# 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.

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**ABSTRACT:**With a few exceptions, phase-field simulations of dendritic growth in cubic materials have been modeled using simple expressions for the interfacial energy anisotropy and with strong anisotropy. However, recent experimental results show that the Dendrite Orientation Transition (DOT) observed in Al-Zn alloys by Gonzales and Rappaz [Met. Mat. Trans. A37 (2006) 2797] occurs at weak anisotropy, and modeling these results requires at least two anisotropy parameters. In the present work, we solve the corresponding phase-field model on an adaptive grid, after measuring and compensating for the grid anisotropy. A systematic scan of equiaxed growth simulations was performed in the range of the anisotropy parameter space where the transition is expected. We find separate domains of existence of ⟨100⟩ and ⟨110⟩ dendrites, similar to those previously reported by Haxhimali et al. [Nat. Mat. 5 (2006) 660] for pure materials. In the so-called hyperbranched regime, lying between the ⟨100⟩ and ⟨110⟩ regions, we observe a competition between ⟨100⟩ and ⟨110⟩ growth directions, but no seaweed structures. Directional solidification simulations showed the stabilizing effect of the thermal gradient on the twofold splitting of ⟨110⟩ dendrites, and the importance of the choice of anisotropy parameters. We also found a strong dependence between the orientation of the crystal axes with respect to the thermal gradient and the actual growth direction. Finally, 3-dimensional seaweed microstructures were modeled for the first time, demonstrating that this pattern is a result of not only the values of anisotropy parameters, but also a consequence of directional solidification.IOP Conference Series Materials Science and Engineering 01/2012; 33(1). - SourceAvailable from: Mathis Plapp[Show abstract] [Hide abstract]

**ABSTRACT:**The structure and dynamics of cellular solidification fronts produced during the directional solidification of dilute binary alloys are studied by phase-field simulations. A quantitative phase-field model in conjunction with a multi-scale simulation algorithm allows us to simulate arrays with 10-40 cells in three dimensions on time scales that are long enough to allow for a significant reorganization of the array. We analyze the geometry of the complex two-phase structure (mushy zone) and extract the fraction of solid and the connectivity of the two phases as a function of depth. We find a transition from stable arrays at high values of the crystalline anisotropy to unsteady arrays at low anisotropy that continuously exhibit tip splitting and cell elimination events.Journal of Crystal Growth 01/2014; · 1.55 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Three-dimensional phase-field simulations are employed to investigate rod-type eutectic growth morphologies in confined geometry. Distinct steady-state solutions are found to depend on this confinement effect with the rod array basis vectors and their included angle (α) changing to accommodate the geometrical constraint. Specific morphologies are observed, including rods of circular cross sections, rods of distorted (elliptical) cross sections, rods of peanut-shaped cross-sections, and lamellar structures. The results show that, for a fixed value of α > 10°, the usual (triangular) arrays of circular rods are stable in a broad range of spacings, with a transition to the peanut-shaped cross sectioned rods occurring at large spacings (above 1.5 times the minimum undercooling spacing λ(m)), and the advent of rod eliminations at low spacings. Furthermore, a transition from rod to lamellar structures is observed for α < 10° for the phase fraction of 10.5% used in the present paper.Physical Review E 07/2011; 84(1 Pt 1):011614. · 2.31 Impact Factor

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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

I. INTRODUCTION

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 [1]. 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 [2], 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 [8] at

large values of ∆, have found agreement with the predictions of this body of work, known as microscopic solvability

theory.

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 [9], and transparent

analog alloys like succinonitrile (SCN) [10] and pivalic acid (PVA) [11]. 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 [16] performed experiments with a flowing alloy of PVA and a seed confined between microscope slides. Saville

and Beaghton [17] presented a theoretical analysis which extended Ivantsov’s solution to consider the superimposed

flow. Jeong et al. [18] performed phase-field simulations of these experiments, and once again found discrepancies

tipVtipwhere σ∗is called the selection constant, and

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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 [19] and, Kurz and Fisher [20]. 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. [21] in samples with slide separation greater than 150 µm have shown excellent agreement with

this theory. However, in recent studies Liu et al. [22] 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 [23].

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.[22], will provide us with an avenue for

testing our numerical predictions.

Here we extend our investigations to directionally

II.MODELING

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 [24], Karma and Rappel [8], and Beckermann et al. [25] 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 [8] 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 [8]. In order to account for the effects of melt convection we adopt the formulation presented by

Beckermann et al. [25], 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

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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. [28] 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

lT

?

.(1)

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 [26].

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)

[29]. Details of the above numerical methods and the finite element formulation are omitted here, as they have been

presented elsewhere [18].

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-

graphic directions.

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.

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Table 1. Pure material simulations

Case∆U∞

δ

1(a)-(f)

2(a)-(f)

3(a)-(e)

4(a)-(f)

5(a)-(d)

6(a)-(e)

0.55

0.55

0.25

0.25

0.15

0.15

0

5

0

5

0

5

128,64,32,16,8,4

128,64,32,16,8,4

128,64,32,16,8

128,64,32,16,8,4

128,64,32,16

128,64,32,16,8

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 [25]). 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

y.

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 [30]. 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 [8] in these

calculations, but work at a grid spacing where its effect is known to be small [13].

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 [8]).

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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. [18], and estimate the mean tip

radius by the formula

ρtip= 2

?

1

ρxy

+

1

ρxz

?−1

. (2)

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 [18]. 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

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FIG. 3: Tip radius vs. δ, corresponding to cases 3 and 4 in Table 1.

(a) δ = 64 corresponding to Case 4(b). u ranges from A ≡ -0.24 to L ≡ 0.008

(b) δ = 8, corresponding to Case 4(e). u ranges from A ≡ -0.24 to L ≡ 0.008

FIG. 4: Temperature contours for two different box heights when ∆ = 0.25 and U∞ = 5.0. In each case, the letter X symbolizes

the dendrite outline (bold contour), under steady growth conditions. Contours are plotted in intervals of 0.025. Notice that

the contours near the dendrite tip are more spread out when δ is smaller.

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(a) δ = 16, 3-D dendrite(b) δ = 4, 2-D dendrite

FIG. 5: 3-D to 2-D dendritic transition at small δ. The shaded surface is the dendrite (φ = 0). For these runs ∆ = 0.55 and

U∞ = 0.

to be a consequence of the ∇φ · n = 0 boundary condition on the upper boundary. The only way for the solid-liquid

interface to match this boundary condition at small δ is for the curvature in the x-z plane to vanish. This is illustrated

in Fig. 5.

It is also interesting to observe the effect that melt flow has on the interaction between the tip and the boundary.

We find that melt convection reduces the value of δ where the tip velocity deviates from its nominal value. A

straightforward explanation for this effect is that advection increases the rate of heat transport from the upstream

dendrite arm. This increases the growth rate (hence Vtip), while compressing the boundary layer, whose thickness

scales as D/Vtip. Thus the dendrite remains three dimensional for a smaller value of δ than was previously possible,

with convection absent. A more negative value of the undercooling also has qualitatively the same effect on the

strength of the tip-boundary interaction, since Vtipagain increases in this case. Fig. 6 summarizes these observations

succinctly. Both melt convection and larger undercooling, cause points on respective curves whereupon interaction

effects become important, to shift to the left.

Using a graph such as Fig. 6, one can derive a semi-quantitative estimate as to when the operating state becomes

affected by the finite height of the container. If one assumes that the tip velocity at δ = 128 for each case is

approximately the tip velocity of a dendrite growing under identical conditions in an infinite domain, it is possible

to quantify the influence on the operating state in terms of a percentage deviation in the true tip velocity from this

nominal value. If, for example, we consider deviations of the order of 3% to constitute a change in the operating state,

a least squares fit to these cut-off points on the respective curves in Fig. 6 yields the criterion

δ ≥ 0.7912

?D

Vtip

?0.9242

. (3)

If this condition is not satisfied, then it is likely that the operating state is being influenced by the boundary. A simple

condition such as the one in Eqn. (3) may be used as a rule of thumb in determining if experimental studies on free

dendrite growth, in geometries similar to ours, are free from contamination. In fact, one may have intuitively guessed

a condition of the type δ ≥ α (D/Vtip) (where α is some constant) to apply based on physical arguments alone, and

Eqn. (3) supports this conjecture.

B. Non-wetting boundary conditions

To underscore the importance of phase field boundary conditions in selecting a particular growth state, we present

results from another simulation with ∆ = 0.55, U∞ = 0 and δ = 4. This time we impose φ = −1 on the upper

boundary, which corresponds to a physical situation where the solidifying material is not allowed to wet the surface.

A three dimensional surface plot of a steadily growing dendrite is shown for this case in Fig. 7.

As expected, the dendrite does not adhere to the top surface at all. We conclude that this will be the case for

any δ if Dirichlet conditions of this nature are imposed. It is evident then, that the 3D-2D transition that we saw

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FIG. 6: Tip velocity as a function of δ for different undercooling and flow conditions. A weak power law relationship emerges

between the δ below which interaction effects are strong, and diffusion length D/Vtip.

FIG. 7: Three dimensional dendritic growth for ∆ = 0.55, U∞ = 0 and δ = 4, with φ = −1 on the upper boundary. Note that

the dendrite does not wet this surface.

previously is a strong function of the boundary conditions imposed on the phase field. It would be interesting to

conduct a detailed investigation of the effect of different boundary conditions on the tip in 3-D. The interested reader

is referred to the article by S´ emoroz et al. [30], which discusses the influence of different contact angles on a dendrite’s

tip velocity, in two-dimensional thin film solidification.

IV.EFFECT OF SMALL δ IN A DIRECTIONALLY SOLIDIFIED ALLOY

In this section, we describe results from our simulations on a directionally solidified alloy, and make comparisons

with the experimental data of Liu et al. [22]. These simulations were conducted for the range of pulling speeds and

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Table 2. Alloy simulations

CaseVp

δ

1

2

3

4

0.8

1.0

1.5

2.0

64,32,16,8,4

64,32,16,8,4

64,32,16,8,4

64,32,16,8,4

Table 3. Physical properties of a SCN-Salol alloy system

|m| (Liquidus slope)

D (Diffusion coefficient)

Γ (Gibbs-Thomson coefficient)

k (Partition coefficient)

G (Thermal gradient)

d0 (Capillary length)

lT (Thermal length)

0.7 K/wt. %

8 × 10−10m2/s

0.64 × 10−7K m

0.2

4 K/mm

3.265 × 10−8m

4.9 × 10−4m

channel depths shown in Table 2. The depth δ was varied from 64 to 4, and in each case, the simulation was continued

until the interface became stationary in the moving frame. From the stable array of cells formed, a few were chosen

as representative of the array, their tip radii were extracted in the two principal planes using a least squares quadratic

polynomial fit, and the mean tip radius of each cell was computed using Eqn. (2). An average over these values was

taken to be the mean tip radius of the interface. We note that this process required some approximation, especially

in cases where we found steadily growing cells of disparate sizes that would have made such averaging inappropriate.

In such cases, our usual approach was to choose cells farthest from the boundaries, and where even this failed to

provide clear-cut choices, smaller cells were chosen because of their lesser likelihood to split. In a majority of our runs

however, the choices were unambiguous. For certain values of δ we found interface evolution to occur in a way that

cells would creep on either the top or bottom surfaces. In those cases, the phase-field boundary conditions on the

respective surfaces allowed us to treat them as symmetry planes for the purpose of calculating tip curvature.

A.Selection of simulation parameters

The following values for the lateral dimensions were seen to yield satisfactory results. Lx= Ly= 256 when δ ≤ 16,

and Lx= 256, Ly= 128, otherwise. We chose our simulation parameters to keep computations tractable. It took

about 90 hours of CPU time on a 3.1 GHz processor to simulate a typical directional solidification experiment for a

chosen set of phase-field parameters on a mesh with about 170 000 elements (δ = 64). The interface required about

250 dimensionless time units to reach steady state in this case. As noted earlier, the use of a moving reference frame

allowed us to cut substantial costs associated with the need for larger domains to prevent the diffusion field from

running out of the domain.

We did not attempt to model a real material in this study as this caused our simulations to become considerably more

expensive. To illustrate this, consider a SCN-Salol system having the properties listed in Table 3. The conditions in a

directional solidification experiment are completely described by the following two dimensionless control parameters,

M = d0/lT = 6.66 × 10−5and S = Vpd0/D = 2.04 × 10−4, where d0is the capillary length, lT is the thermal length

and D is the solute diffusivity in the liquid phase; for a pulling velocity of Vp = 5µm/s. To get converged results

with the phase-field model we require that the solution become independent of the parameter ǫ = W0/d0. After

ensuring vanishing interface kinetics, the following relationships involving the phase-field parameters are realized:

Dτ0/W2

arise in the phase-field formulation. [28]

Echebarria et al. [28] have shown that mesh converged results can be obtained with ǫ as large as 50. Setting

ǫ = 50, gives us an under-determined system of three equations with the five unknowns D, Vp, lT, τ0and W0. Making

arbitrary choices for two of these parameters by setting W0= τ0= 1, we obtain D = 27.7, Vp= 0.2825, and lT= 300.

A large value of lT implies Lxneeds to be very large at steady state, even in a moving reference frame, to contain

the diffusion field. To avoid this, if we choose a more tractable value for lT (say 100), and fix τ0= 1, we now get

0= a1a2ǫ, Vpτ0/W0= Sa1a2ǫ2, and lT/W0= 1/(ǫM). Here, a1= 0.8839 and a2= 0.6267, are constants that

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W0 = 0.3333, D = 3.077 and Vp = 0.094. Thus, the smallest element in our mesh needs to be at the very least

∆x = 0.3333. Stability considerations now place a severe restriction on the size of the time step (∆t) needed for

solving the phase-field equations by the Forward-Euler method, as ∆t ∼ ∆x2. Since it is clearly impossible to choose

both lT and W0independently, calculations involving real materials are typically more expensive.

We choose instead a more computationally favorable set of dimensionless parameters M and S for our study. To

achieve our primary objective, which is to study the effect of δ on interface morphology, we anticipate, and in the

following paragraphs demonstrate, that this hypothetical treatment will not obscure any physics. The parameters

used in our study are: τ0= W0= 1, D = 20, k = 0.8, ǫ4= 0.05, and lT = |m|(1 − k)C∞/kG = 50; where m is the

liquidus slope, G is the imposed thermal gradient, and C∞ is the far field solute concentration. The condition for

negligible interface kinetics gives d0= 0.0277 and therefore ǫ = 36.1, which is sufficiently small to ensure convergence.

The size of the smallest element in our adaptive mesh is ∆x = 1, when δ ≥ 8, and ∆x = 0.5, when δ = 4, while

∆t = 0.005 is the size of the time-step. For these parameter choices, the dimensionless control parameters work out

to be M = 5.54 × 10−4and S ∼ 1.385 × 10−3.

B.Interface morphology and comparison with experiments

In order to test the model, we initially performed a set of runs to verify the Mullins and Sekerka stability limit of

a planar interface [31], perturbed by small sinusoidal perturbations. We found that the model captures the stability

spectrum correctly. Having convinced ourselves that this fundamental requirement was met, we proceeded with our

study. Comparisons with the experimental data in this section offer a better validation of the model.

Fig. 8 shows the computed interface morphology at different values of δ. For small values of pulling speed (Vp≤ 2),

the steady state consists of a stationary array of cells as in Fig. 8. However, unlike the cells observed in experiments,

that are usually characterized by blunt tips, these appear to have sharper and better defined tips, giving the impression

of dendrites. It is conceivable that ignoring thermal noise in our calculations is responsible for the absence of side-

branches on these structures, that are typical of dendrites. At large values of δ, we find that the tip radii of these

cells, measured on the two principal planes, are almost identical. However, as δ decreases, the in plane radii diverge

from one another. In particular, the radius in the x-z plane becomes significantly smaller, and cross sections of the

cells look elliptic. At δ = 4, for small pulling speeds (Vp≤ 1), we get a two-dimensional interface (Fig. 8(c)). The

inter-cellular spacing also increases as δ is decreased.

As pulling velocity is increased, the morphology becomes finer, with sharper and more tightly packed cells. This

behavior is consistent with that seen of both cellular and dendritic arrays in directional solidification experiments

[21, 22, 32], where the primary spacing decreases with Vp.

For the set of phase-field parameters we have chosen, if Vp ≥ 2.5, the interface does not reach steady state in a

reasonable amount of CPU time, due to repeated tip-splitting of the cells. Splitting is initiated by oscillations that

appear at the tip and propagate downward along the trunk of the cell. Cell spacing and shapes change very rapidly

in this regime. We did not continue these runs any further to check if steady state is reached eventually. Instead,

we set Vp= 2.0 as an upper bound on the pulling velocity, below which a stationary state was always the outcome.

Fortunately, this still left us with sufficient sample space to conduct our study and make effective comparisons.

To enable plotting of our results on the same graph with the experimental data of Liu et al., which corresponds to

a SCN - 0.7 % wt. Salol system (properties in Table 3), we non-dimensionalized the axes as follows. The abscissa

is the pulling speed (i.e. the tip velocity at steady state) Vtip, scaled by a characteristic velocity D/d0k, while the

ordinate is the tip radius ρtip, scaled by the diffusion length D/Vtip. One may appreciate the fact that the abscissa is

in fact the dimensionless parameter that we had earlier denoted by S, multiplied by k. Fig. 9 shows a comparison of

the data. The open symbols correspond to the experimental data of Liu et al., while the solid symbols correspond to

our calculations. A comparison between our data and theirs holds up surprisingly well. Of special significance are the

following two observations: 1) Although we conducted our simulations at values of S and M that were each about

an order of magnitude off theirs, the two sets of data correlate very well, i.e. appear to collapse on parallel curves

that are not significantly different by way of intercept. This tells us that our choices of parameters for scaling the

axes are appropriate. 2) Since their experimental data correspond to dendritic arrays, the cell-like structures we have

computed are likely branchless dendrites.

In their experiments, Liu et al. note that tip radius data for dendritic arrays agree quite nicely with a relationship

of the form ρ2

tipVtip= C, where C is a constant dependent on d0, k and D, as postulated by theoretical models of

constrained growth [19, 20]. However, when δ is of the order of inter-dendritic spacing λ1, this agreement deteriorates.

This is evident in Fig. 9, where the ρ2

tipVtip = C line is shown as a guide to the eye. The open circles, which are

data for δ = 12.5 µm deviate in both slope and intercept from this line, which passes through the rest of their data,

indicating a breakdown in the relationship. We observe similar trends in our data, viz., the line ρ2

data at δ = 32 and 64 reasonably well, but as δ decreases from 16 to 4, this agreement deteriorates.

tipVtip= C fits our

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(a) δ = 64, Vp= 1.5, three-dimensional cells

(b) δ = 16, Vp= 1.5, three-dimensional cells

(c) δ = 4, Vp= 0.8, two-dimensional cells

FIG. 8: Interface morphology in a directionally solidified alloy. At steady state, stable arrays of three-dimensional cells appear.

Note that at δ = 4, the array comprises of two-dimensional cells.

As in the numerical experiments with the pure material, we find that decreasing δ has a pronounced effect on

interface morphology. Dendritic arrays seen in experiments have a certain structure/periodicity to them, that arises

from underlying crystalline symmetries.For example, in our simulations we observe that the cells constitute a

hexagonal array. When δ is large, away from the boundaries the diffusion field surrounding each cell tip obeys this

symmetry, and the optimal λ1is selected. As δ decreases however, the diffusion field becomes increasingly asymmetric

due to interaction with the boundaries at x = 0 and x = δ. In particular, solute rejection decreases in the vertical

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FIG. 9:

symbols correspond to the phase-field model and the open symbols are experimental data. The line denotes a relationship of

the form ρ2

Comparison of binary alloy simulations with the phase-field model, and experimental data of Liu et al. [22]. Solid

tipVtip = C between tip radius and velocity, where C can be expressed in terms of process parameters.

plane x-z, while increasing in the horizontal plane x-y. Increased solute accumulation between cells in x-y, contributes

to an increase in λ1. However, since Vtipis fixed by the pulling speed, and a certain rate of solute rejection needs to

be maintained, the tips tend to grow sharper as λ1increases. It is precisely this effect that causes the operating state

to deviate from theoretical predictions.

To check for what value of δ our results deviate from theory, let us assume the steadily growing array at δ = 64,

Vp= 1.5 (Fig. 8(a)) to be one in which cells close to the plane z = 32 are “free” from boundary effects. We estimate

the cell spacing in that plane to be λf

1≈ 18, where the superscript ‘f’ denotes “free”. When δ = 16 and Vp= 1.5

(Fig. 8(b)), we notice that our data points start departing from the theoretical prediction, viz. we see a relationship

of the type ρa

what Liu et al. concluded from their experiments. A more precise form of the above criterion can be obtained my

making a careful study of λ1as a function of δ, for different Vp, and obtaining a criterion based on a least squares

fit to the deviation points (as we did with the pure material). Increasing pulling velocity suppresses tip-boundary

interaction by reducing the thickness of the diffusion boundary layer D/Vtip, similar to the effect of melt convection

in pure materials, and should induce a leftward shift in the curves.

tipVtip= C, where a > 2. This suggests that agreement with theory deteriorates as δ ∼ λf

1, which is

V.CONCLUDING REMARKS

We have investigated the role of confinement on solidification in both pure materials and binary alloys. Our

simulations show that, for equi-axed growth in a pure material, the dendrite’s operating state is affected when the

container dimension δ, approaches the scale of the diffusion field D/Vtip near the tip. For directionally solidified

binary alloys, confinement effects become important when δ is of the order of the primary dendrite spacing λ1. Where

applicable, one needs to consider the influence of these interactions when comparing experimental data with theoretical

models that do not account for confinement effects.

It is notable that we were able to make meaningful comparisons with real experimental results using the phase-field

model for the alloy. In particular, the agreement obtained in the trends shown by the dendrite tips for different δ and

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Vpis very encouraging, and is a testament to the power of phase-field modeling. Some ambiguity remains in classifying

the computed microstructure as cells or dendrites. Since our results correlated well with dendrite data, we would like

to think of them as dendrites. Perhaps, incorporating random fluctuations in the phase-field model will resolve this

issue. We also found tip-splitting inducing oscillations above certain values of the pulling speed. We are unsure as to

whether this instability has a physical meaning. In experiments, it is seen that increasing Vpcauses a decrease in λ1

and ρtipfor a steadily growing interface, and our simulations capture this effect, viz. as Vpincreases the cells split in

a manner that produces a more stable configuration with a smaller λ1and ρtip. However, when Vp≥ 2.5, it appears

that an optimal configuration is not possible in our system, given the constraints on its size and boundary conditions.

We speculate that larger domains should allow for more stable cell configurations at higher pulling speeds. This issue

too needs further investigation.

We observed a change in dimensionality of the liquid-solid interface for certain values of δ and Vtip, when zero

flux boundary conditions were imposed on the phase-field variable. There is some experimental evidence of this

phenomenon in the literature. Liu and Kirkaldy [32] reported a 2-D to 3-D transition in their experiments on a

SCN-Salol mixture. In their directional solidification experiments in a cell of fixed height (δ = 28µm), they found

this transition to occur at a driving velocity of 10.8 µm/sec. At a lower driving velocity of 7.6 µm/sec, the dendrites

looked two dimensional. In our analysis of directional solidification, we found at δ = 4, the cells underwent a 2-D to

3-D transition as the pulling velocity was changed from 1 to 1.5. The significance of this result is that through an

appropriate selection of δ and Vp in experiments, it should be possible to obtain almost two dimensional dendritic

arrays in materials that favor wetting. Such experiments will permit more favorable comparisons with 2-D dendrite

growth theories, since finite boundary effects along the z axis cease to impact the growth.

Acknowledgments

The authors gratefully acknowledge support for this work from NASA under Grant NAG 8-1657, and from the

National Science Foundation under Grant DMR 01-21695. We thank Jun-Ho Jeong and Navot Israeli for their

significant contributions in the development and implementation of the adaptive grid algorithm which made this

study possible, and the Computational Science and Engineering program at UIUC for availing us their computing

facilities.

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