FR39 · Crystal defects

Alloy Dendrite Growth: Why Phase-Field Models Need Anti-Trapping

How does a dendrite grow into a supersaturated alloy melt, and why does a diffuse interface trap too much solute? The solid takes in less solute than the liquid, so the rejected solute piles up ahead of the crystal and any bump that reaches fresher liquid grows faster into an arm. A simulated interface is far wider than a real one, which lets the solid keep too much solute; an extra current that pushes solute back into the liquid restores the right amount.

Grow a four-armed dendrite with Karma’s 2001 phase-field model. Switch the anti-trapping current off and watch the solid keep far too much solute.

Tip speed $V d_0/D=0.0253$

Tip radius $\rho/d_0=45.4$

Solute left in the solid $c_{\mathrm{s}}/c_l^0=0.146$

Time since the seed was placed $t D/d_0^2=1334$

Grow a dendrite from a supersaturated melt

Interface model

Supersaturation of the melt

0.55

\Omega

Strength of the four-fold anisotropy

0.02

\epsilon_4

Partition coefficient

0.15

k

Drag the ring into the solid arm or the melt ahead of it to read the solute concentration there.

Playback speed

Time steps per frame

24

n_{\mathrm{step}}

Model, method and sources

$\vec j=-D\,q(\phi)\,c\,\nabla u-a_tW(1-k)\,e^u\,\partial_t\phi\,\hat n,\qquad a_t=\frac{1}{2\sqrt2},\quad \lambda=\frac{a_1W}{d_0}$

This is the isothermal dilute-alloy phase-field model of Karma (2001): a smooth interface of width W, a coupling chosen so that the interface reproduces the sharp-interface laws of solidification, and an extra anti-trapping solute current that removes the spurious solute trapping of a diffuse interface. The quadrant is computed with mirror edges and drawn four times. The Lab uses a grid twice as coarse as the paper. In the route benchmark this lowers the tip speed by about a tenth and leaves the solid concentration unchanged; the run at the paper's grid is in the route records. Late in a run the small box slows the tip by about 2 per cent more.

Karma (2001), arXiv:cond-mat/0103289v1 · Echebarria, Folch, Karma & Plapp (2004), arXiv:cond-mat/0404164v2

Read why anti-trapping is needed

Model assumptions

The crystal grows isothermally in a dilute binary alloy using the thin-interface phase-field model of Karma (2001), with his partition coefficient, supersaturation, anisotropy and interface width. Ahead of each tip the rejected solute piles up and must diffuse away, which sets the tip speed. The anti-trapping current removes a spurious solute jump that a diffuse interface would otherwise leave in the solid. The comparison marker on the tip-speed chart is read by eye from Fig. 1 of the paper, which used a grid twice as fine as this Lab; in the route benchmark the Lab grid gives a tip about a tenth slower and the same solid concentration. Everything is simulated; no micrograph is shown.

Simulated tip speed over time

Solute along the arm axis

More numbers and how they are computed

Local-equilibrium solid concentration at the tip: $0$
Excess solute in the solid: $0$
Solute far from the crystal: $0$
Tip position: $0$
Interface model: $0$
Change in total solute: $0$

The tip is where the interface crosses the growth axis, found by linear interpolation; its radius comes from a parabola through the interface on the rows nearest the axis. The solid concentration is averaged along the arm axis from 30 capillary lengths out from the crystal centre to 20 behind the tip. The dashed reference is local equilibrium at the measured tip curvature, not a fit. The exported file lists the settings and every sample.

Physics tutorial

A growing crystal must push its solute away

BackgroundWhen an alloy solidifies, the solid usually accepts less solute than the liquid it grows from. The rejected solute piles up ahead of the interface and has to diffuse away before the crystal can advance, so any bump that sticks out into fresher liquid grows faster. That instability turns a round seed into a dendrite.

Why it mattersDendrite spacing and the solute left between the arms set the microstructure of castings and welds. Phase-field models are the standard way to compute it, but only if their diffuse interface reproduces the sharp-interface physics.

Start with the essentials

Focus question: How does a dendrite grow into a supersaturated alloy melt, and why does a diffuse interface trap too much solute?
One-sentence intuition: At slow growth the solid should hold $c_{\mathrm s}=k\,c_{\mathrm l}$ with $k=0.15$ . A phase-field interface of width $W$ far wider than a real one lets solute diffuse through it at the wrong rate and leaves extra solute in the solid. Karma's anti-trapping current pushes that solute back into the liquid and restores local equilibrium.

Core mathematical model

Solute flux with the anti-trapping current

\vec j=-D\,q(\phi)\,c\,\nabla u-a_t\,W\,c_l^0(1-k)\,e^{u}\,\partial_t\phi\;\frac{\nabla\phi}{|\nabla\phi|},\qquad a_t=\frac{1}{2\sqrt2}

The first term is ordinary diffusion in the liquid, switched off in the solid. The second points from solid to liquid and acts only where the interface is moving, with the strength $a_t$ fixed so that the spurious solute jump vanishes.

Choosing the coupling

d_0=a_1\,\frac{W}{\lambda},\qquad \beta=a_1\Bigl(\frac{\tau}{\lambda W}-a_2\,\frac{W}{D}\Bigr)=0

With $a_1=0.8839$ and $a_2=0.6267$ , choosing $D=a_2\lambda W^2/\tau$ removes interface kinetics, so the model describes growth controlled by diffusion and capillarity alone.

Common difficulties

A thinner interface is not the fix

Typical misconceptionSolute trapping is just a resolution problem; make W small enough and it goes away.

Better mental modelShrinking W to a real interface width would need a grid far too fine for any dendrite calculation. The anti-trapping current removes the error at the interface width the computation can afford.

Run the experiment

01
Grow a dendrite
Start with A four-armed dendrite and read the tip speed and the solute left in the solid.
What to observe: Four arms grow along the crystal axes, the tip speed settles, and the solid behind the tip holds close to $k\,c_l^0=0.15\,c_l^0$ .
02
Switch the current off
Choose Solute gets trapped, which keeps every parameter but drops the anti-trapping current.
What to observe: On the Lab grid the solid now holds about $0.31\,c_l^0$ near the seed and $0.21\,c_l^0$ further along the arm, against $0.145\,c_l^0$ with the current. That excess is the spurious trapping Karma removed.
03
Take away the anisotropy
Choose No anisotropy, no clean tip and compare the tip shape with the four-armed dendrite.
What to observe: Without a preferred direction the tips flatten and split instead of growing as steady needles.

Alloy Dendrite Growth: Why Phase-Field Models Need Anti-Trapping

A growing crystal must push its solute away

Core mathematical model

Solute flux with the anti-trapping current

Choosing the coupling

Common difficulties

A thinner interface is not the fix

Run the experiment

Grow a dendrite

Switch the current off

Take away the anisotropy

Related experiments and articles

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