On the role of confinement on solidification in pure materials and binary alloys
ABSTRACT We use a phase-field model to study the effect of confinement on dendritic growth, in a pure material solidifying in an undercooled melt, and in the directional solidification of a dilute binary alloy. Specifically, we observe the effect of varying the vertical domain extent ($\delta$) on tip selection, by quantifying the dendrite tip velocity and curvature as a function of $\delta$, and other process parameters. As $\delta$ decreases, we find that the operating state of the dendrite tips becomes significantly affected by the presence of finite boundaries. For particular boundary conditions, we observe a switching of the growth state from 3-D to 2-D at very small $\delta$, in both the pure material and alloy. We demonstrate that results from the alloy model compare favorably with those from an experimental study investigating this effect.
arXiv:cond-mat/0606388v1 [cond-mat.mtrl-sci] 14 Jun 2006
On the role of confinement on solidification in pure materials and binary alloys
B. P. ATHREYA†, J. A. DANTZIG†, S. LIU‡, and R. TRIVEDI‡
†Department of Mechanical and Industrial Engineering,
University of Illinois, Urbana, IL 61801.
‡Department of Materials Science and Engineering,
Iowa State University, Ames, IA 50011.
We use a phase-field model to study the effect of confinement on dendritic growth, in a pure
material solidifying in an undercooled melt, and in the directional solidification of a dilute binary
alloy. Specifically, we observe the effect of varying the vertical domain extent (δ) on tip selection, by
quantifying the dendrite tip velocity and curvature as a function of δ, and other process parameters.
As δ decreases, we find that the operating state of the dendrite tips becomes significantly affected
by the presence of finite boundaries. For particular boundary conditions, we observe a switching
of the growth state from 3-D to 2-D at very small δ, in both the pure material and alloy. We
demonstrate that results from the alloy model compare favorably with those from an experimental
study investigating this effect.
Keywords: Dendrites, Phase field, Adaptive grid, Confined growth
Dendrites are one of the basic microstructural patterns seen in solidified metals. The mechanical behavior of the
solidified product is often decided by the length scales set by these patterns. Study of dendritic growth is therefore
motivated by the need to predict these length scales. The fundamental quantities that completely describe the
growth of a dendrite at steady state under a given set of external conditions are its tip velocity and radius, which
together define the so called “operating state”. Despite considerable advances in the understanding of solidification
science, discrepancies still arise when one attempts to compare theoretical predictions of dendrite operating states
with experimental observations. We find that some of these discrepancies derive from differences between the ideal
conditions assumed in theoretical treatments, and those experienced by materials under actual experimental situations.
The first theoretical treatment of the “free” dendrite growth problem was presented by Ivantsov . He considered
a pure dendrite, modeled as a paraboloid of revolution with tip curvature ρtip, growing into an infinite undercooled
melt with temperature T → T∞far from the advancing tip. The dendrite was assumed to be isothermal at the melting
temperature Tm, and to be growing along its axis at constant velocity Vtipin a shape-preserving way. Ivantsov found a
solution to the thermal transport problem, in which the dimensionless undercooling ∆ = (Tm−T∞)/(Lf/cp) = I(Pe)
where Pe = ρtipVtip/2D is the P´ eclet number, D is the thermal diffusivity, and I is the Ivantsov function. The
temperature has been scaled by the characteristic temperature Lf/cp, with Lf being the latent heat of fusion and cp
the specific heat.
This solution presented a conundrum, because it showed that the transport problem alone did not uniquely specify
the operating state of the dendrite, i.e., the single combination of ρtipand Vtipobserved in experiments. Additional
considerations, such as the effect of curvature on the melting point , stability [3, 4] and eventually the anisotropy of
the surface tension [5, 6, 7], led to a second condition σ∗= 2d0D/ρ2
d0is the capillary length. The combination of Ivantsov’s solution (modified for surface tension and its anisotropy) and
the condition σ∗is constant gives a unique operating state. Numerical simulations using the phase-field method  at
large values of ∆, have found agreement with the predictions of this body of work, known as microscopic solvability
Glicksman and co-workers developed experimental techniques for studying the solidification of pure materials, with
the objective of observing the operating state. They performed experiments with phosphorous , and transparent
analog alloys like succinonitrile (SCN)  and pivalic acid (PVA) . The results of these careful experiments found
some areas of agreement with microscopic solvability theory, in particular, the value of σ∗was found to be constant,
but the operating combination of ρtipand Vtipdid not agree. Provatas, et al. were able to explain this discrepancy by
showing that for the low undercooling conditions found in the experiments, interaction between neighboring dendrite
branches [12, 13] affected the operating state.
Experiments have also been performed to examine the role of superimposed fluid flow on dendritic growth. Gill
and coworkers [14, 15] used SCN in a special cylindrical chamber with a bellows to effect fluid flow. Bouissou and
Pelc´ e  performed experiments with a flowing alloy of PVA and a seed confined between microscope slides. Saville
and Beaghton  presented a theoretical analysis which extended Ivantsov’s solution to consider the superimposed
flow. Jeong et al.  performed phase-field simulations of these experiments, and once again found discrepancies
tipVtipwhere σ∗is called the selection constant, and
with theory. They conjectured that the differences arose because of the effect of finite containers in the experiments,
leading to boundary conditions which differed from the assumptions of infinite media used in the theory.
Dendrite tip theories for constrained growth, such as directional solidification of dilute binary alloys between
microscope slides, have been developed by Trivedi  and, Kurz and Fisher . They have shown that a relationship
of the form ρ2
tipVtip= constant, should hold for constrained growth just as in free dendrite growth. Early experiments
by Somboonsuk et al.  in samples with slide separation greater than 150 µm have shown excellent agreement with
this theory. However, in recent studies Liu et al.  have demonstrated that experimental results start to deviate
significantly from theory when the slide separation approaches the scale of the primary dendrite spacing.
In this article, we systematically study the role of confinement on dendritic growth. Since every experiment is
performed in a finite container, we feel that this effect cannot be ignored. The first two authors have previously
reported a study on confined growth in pure materials .
solidified binary alloys. For purposes of continuity and completeness, we will again present results from our study on
pure materials. For the binary alloy, recent experimental data from Liu et al., will provide us with an avenue for
testing our numerical predictions.
Here we extend our investigations to directionally
A. Phase field model, adaptive grids and numerical methods
The objective in a general solidification problem is to solve the equations governing thermal and solute transport,
subject to boundary conditions on the solid-liquid interface (moving boundary) and elsewhere. If melt convection is
to be modeled, one needs to solve the momentum equations for fluid flow simultaneously with the above transport
equations. Imposing the interface boundary conditions upon discretizing the governing equations poses a difficulty
however, since the interface, as it evolves, will not in general align itself with a fixed set of mesh points.
The phase-field method eliminates the sharp liquid-solid boundary by introducing evolution equations for a continu-
ous order parameter φ ∈ [−1,1], where φ = −1,+1,0 corresponds to liquid, solid and interface respectively. Thus, the
arduous task of solving the transport equations separately in liquid and solid domains while simultaneously satisfying
boundary conditions on arbitrarily shaped interfaces, is replaced by that of solving a system of coupled differential
equations; one for the evolution of φ and one for each of the transport variables (temperature, concentration and ve-
locity). Phase-field modeling has been an active area of research in the past decade, and we refer the interested reader
to original work by Langer , Karma and Rappel , and Beckermann et al.  for derivations of the phase-field
equations and selection of phase-field parameters ensuring convergence to the original sharp interface problem.
The phase-field model introduces a parameter W0 that connotes the finite width of the now ‘smeared’ interface.
Karma and Rappel  showed that the model converges to the sharp interface equations when p = W0Vtip/D ≪ 1,
where D is the thermal or concentration diffusion coefficient, and Vtip is the nominal tip velocity of the dendrite.
Resolving the interface on a discrete mesh requires that the mesh spacing ∆x ∼ W0, while demanding that the
diffusion field not interact with the boundaries leads to the domain size LB≫ D/Vtip. Satisfying these requirements
causes calculations on regular meshes to quickly reach the limit of available computing resources. For example, if we
choose p = 0.01, fix LBVtip/D = 10, and enforce ∆x = W0, then we find that the number of grid points per dimension
on a regular mesh should be at least LB/W0= 1000. This makes computations challenging on regular meshes even
in 2-D, while 3-D computations may not be practical at all, depending on available computing power.
We have mitigated this problem successfully by solving the equations on an adaptive finite element mesh [13, 18].
In three dimensions, we use eight-noded trilinear brick elements stored using an octree data structure. A local
error estimator indicates refinement or coarsening of the mesh, and this permits tracking of the interface as well as
resolution of gradients in the other fields. There are six degrees of freedom at each node (three velocities, pressure,
temperature/concentration and φ), and a typical computation reaches well over one million unknowns. The finest
elements (∆xmin), which are distributed near the interface, now need to be order of W0.
For our studies on pure materials, we have used a finite element discretization of the 3-D phase field model developed
by Karma and Rappel . In order to account for the effects of melt convection we adopt the formulation presented by
Beckermann et al. , who use an averaging method for the flow equations coupled to the phase-field. By appropriate
choice of phase-field parameters we have ensured zero interface kinetics, which is a valid assumption for the range of
undercooling we are concerned with.
For our alloy simulations, we have used a one sided (vanishing solid diffusivity) phase-field model [26, 27, 28], with
a frozen temperature approximation. In a directional solidification arrangement, for certain values of the problem pa-
rameters (particularly when simulating real materials), a considerable amount of time can elapse before the transients
vanish and the solid-liquid interface reaches steady state. In that time, the interface can encounter the end of the
simulation box if the equations are solved in a reference frame that is fixed globally, and if the box is not large enough
to contain the diffusion field. To alleviate this difficulty, we have solved the phase-field equations in a coordinate
frame translating with the pulling speed. This saves some computational expense by allowing us the use of smaller
boxes. We have not investigated the effect of melt convection in our numerical experiments with alloys.
In a recent article, Echebarria et al.  have emphasized that the same choice of phase-field parameters that
produced zero interface kinetics in the pure material cannot also ensure this condition in the alloy model. This
is due to the presence of certain additional terms in the kinetic parameter β, that arise out of accounting for the
discontinuity in the concentration field at the interface in a model with vanishing solid diffusivity. To ensure that
the kinetic coefficient is negligible at the interface, the phase-field relaxation time τ needs to be made temperature
dependent in this region by setting
τ = τ0
1 − (1 − k)z − Vpt
Here τ0is the usual relaxation time, k is the partition coefficient, z is the distance from the interface, Vpis the pulling
velocity, t is time and lT is the thermal length .
We evolve the nonlinear order parameter equation using a Forward-Euler time stepping scheme, while the linear
thermal/solute transport equations are solved using the Crank-Nicholson scheme with a diagonally preconditioned
conjugate gradient solver. The transport equations typically converge in fewer than five iterations per time step. The
3-D flow equations for the pure material are solved using the semi-implicit approximate projection method (SIAPM)
. Details of the above numerical methods and the finite element formulation are omitted here, as they have been
presented elsewhere .
B. Geometry, initial and boundary conditions
Our three dimensional simulation domain is the rectangular parallelepiped illustrated in Fig. 1, with edge lengths
along the x, y and z axes; Lx, Lyand δ respectively. The edges of the box are oriented along ?100? cubic crystallo-
FIG. 1: Simulation Domain. All surfaces are modeled as symmetry planes in pure material simulations, whereas the surfaces
y = 0 and y = Ly are periodic boundaries in the alloy simulations.
Table 1. Pure material simulations
For the pure material, the initial condition is a spherical solid seed with a radius greater than the critical nucleation
radius, centered at the origin depicted in Fig. 1. Because of the inherent symmetry in the growth of the seed, it is
usually sufficient to model one octant in three dimensional space. However, if forced fluid convection is incorporated
along a particular direction, then the solidification rates into and counter to this direction become unequal. For
example, if there is a flow parallel to the x axis, x = 0 is no longer a plane of symmetry. To account for this break in
symmetry, we have to model at least a quadrant of space, with z = 0 and y = 0 as planes of symmetry.
In the simulations with the pure material, the dimensionless thermal field u is subjected to zero flux (∇u · n = 0)
boundary conditions on all surfaces. Fluid flow, when it is included in our study, is imposed as an inlet boundary
condition U∞, normal to the face x = −Lx/2. The velocity field is subjected to symmetry boundary conditions on the
domain walls, and is forced to vanish in the solid (φ = 1) by an appropriate formulation of the momentum equations
(see ). We fix the lateral dimensions of the simulation box (Lx= 512 and Ly= 256 are typical values), and study
the interface evolution as a function of δ, which is varied from 128 to 4. Here, Lx, Lyand δ are in units of the interface
width W0. For very small δ (≤ 8), steady growth conditions are reached relatively quickly for large undercooling.
To save on computational cost in these runs (where ∆xmin= 0.5), we sometimes use shorter lengths for Lxand Ly,
chosen to ensure that the diffusion field does not interact with the ends (in the x and y directions) of the box. For
smaller ∆ however, it typically takes much longer to reach steady conditions, and when melt convection is included,
it can take impractically long CPU times to get converged results. For these cases, we terminate our runs when the
tip radius/velocity versus time curves start to even out. Fortunately, it turns out that the behavior we are interested
in appears for combinations of δ and ∆ where steady state conditions are always achieved.
In the alloy simulations, the initial condition is a planar interface at x = X0, perturbed by randomly spaced finite
amplitude fluctuations. The box in these simulations is taken to represent the shallow channel between microscope
slides where directional solidification conditions are imposed, viz. a fixed thermal gradient moving at a constant speed
Vp. Once again, in this arrangement we study the influence of the depth of the channel δ (or equivalently the sample
film thickness), on interface morphology. To minimize the diffusion field’s interaction with the lateral boundaries, Lx
and Lyare chosen to be relatively large (∼ 256). We enforce zero flux boundary conditions on the concentration field,
on the surfaces x = Lx/2 = −Lx/2 and z = 0 = δ, while periodic boundary conditions are imposed on the boundaries
y = 0 and y = Ly. The rationale behind periodic boundary conditions is to be able to simulate an infinite domain in
Unless otherwise stated, on each boundary, we employ the same type of boundary condition on the phase-field
variable φ, as we do on the transport variable. Where ∇φ · n = 0, the material “wets” the boundaries, and the
corresponding contact angle is 90o. S´ emoroz et al. have previously used this technique to capture wetting of solid
surfaces, with a two-dimensional phase-field model for binary alloys . We also show a calculation with φ = −1
on the boundaries, which is equivalent to making the material “non-wetting” (contact angle = 0o). The real contact
condition probably lies somewhere in between these two extremes.
III.EFFECT OF SMALL δ IN A PURE MATERIAL
In this section, we report the effect of changing δ on the tip of a pure material dendrite, evolving along the negative
x axis (upstream direction when flow is present). We simulated the cases shown in Table 1. We used a fixed value
for the four-fold anisotropy in all our simulations (ǫ4= 0.05). We have not corrected for grid anisotropy  in these
calculations, but work at a grid spacing where its effect is known to be small .
We use the following values for parameters in our calculations: interface width W0= 1, time scale for interface kinetics
τ0= 1, coupling constant λ = 6.383, thermal diffusivity D = 4, capillary length d0= 0.1385W0(which leads to zero
interface kinetics), and Prandtl number Pr = 23.1, where W0, τ0and D are in dimensionless units (see ).
A.“Wetting” boundary conditions, ∇φ · n = 0
To make ideas more concrete, we choose cases 3 and 4 as a representative subset of our computations and present
detailed analyses on those runs. Figures 2 and 3 show the upstream dendrite’s tip velocity and radius respectively, as
functions of the box height δ.
FIG. 2: Tip velocity vs. δ, corresponding to cases 3 and 4 in Table 1.
In these plots, δ and ρtipwere made dimensionless by scaling them with d0, and Vtipis scaled by D/d0. We compute
the tip radii ρxzand ρxyalong two principal planes using the method of Jeong et al. , and estimate the mean tip
radius by the formula
We can see from Fig. 2 that for large values of δ, the tip velocity remains relatively unaffected by the box height.
However, as we go to very small heights Vtipdecreases quite dramatically. As δ is decreased, there is also a gradual
decrease in the tip radius ρtip. Clearly enough, box height has a pronounced effect on tip dynamics. Fluid flow
induces a parallel shift in these curves. The dendrite tip velocity increases uniformly in the presence of flow . On
the other hand, the tip radius is lower than the case with pure diffusion.
The observed trends can be explained as follows. As long as δ is sufficiently large, the thermal field enveloping
the dendrite will interact with the upper boundary at a distance that is relatively far behind the tip. In particular,
the thickness of the thermal boundary layer near the tip remains unaffected by this interaction. However, as δ is
decreased, this thickness can grow quite rapidly. We illustrate this effect by examining the temperature profile in the
x-z plane, as shown in Fig. 4. It is evident that the temperature contours are more spread out in Fig. 4(b) where
δ = 8, compared to those in 4(a), where δ = 64. The increased boundary layer thickness, decreases the thermal
gradient into the liquid at the liquid-solid interface, which in turn retards the growth rate as a direct consequence
of the Stefan condition. Due to the zero flux boundary condition on the plane z = δ, further reduction in δ makes
heat transfer in the vertical direction almost completely ineffective. Tip curvature in the x−z plane vanishes and the
dendrite switches morphology from 3-D to 2-D. We note that once the dendrite goes 2-D, ρtip= ρxy.
An interesting result here is the 3-D to 2-D transition. We have performed tests with finer meshes (more elements in
the vertical direction) to ensure that it is not simply an artifact of poor grid resolution. We believe this phenomenon