulated with the phase field model coupling with the fluctuation. The effect of fluctuation intensity on the dendrite morphology and that of the thermal fluctuation together with the phase field fluctuation on the forming of side branches were investigated. The results indicate that with the decrease of thermal fluctuation amplitude, the furcation of dendrite tip also decreases, transverse dendrites become stronger, longitudinal dendrites become degenerated, Doublon structure disappears, and a quite symmetrical dendrite structure appears finally. Thermal fluctuation can result in the unsteadiness of dendrites side branches, and it is also the main reason for forming side branches. The phase field fluctuation has a little contribution to the side branches, and it is usually ignored in calculation. When the thermal fluctuation amplitude (Fu) is appropriate, the thermal noise can result in the side branches, but cannot change the steady behavior of the dendrites tip. Key words: fluctuation; dendrites; phase field method; side branches

1 Introduction Dendrite growth is an un-equilibrium process. The side branch plays an important role in solidification microstructure. Phase field method is a very important means in simulating the morphology of dendrites[1−3]. With the development of phase field method, studying the forming of side branch in solidification becomes one of the popular topics at present[4−5]. But the main problem that concerns side branch growth frequency and amplitude has not been absolutely solved for the moment. BARBER et al[6] approximately studied the dendrite growth with planar symmetrical solidification model in little Peclet value by using WKB. The result indicates that partial amplitude of wave packet increases according to the exponential function Z1/4, where Z is the distance between dendrite symmetrical axis and dendrite tip. PIETERS[7] got the same result with BARBER by analyzing and calculating the model of boundary layer. LANGER[8] studied the growth of three-dimensional dendrites with the similar method, and figured out that noise is the reason for creating side branch, but noise cannot explain this phenomenon entirely. BRENER and TEMKIN[9] studied the 3D dendrites with anisotropy, and considered that the side branch that can be observed

through experiments can be explained by using thermal noise. At the same time, they found that the amplitude of side branch increases according to the exponential function: Z2/5. It is larger than the case under the axial symmetry, the function of which is Z1/4. But the change of wave length is similar to axis symmetry case, and they all change according to the exponential function Z1/5. The noise can be divided into two kinds[10−11]: one is thermal noise, the other is interface noise. Thermal noise comes from heat exchange of the solid/liquid interface. And interface noise derives from atom exchange of the two phases, which is the atom adsorption and disengaging. In this work, the thermal noise was introduced into the phase model and the side branch was produced.

2 Phase field model 2.1 Phase field controlling equation The phase equation is constructed based on entropy function[12]: 2

Wf εf 2 ⎛ 1 1⎞ − ⎟⎟ p ′(φ ) − g ′(φ ) + ∇ φ T 2 ⎝ TM T ⎠

τφ& = L0 ⎜⎜

Corresponding author: LIU Xin-mei, PhD; Tel: +86-351-3921479; E-mail: [email protected]

(1)

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c v T& + L0 p ′(φ )φ& = k∇ 2T

(2)

KOBAYASHI[13] introduced fluctuation into phase model. Function (1) introduces an item A φ (1− φ )R, where R is a stochastic variable of homogeneous distribution which obeys [−1, 1], A is an intensity factor that is used to adjust disturbing intensity. A stochastic item was added in Eqn.(2), where ＜f(x, t)＞=0,

~ ＜f(x, t)f(x′, t′)＞= A 2φ 2 (1 − φ ) 2 δ ( x − x ′)δ (t − t ′)

(3)

From microcosmic sense, the noise mainly derives from thermal flow disturbance of solid/liquid phase, and the interface noise is the result of atom changing between the two phases. There are two main methods of introducing noise into phase field[13]: one is to introduce a Gauss distribution stochastic variable matrix q(r, t) to thermal or solute function, in which the average related to the space and time is zero. The noise produced by this method is called thermal noise. The introducing way is shown in Eqn.(4): ⎧ ∂T 2 δF ⎪⎪ ∂t = DT ∇ δT + ∇ ⋅ q (r , t ) ⎨ ⎪ ∂c = M ∇ 2 δF − ∇ ⋅ q(r , t ) c ⎪⎩ ∂t δc

(4)

The other method is to introduce a stochastic variable which is homogeneous distribution to the phase field equation. The noise produced by this method is called interface noise, which is shown in Eqn.(5): ∂φ δF = −M φ + αr[16 g (φ )][(1 − c)] f ∂t δφ

A

+ cf

B

Let Fu =

2k B TM2 c L2 l 3

δ ij ⋅ δ ( x − x′)δ (t − t ′) (6)

2k BTM2 c

, where Fu is the thermal fluctuation L2 l 3 amplitude, and kB is Boltzmann constant. After adding the fluctuation item to Eqns.(1) and (2), the equation can be obtaied as follows:

τφ& = L0 ( ε 2f T

Wf 1 1 g ′(φ ) + − ) p ′(φ ) − TM T 2

∇ 2φ + arφ (1 − φ )

(8)

2.2 Initial and boundary conditions To the crystal nucleus whose radius is r0, the initial condition of the calculation is as follows:

When x2+y2≤ r02 , T=T0, c=c0,φ=1, When

2

＞ r02

2

x +y

(9)

, T=T0, c=c0,φ=−1,

(10)

where x and y are the coordinates, T0 is the initial temperature of the superfusion. At the boundary of calculation district, the Zero-Neumann boundary condition was selected to the phase field and temperature field. 2.3 calculation parameter Pure nickel was selected to perform the simulation; and the calculation parameters are listed in Table 1. Table 1 Thermo-physical parameters of pure nickel L/(J·m−3)

c/(J·m−3·K−1)

σ/(J·m−2)

2.35×109

5.42×106

0.37

DT/(m2·s−1)

β/(S·m−1)

d0/m

1.55×10−5

0.5

0.627×10−9

During the simulation, Eqns.(6) and (7) must be dealt with dimensionlessly. Let x = x / w, t = tD L / w 2 , selecting w=2.3 × 10−8m, the mesh dimension as dx=dy=0.4, dt=0.005.

(5)

where r is the stochastic value between −1 and 1, α is the fluctuation intensity parameter. The phase field noise item was introduced to Eqn.(1) by adopting the second method; and the item ∇ · qst was introduced to Eqn.(2), which follows the Gaussian distribution: q ist (r , t )q stj (r ′, t ′) =

cvT& + L0 p ′(φ )φ& = k∇ 2T + ∇ ⋅ q st

(7)

2.4 Noise item When calculating the function, Eqn.(6) adopts Euler arithmetic, and Eqn.(7) adopts ADI arithmetics. In Eqn.(5): δ ( x − x ′) difference is expressed as difference is expressed as δii′/∆x; r δ (t − t ′) δnn′/∆t; ∇ ⋅ q(r , t ) |i , j difference is expressed as [qx(i+1, j)−qx(i, j)]/∆x+[qy(i, j+1)−qy(i, j)]/∆y.

3 Simulation results 3.1 Dendrite morphology with and without fluctuation From Fig.1, it can be seen that without noise, the growth presents a dendrite growth mode which is mainly composed of trunk. In this case, the growth direction is along with the trunk, and it is sequential and slick, and also has no square dendrites, as shown in Fig.1(a). In Fig.1(b), with noise, the growth presents abundant side branches. The experimental results of the dendrites morphology, shown in Figs.2 and 3 [14] are approximate to the simulation. The tip appearance is very similar in

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Fig.3 Experimental results of dendrite growth with noise

Fig.1 Dendrite morphologies under different conditions: (a) Without noise; (b) With noise

Fig.2 Dendrite growth of ammonium bromide

the two states: with and without noise. And their tip curvature radius is alike. But with noise, the beginning shaped dendrites are longer, namely the growth speed of them is faster than that without noise.

3.2 Effect of thermal noise amplitude on side branches Fig.4 shows the morphologies at t=100 000 and different Fu values. With noise, the hot disturbing amplitude takes a great effect on the simulation result. Although the introduction of thermal noise does not change the temperature distribution rule, the temperature field presents a random undulation. The larger the Fu, the greater the temperature field undulation. After adding noise, interface foreland forms the sustained disturbing and side branches appear. Fig.4 also shows that the tips of dendrite morphologies form obvious branches when the amplitude Fu varies in the range of 1×10−2−5× 10−5, which is the Doublon structure[15]. The extents of branches diminish with the amplitude decreasing. When Fu=1 × 10−3, the dendrites morphologies form square dendrites, all of which grow along the same distance of one symmetrical axis. When it is closer to the symmetrical axis, the amplitude of square dendrites becomes larger; when it is further from the symmetrical axis, the amplitude of square dendrites becomes smaller. When Fu=5×10−4, the growth direction of the square dendrites transforms from the transverse symmetrical axis to the longitudinal symmetrical axis. Inside square dendrites, they grow along the transverse symmetrical axis; outside the square dendrites, they grow along the other symmetrical axis. When Fu=5×10−5, all of the square dendrites grow along another symmetrical axis; when Fu=1×10−5, the dendrites grow along two symmetrical axes, and the square dendrite arms of the two distances are basically the same.

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Fig.4 Dendrite morphologies at different fluctuation amplitudes: (a) Fu =1×10−2; (b) Fu =5×10−3; (c) Fu =1×10−3; (d) Fu =5×10−4; (e) Fu =5×10−5; (f) Fu =1×10−5

3.3 Effect of phase field noise and thermal noise on side branches The phase field functions with and without noise were simulated respectively, and the dendrite morphologies were compared under the same growth time. Fig.5 shows the dendrite morphologies with different noises; Fig.6 shows the contrast of dendrite morphologies without noise and with phase field noise (noise factor a = 1.0 ); Fig.7 shows the contrast of dendrite morphology without noise and with thermal noise(Fu=1×10−3). From the

contrastive analysis of the dendrite morphology, it can be got from Figs.5−7 that the thermal noise, which is the main reason for forming side branches, can cause unstable side of dendrites and the phase field noise has little contribution to forming the side branches. 3.4 Effect of noise on stable behavior of dendrites Fig.8 shows the relationship between dendrite tip growth speed and time at Fu=1×10−5; Fig.9 shows the relationship between dendrite tip radius and time at

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Fig.6 Dendrite morphology with and without noise

Fig.7 Dendrite morphology with phase field noise and thermal noise

Fig.5 Dendrite morphologies with different noises: (a) Without noise; (b) Phase field noise; (c) Thermal noise

Fu=1×10−5. From Figs.8 and 9, it can be seen that tip growth speed becomes convergent, tip growth radius fluctuates within a definite range, and the growth comes into a steady state with time. It can be concluded that

Fig.8 Relationship between dendrite tip growth velocity and time

when Fu is given an appropriate value, the introduction of noise causes the side branches to form, but the steady state of dendrites tip does not change.

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4 Conclusions

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1) The method of introducing noise into the phase field model is directly related to the forming of dendrites side branches. 2) Thermal noise can cause the non-steady state of dendrites side branches to form, and it is the main reason for forming side branches. Phase field noise has little contribution to the side branches. 3) With the decrease of the thermal fluctuation amplitude, the furcation of dendrite tip also decreases, transverse dendrites become strong, longitudinal dendrites become degenerated, Doublon structure disappears, and a quite symmetrical dendrite structure forms finally. When Fu is appropriate, thermal noise can cause side branches to form, but it does not change the steady behavior of the dendrites tip.

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(Edited by CHEN Wei-ping)