PHYSICAL REVIEW B 79, 014204 共2009兲 Phase field approach to heterogeneous crystal nucleation in alloys James A. Warren,1 Tamás Pusztai,2 László Környei,2 and László Gránásy3 1Metallurgy Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA 2Research Institute for Solid State Physics and Optics, P.O. Box 49, H-1525 Budapest, Hungary 3Brunel Centre for Advanced Solidification Technology, Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom 共Received 25 October 2008; published 15 January 2009兲 We extend the phase field model of heterogeneous crystal nucleation developed recently 关L. Gránásy et al., Phys. Rev. Lett. 98, 035703 共2007兲兴 to binary alloys. Three approaches are considered to incorporate foreign walls of tunable wetting properties into phase field simulations: a continuum realization of the classical spherical cap model 共called model A herein兲, a nonclassical approach 共model B兲 that leads to ordering of the liquid at the wall and to the appearance of a surface spinodal, and a nonclassical model 共model C兲 that allows for the appearance of local states at the wall that are accessible in the bulk phases only via thermal fluctuations. We illustrate the potential of the presented phase field methods for describing complex polycrystalline solidification morphologies including the shish-kebab structure, columnar to equiaxed transition, and front-particle interaction in binary alloys. DOI: 10.1103/PhysRevB.79.014204 PACS number共s兲: 64.60.qe, 64.70.D⫺, 82.60.Nh I. INTRODUCTION When one cools a liquid below its melting temperature, it is no longer stable and it will freeze eventually.1 However, the liquid will exist in a metastable state until a nucleation event occurs. In the study of nucleation, a distinction is made between homogeneous and heterogeneous nucleation.1,2 Homogeneous nucleation occurs in an idealized pure material, where the only source of nucleation in an undercooled melt is due to fluctuation phenomena.1,2 On the other hand, heterogeneous nucleation occurs in “impure” materials, where walls or some agent usually particles substantially larger than the atomic scale introduced to the melt 共either intentionally or not兲 facilitate nucleation by reducing the energy barrier to the formation of the stable phase. This reduction occurs when the impurities induce ordering in the liquid3 that enhances the formation of the solid phase. Heterogeneous nucleation is not only a phenomenon of classic importance in materials science but also remains one of continuously growing interest due to the emerging technological interest in nanopatterning techniques and control of related nanoscale processes.4 In spite of its technological importance, heterogeneous nucleation is poorly understood due to difficulties in describing the interaction between the foreign matter and the solidifying melt. In classical theory, the action of the impurity to enhance or suppress the solid phase can be formulated within the language of wetting. That is, given the surface energies for liquid-solid 共␥SL兲, wall-liquid 共␥WL兲, and wall-solid 共␥WS兲 boundaries, we may calculate the contact angle of a solidliquid-wall triple junction 共assuming isotropic surface energies5兲. Using Fig. 1 as a schematic guide 共where the drop is imagined to be solid in liquid and not liquid in gas兲 to determine the contact angle between the solid-liquid surface and the wall 共with the angle subtending the solid material兲, we find a version of the Young-Laplace equation, cos = ␥WL − ␥WS . ␥SL 共1兲 Clearly, within this framework, if = 0 then the surface will be wet with the solid phase and there will be no barrier to 1098-0121/2009/79共1兲/014204共16兲 nucleation. In the case where = the liquid phase is preferred at the interface, and the system behaves as if the particle was not there. Within the framework of the classical “spherical cap” model, the nucleation barrier is simply reduced by the catalytic potency factor f共兲 as follows: Whetero = Whomo f共兲, where f共兲 = 关 − 21 sin共2兲兴 / and f共兲 = 41 关2 – 3 cos共兲 + cos共兲3兴 for two and three dimensions, respectively.1,5 The above argument becomes more complex if the surface energies are anisotropic5 but are not changed in qualitative detail. Wetting of a foreign wall by fluids/crystals has been studied extensively7 including such phenomena as critical wetting and phase transitions at interfaces.8 Various methods have been applied to address these problems such as continuum models9 and atomistic simulations.10 Despite this inventory, recent studies11 addressing heterogeneous crystal nucleation rely almost exclusively on the classical spherical cap model, which assumes mathematically sharp interfaces.1,12 While this approach may quantitatively describe wetting on the macroscale, it loses its applicability2,13 when the size of nuclei is comparable to the interface thick- FIG. 1. Definition of contact angle : glycerin droplet on glass surface 共Ref. 6兲. 014204-1 ©2009 The American Physical Society PHYSICAL REVIEW B 79, 014204 共2009兲 WARREN et al. ness 共the nanometer range, according to experiment and atomistic simulations14兲. Such nanoscale nuclei are essentially “all interface.” Recent investigations show13 that the phase field approach 关共PFT兲; for recent reviews see Refs. 15 and 16兴 can describe such nonclassical nuclei. Indeed, the PFT can quantitatively predict the nucleation barrier for systems 共e.g., hard sphere, Lennard-Jones, and ice water兲 where the necessary input data are available.13,17 We therefore adopt this approach to describe heterogeneous nucleation. Experimentally, the details of the wall-fluid interaction are embedded in more directly accessible quantities such as the contact angle in equilibrium. It is thus desirable to develop a model that describes the wall in such phenomenological terms. To address heterogeneous nucleation within the phase field approach, we need to include foreign walls. Ideally, we may regard the foreign “wall” as a phase with all its chemical and wetting properties known. This is the case in previous studies addressing solidification in eutectic and peritectic systems, where the secondary crystalline phase appears via heterogeneous nucleation on the surface of the first-nucleated primary phase. Nucleation and subsequent growth on such intrinsic walls have been addressed in some depth in previous work.18 More often, however, we do not have such detailed information on foreign walls and we have to satisfy ourselves with knowing only their wetting properties 共e.g., the contact angle兲. It would be, therefore, desirable to work out PFT techniques for such cases. In order to distinguish this case from the fully characterized walls and because of the fact that they can be represented in the PFT by boundary conditions, we are going to term them as external walls. Indeed, as we will see, to achieve this, we have to specify appropriate boundary conditions at the wall represented by a mathematical surface. Previous work along this line incorporates numerical approaches designed to ensure the desired contact angle19 or either fixing the value of the phase field at the wall20 or the normal component of the phase field gradient.16,20–23 Early work in this area addressed only the nonwetting case 共 = 0 corresponding to = 兲20 or the semiwetting case 共 = / 2兲 realized by the no-flux boundary.16,20–22 Recently, however, we have shown that either fixing the normal component of the phase field gradient 共model A兲 or the value of the phase field 共model B兲 appropriately at the wall, one can realize all kinds of contact angles.24 It is appropriate to mention that ideas similar to those presented in our paper24 seemed to be “in the air” in other branches of field-theoretic modeling. For example, a simulation by Jacqmin25 performed for a liquid-liquid interface forming contact angles of = ⫾ / 4 with opposite walls suggests that he might have been aware of model A, although neither a derivation of the model nor its general formulation valid for other contact angles was presented in his paper. In fact, model B had already been used in an earlier study;9 however, for describing the wetting of solid surfaces by fluids, yet not for a structural order parameter. Finally, a few days before our prior paper on this topic26 was published electronically, a similar work had been submitted, which outlines model A for interfaces between two fluids. These techniques have been worked out for single fields and they have yet to be generalized to cases where the structural order parameter is coupled to other fields. Herein, we generalize the approaches described in Ref. 24 for the solidification of binary alloys 共structural order parameter coupled to a concentration field兲. It will be shown that with a specific parabolic approximation of the free-energy surface, the contact angle vs boundary condition relationships described in Ref. 24 remains valid. After developing the model for isotropic binary alloys, we extend the model 共adding noise, grain-boundary effects, and interfacial anisotropy兲 allowing us to perform simulations of heterogeneous nucleation during the growth of a polycrystalline material. II. PHASE FIELD THEORY FOR WETTING AND HETEROGENEOUS NUCLEATION Recently, a rich array of phenomena has been modeled using a phase field theoretic approach that has a fairly simple form 共see the appendixes of Ref. 16兲, F= 冕 冋 dV f共,c兲 + 册 2 2 ⌫ 共ⵜ · R兲 + f ori共ⵜR兲 , 2 共2兲 where f共 , c兲 has the form of a skewed double well, with minima in the two phases at = 0 共liquid兲 and = 1 共solid兲, and the difference in height being controlled by the thermodynamic variables such as temperature T and concentration c. In this model T is assumed uniform. The gradient coefficient sets the interface width; while the form of ⌫, a homogeneous degree one function of its argument, determines the anisotropy. The contribution from the orientation due to grain boundaries is embedded in the local orientation matrix R. In general, R is an SO共3兲 object and thus transforms in a manner consistent with this group. There are a number of equivalent representations of R,27,28 but here we will use a quaternion form from Ref. 29. ⵜ · R rotates the vector ⵜ into the frame of the orientation of the crystal. The function then ⌫ determines the penalty for gradients in this direction. It thus represents the local interface energy anisotropy. As a homogeneous degree one function, ⌫共v兲 can be written as ⌫共v兲 = 兩v兩关1 + 共n兲兴, where n = v / 兩v兩 is the normal to the level sets of , in the coordinate system of the crystal, and thus is the natural extension of the surface normal in classical theory to a phase field model, and  is necessarily a homogeneous degree zero function of its argument. For fourfold symmetry, a common choice for  = ␣共n4x + n4y + nz4兲, where ␣ is a constant. When ⌫共ⵜ兲 = 兩ⵜ兩 共or  = 0兲, the interface energy will be independent of the orientation of the phase boundary. Finally the orientation penalty taken to have the simple form f ori = HTu共兲兩ⵜR兩 + O共兩ⵜR兩2兲, where we use the standard L2 norm of the gradient of the orientation, which is a metric that yields the local misorientation. The function u共兲 is zero in the liquid and increases in the solid, where grain-boundary energies are well defined. In quaternion notation 共where qi are the components of the quaternion representation of the orientation matrix兲, 014204-2 PHYSICAL REVIEW B 79, 014204 共2009兲 PHASE FIELD APPROACH TO HETEROGENEOUS CRYSTAL… 兩ⵜR兩 = 冋兺 册 共ⵜqi兲2 1/2 A. Defining “external” walls i as in Refs. 27 and 29. Until we return to the more complex case of polycrystalline growth later in the paper, we will focus on the isotropic case. We thus drop f ori and assume ⌫共v兲 = 兩v兩, yielding a standard model phase field model of a binary alloy, such as that found by Warren and Boettinger30 and many others,31 with the form F= 冕 冋 册 2 dV f共,c兲 + 兩ⵜ兩2 . 2 冉 冊 2 2 x 共3兲 We should note that Eq. 共3兲 does not include a gradient term in the concentration 共as was found by Cahn8兲; although there is no fundamental reason why such a term should be discounted, only a desire for a minimal model that yields nontrivial behaviors for the chosen variable set. Equations of motion then have the form 冋 There are several mathematical approaches to modeling a three phase system. The first we consider involves treating the inert material as a sharp wall; the walls influence is controlled by the behavior of the alloy abutting the wall. The two most natural choices to consider are either specifying ⵜ · n or on the boundary. We explore both below. In a steady state, the one-dimensional 共1D兲 equations describing the system can be integrated to find 册 f ␦F = − M = M 2ⵜ 2 − , ␦ t 共4兲 c f ␦F = ⵜ · Mc ⵜ = ⵜ · Mc ⵜ , t ␦c c 共5兲 where M and M c are mobilities. As in classical theory, the critical fluctuation 共nucleus兲 represents an extremum of the free energy. Thus it can be found by solving the appropriate Euler-Lagrange 共EL兲 equations. Mass conservation needs to be taken into account here; a constraint that can be enforced by adding ⌳兰c共r兲dV to the free energy, where ⌳ is a Lagrange multiplier, ␦F⬘ = 0, ␦ 共6兲 ␦F⬘ = 0, ␦c 共7兲 where F⬘ = F + ⌳兰c共r兲dV is the free-energy functional that includes the term with the Lagrange multiplier. We wish now to supplement this model in a manner that will allow for a physical representation of the influence of an additional material on the statics and dynamics of crystallization. There are several ways that this can be done: 共i兲 the imposition of appropriate boundary conditions on the above equations or 共ii兲 the addition to the model of a second phase field to directly model the impurities. Both of these approaches will allow us to model the wetting of a chemically inert phase embedded in the solid-liquid matrix and, concomitantly, develop a physical model of heterogeneous nucleation. Having developed these two approaches and examining their respective benefits and costs, we will then discuss their relationship to the “full” three-phase field model and demonstrate the common mathematical basis for all of models developed herein. We now present, in substantial detail, the specifics of these assertions and then use the model to examine several relevant examples. 2 = ⌬f共,c兲 = f共,c兲 − 共c − c⬁兲 − f ⬁ , const ⬅ = f , c 共8兲 where is the chemical potential, f ⬁ = f共⬁ , c⬁兲 and ⬁ and c⬁ are the far-field values of and c, respectively. From Eq. 共8兲 we see immediately that specifying the value of either , f / x, or c at the boundary immediately determines the other two 共in steady state兲. In this paper we will examine the consequences of three possible choices: 共i兲 specification of at the boundary, 共ii兲 specification of a normal gradient ⵜ · n at the boundary 共n is the surface normal兲, and finally 共iii兲 introduction of an additional bulk field term W—an auxiliary field that takes the value of 1 in the wall material 共where = 0兲. We explore this last approach in Sec. II C, as it will become the foundation for implementing all of the methods described herein. We follow the ideas of Cahn,8 who examined the problem of introducing a wettable interface into a binary-alloy liquid with a miscibility gap 共the system therefore had a critical point兲. Cahn8 imposed an “interface function” that determined a boundary condition on the concentration field and then examined the behavior of the system near the critical point. We now propose to do the same analysis in the context of the nonconserved phase field model of an ideal binary solution. We should note that the use of a structural order parameter in this analysis has specific physical implications that are nontrivially different from those realized under the analysis by Cahn8 of a conserved order parameter. In particular, as the dynamics of these two types of flows is substantially different, we expect the behavior of nucleation in these systems to be qualitatively different. The analysis of equilibrium will, however, be quite analogous, and we will adapt considerable material developed by others for our own use. We assume a free energy of the following form, where there is a boundary S, which specified the presence of an “inert” wall: F= 冕 S Z共兲dS + 冕 冋 dV f共,c兲 + 册 2 2 ⌫ 共ⵜ兲 . 2 共9兲 Minimization of the total free energy ␦F = 0 yields both Eqs. 共4兲 and 共5兲 as well as the boundary condition ␦关z共兲 + 2 ⵜ · n兴 = 0 on S, 共10兲 where n is the outward pointing normal to the surface S and z共兲 ⬅ Z / . This boundary condition can be met in one of 014204-3 PHYSICAL REVIEW B 79, 014204 共2009兲 WARREN et al. two ways: either −2 ⵜ · n = z共兲 共model A兲 or = const 共model B兲 on the boundary. These names follow the nomenclature of Ref. 24, but we will allow another way of setting the normal component of the phase field gradient at the interface 共model C兲. For concreteness we will explore herein both the models found in Ref. 24, where a special choice for z共兲 is used; namely, that along the inert boundary, all phase field contours have the same form as the one-dimensional profile rotated by an angle , as well as a simple quadratic model, where Z = 6␥SL共 21 g2 + h兲, with g and h as specified constants. Specifically we have for models A and C on surface S the following: profiles across the solid-liquid interface via the integral form of the EL equation that holds only for the planar interface, 冉 冊 2 2 nSL 2 = ⌬f关,c共兲兴, 共15兲 where nSL is a spatial coordinate normal to the solid-liquid interface, while the component of ⵜ normal to the wall is then 共n · ⵜ兲 = 共 / nSL兲cos共兲 = 共2⌬f / 2兲1/2cos共兲. Remarkably, if the parabolic groove approximation by Folch and Plapp34 is applied for the free-energy surface, one finds that conveniently ⌬f关 , c共兲兴 ⬀ 2共1 − 兲2. model A, C. Three-phase model versus model B z = − 2共ⵜ · n兲 = − 6␥SL共1 − 兲cos共兲, 共11兲 = const, 共12兲 z = − 2共ⵜ · n兲 = 6␥SL共g + h兲, 共13兲 model B, Typically to model three phases 共solid, liquid, and wall兲, we would introduce three-phase field variables 共S, L, and W, respectively兲, where each variable takes on the value of 1 in its named phase and 0 elsewhere, and impose the constraint that and model C, where is the imposed contact angle, and the specific choice of z共兲 in model A will become self-evident in Sec. II B. Models A–C are all legitimate solutions to the variational problem, but they may have different consequences on the dynamics of the system. Note that if there are multiple interfaces, then the boundary condition applies at each instance of the surface with in or surrounding the alloy. We see that the boundary condition generally relates to 共ⵜ · n兲 on surface S. Next, we briefly deduce models A–C. Along these lines, we demonstrate that under certain circumstances, a threephase field model 共liquid-solid-wall兲 can be reduced to a single phase field model with a boundary condition = 0 = const at the inert interfaces 共model B兲 and show that how model A can be nested into a three-phase field model. B. Derivation of model A We wish to ensure in equilibrium that the solid-liquid interface has a fixed contact angle with a foreign wall placed at z = 0. To achieve this, we prescribe the following boundary condition at the wall, which can be viewed as a binary generalization of model A presented in Ref. 24: 共n · ⵜ兲 = 再 2⌬f关,c共兲兴 2 冎 1/2 cos共兲. 共14兲 The motivation for this boundary condition is straightforward in the case of a stable triple junction, in which the equilibrium planar solid-liquid interface has a contact angle with the wall. The wall is assumed to lead to an ordering of the adjacent liquid: an effect that extends into a liquid layer of thickness d, which is only a few molecular diameters thick 共see, e.g., Refs. 32 and 33兲. If we take a plane z = z0, which is slightly above this layer, i.e., z0 ⬎ d, on this plane, the structure of the equilibrium solid-liquid interface remains unperturbed by the wall. Then, at z = z0, the phase field and concentration profiles are trivially related to the equilibrium S + L + W = 1. 共16兲 This constraint requires that phases evolve only by transforming into another phase 共no holes can develop兲. The constraint reduces the problem to only two independent phase field variables. For specificity, we will assume a liquid/solid binary alloy system at a uniform temperature 共similar to the one employed by Warren and Boettinger30 and many others31兲. We then postulate a free energy of the following form: F= 冕 dV关f共S, L, W,c兲 + S兩ⵜS兩2 + L兩ⵜL兩2 + W兩ⵜW兩2兴, 共17兲 where f is a local free-energy density with minima at j = 0 and 1 for all three phases j = L , S , W. The stability or metastability of each of these minima is dependent on the particulars of the form of f, which is in turn dependent upon the thermodynamic particulars of the alloy in question. We will specify a particular choice of f below. The gradient terms have a form that yields an isotropic interface energy, a choice which can be remedied in a number of ways; but this choice in no way effects the gist of the argument below. Using constraint 共16兲, we can eliminate one of the variables 共in this case we will choose the liquid兲, relabel the solid variable S → , and obtain F= 冕 dV关f共, W,c兲 + 共S + L兲兩ⵜ兩2 + 2L ⵜ · ⵜW + 共W + L兲兩ⵜW兩2兴. 共18兲 In a typical phase field treatment, at this stage we would minimize the free energy with respect to our two remaining phase field variables and postulate dynamics according to a law of gradient flow. However, in this instance we take a different approach; namely, that the profile of W is determined by the free energy in steady state and it is comprised of a series of sharp jumps between regions where W = 1 共inert wall兲 and W = 0 共liquid or solid兲. In principle, this 014204-4 PHYSICAL REVIEW B 79, 014204 共2009兲 PHASE FIELD APPROACH TO HETEROGENEOUS CRYSTAL… implies some singular behavior for the total free-energy density, if it is to yield a sharp interface in W 共and therefore discontinuous jumps in the other variables兲. However, as we demonstrate below, the particular nature of these singularities is irrelevant for practical computations. Given this constraint, we write the dynamical equations as 冋 册 f ␦F = − M − 2共S + L兲ⵜ2 − 2Lⵜ2W , = − M ␦ t 共19兲 c f ␦F = ⵜ · Mc ⵜ = ⵜ · Mc ⵜ . t ␦c c 共20兲 Note that the form of Eq. 共19兲 is—excepting the term proportional to ⵜ2W—identical to Eq. 共4兲. We assume that W is time independent and makes discontinuous jumps between regions of inert wall 共W = 1兲 and the regions of liquid or solid 共W = 0兲. As = 0 in regions where W = 1, it is clear that must make discontinuous jumps that mirror W. Given this, we assume the existence of an auxiliary function R共x , t兲 and write = R共x,t兲关1 − W共x,t兲兴, 共21兲 where R共x , t兲 is differentiable everywhere 共C 兲 and R共x , t兲 = 0 at the inert wall. If we insert these expressions into Eq. 共19兲 and equate orders in the divergences associated with spatial derivatives in W, we find that at the wall 1 0 = L , S + L In addition to allowing the solution to model B, the auxiliary field W is a useful numerical convenience for implementing the classical approach described above in Sec. II B for model A. Specifically, we can extend the integrals over the inert wall and the containing volume to all of space 共the space containing both the liquid/solid and the wall material兲 by the following modifications to 再 冋 dV Z共兲兩ⵜW兩 + f共,c兲 + 冉 册 冎 2 2 ⌫ 共ⵜ兲 共1 − W兲 , 2 共23兲 where 兩ⵜW兩 is a Dirac ␦ function that locates Z共兲 to the interface, while the factor 共1 − W兲 locates the free-energy 冊 1 ␦F f = = − 2ⵜ2 共1 − W兲 M t ␦ + 关z共兲 + 2 ⵜ · n兴兩ⵜW兩, 共24兲 where we have used n = ⵜW / 兩ⵜW兩. This expression is in some sense “obvious” since all we are doing is adding the model A boundary condition multiplied by a ␦ function to the original variation over the volume bounded by the inert wall. Thus, in prescribing the auxiliary field W, we may perform computations over all of space and need not explicitly impose boundary conditions on the liquid-solid material at the inert wall. E. Comment on grain boundaries At this juncture it is useful to briefly consider the grainboundary model mentioned in Sec. II D, as there is a mathematical analogy between the introduction of the boundary condition in the phase field model through 兩ⵜW兩 and the grain-boundary energy penalty term 兩ⵜR兩. As noticed by Tang et al.,35 this grain-boundary model is mathematically identical to the model of Cahn8 for critical wetting. Now, with the above analysis, the reasons for this mathematical equivalence become obvious. Specifically, if R has a step discontinuity at a point in space, a term of the form, 冕 V D. Three-phase model versus model A 冕 − 共22兲 for all time. Note that the value of 0 is independent of far-field boundary conditions. Thus, we see that these equations effectively yield a boundary condition on = 0 共model B兲 at inert particle interfaces. Hence, we propose that an essentially equivalent approach to the above problem is to simply impose this boundary condition and drop the additional equations associated with the other phase fields. Either method will be equally effective; however implementation may be easier for one or the other method depending on the particulars of the problem to be considered. F= density for the liquid/solid phase to those regions where W = 0. Computing ␦F yields a modified equation for , specifically, u共兲兩ⵜR兩dV = ⌬ 冕 u共兲dS, S where ⌬ is the misorientation across the grain boundary. Thus, the model including grain-boundary effects described above and solved in Secs. III–V includes two effective boundarylike terms: one static 共the inert particles described by W兲 and one dynamic 关the grain boundaries described by R共q兲兴. F. Physical interpretation of models A–C At this stage, it is appropriate to discuss the physical pictures underlying models A–C. Model A places the mathematical surface at which the boundary condition acts slightly beyond the boundary layer influenced by the wall. Thus the bulk liquid and solid phases in contact with the wall are connected through an unperturbed solid-liquid interface profile, and the derivation of the interface function for the desired contact angle is straightforward. All effects of liquid ordering and solid disordering due to the wall are buried into the contact angle 共realized by the particular surface function兲 as in the classical sharp interface model. Then, the total free energy of the system incorporates both a volumetric and a surface contribution. Model B prescribes liquid ordering and solid disordering at the wall explicitly. A shortcoming of Model B, however, is its implicit assumption that the wall enforces the formation of a specific layer of the solid-liquid interface 共corresponding to 0兲, simplifying considerably the nature of the wall-liquid/wall-solid interactions. Here, the 014204-5 PHYSICAL REVIEW B 79, 014204 共2009兲 WARREN et al. free energy of the system originates exclusively from the volumetric contribution. Relying on a model parameter 共h兲 of less straightforward physical interpretation than either the contact angle or the value of 0, model C is able to prescribe local conditions that are not present inside the solid-liquid interface. For example, values may appear along the wall that fall outside of the 共0, 1兲 range. Note that the appearance of such states is not unnatural: they are also present in phase field models, when Langevin noise is added to the equations of motion. These values of may be viewed as local states that are either more ordered or disordered than the bulk crystal and liquid phases 共e.g., for ⬎ 1, atoms are more localized that in the bulk solid; while ⬍ 0 might indicate a liquid with density deficit兲. Further work is, however, desirable that relates the model parameters to microscopic properties 共such as molecular interaction or molecular scale misfit at the wall兲. Remarkably, model C incorporates both structural change at the interface and at a surface function: both of which contribute to the total free energy of the system. In this respect, model C can be viewed as a generalization of the other two models. Nevertheless, we note that generally model A cannot be obtained as a limiting case of model C. It is remarkable, however, that setting g = h = 0 in model C, one can recover the approach by Castro21 that uses the “no-flux” boundary condition 关共n · ⵜ兲 = 0兴 to realize a 90° contact angle. This specific case can also be recovered in model A by prescribing = 90°. In these specific limits, solutions of models A and C coincide. III. MATERIAL PROPERTIES In this work we employ a parabolic well approximation to the free-energy density based on the work of Folch and Plapp34 共FP free-energy henceforth兲. Specifically, we select f共 , c兲 to be appropriate for an ideal solution, f共,c兲 = wG共兲 + X 再 1 兵c − c̄ − ⌬c共T兲关1 − p共兲兴其2 2 冎 + 关cS共T兲 − c̄兴⌬c共T兲关1 − p共兲兴 , 共25兲 where G共兲 = 2共1 − 兲2 is a double well with minima at = 0 and 1 共the common “4 potential”兲, w is the scale of the height of the double well, X is an energy scale associated with chemical changes in the system, and p共兲 = 3共10 − 15 + 62兲 is an interpolating function between phases with p共0兲 = 0 and p共1兲 = 1. The functions cL共T兲 and cS共T兲, which determine ⌬c共T兲 = cL − cS are the concentrations of liquid/solid coexistence 共the liquidus and solidus兲, which in turn depend on the temperature T 共which has been assumed uniform兲. Finally, c̄ is a concentration locating the minimum in the solid free energy. This free-energy model has the advantage of reproducing a variety of phase diagrams while allowing for a significant amount of analysis in one dimension, as will be discussed below. This parabolic well approximation to the free-energy surface has, furthermore, the interesting property that ⌬f关 , c共兲兴 = wG共兲, where c = c共兲 is the explicit relationship between c and emerging from the Euler-Lagrange equation 关Eq. 共7兲兴 for the concentration field.36 This feature means that in equilibrium 共whether stable or unstable, i.e., planar surface or nuclei兲, there is no chemical contribution to the interfacial free energy and that the Gibbs absorption of solute is of a particularly straightforward form 共as the concentration through the interface is an interpolation between the two equilibrium phase concentrations, with no excursion above or below these values permitted兲. This also means that a single solution of the EL equation for the one-component case can be transformed into an infinite number of binary solutions using the explicit relationships c = c共兲 emerging from Eq. 共7兲 provided that the latter does not contain a ⵜ2c term. 关We note that for nuclei the c = c共兲 relationship depends on the initial composition of the liquid.13兴 Since these features simplify the analytical calculations considerably, we use the approximate thermodynamics given by Eq. 共25兲 throughout this work. We note, however, that in general ⌬f关 , c共兲兴 has a more complex form. In order to do numerical calculations, we need to specify a number of parameters in the theory: , X, w, cL共T兲, cS共T兲, and T, and for dynamics, the mobilities M and M c. Herein, these model parameters are chosen so that our computations are comparable to the Cu-Ni ideal solution applied in many earlier studies.16,22,30,37 With this in mind, we choose a temperature T = 1574 K, where cS = 0.399 112, cL = 0.466 219, and X = 7.0546⫻ 109 J / m3. Next, we chose the interfacial parameters. In the case of nucleation studies relying on solving the EL equations, we have used d10%–90% = 1 nm for the 10%–90% interface thickness, as expected on the basis of atomistic simulations.14,15 The interfacial free energy has been chosen as the average 共␥SL = 0.2958 J / m2兲 of the experimental values for Cu 共0.223 and 0.232 J / m2兲 and Ni 共0.364 J / m2兲 from the grain-boundary groove and dihedral techniques 共data compiled in Ref. 38兲. Accordingly, 2 = 3␥SLd0 = 4.038 ⫻ 10−10 J / m and w = 6␥SL / d0 = 3.899⫻ 109 J / m3, where d0 = d10%–90% / ln共0.9/ 0.1兲. These calculations can be regarded as quantitative. As our illustrative computations for complex structures forming via heterogeneous nucleation are intended to be merely technology demonstrators, we aimed at only qualitative modeling. For example, in describing the shish-kebab morphology appearing in polymer-carbon nanotube systems,39 we have used an ideal solution approximant of the Cu-Ni alloys applied in several previous works of us.22,37 However, to mimic polymers, a high anisotropy of sixfold symmetry for the kinetic coefficient 关see Fig. 2共a兲兴 has been applied, 再 M = 1 – 3 M ,0 + 42z4 + 共2x + 2y 兲2兵3 + cos关6 atan共y/x兲兴其 共2x + 2y + z2兲2 冎 , 共 = 31 and = 0.001兲, a combination that mimics the behavior of polymeric systems in that the asymptotic growth form 共kinetic Wulff shape calculated according to Ref. 40兲 is a 014204-6 PHYSICAL REVIEW B 79, 014204 共2009兲 PHASE FIELD APPROACH TO HETEROGENEOUS CRYSTAL… Finally, a few illustrative computations have been performed to model the columnar to equiaxed transition as a function of contact angle in the Al55Ti45 alloy. The thermodynamic properties have been taken from a CALPHAD-type 共CALculation of PHAse Diagrams; a standard, thermodynamically consistent, fitting technique兲 assessment of the phase diagram.41 For further details see Ref. 42. Here, the same anisotropy function has been used for M as in the case of the polymeric system, however, now with = 0.3 and = 1.0. The respective orientation dependence of M and the respective asymptotic growth form 共octahedron兲 are shown in Fig. 3. (a) IV. RESULTS AND DISCUSSION (b) FIG. 2. 共Color online兲 共a兲 Kinetic anisotropy used in simulations for the polymer-carbon nanotube mixture and 共b兲 the respective asymptotic growth form 关kinetic Wulff shape 共Ref. 39兲兴. hexagonal plate 关see Fig. 2共b兲兴. 关Here we use the notation 共ⵜ兲2 = 2x + 2y + z2.兴 In the simulations, we have used the following parameter set for the free energy: 2 = 1.65 ⫻ 10−8 J / m and w = 5.28107 J / m3, while X is the same as above, and assuming that the diffusion coefficients in the liquid and solid are Dl = 10−9 m2 / s and Ds = 0, respectively, while the phase field mobility is M = 0.05 m3 / 共J s兲. Other simulations for solidification in the presence of foreign particles have been performed for a 10%–90% interface thickness of 5 nm and a slightly higher interfacial free energy 共0.360 J / m2兲. The respective values for the model parameters are 2 = 4.95⫻ 10−9 J / m and w = 3.96⫻ 108 J / m3. (a) Before performing numerical solutions to the equations it is useful to determine those cases where analytic calculations are tractable. With this in mind we examine threes cases of interest in steady state: 共i兲 a triple junction 共solid-liquid-wall兲 of three flat interfaces, 共ii兲 an undercooled liquid in contact with an inert wall, and 共iii兲 solid droplet 共spherical-cap兲 in contact with an inert wall. A. Wetting properties of external walls In order to compare these phase field models with Young’s equation 关Eq. 共1兲兴, we must compute the surface energies and other relevant quantities. The surface energies can be computed using the first integral and arguments found in a number of sources.9,43 In general the surface energy between any two semi-infinite phases A and B will be ␥AB = 2 冕 ⬁ dx⌬f关,c共兲兴. 共26兲 −⬁ We are going to address wetting properties using this expression valid for far-field behavior and utilizing the specific form of the free-energy surface given by Eq. 共25兲. (b) FIG. 3. 共Color online兲 共a兲 Anisotropy of the interfacial free energy used in simulations for columnar to equiaxed transition in the Al-Ti alloy and 共b兲 the respective asymptotic growth form 关kinetic Wulff shape 共Ref. 39兲兴. 014204-7 PHYSICAL REVIEW B 79, 014204 共2009兲 WARREN et al. 1. Triple junction of three flat interfaces In order to examine all of these cases, it is useful to consider flat interfaces in equilibrium. The three phases will coexist at the melting point. Utilizing the properties of the FP free energy, the liquid/solid interface free energy ␥SL far from the junction is ␥SL = 冕 1 d冑2⌬f关,c共兲兴 = 冑2w 0 = 冕 1 共1 − 兲d 0 冑2w . 6 共27兲 Regardless of whether we impose a condition on or ⵜ · n, the interface boundary condition establishes a value for . As discussed above, is either specified 共␦ = 0兲 on the boundary or, if an interface function is specified, then we may combine Eq. 共8兲 with the interface boundary condition to find the roots of 22⌬f关,c共兲兴 = 共6␥SL兲2G共兲 = z2共兲, 共28兲 which has the simpler form 6␥SL共1 − 兲 = ⫾ z共兲, 共29兲 Eq. 共28兲, combined with f / c = , is sufficient to determine the interface values of and c. In general, these expressions will have multiple roots and the system’s selection of a particular root will be determined such that the free energy is minimized. We denote the roots selected when either liquid or solid abuts the inert wall as L and S, respectively. Given at the wall, we may determine the energies of wall-liquid and wall-solid boundaries as ␥WL = Z共L兲 + ␥SL关3L2 − 2L3 兴, ␥WS = Z共S兲 + ␥SL关1 − 32S + 23S兴. 共30兲 共31兲 Thus the expression for the contact angle reads as cos共兲 = simplifies to cos共兲 = 220共3 − 20兲 − 1. In this case we see that as 0 ranges from 0 to 1 then ranges from 0 共total liquid wetting, solid dewetting兲 to 共total solid wetting, liquid dewetting兲. This is not surprising, since making the interface “solidlike” causes solid to wet the surface, while when the surface is “liquidlike” the reverse is true. For model C, things are a bit more complicated 共not surprisingly兲, but the analysis is revealing. We have at the boundary that 共1 − 兲 = ⫾ 共h + g兲, which can be quickly solved to find up to four real roots Z共L兲 − Z共S兲 + 共3L2 − 2L3 兲 − 关1 − 32S + 23S兴. ␥SL 共32兲 This expression includes the mild assumption that the free energies of flat isolated interface can be used in constructing a Young’s equation. While this is exact for the stable triple junction in an infinite system, it is only an approximation in the undercooled state, as all the field variables interact in the triple junction region, but the approximation may hold under a variety of circumstances. We will test the accuracy of this through simulations shown below. For model A, the above analysis is nearly trivial, as WL = 0 and WS = 1, and the contact angle is actually the control parameter of the model. Accordingly, the surface function can be expressed as Z共兲 − ␥WL = ␥SL cos共兲关23 − 32兴 = 共␥WS − ␥WL兲关23 − 32兴, yielding 0 for the bulk liquid phase 共 = 0兲 and 共␥WS − ␥WL兲 for the bulk solid 共 = 1兲 phase at the interface. For the case, when is specified at the interface 共model B兲 then L = S = 0 and the expression for the contact angle 1 1,2 = 关1 − g ⫾ 冑共1 − g兲2 − 4h兴, 2 1 3,4 = 关1 + g ⫾ 冑共1 − g兲2 + 4h兴. 2 共33兲 It simplifies the analysis of these roots to consider the case g = 0, as this assumption does not change the character in the solutions, only the particulars. 关Note that this case 共n · ⵜ兲 = −const, which can be viewed as a straightforward generalization of the “no-flux” condition by Castro21 for establishing a contact angle of / 2.兴 In this case, one finds that the minimum free energy solutions for are 1 S = 关1 + 冑1 − 4h兴, 2 1 L = 关1 − 冑1 + 4h兴, 2 共34兲 where we note that will take values outside the range 关0,1兴 at the wall. As is a structural order parameter and not a concentration, this is not necessarily unphysical as we mentioned above. This changes some of the signs for some of the terms in the expression for the surface energy and care must be taken. Using these values of 0, we can calculate the contact angle to be 1 cos = 关共1 − 4h兲3/2 − 共1 + 4h兲3/2兴. 2 共35兲 Note that if h ⬎ 1 / 4关2 ⫻ 31/2 − 3兴1/2 ⬇ 0.1703, then the contact angle is 共complete wetting by the liquid兲, while if h ⬍ −1 / 4关2 ⫻ 31/2 − 3兴1/2 ⬇ −0.1703 the contact angle is 0 and the solid “wets” the interface. A plot of the contact angle as a function of h is given in Fig. 4. 2. Undercooled liquid next to an inert wall and “critical” wetting: 1D solutions In this section, we consider a semi-infinite supersaturated 共undercooled兲 liquid 关c⬁ ⫽ cL共T兲 , ⬁ = 0兴 in contact with a planar wall placed at z = 0. Then the first integral of the respective 1D Euler-Lagrange equation for the phase field reads as 014204-8 PHYSICAL REVIEW B 79, 014204 共2009兲 PHASE FIELD APPROACH TO HETEROGENEOUS CRYSTAL… FIG. 4. 共Color online兲 A plot of contact angle as a function of parameter h in model C 共g = 0兲. For h ⬍ −0.1703 the contact angle is 0, while for h ⬎ 0.1703, it is . 冉 冊 2 2 z 2 = f − f ⬁ − 共c − c⬁兲 = wG共兲 − X⌬c共cL − c⬁兲p共兲 = ⌬f关,c共兲兴. 共36兲 Here, the FP choice of the free-energy density yields a skewed double well as a function of . Model A shows a classical behavior: neither liquid ordering nor critical wetting is predicted at the interface. In model B liquid ordering is inherent and a spinodal-like behavior can be seen at high enough driving force. Here, we have 0 = const苸 关0 , 1兴 at the wall. Under such conditions, the 1D Euler-Lagrange equation can be integrated to obtain 共x兲, yielding a solution representing a metastable equilibrium 共supersaturated liquid in contact with the wall兲. Remarkably, Eq. 共36兲 can only be integrated to yield a real solution in the region, where wG共兲 − X⌬c共cL − c⬁兲p共兲 ⱖ 0. For wG共兲 − X⌬c共cL − c⬁兲p共兲 ⬍ 0, only a time-dependent solution exists: a propagating solidification front. The critical supersaturation that separates these two types of solutions, while prescribing a fixed 0 value at the wall, is given by the condition wG共0兲 − X⌬c共cL − c⬁兲p共0兲 = 0, yielding Scrit = wG共兲 / 关X⌬c2 p共兲兴. 共It is the binary analog of the critical undercooling of the unary systems discussed in Ref. 24.兲 The critical supersaturation vs 0 relationship corresponding to the FP parameters specified in Sec. III is shown in Fig. 5. It can also be shown 共see Sec. IV B兲 that the nucleation barrier—the solid phase has to pass to start solidification— tends to zero in this limit, and the solid phase wets ideally the wall. This phenomenon is analogous to the critical wetting of a solid wall seen in two-fluid systems near the critical point. However, we have here solid and liquid phases, instead of the two fluids. For g = h = 0, model C coincides with model A at = / 2; therefore, under such conditions no surface ordering/ disordering or spinodal are observed. Despite surface ordering/disordering, for h ⬎ 0, no surface spinodal exists in model C 共g = 0兲. However, for h ⬍ 0, where 苸 关0 , 1兴 at the wall, model C 共g = 0兲 predicts both surface ordering and a spinodal. The relationship between h and the critical supersaturation can be computed using the expressions Scrit = wG⬘共兲 / 关X⌬c2 p⬘共兲兴 and h = 共兲 = −兵G共兲 − p共兲X⌬c共cL − ccrit兲 / w其1/2, where ccrit = cL − Scrit⌬c, where the expression for Scrit originates from the condition that the critical state FIG. 5. 共Color online兲 Critical liquid supersaturation corresponding to ideal wetting as a function of phase field value 0 at the wall in model B at T = 1574 K for Cu-Ni with a physical interface thickness of 1 nm. The horizontal dashed line shows the maximum possible supersaturation Smax = 6.9474 共corresponding to c⬁ = 0兲. Results above this line are unphysical. corresponds to the extremum of the loop in 共兲 that incorporates the point = 0 and = 0. 共Note that the expression for Scrit is the condition both for the maximum gradient / z of the 1D solution and for the location of the central hill of the double-well free energy.兲 The respective Scrit vs −h relationship is shown for the Cu-Ni system with d10%–90% = 1 nm interface thickness at T = 1574 K in Fig. 6. We note that with the actual choice of the model parameters 共as for other continuum models兲, the spinodal point between the solid and supersaturated liquid falls into the physically inaccessible region of negative concentrations 共see the discussion in Sec. IV B兲. B. Heterogeneous nucleation on external walls in three dimensions In our previous work,24 we have investigated heterogeneous nucleation in two dimensions using models A and B for a single-component system. Herein, assuming isotropic interfacial free energy and utilizing the respective cylindrical symmetry, we extend our study to three dimensions and binary alloys using the FP thermodynamic model 关Eq. 共22兲兴. FIG. 6. 共Color online兲 Critical liquid supersaturation corresponding to ideal wetting as a function of −h at the wall in model C 共g = 0兲 at T = 1574 K for Cu-Ni with a physical interface thickness of 1 nm. The horizontal dashed line shows the maximum possible supersaturation Smax = 6.9474 共corresponding to c⬁ = 0兲. Results above this line are unphysical. 014204-9 PHYSICAL REVIEW B 79, 014204 共2009兲 WARREN et al. The respective form of the Euler-Lagrange equation for the phase field is ψ = 30° ψ = 45° ψ = 60° ψ = 90° ψ = 120° ψ = 170° 冉 冊 2 G⬘共兲w − p⬘共兲X⌬c共cL − c⬁兲 1 r + 2 = , r r r z 2 共37兲 WA,C = F关ⴱ1共r兲兴 − F关ⴱ0共r兲兴 − 冕 dA兵␥W共兲 − ␥WL其, S 共38兲 where the first two terms represent the volumetric contribution, while the third term accounts to the change in the surface function. Here ␥W共兲 − ␥WL = −␥SL cos共兲关23 − 32 + 1兴,24 while ⴱ1共r兲 is the solution corresponding to the nucleus, and ⴱ0共r兲 is the solution without nucleus 共liquid of the initial composition in contact with the wall兲. The latter solution has been obtained the same way as the one for the nucleus, however, using a homogeneous bulk liquid in contact with the wall as the starting condition. In model B, there is no contribution from the interface function, thus WB = F关ⴱ1共r兲兴 − F关ⴱ0共r兲兴 共39兲 applies. We have investigated the properties of nuclei at a high supersaturation 共S = 5.0兲. The free energy of formation of the heterogeneous nuclei relative to the free energy of formation of the respective classical 共sharp interface兲 homogeneous solution is shown for models A–C 共g = 0兲 in Figs. 7–9, respectively. One observes remarkable differences in the shape of the contour lines which the three models predict. In model A 1 0.8 W/WCNT where prime stands for differentiation with respect to the argument of the function. This equation has been solved numerically under boundary conditions given by models A–C 共g = 0兲 using the PDE 共Partial Differential Equation兲 Toolbox of MATLAB 共@The MathWorks Inc., 1984–2008兲 that relies on a combination of the finite element and relaxation methods.44 This approach needs a reasonable guess for the phase field distribution that is sufficiently close to the solution to be used as the starting distribution for relaxation. In mapping the properties of nuclei, we have used the following strategy. First, the solution corresponding to semiwetting case 关 = / 2, 0 = 0.5, h = 0, respectively, in models A, B, and C 共g = 0兲兴 has been determined. The initial phase field distribution used here was 共r兲 = 1 / 2兵1 − tanh关共r − RCNT兲 / d0兴其, where RCNT = 2␥LS / 关X⌬c共cL − c⬁兲兴 and d0 = d10%–90% / ln共0.9/ 0.1兲 are the classical radius of nuclei and an interface thickness parameter expressed in terms of the 10%–90% interface thickness. Having found the respective solution by the relaxation method, the mapping property 共, 0, h, supersaturation, etc.兲 has been changed in small increments, so that the solution for the previous computation could be used as a suitable starting distribution for the next computation. For models A and C 共g = 0兲, the free energy of formation of nuclei has been calculated as 0.6 0.4 0.2 0 0 45 90 ψ (deg) 135 180 FIG. 7. 共Color online兲 Structure of heterogeneous nuclei at S = 5.0 in model A at various contact angles 共upper and central row兲. There is a symmetry plane on the left edge. The contour lines vary between 0.1 and 1.0 by increments of 0.1: = 0.1, 0.2, . . . , 0.9. Note that the corresponding local supersaturation, s共兲, can be obtained by s共兲 = S + p共兲. The lowermost panel shows the ratio of the PF prediction for the nucleation barrier height 共circles兲 normalized by the barrier height for the homogeneous nucleus in the droplet model of the classical nucleation theory. For comparison, the catalytic potency factor f共兲 from the spherical cap model is also shown 共solid line兲. 共Fig. 7兲, the contour lines corresponding to phase field levels of 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9 共when they appear兲 are roughly concentric circles, of which those for ⱕ 0.5 intersect the wall. The contour line = 0.5 approximates well the nominal 共equilibrium兲 contact angle. Accordingly, model A can indeed be viewed as a diffuse interface realization of the classical spherical cap model 共a diffuse solid-liquid interface combined with a sharp wall兲. At this undercooling, the radius of curvature of the particle is several times larger than the interface width. Accordingly, the behavior of the classical spherical cap model is recovered quite accurately. For example, W / WCNT from the phase field computations approximates closely the catalytic potency factor f共兲 共see lowermost panel in Fig. 7兲. In model B 共Fig. 8兲, only a single contour line intersects the wall: the one corresponding to 0; while the others are either closed 共 ⬎ 0兲 or open 共 ⬍ 0兲. Accordingly, one can define a contact angle for only the contour line = 0. The contact angle defined this way, however, depends strongly on the applied supersaturation and converges to = 0 as the respective critical liquid composition 共that depends 014204-10 PHYSICAL REVIEW B 79, 014204 共2009兲 PHASE FIELD APPROACH TO HETEROGENEOUS CRYSTAL… φ = 0.7 φ = 0.6 φ = 0.5 φ = 0.4 φ = 0.3 φ = 0.1 0 0 0 0 h = −0.2 h = −0.15 h = −0.1 h=0 h = 0.1 h = 0.15 1 1 0.8 0.8 W/WCNT W/W CNT 0 0 0.6 0.4 0.2 0 0.6 0.4 0.2 0 0.2 0.4 φ 0.6 0.8 0 −0.2 1 0 FIG. 8. 共Color online兲 Structure of heterogeneous nuclei at S = 5.0 in model B at various phase field values at the wall 共upper and central row兲. The contour lines vary between 0.1 and 1.0 by increments of 0.1: = 0.1, 0.2, . . . , 0.9. Note that the corresponding local supersaturation, s共兲, can be obtained by s共兲 = S + p共兲. The lowermost panel shows the ratio of the PF prediction for the nucleation barrier height 共circles兲 normalized by the barrier height for the homogeneous nucleus in the droplet model of the classical nucleation theory. For comparison, the catalytic potency factor f共兲 from the spherical cap model is also shown 共solid line兲. on 0 as shown in Fig. 4兲 is approached. Accordingly, at a fixed supersaturation, the W / WCNT vs 0 curve reaches zero 共ideal wetting兲 at a finite 0 value 共see lowermost panel in Fig. 8兲, where the actual liquid composition is the critical composition 共given by the relationship shown in Fig. 4兲. In the case of model C 共g = 0兲 共Fig. 9兲, the situation resembles to that seen for model B, though it is somewhat more complex: there are closed contour lines and also open ones; however, they are separated by not a single contour line that intersects the wall, as in model B, but by a range of such contour lines. Phase field values out of the 关0,1兴 range can be observed at the interface if h ⫽ 0 as predicted in Sec. IV A 2 共 ⬍ 0 values at the wall-liquid interface if h ⬎ 0; ⬎ 0 values at the wall-solid interface if h ⬍ 0兲. These local states at the wall cannot be found in the bulk solid and liquid phases, though they are temporarily accessible in the bulk phases via thermal fluctuations. The W / WCNT vs h relationship is shown in the lowermost panel of Fig. 9. Interestingly, in the supersaturated state h can reach lower values than allowed in equilibrium. Also, the maximum value for the nucleation barrier may exceed that for homogeneous nucleation. The latter finding suggests that −0.1 h 0 0.1 FIG. 9. 共Color online兲 Structure of heterogeneous nuclei at S = 5.0 in model C 共g = 0兲 at various values of the model parameter h 共upper and central rows兲. The contour lines vary between 0.1 and 1.0 by increments of 0.1: = 0.1, 0.2, . . . , 0.9. Note that the corresponding local supersaturation, s共兲, can be obtained by s共兲 = S + p共兲. The lowermost panel shows the ratio of the PF prediction for the nucleation barrier height normalized by the barrier height for the homogeneous nucleus in the droplet model of the classical nucleation theory 共circles兲. For comparison, the catalytic potency factor f共兲 from the spherical cap model is also shown 共solid line兲. The deviation of the background hue from white in panels for h ⬎ 0 indicates that negative and S values appear in the vicinity of the wall, where it is contact with the liquid phase. model C 共g = 0兲 can capture walls that prevent nucleation in their neighborhood.45 Such walls represent a foreign matter that enforces a local structure on the liquid, which is incommensurable with the crystal structure to which the liquid structure transforms during freezing. This might have interesting consequences: nanoporous materials of walls of this kind could stabilize the liquid state in the pores at temperatures, where otherwise the liquid would freeze. For fixed model parameter values 共, 0, and h兲 corresponding to the same equilibrium contact angle, we have computed the nucleation barrier as a function of supersaturation. The results are compared in Fig. 10. For all the models, we find that for S → 0 the ratio of the nucleation barrier to the corresponding classical spherical cap result 共W / WSC兲 tends to 1, i.e., with an increasing size the phase field results converge to the classical spherical cap model. In the cases of models A and B, the nucleation barrier decreases monotonically with an increasing driving force and for model B it converges to 0 at a 0-dependent critical supersaturation 共for 014204-11 PHYSICAL REVIEW B 79, 014204 共2009兲 WARREN et al. FIG. 10. 共Color online兲 Nucleation barrier height 共W兲 normalized by that from the classical spherical cap model 共WSC兲 vs supersaturation 共S兲 for models A 共square兲, B 共triangle兲, and C 共diamond兲 at interface parameters = 60°, 0 = 0.673648, and h = −0.083733, respectively, which all realize the same equilibrium contact angle. Note that for both models B and C 共for h ⬍ 0兲, there exists a critical supersaturation, where ideal wetting switches in 共the nucleation barrier disappears兲. This critical supersaturation Sc depends on the respective interfacial parameter 共0 or h兲. Such surface spinodal-like behavior has not been observed for model A that realizes the nominal contact angle fairly accurately even at high supersaturations 共for model A the deviation from W / WSC = 1 originates dominantly from the fact that at large supersaturation, the height of heterogeneous nuclei becomes comparable to the interface thickness, and thus there are no bulk crystal properties in the nuclei兲. The vertical dash-dotted line indicates the border Smax of the physically accessible region 共S ⱕ Smax兲. the dependence, see Fig. 5兲. In contrast, in model C 共g = 0兲, the W vs S relationship shows a maximum before the barrier height decreases to 0 at an h-dependent critical supersaturation 共for the dependence, see Fig. 6兲. For h ⬎ 0 there is no critical supersaturation in model C and W decreases monotonically with increasing driving force S, although W / WSC increases. We note here that in many gradient theories, one has a spinodal point between the highly undercooled bulk liquid and the crystalline phase46 共though usually it falls into the nonphysical regime, e.g., to a negative temperature兲. The existence of such a spinodal point and its influence on nucleation has been the subject of extensive discussions,47,48 especially for short-range interactions 共see Ref. 47 for review兲. Recent atomistic simulations seem to imply that no convincing evidence is available for the existence of such a spinodal point.49 With the present choice of G共兲 and p共兲, this crystal-liquid spinodal falls to the S → ⬁ limit, well beyond the boundary of the physically accessible range Smax = 6.9474 corresponding to c⬁ = 0. C. Formation of complex structures via heterogeneous nucleation 1. Shish-kebab structure in model A Here, we present polycrystalline structures obtained relying on the quaternion representation of local crystallographic orientation, which Pusztai et al.27 proposed recently. For the sake of illustrating the capabilities of advanced phase field modeling that relies on noise-induced heterogeneous nucle- ation on external walls, we simulate the shish-kebab structure seen to form on carbon nanotubes in polymeric systems.39 To accomplish this, we have introduced curved tubes into the simulation box generated so that the local 共gradually changing兲 crystallographic orientation lies in the axis of the tube, whose shape has been constructed stepwise, so that its direction in the next segment might deviate from the orientation of the previous segment by only a small random angle. The contact at the wall of the nanotubes is characterized by Eq. 共14兲, while prescribing a contact angle of = / 4. Illustrative simulations have been performed for a hypothetic binary system, whose phase diagram and thermodynamic properties are similar to those of the Ni-Cu system approximated by the ideal solution model applied in previous work.22 共Application for real polymer blends of known Gibbs free-energy functions should be straightforward.兲 Unlike the metallic systems, polymers often crystallize in the form of disklike flakes. To mimic this behavior, we have introduced an anisotropic form for the phase field mobility, which prefers the formation of disklike growth forms 共see Fig. 2兲. The simulation has been performed on a 200⫻ 200 ⫻ 300 cubic grid with spatial and time steps of ⌬x = 10 nm and ⌬t = 10 ns at the initial liquid concentration of cNi = 0.4192. Snapshots of the simulation are shown in Fig. 11. Note the similarity to the experimental structures reported in Ref. 39. 2. Columnar to equiaxed transition in model A Another illustration that shows the capabilities of phase field simulations incorporating heterogeneous nucleation is the application of model A for describing the columnar to equiaxed transition 共a work done in the framework of the EU FP6 IMPRESS project50兲. Here, we have combined model A with a three-dimensional 共3D兲 model of polycrystalline solidification relying on the quaternion representation of the crystallographic orientations27 and adopted it to constanttemperature gradient and a moving frame. To enable large scale simulations, we have used a broad interface 共of thickness 65.6 nm兲 and included an antitrapping current51 to ensure a quantitative description of growth kinetics. In the simulation window, the material is made to move with a homogeneous velocity from right to left, while a fixed temperature gradient is prescribed in the horizontal direction. Particles of given number density of random orientation and size and of given contact angle enter into the simulation window at the right edge, while periodic boundary condition is used on the horizontal edges. Snapshots of the chemical and orientation maps illustrating polycrystalline solidification under such conditions are presented in Fig. 12. As a result of the diminishing nucleation rate due to the increasing contact angle, we observe a gradual transition from the equiaxed polycrystalline structure to a columnar structure. A more detailed analysis of this phenomenon will be presented elsewhere.52 3. Liquid-solid meniscus position in model B As demonstrated above, if we fix the value of the phase field at a wall to = 0, a contact angle will result. Specifi- 014204-12 PHYSICAL REVIEW B 79, 014204 共2009兲 PHASE FIELD APPROACH TO HETEROGENEOUS CRYSTAL… (a) (b) (c) FIG. 11. 共Color online兲 Formation of shish-kebab structure by noise-induced heterogeneous nucleation on tubular walls of contact angle = / 4 in model A 关Eq. 共14兲兴 at T = 1574 K and cCu = 0.4192, in a hypothetical system whose thermodynamic properties are given by an ideal solution approximation of the Cu-Ni system, while its kinetic anisotropy 共anisotropy of the phase field mobility兲 and growth shape are shown in Fig. 2. Snapshots taken at times t = 30, 40, 50, and 60 s show the walls and the solidification front 共 = 0.5兲. The computations have been performed on a 200⫻ 200⫻ 300 grid 共2 ⫻ 2 ⫻ 3 m3兲. cally, at the wall, the expression for the contact angle is cos共兲 = 2共兲20关3 − 2共兲0兴 − 1. This can be realized numerically by fixing the value of the phase field at = 0 everywhere in the “wall” material. To illustrate this approach we have done a few sample calculations of capillary rise under circumstances that favor either wetting or dewetting. Using our model, we are able to investigate the evolution of a column of liquid-solid and between two wetting interfaces and compare the predictions with the analysis made above. We choose to do calculations in an insulating box 共no change in the total mass of the system兲, with 2 = 4.95⫻ 10−9 J / m and w = 3.96⫻ 108 J / m3, as is done in the simulations with par- ticles shown above. Figure 13 shows two typical simulations with box sizes of 22.8⫻ 22.8 nm2, with 100⫻ 100 grid resolution, and 20% of the box on the left and right occupied by wall material. We consider the symmetric cases of 0 = 0.3 关Figs. 13共a兲 and 13共b兲兴 and 0 = 0.7 关Figs. 13共c兲 and 13共d兲兴, corresponding to solid-wall contact angles of = 55.39° and = 124.61°, respectively. We start with a system at a uniform concentration that guarantees that the final interface position remains in the box, which, for these choices of contact angle, is satisfied by S = 1.0 and 0.2, respectively. The figure shows the calculation initially and after it has come to equilibrium 共1 ms兲. FIG. 12. 共Color online兲 Phase field simulation of columnar to equiaxed transition in the Al0.45Ti0.55 alloy as a function of contact angle of foreign particles in a moving frame 共V = 1.26 cm/ s兲 and a constant-temperature gradient 共ⵜT = 1.12⫻ 107 K / m兲 in model A 关Eq. 共14兲兴. Composition 共on the left兲 and orientation maps 共on the right兲 corresponding to contact angles of = 30°, 60°, 90°, and 120° 共from top to bottom, respectively兲 are shown. The computations have been performed by solving the 3D phase field model of polycrystalline solidification 共Ref. 27兲 in two dimensions on a 600⫻ 3000 grid 共3.93⫻ 19.68 m2兲. White spots in the chemical map indicate the foreign particles, whose diameter varies in the 13–66 nm range. In order to be able to distinguish the orientation of the foreign particles, the fluctuating orientation field of the liquid is not shown in the orientation map 关color map is multiplied by p共兲兴. 014204-13 PHYSICAL REVIEW B 79, 014204 共2009兲 WARREN et al. h=-0.5 c a a b d FIG. 13. 共Color online兲 Time evolution of solid-liquid meniscus at vertical walls in model B for 关共a兲 and 共b兲兴 wetting 共 = 55.39°兲, and 关共c兲 and 共d兲兴 nonwetting 共 = 124.61°兲 walls. The phase field map is shown 关black: bulk solid 共 = 1兲; white: bulk liquid 共 = 0兲兴. The computations have been performed for Cu-Ni, assuming a 5-nm-thick solid-liquid interface. The software tool FIPY was used for the calculations 共Ref. 53兲. To get an estimate of the equilibrium configuration of the meniscus, one can use a simple mass conservation argument to show that 冋 b t=600 h HS = WS + t=0 册 L − cos sin , 4 sin2 共40兲 where HS is the height at the center of the solid meniscus from the bottom of the column, W is the width of the column, and L is the height of the entire column, the bracketed quantity accounts for the circular cap of liquid or solid, and S is here the supersaturation, uncorrected for curvature. To improve on the estimate, the supersaturation can be corrected to linear order by replacing S by S + 2␥SL sin / 共X⌬c2L兲. Using these relationships the calculations shown in the figure agree with this analysis within 5%, with no substantial improvement if the grid is refined. For 0 = 0.3, the measured value of HS is 0.625 while the analysis yields 0.603; for 0 = 0.7 HS converges to 0.628, while the analysis yields 0.597. The estimate could be improved by using a full nonlinear correction to the compositions, as well as further consideration of the nonclassical influence of the finite interface thickness. Clearly, the method is a convenient approach for capturing complex phenomena. 4. Particle-front interaction in model C Finally we examine an application of model C. We consider the same parameters 共except that the initial supersaturation is S = 0.86兲 used in Sec. IV C 3 but now examine the passage of an interface through a distribution of interacting particles. The analysis of such a phenomenon is essentially similar to Zener pinning,54 and it is not the intent of this paper to fully explore this phenomenon but instead to demonstrate the flexibility and generality of our approach. Figure 14 shows a series of calculations that show the phenomenological richness available within this relatively simple model. We discuss the images in a clockwise se- t=100 c t=600 g t=600 d t=600 f t=600 t=600 e h=-0.025 FIG. 14. 共Color online兲 Particle-front interaction in model C at a fixed initial liquid supersaturation S = 0.86. Pinning of solidification front to foreign particles: 关共a兲–共c兲兴 Propagation and pinning of the solidification front in the presence of circular foreign particles. 关共d兲 and 共e兲兴 The effect of shape and contact angle on front pinning. 关共f兲–共h兲兴 The effect of particle size on the front pinning. For discussion, see the text. The computations have been performed for CuNi, assuming a 5-nm-thick solid-liquid interface. Images 关共a兲–共c兲兴 have been computed with h = −0.05, while 关共e兲 and 共f兲兴 with h = −0.025 共black: solid; white: liquid; green 共gray兲 circles or bars: foreign particle兲. The software tool FIPY was used for the calculations 共Ref. 53兲. quence. Panels 共a兲–共c兲 show the propagation of the solidification front in the presence of foreign particles of circular shape, starting with 共a兲 and proceeding for 6 s, with a contact angle on the particles set in model C using h = −0.05 关so the equilibrium contact angle defined by Eq. 共35兲 is ⬇ 73°兴. The interface eventually arrests but much of the box solidifies 共⬇71%兲. In order to examine the influence of shape and wettability, we first examine changing the shape of the drops to “sticks” 关seen in panels 共d兲 and 共e兲兴 but with the same distribution of particles. We see that for the sticks, the interface arrests much more quickly at a solid fraction of ⬇39%. Solidification at a reduced wettability h = −0.025 共corresponding to ⬇ 81°兲 is shown in panels 共e兲–共h兲. Apparently, this does not alter substantially the solidification front 关cf. panels 共d兲 and 共e兲兴. Thus we infer that the presence of right-angle corners strongly influences the pinning of the interface. In contrast, in frame 共f兲 all that is changed from frame 共c兲 is the contact angle, and we see that the interface now arrests with this modestly higher angle 共a solid fraction of ⬇29%兲 and increasing the size of the droplets 关see panel 共g兲兴 does not influence the profile substantially 共25%, however, much of the difference is due to the increased percentage of impurities兲. Finally, in panel 共h兲, we reduce the size of the particles, and the interface once again is substantially less impeded with 73% of the liquid solidifying. Clearly, a substantial numerical exploration of this phenomenon could yield further insights into such pinning behavior in real systems, particularly, if a physically motivated wall function could be established through either measurement or ab initio calculations. 014204-14 PHYSICAL REVIEW B 79, 014204 共2009兲 PHASE FIELD APPROACH TO HETEROGENEOUS CRYSTAL… V. SUMMARY Models A–C represent different levels of abstraction as we discussed above and can be used to address a broad variety problems of including the formation of complex solidification structures such as the shish-kebab morphology in carbon nanotube filled polymers, the columnar to equiaxed transition, and the front-particle interaction in alloys. Any of these models can be used to describe interfaces that are characterized by a given contact angle in equilibrium; however, the behavior predicted in the supersaturated state depends on the individual model. Comparative studies relying on combined phase field and atomistic simulations are planned to identify the validity range of the individual models and the predicted complex behavior 共e.g., the appearance of surface spinodal兲. We have presented three possible approaches to model the wetting properties of foreign walls in the framework of phase field simulations for the solidification of binary alloys. These approaches differ in the treatment of the foreign surfaces. 共a兲 Model A is a diffuse interface realization of the classical spherical cap model with a contact angle that is essentially independent of the driving force ensured by a specific surface function. 共b兲 Model B is a nonclassical formulation that assumes a fixed phase field value at the interface, leading to surface ordering/disordering, a strongly supersaturation-dependent contact angle, and to a surface spinodal 共ideal wetting beyond a critical supersaturation兲. In this model, only such local states can be realized at the wall, which are present in the solid-liquid interface. 共c兲 Model C is a nonclassical approach, which in its simplest form 共g = 0兲 fixes the normal component of the phase field gradient, leading to surface ordering/disordering, a supersaturation-dependent contact angle, and to a surface spinodal; the latter restricted to the h ⬍ 0 region 共 ⬍ / 2兲. 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