L4 · Chapter 7

Reading Karma 2001 on alloy phase fields and reproducing its dendrite figures

Sandbox Physics · 2026-10-30

Karma’s 2001 letter claims that one extra solute current lets a phase-field model of a dilute alloy use an interface wide enough to compute dendrites, yet still obey local equilibrium at the interface. The evidence is a convergence test in two figures. This chapter is a guide to reading the paper with the Lab beside you: where each claim sits, what the figures can and cannot show, what you have to choose yourself to reproduce them, and which of their values our runs match.

The claim in one paragraph

Earlier phase-field models of alloys needed an interface not much thicker than the capillary length to avoid spurious trapping, surface diffusion and interface stretching, which made them too expensive for slow growth. Karma adds an anti-trapping current and picks the interpolation functions so that all three effects vanish in the thin-interface limit. The model then reduces to the classic free-boundary problem with local equilibrium. The claim is supported in two ways: a sketch of the thin-interface analysis, and two-dimensional dendrite runs whose tip speed and solute profile stay the same when the interface width changes.

Source: A. Karma

Where to look first

The first half of the letter, Eqs. (1)–(12), treats an idealised alloy whose liquidus and solidus are parallel. It shows how the current removes the chemical-potential jump and gives the constants $a_1$ and $a_2$ . The dilute alloy starts at Eq. (13). Eqs. (13)–(15) are the sharp-interface target, Eqs. (16)–(18) the model, Eq. (20) the interpolation function, and Eq. (22) the four-fold anisotropy.

One paragraph near the end lists every number you need: $\Delta x=0.4$ , $\Delta t=0.008$ , $W=\tau=1$ , $\epsilon_4=0.02$ , $k=0.15$ , $\Omega=0.55$ , and a seed of radius $22\,d_0$ . Fig. 1 plots the tip speed $Vd_0/D$ against $tD/d_0^2$ for both models at $d_0/W=0.277$ and $0.544$ . Fig. 2 plots the solute left in the solid along the dendrite axis. The thin-interface derivation itself is promised for later; Echebarria and co-workers published it in 2004.

Source: A. Karma · B. Echebarria

How the paper tests convergence

There is no exact dendrite solution to compare with. Instead Karma runs each model at two interface widths. If a model has reached its thin-interface limit, its curves must lie on top of each other. The present model passes: its speeds and solute profiles at the two ratios superimpose. The standard model fails, most clearly in the solute profile, which is still far from converged at the finer ratio.

A second, independent check is self-consistency. With the tip radius $\rho$ measured in the run, local equilibrium predicts how much solute the solid behind the tip should hold. Karma reports agreement within a few per cent for the present model. Note what the cost argument implies: halving $d_0/W$ makes a run about $32$ times cheaper, so the convergence test is also a test of how coarse one can afford to be.

V\,\frac{d_0}{D}\Big|_{d_0/W=0.277}\;\overset{?}{=}\;V\,\frac{d_0}{D}\Big|_{d_0/W=0.544},\qquad \frac{c_s(x)}{c_l^0}\;\overset{?}{=}\;k\Bigl[1-(1-k)\frac{d_0}{\rho}\Bigr]

The two checks behind Figs. 1 and 2: agreement between interface widths, and agreement of the solid concentration with local equilibrium at the measured tip radius.

\text{cost}\propto\Bigl(\frac{d_0}{W}\Bigr)^{5}:\qquad \frac{\text{cost}(0.544)}{\text{cost}(0.277)}\approx2^5=32

Karma’s estimate of how the cost of a run grows with the ratio of capillary length to interface width.

Source: A. Karma

What you have to choose yourself

The letter does not give the size of the box, the boundary conditions, where the seed sits, or how the tip speed and radius are measured. Our runs grow one quadrant of the crystal from a seed in the corner with mirror edges, on $128\,W$ square boxes, so the box spans $462\,d_0$ at the coarser ratio. The tip is located where $\phi$ crosses zero on the axis, its radius comes from a parabola through the zero contour on the four rows nearest the axis, and speeds are averaged over $5000$ to $6000\,d_0^2/D$ .

Fig. 1 and Fig. 2 give values only as curves, so every reference number below is our reading by eye, good to about $0.001$ in speed and $0.005$ in concentration. The solute plateau is read between $150$ and $400\,d_0$ , where the curves are flat.

Source: A. Karma

Worked example: reproducing Figs. 1 and 2 at the coarser ratio

Problem. Run both models at $d_0/W=0.277$ on Karma’s grid and compare the plateau speed and the solid concentration with his figures.

Present model. The tip speed is $0.0180$ , against $\approx0.018$ in Fig. 1. The solid plateau is $0.1447\,c_l^0$ , against $\approx0.145$ in Fig. 2. The tip radius is $21.3\,d_0$ , for which local equilibrium predicts $0.144$ , so the run is self-consistent to $0.5\%$ .

Standard model. The tip runs at $0.0244$ , against $\approx0.025$ . The solid holds $0.2247\,c_l^0$ , against $\approx0.225$ , with a peak of $0.3185$ at $50\,d_0$ against $\approx0.315$ near $50\,d_0$ . With a tip radius of $16.9\,d_0$ , local equilibrium predicts $0.142$ , so this solid holds $58\%$ too much.

Finer ratio, coarser grid. We ran $d_0/W=0.544$ only on a grid of $\Delta x=0.8$ , twice Karma’s spacing. The standard solid plateau is $0.181$ against $\approx0.183$ , and its peak $0.262$ at $30\,d_0$ against $\approx0.265$ near $27\,d_0$ . The present model gives $0.0166$ for the speed, $8\%$ below the figure; on this grid the speed is not converged.

Answer. At $d_0/W=0.277$ on Karma’s grid both models reproduce the figure readings within the reading error. The present model agrees with local equilibrium, and the standard one leaves too much solute in the solid.

Source: A. Karma

What this reproduction does not show

These are single deterministic runs, so there is no seed scatter to quote; the spreads in the benchmark file are fluctuations over the averaging window. The references are readings by eye, not published numbers. We did not rerun the finer ratio on Karma’s grid, so we have not reproduced his convergence test itself, only the coarser half of it plus a grid-limited look at the finer half.

The Lab runs the coarser ratio on a grid of $\Delta x=0.8$ to stay interactive. There the tip runs at $0.0161$ , $11\%$ slower than on Karma’s grid, while the solid still holds $0.1446\,c_l^0$ . Solute partition survives the coarse grid; tip selection does not. Nothing here is compared with an experiment, and we have not rederived the thin-interface asymptotics, which Echebarria and co-workers give in full. The runs are two-dimensional, isothermal and noise-free.

Source: A. Karma · B. Echebarria

Try it in the Lab

Hold the Lab against Fig. 1

Choose A four-armed dendrite and let it grow. Predict the tip-speed readout late in the run from Fig. 1, then from the route benchmark for the Lab’s coarser grid.
Switch the interface model to Standard, no anti-trapping. Predict where the tip speed and solute readouts go relative to Karma’s standard-model curves.
Explain why a converged model must give the same curves at two interface widths, and why the Lab’s coarse grid can slow the tip while leaving the solute readout almost unchanged.

Exercises with explained answers

Explain Karma’s estimate that the cost of a run grows as $(d_0/W)^5$ at fixed grid spacing and time step in units of $W$ and $\tau$ .
Show the answer
The box is fixed in units of $d_0$ , so its side in grid cells grows as $d_0/W$ and the number of cells in two dimensions as $(d_0/W)^2$ . The time unit $d_0^2/D$ in units of $\tau$ is $(d_0/W)^2/(a_2a_1W/d_0)$ , which grows as $(d_0/W)^3$ , and so does the number of steps. The product grows as $(d_0/W)^5$ .
How much cheaper is the Lab’s grid, $\Delta x=0.8$ and $\Delta t=0.032$ , than Karma’s for the same box and time, and what does it cost in accuracy?
Show the answer
Four times fewer cells and four times fewer steps, so about $16$ times cheaper. The tip speed drops by $11\%$ to $0.0161$ ; the solid concentration stays at $0.145$ .
Two curves superimpose at two interface widths. What does that prove about the model, and what does it not prove?
Show the answer
It shows that the results no longer depend on the interface width, which is what the thin-interface limit promises. It does not show that they agree with experiment, nor that they are free of errors that do not depend on the width, such as the grid spacing in units of the capillary length, the box size or the way anisotropy is discretised. That is why the local-equilibrium check, which uses an independent prediction, matters.

Sources and what we recomputed

A. Karma, Phase-field formulation for quantitative modeling of alloy solidification, arXiv:cond-mat/0103289v1 (2001)
Version 1, full text. Eqs. (16)–(18) and (22): the isothermal dilute-alloy model with the anti-trapping current. Eqs. (19)–(20): the equilibrium profile and the choice of $q(\phi)$ that removes surface diffusion. The parameter paragraph: $\Delta x=0.4$ , $\Delta t=0.008$ , $k=0.15$ , $\Omega=0.55$ , $\epsilon_4=0.02$ , $a_1=0.8839$ , $a_2=0.6267$ , a seed of radius $22\,d_0$ . Fig. 1: tip velocity against time. Fig. 2: solute in the solid along the dendrite axis.
B. Echebarria, R. Folch, A. Karma and M. Plapp, Quantitative phase-field model of alloy solidification, arXiv:cond-mat/0404164v2 (2004)
Version 2. Section I B, Eq. (2): with a mesoscopic interface thickness, solute trapping, surface diffusion and interface stretching all grow in proportion to $W$ . Section I D: the anti-trapping current as a non-variational fix. Section I E, Eqs. (7)–(13): the full model, the current of Eq. (10) with $a=1/(2\sqrt2)$ , and the relations $d_0=a_1W/\lambda$ and $\beta$ of Eq. (13). Section II A, Eqs. (17)–(21): the sharp-interface problem for isothermal growth, with the Stefan condition and the Gibbs–Thomson relation. Section IV gives the thin-interface analysis; it was skimmed, not rederived.