Sharp-interface projection of a fluctuating phase-field model

Sharp-interface projection of a fluctuating phase-field model R. Benítez and L. Ramírez-Piscina Departament de Física Aplicada, Universitat Politècnica de Catalunya, Doctor Marañón 44, E-08028 Barcelona, Spain 共Received 18 October 2004; revised manuscript received 18 March 2005; published 23 June 2005兲 We present a derivation of the sharp-interface limit of a generic fluctuating phase-field model for solidification. As a main result, we obtain a sharp-interface projection which presents noise terms in both the diffusion equation and in the moving boundary conditions. The presented procedure does not rely on the fluctuationdissipation theorem, and can therefore be applied to account for both internal and external fluctuations in either variational or nonvariational phase-field formulations. In particular, it can be used to introduce thermodynamical fluctuations in nonvariational formulations of the phase-field model, which permit to reach better computational efficiency and provide more flexibility for describing some features of specific physical situations. This opens the possibility of performing quantitative phase-field simulations in crystal growth while accounting for the proper fluctuations of the system. DOI: 10.1103/PhysRevE.71.061603 PACS number共s兲: 81.10.Aj, 05.40.⫺a, 64.70.Dv, 68.08.⫺p I. INTRODUCTION In recent years, phase-field models have emerged as an efficient technique to simulate interfacial phenomena in nonequilibrium systems 关1兴. This method has mainly been developed for solidification 关2–4兴, but has also successfully been applied to other problems, such as grain boundaries 关5兴, crack propagation 关6兴, viscous fingering 关7兴or vesicle dynamics 关8兴. The phase-field approach introduces an equation for a continuous variable ␾ 共r,t兲, which appears as an order parameter, and takes distinct, constant values in the different phases. The interface is then described by the level set ␾ =constant, and the transition between both phases takes place in a diffuse interface of thickness W. The model is completed by coupling the ␾ equation with a diffusion field which acts as a driving force for the motion of the front. The behavior of the diffuse interface can then be computed by the integration of a set of partial differential equations for the whole system, therefore avoiding the explicit tracking of the interface position. This has practical advantages over using the free boundary conditions that are characteristic of a moving boundary description. Phase-field models are usually constructed to recover the classical moving boundary dynamics in the so-called sharp-interface limit as W→0关9兴. This limit is taken by means of a systematic asymptotic expansion on the interface width, and allows the model parameters to be determined in terms of the physical properties of the system. In early phase-field formulations, the model equations were derived from the variational minimization of a global free-energy functional for the heterogeneous system. Such variational formulations, however, in spite of their appealing structure, presented poor computational efficiency and did not permit to obtain truly quantitative results. For this reason, recently proposed phase-field models are not derived from a variational principle, but are specifically constructed to recover a certain moving boundary problem in the sharpinterface limit 关7,10兴. Besides presenting a better computational behavior, nonvariational phase-field formulations provide for more flexibility in the description of some particular features such as different transport properties in the solid and liquid phases 关11,12兴. On the other hand, the presence of fluctuations has always been an important issue in the study of pattern-forming instabilities in crystal growth 关13,14兴. Indeed, internal or external noises play the role of an initiation mechanism for the morphological deformations of the interface 关15,16兴. Thermal or solute fluctuations, for instance, must be taken into account in order to study important problems such as the dynamical selection of the primary spacing in directional solidification 关16兴or the formation of secondary instabilities 共side branches兲in dendritic growth 关17兴. Fluctuations were soon introduced into phase-field models in an ad hoc way as a controlled source of interfacial perturbations 关18兴. However, phase field models accounting for internal thermodynamical fluctuations have not been proposed until recently, and in the context of variational formulations 关19–22兴.In such variational cases, the statistical properties of the fluctuating terms can straightforwardly be determined by using the fluctuation-dissipation theorem, following the lines applied by Hohenberg and Halperin within the context of critical dynamics 关23兴. In nonvariational formulations, however, the fluctuation-dissipation relation becomes useless for this purpose because the dynamics of the system cannot be derived from a single free-energy functional. The aim of this work is to present a systematic procedure to account for the introduction of generic sources of noise in either variational or nonvariational phase-field models. To this end, we will perform the sharp-interface limit of a fluctuating phase-field model for solidification and explicitly obtain the properties of the projected noise terms that will appear in the moving boundary equations. This projection, which does not rely on the fluctuation-dissipation theorem, will be carried out by means of a hybrid asymptotic expansion which combines a standard sharp-interface limit with a small noise assumption for the intensities of the noise terms in the model. The structure of the resulting sharp-interface projection takes the form of a moving boundary problem, which now includes bulk and interfacial stochastic terms. The statistical properties of these terms are related to those of the noises appearing in the starting phase-field equations. PHYSICAL REVIEW E 71, 061603 共2005兲 1539-3755/2005/71共6兲/061603共9兲/$23.00 ©2005 The American Physical Society061603-1 The extension of our procedure to thin-interface asymptotics 关4兴is straightforward and is not presented here for the sake of clarity. As a particular case, this analytical technique will enable a prescription for the introduction of internal thermodynamical fluctuations in nonvariational phase-field models, subject only to the constraint of providing the correct interface equilibrium fluctuations. This approach will also allow for the consideration of more general noise sources of an external origin, such as experimental imperfections or controlled perturbations, which do not follow equilibrium statistics. It is worth pointing out that while the calculations will be performed within the framework of the symmetric solidification model, the approach can be easily extended to one-sided formulations 关12兴. This work has been organized as follows: The stochastic model equations are presented in Sec. II. The asymptotic stochastic procedure is developed in Sec. III, which has been divided into four different sections: Secs. III A and III B are dedicated to find solutions of the equations in the inner and outer asymptotic regions, respectively. The solvability conditions for the inner expansion are imposed in Sec. III C, whereas in Sec. III D we perform the asymptotic matching between the inner and outer stochastic fields in order to obtain the form of the projected equations. The projected problem is then compared in Sec. IV with the standard Lanvegin formulation for solidification 关13,14兴, allowing for the determination of the model parameters in the case of having internal noises of a thermodynamical origin. A numerical test for the validity of the approach is reported in Sec. V, and Sec. VI is devoted to present some discussion and concluding remarks. II. MODEL EQUATIONS Our approach starts from a generic nonvariational phasefield model, which applies for both the solidification of a pure substance and for the symmetric solidification of a dilute alloy with a constant miscibility gap 关4兴, ␣ ␧2 ⳵ t ␾ =␧2ⵜ2 ␾ −f⬘共 ␾ 兲−␧␭g⬘共 ␾ 兲u+␧3/2 ␩ 共r,t兲, 共2.1兲 ⳵ tu=ⵜ2u+1 2 ⳵ th共 ␾ 兲−⵱·q共r,t兲,共2.2兲 where ␣ is a parameter determining the time scale of the phase-field dynamics and ␭accounts for the coupling strength between ␾ and the diffusion field u. We choose g共 ␾ 兲 and h共 ␾ 兲to be odd polynomial functions of ␾ satisfying the limiting conditions g⬘共±1兲=0 and h共±1兲= ±1, and f共 ␾ 兲to be given by the standard double-well potential f共 ␾ 兲=1 4 ␾ 4−1 2 ␾ 2.共2.3兲 In the model equations Eqs. 共2.1兲and 共2.2兲,uis a reduced diffusive field defined by u=共T−TM兲/共L/c兲in the case of pure substances and by u=关c−1 2共cS 0+cL 0兲兴/⌬c0+1 2g共 ␾ 兲for symmetric alloys, where TMis the melting temperature, Lthe latent heat per unit volume, cthe specific heat per unit volume, and ⌬c0⬅cL 0−cS 0, being cS 0,cL 0the solid and liquid equilibrium concentrations of the alloy, respectively. The two minima ␾ =±1off共 ␾ 兲in Eq. 共2.3兲correspond, respectively, to the solid and liquid phases of the system, so the interface will be represented by the transition zone between these two values. Space and times in Eqs. 共2.1兲and 共2.2兲have been scaled out using a characteristic length land a time scale ␯ =l2/D, where Dis the thermal or chemical diffusivity of the substance. The control parameter ␧=W/lis the scaled interface thickness, and will be the small parameter in which the formal expansions will be carried out. Fluctuations appear in the model as a nonconserved noise term ␩ in the equation for the phase field, and as a conserved stochastic current qin the diffusion equation. These fluctuations account for generic noise sources of either an internal or an external origin. We assume that the noises are white and Gaussian with correlations given by 具 ␩ 共r,t兲 ␩ 共r⬘,t⬘兲典 =2 ␴ ␾ 2 ␦ 共r−r⬘兲 ␦ 共t−t⬘兲,共2.4兲 具qi共r,t兲qj共r⬘,t⬘兲典 =2 ␴ u 2 ␦ ij ␦ 共r−r⬘兲 ␦ 共t−t⬘兲.共2.5兲 In the proposed phase field model, parameters such as ␣ ,␭, and the noise amplitudes ␴ ␾ , ␴ u, are intended to represent 共or to be directly related to兲physical parameters. On the contrary, the scaled interface width ␧has been introduced as an expansion parameter. As a matter of fact, Eqs. 共2.1兲and 共2.2兲 have been constructed so that the resulting dynamics 共in the double limit of sharp interface and small noise兲will be independent of ␧. In particular, the scaling factor ⑀ 3/2 of the noise term in Eq. 共2.1兲has been introduced in order to make the fluctuations of the interfacial dynamics, as will be obtained below, independent of ⑀ . The details of this calculation and the presentation of the results are given in the next section. III. HYBRID ASYMPTOTIC EXPANSION In order to deal with fluctuating phase-field models, the standard asymptotic expansion, performed in terms of a small interface thickness, should be complemented with a small noise assumption. The combination of these approaches will give rise to a hybrid asymptotic procedure. To this end, the small noise assumption will be imposed by assuming that ␴ ␾ , ␴ uobey some order relations with the interface thickness ␧. Namely, we take ␴ ␾ ⬃O共␧3/2兲,共3.1兲 ␴ u⬃O共␧2兲,共3.2兲 which will permit along the expansion procedure to maintain the fluctuating terms as small perturbations at the desired order in a consistent way. Relations 共3.1兲and 共3.2兲should not be understood as any explicit dependence of these parameters on ␧, but only as a way to formalize a double expansion in terms of a single vanishing parameter, the interface thickness, ␧→0. Our method closely follows the standard asymptotic procedure described in Ref. 关11兴. We start by dividing the system into two different regions, an outer region far from the interface at distances much greater than ␧, where the phase R. BENÍTEZ AND L. RAMÍREZ-PISCINA PHYSICAL REVIEW E 71, 061603 共2005兲 061603-2 field presents the two constant values ␾ = ±1 representing the solid and liquid phases at each side of the interface, and an inner region located around the interface up to distances of order ␧, where the phase field varies between these two values. In the limit ␧→0, solutions for the fields in both regions should match order by order in ␧at some intermediate distance rM, which can be taken of order rM⬃␧1/2. A. Outer region In the outer region, the equations can be solved at each order by expanding the fields in powers of ␧as u=u0+␧u1+␧2u2+O共␧3兲,共3.3兲 ␾ = ␾ 0+␧ ␾ 1+␧2 ␾ 2+O共␧3兲,共3.4兲 and by expanding in Taylor series around ␾ = ␾ 0the functions f,gappearing in the model equations. If we use the order relations Eqs. 共3.1兲and 共3.2兲, the noise terms can be assumed to be of orders ␩ ⬃O共␧3/2兲,共3.5兲 q⬃O共␧2兲,共3.6兲 and the outer equations can then be obtained at each order in ␧. 1. Zero order At the leading order 共␧0兲, the outer equations are given by f⬘共 ␾ 0兲=0, 共3.7兲 ⳵ tu0=ⵜ2u0+1 2 ⳵ th共 ␾ 0兲.共3.8兲 Introducing the function f共 ␾ 兲into Eq. 共3.7兲, we obtain ␾ 0 = ±1, and using that h共±1兲= ±1, Eq. 共3.8兲adopts the form ⳵ tu0=ⵜ2u0.共3.9兲 2. First order At first order in ␧,wefind f⬙共 ␾ 0兲 ␾ 1=−␭g⬘共 ␾ 0兲u0,共3.10兲 ⳵ tu1=ⵜ2u1+1 2 ⳵ t关 ␾ 1h共 ␾ 0兲兴.共3.11兲 From Eq. 共3.10兲we determine ␾ 1=0 by noting that the functions f,gsatisfy f⬙共±1兲⫽0 and g⬘共±1兲=0, and introducing ␾ 1=0 into Eq. 共3.11兲we get ⳵ tu1=ⵜ2u1.共3.12兲 3. Second order At second order 共␧2兲, and using that ␾ 1=0, the random current qappears in the equation for the outer diffusive field f⬙共 ␾ 0兲 ␾ 2=−␭g⬘共 ␾ 0兲u1,共3.13兲 ⳵ tu2=ⵜ2u2+1 2 ⳵ t关h⬘共 ␾ 0兲 ␾ 2兴−⵱·q.共3.14兲 Using that g⬘共±1兲=0 and f⬙共±1兲⫽0, Eq. 共3.13兲is solved by ␾ 2=0, and the second term on the right-hand side of Eq. 共3.14兲can be neglected. Collecting the results obtained at the three first orders, the outer fields are given, up to second order in ␧,by ␾ = ±1+O共␧3兲,共3.15兲 ⳵ tu=ⵜ2u−⵱·q共r,t兲+O共␧3兲.共3.16兲 B. Inner region For the inner region, we write Eqs. 共2.1兲and 共2.2兲in a curvilinear coordinate system centered at the interface. The idea is that the solvability condition for the very existence of solutions of these transformed equations will provide the evolution of the coordinate system, i.e., of the interface, which in fact constitutes the solution we are looking for. To define this coordinate system by maintaining it smooth at small scales, we use an auxiliary coarse grained field defined as a local spatial and temporal average of the fluctuating field ␾ . The surface corresponding to the level set of this coarse grained field 具 ␾ 共r,t兲典=0 allows to define the 3D orthogonal curvilinear coordinate system 共r,s1,s2兲, where ris a normal distance from the surface and s1,s2are the arclength distances measured along the principal curvature directions of the surface. Furthermore, we introduce in the inner region the scaled normal coordinate ␳ =r/␧and the scaled time ␶ =t/␧. We use capital letters to refer to all the fields when written in the inner region. After some manipulation, and keeping terms up to second order in ␧, we obtain the inner equations in the frame of the moving interface ␣ ␧ 冉 d d ␶ −v ⳵ ␳ 冊 ⌽= ⳵ ␳ 2⌽+␧ ␬ ⳵ ␳ ⌽−␧2 ␳ 共 ␬ 2−2⌸兲 ⳵ ␳ ⌽ +␧2兺 i=1,2 ⳵ si 2⌽−f⬘共⌽兲−␧␭g⬘共⌽兲U +␧1/2H共 ␳ ,s, ␶ 兲,共3.17兲 1 ␧ 冉 d d ␶ −v ⳵ ␳ 冊 U=1 ␧2 ⳵ ␳ 2U+1 ␧关 ␬ −␧ ␳ 共 ␬ 2−2⌸兲兴 ⳵ ␳ U +兺 i=1,2 ⳵ si 2U−v 2␧ ⳵ ␳ h共⌽兲+1 2␧ dh共⌽兲 d ␶ −1 ␧2 ⳵ ␳ Q ␳ 共 ␳ ,s, ␶ 兲,共3.18兲 where v=v共s, ␶ 兲is the local normal velocity of the interface, and we have introduced ␬ = ␬ 1+ ␬ 2and ⌸= ␬ 1 ␬ 2as the mean and Gaussian curvatures of the surface, being ␬ 1共s, ␶ 兲, ␬ 2共s, ␶ 兲its principal curvatures. The fluctuating functions H共 ␳ ,s, ␶ 兲=␧ ␩ 共r,t兲and Q共 ␳ ,s, ␶ 兲=␧q共r,t兲in Eqs. 共3.17兲and 共3.18兲stand for the renormalized noises in the inner region, and Q ␳ corresponds to the normal component of the stochastic current Q. The SHARP-INTERFACE PROJECTION OF A FLUCTUATING…PHYSICAL REVIEW E 71, 061603 共2005兲 061603-3 correlations of these noise terms are given by 具H共 ␳ ,s, ␶ 兲H共 ␳ ⬘,s⬘, ␶ ⬘兲典 =2 ␴ ␾ 2 ␦ 共 ␳ − ␳ ⬘兲 ␦ 共s−s⬘兲 ␦ 共 ␶ − ␶ ⬘兲, 共3.19兲 具Qi共 ␳ ,s, ␶ 兲Qj共 ␳ ⬘,s⬘, ␶ ⬘兲典 =2 ␴ u 2 ␦ ij ␦ 共 ␳ − ␳ ⬘兲 ␦ 共s−s⬘兲 ␦ 共 ␶ − ␶ ⬘兲, 共3.20兲 so that the orders in ␧of H共 ␳ ,s, ␶ 兲and Q共 ␳ ,s, ␶ 兲are those of ␴ ␾ and ␴ u, respectively. Note that the renormalization of the noise terms is a direct consequence of the scaling of the t,r coordinates in the inner region. Indeed, noise terms give rise to an ⑀ factor when written in the inner region due to the rescaling in both time and normal distances of the delta functions ␦ 共 ␳ 兲=␧ ␦ 共r兲and ␦ 共 ␶ 兲=␧ ␦ 共t兲. Now we can see how the small noise assumption has been implemented in our approach. With the choice given by Eqs. 共3.1兲and 共3.2兲for the orders in ␧of the noise amplitudes, ␧1/2His O共␧2兲in Eq. 共3.17兲and −␧−2 ⳵ ␳ Q ␳ is O共␧0兲in Eq. 共3.18兲, i.e., one order higher than the temporal derivatives in these equations. In other words, both noise terms are first order perturbations for the dynamics in their respective inner equations. At this point, we proceed as in the outer region by expanding the inner fields and parameters in powers of ␧, U=U0+␧U1+␧2U2+O共␧3兲,共3.21兲 ⌽=⌽0+␧⌽1+␧2⌽2+O共␧3兲,共3.22兲 ␬ i= ␬ i0+␧ ␬ i1+O共␧2兲,i=1,2, 共3.23兲 ⌸=⌸0+␧⌸1+O共␧2兲,共3.24兲 v=v0+␧v1+O共␧2兲,共3.25兲 and inserting the expansions into the inner equations Eqs. 共3.17兲and 共3.18兲. The inner solutions will be obtained by matching with the outer solutions for ␳ →±⬁and r→0±, respectively. In the phase field equations, direct matching with the outer ␾ isolutions Eq. 共3.15兲provides the limiting boundary conditions for the ⌽iterms of the inner expansion ⌽0共 ␳ →±⬁兲=⫿1, 共3.26兲 ⌽i共 ␳ →±⬁兲= 0 for i=1,2. 共3.27兲 Similarly, the matching condition for the inner diffusion field requires that, at leading order, the gradients of U0vanish lim ␳ →±⬁ ⳵ ␳ U0=0. 共3.28兲 At higher orders, the matching conditions for the diffusive field present some additional difficulties due to the apparition of random terms, and will be discussed in detail in Sec. III D. 1. Zero order At leading order 共␧0for the ⌽equation, ␧−2 for the U equation兲, the inner equations are given by ⳵ ␳ 2⌽0−f⬘共⌽0兲=0, 共3.29兲 ⳵ ␳ 2U0=0. 共3.30兲 Inserting the double-well potential Eq. 共2.3兲into Eq. 共3.29兲, we obtain the standard kink solution for the phase field at zero order, ⌽0共 ␳ 兲= − tanh 冉 ␳ 冑2 冊 ,共3.31兲 which satisfies the matching condition Eq. 共3.26兲for ␳ →±⬁. Integrating Eq. 共3.30兲twice over ␳ , we have U0共 ␳ ,s, ␶ 兲=A共s, ␶ 兲+B共s, ␶ 兲 ␳ ,共3.32兲 where Aand Bare integration constants. Imposing the matching condition Eq. 共3.28兲, we determine B共s, ␶ 兲=0 and obtain a ␳ -independent solution for the diffusion field at zero order U0共s, ␶ 兲=A共s, ␶ 兲.共3.33兲 2. First order Using the solutions obtained at zero order, the first-order inner equations 共␧1for the ⌽equation, ␧−1 for the Uequation兲read ⍀⌽1=−共v0 ␣ + ␬ 0兲 ⳵ ␳ ⌽0+␭g⬘共⌽0兲U0,共3.34兲 ⳵ ␳ 2U1=dU0 d ␶ +v0 2 ⳵ ␳ h共⌽0兲,共3.35兲 where ⍀is the self-adjoint operator ⍀⬅ ⳵ ␳ 2−f⬙共⌽0兲and we have used that d⌽0/d ␶ =0 from Eq. 共3.31兲. As described by Almgren 关11兴, an expression for ⌽1can be obtained from Eq. 共3.34兲by inverting the operator ⍀, leading to ⌽1=⍀−1关−共v0 ␣ + ␬ 0兲 ⳵ ␳ ⌽0+␭g⬘共⌽0兲U0兴.共3.36兲 Since ⍀is an even operator and ⳵ ␳ ⌽0,g⬘共⌽0兲are even functions of ␳ ,⌽1is an even function of ␳ . Integrating Eq. 共3.35兲 twice over ␳ ,weget U1=D共s, ␶ 兲+C共s, ␶ 兲 ␳ +v0 2 冕 0 ␳ d ␳ ⬘h共⌽0兲+1 2 dU0 d ␶ ␳ 2, 共3.37兲 where Dand Care integration constants and we have used that ⳵ ␳ U0=0 关cf. Eq. 共3.33兲兴. 3. Second order The stochastic terms appear in the inner equations at second order 共␧2for the ⌽equation, ␧0for the Uequation兲, which are given by ⍀⌽2=−共 ␣ v1+ ␬ 1兲 ⳵ ␳ ⌽0−共 ␣ v0+ ␬ 0兲 ⳵ ␳ ⌽1+ ␣ d⌽1 d ␶ +1 2f⵮共⌽0兲⌽1 2+ ␳ 共 ␬ 0 2−2⌸0兲 ⳵ ␳ ⌽0+␭g⬘共⌽0兲U1 +␭g⬙共⌽0兲⌽1U0−␧−3/2H,共3.38兲 R. BENÍTEZ AND L. RAMÍREZ-PISCINA PHYSICAL REVIEW E 71, 061603 共2005兲 061603-4 ⳵ ␳ 2U2=−共v0+ ␬ 0兲 ⳵ ␳ U1+dU1 d ␶ +v1 2 ⳵ ␳ h共⌽0兲 +v0 2 ⳵ ␳ 关h⬘共⌽0兲⌽1兴−兺 i=1,2 ⳵ si 2U0−1 2h⬘共⌽0兲d⌽1 d ␶ +1 ␧2 ⳵ ␳ Q ␳ .共3.39兲 The first equation Eq. 共3.38兲will be used in the next section when imposing the second order solvability condition of the problem. Integrating Eq. 共3.39兲twice over ␳ ,wefind U2=F共s, ␶ 兲+E共s, ␶ 兲 ␳ −共v0+ ␬ 0兲 冕 0 ␳ d ␳ ⬘U1 + 冕 0 ␳ d ␳ ⬘ 冕 0 ␳ ⬘ d ␳ ⬙dU1 d ␶ −1 2 ⳵ s 2U0 ␳ 2+v1 2 冕 0 ␳ d ␳ ⬘h共⌽0兲 +v0 2 冕 0 ␳ d ␳ ⬘h⬘共⌽0兲⌽1+1 ␧2 冕 0 ␳ d ␳ ⬘Q ␳ 共 ␳ ,s, ␶ 兲,共3.40兲 where Fand Eare again ␳ -independent integration constants. C. Solvability conditions We impose now the solvability conditions for the inner problem, which at first and second orders are, respectively, given by 冕 −⬁ ⬁ 共 ⳵ ␳ ⌽0兲⍀⌽jd ␳ = 0 for j=1,2. 共3.41兲 Inserting Eq. 共3.34兲into the first order solvability condition, we get −共 ␣ v0+ ␬ 0兲I1−␭I2U0=0, 共3.42兲 which allows to determine U0as U0共s, ␶ 兲=− ␣ I1 ␭I2 v0−I1 ␭I2 ␬ 0,共3.43兲 where I1and I2are integral constants given by I1= 冕 −⬁ ⬁ d ␳ 共 ⳵ ␳ ⌽0兲2,共3.44兲 I2=− 冕 −⬁ ⬁ d ␳ g⬘共⌽0兲共 ⳵ ␳ ⌽0兲.共3.45兲 Imposing the second order solvability condition Eq. 共3.41兲, and taking into account the parity of the potentials f,g,hand of the inner solutions ⌽0,⌽1, we determine an expression for the constant Din Eq. 共3.37兲, D共s, ␶ 兲=−共 ␣ v1+ ␬ 1兲I1 ␭I2 +v0 I3 2I2 +I4 2I2 + ␣ I5 ␭I2 −␧−3/2Z共s, ␶ 兲 ␭I2 , 共3.46兲 where I3,I4, and I5are defined by I3= 冕 −⬁ ⬁ d ␳ 共 ⳵ ␳ ⌽0兲g⬘共⌽0兲 冕 0 ␳ d ␳ ⬘h共⌽0兲,共3.47兲 I4=dU0 d ␶ 冕 −⬁ ⬁ d ␳ 共 ⳵ ␳ ⌽0兲g⬘共⌽0兲 ␳ 2,共3.48兲 I5= 冕 −⬁ ⬁ d ␳ 共 ⳵ ␳ ⌽0兲d⌽1 d ␶ ,共3.49兲 and Zis a stochastic term given by Z共s, ␶ 兲= 冕 −⬁ ⬁ d ␳ 共 ⳵ ␳ ⌽0兲H共 ␳ ,s, ␶ 兲,共3.50兲 whose statistical properties are given by 具Z共s, ␶ 兲Z共s⬘, ␶ ⬘兲典 =2I1 ␴ ␾ 2 ␦ 共s−s⬘兲 ␦ 共 ␶ − ␶ ⬘兲.共3.51兲 D. Matching of fluctuating fields At this point, we continue by imposing the remaining asymptotic matching conditions of the problem. However, the matching of the diffusion field presents some subtleties due to its fluctuating character. The main problem is that, at second order in ␧, the Ufield fluctuates in the normal direction 关cf. Eq. 共3.40兲兴, and hence cannot be written as a simple asymptotic expansion for ␳ →±⬁, preventing the matching with the outer field. This difficulty can be overcome by introducing an auxiliary matching function defined in both regions as ␹ 共r,s,t兲=u共r,s,t兲− 冕 0 r dr⬘qr共r⬘,s⬘,t兲,共3.52兲 X共 ␳ ,s, ␶ 兲=U共 ␳ ,s, ␶ 兲− 冕 0 ␳ d ␳ ⬘Q ␳ 共 ␳ ⬘,s, ␶ 兲.共3.53兲 In view of Eq. 共3.40兲, it is easy to see that the inner auxiliary function Xintroduced in Eq. 共3.53兲is smooth up to order ␧2 in the matching region rM. Explicitely, if Xis asymptotically expanded for ␳ →±⬁as X⬃T+S ␳ +R ␳ 2+O共 ␳ 3兲,共3.54兲 and the outer matching function ␹ is expanded in Taylor around r=0±by ␹ ⬇ ␹ 共0±兲+ ⳵ r ␹ 共0±兲·r+1 2 ⳵ r 2 ␹ 共0±兲·r2+O共r3兲, 共3.55兲 the inner and outer terms can be matched at rM⬃␧1/2 in the limit ␧→0 to obtain the matching relations T= ␹ 共0±兲,共3.56兲 S=␧ ⳵ r ␹ 共0±兲,共3.57兲 SHARP-INTERFACE PROJECTION OF A FLUCTUATING…PHYSICAL REVIEW E 71, 061603 共2005兲 061603-5 R=␧2 2 ⳵ r 2 ␹ 共0±兲.共3.58兲 The last step is to expand the previous equations 共3.56兲–共3.58兲in powers of ␧to complete the matching at each order in ␧. 1. First order At first order in ␧, the inner field U1given by Eq. 共3.37兲 can be asymptotically expanded for ␳ →±⬁as U1⬃1 2 dU0 d ␶ ␳ 2+ 冉 C⫿v0 2 冊 ␳ +D+v0 2J1 ±,共3.59兲 where Dis given by Eq. 共3.46兲and J1 ±= 冕 0 ±⬁ d ␳ 兵h关⌽0共 ␳ 兲兴 ±1其,共3.60兲 where we have used that hsatisfies h共±1兲= ±1 and the far field condition Eq. 共3.26兲. Since h共⌽0兲is an odd function of ␳ , we have J1 +=J1 −⬅J1.共3.61兲 2. Second order Similarly, the second order inner solution for the diffusive field Eq. 共3.40兲can be expanded asymptotically for ␳ →±⬁as U2⬃F+E ␳ −共v0+ ␬ 0兲 冕 0 ␳ d ␳ ⬘dU1 d ␶ + 冕 0 ␳ d ␳ ⬘ 冕 0 ␳ ⬘ d ␳ ⬙dU1 d ␶ −1 2 ⳵ s 2U0 ␳ 2− 冕 0 ␳ d ␳ ⬘ 冕 0 ␳ ⬘ d ␳ ⬙h⬘共⌽0兲d⌽1 d ␶ +v0 2J2 ± +v1 2J1⫿v1 2 ␳ +1 ␧2 冕 0 ␳ d ␳ ⬘Q ␳ 共 ␳ ,s,t兲,共3.62兲 where J2 ±= 冕 0 ␳ d ␳ ⬘h⬘共⌽0兲⌽1,共3.63兲 and we have used the far field conditions Eqs. 共3.26兲and 共3.27兲. Inserting the expressions Eqs. 共3.43兲,共3.59兲, and 共3.62兲into the right-hand side of Eq. 共3.53兲, we can determine the parameters R,S, and Tin the far field expansion of the matching function X关cf. Eq. 共3.54兲兴 and perform the matching with the outer function ␹ 关cf. Eq. 共3.55兲兴. Imposing the third matching condition Eq. 共3.58兲at first order in ␧, we determine that dU0 d ␶ =0, 共3.64兲 which, using the relation Eq. 共3.36兲, brings to d⌽1 d ␶ =0, 共3.65兲 and therefore the integral constants I4and I5defined in Eqs. 共3.48兲and 共3.49兲vanish, I4=I5=0. 共3.66兲 From the two first orders of Eq. 共3.56兲, we get an expression for the outer diffusive field at the interface valid up to first order, u共0±兲=− I1 ␭I2 共 ␣ v+ ␬ 兲+␧v0 2 冉 I3 I2 +J1 冊 −z共s,t兲 ␭I2 +O共␧2兲, 共3.67兲 where z共s,t兲=Z共s, ␶ 兲␧−1/2 is a stochastic term whose statistical properties can be determined from Eq. 共3.51兲and are given by 具z共s,t兲z共s⬘,t⬘兲典 =2I1 ␴ ␾ 2 ␦ 共s−s⬘兲 ␦ 共t−t⬘兲.共3.68兲 Note that the projected interfacial noise term has neither in Eq. 共3.67兲nor in Eq. 共3.68兲any explicit dependence in ␧, which is a direct consequence of the ␧3/2 factor introduced in the noise term of Eq. 共2.1兲. Indeed, this is the reason why such factor was introduced in the formulation of the model. The calculation is completed by imposing the matching condition Eq. 共3.57兲up to second order, which can be written as v0+␧v1=兩 ⳵ r ␹ 兩+ −+O共␧2兲.共3.69兲 Inserting Eq. 共3.52兲into Eq. 共3.69兲, we get a heat/mass conservation equation valid up to first order in ␧, v=v0+␧v1=关 ⳵ ru兴+ −−关qr兴+ −+O共␧2兲,共3.70兲 where qraccounts for a normal stochastic current across the interface. This term, although being of order ␧2, has not been neglected in Eq. 共3.70兲in order to not break mass conservation in the stochastic diffusion equation 共3.16兲, which is valid up to second order. This last equation completes the sharp-interface projection of the stochastic phase-field model of Eqs. 共2.1兲–共2.5兲. This projection constitutes the main result of this paper, and is given by the diffusion equation Eq. 共3.16兲, with the noise of Eq. 共2.5兲, supplemented with two moving boundary conditions at the interface, the conservation condition Eq. 共3.70兲, and the Gibbs-Thomson equation 共3.67兲, where a projected interfacial noise appears with correlation given by Eq. 共3.68兲. Note that the projected boundary conditions at the interface equations 共3.70兲and 共3.67兲are obtained at the order immediately lower than the order at which the asymptotic expansion is performed. While the general lines of the calculation follow the standard sharp-interface asymptotics, we have included the fluctuation terms during all the procedures, which have been projected in the weak noise limit. This calculation is thus similar to the front dynamics projection performed in Ref. 关24兴. Indeed, the projected interfacial noise appearing in Eq. 共3.67兲is the analogous counterpart of the noise term of the projected eikonal front equation of Ref. 关24兴. R. BENÍTEZ AND L. RAMÍREZ-PISCINA PHYSICAL REVIEW E 71, 061603 共2005兲 061603-6 IV. INTERNAL FLUCTUATIONS IN A GENERIC PHASE FIELD MODEL Thus far, the noises considered in this work are intended to account for both external and internal sources of fluctuations. Nevertheless, it is worth pointing out that the resulting stochastic sharp-interface equations are similar to those postulated in the Langevin formulation of solidification due to Karma 关13,14兴共see also Ref. 关25兴兲, which was constructed to follow equilibrium statistics. This offers the possibility of using the results above to provide generic 共not necessarily variational兲phase-field models with the correct equilibrium fluctuations. To illustrate this, let us consider the Langevin sharp-interface equations 关13,14兴 ⳵ tuSI =ⵜ2uSI −⵱·qSI共r,t兲,共4.1兲 vSI =关 ⳵ ruSI兴+ −−关qr SI兴+ −,共4.2兲 uSI共0兲=−d0 ␬ − ␤ v+ ␪ 共r,t兲,共4.3兲 where qSI and ␪ are fluctuating terms with correlations given by 具qi SI共r,t兲qj SI共r⬘,t⬘兲典 =2KBTM 2c L2ld ␦ ij ␦ 共r−r⬘兲 ␦ 共t−t⬘兲, 共4.4兲 具 ␪ 共s,t兲 ␪ 共s⬘,t⬘兲典 =2KBTM 2c ␤ L2ld ␦ 共s−s⬘兲 ␦ 共t−t⬘兲.共4.5兲 The Gibbs-Thompson equation Eq. 共4.3兲can be compared with Eq. 共3.67兲and the diffusion equation 共4.1兲with Eq. 共3.16兲. This comparison enables the determination of the phase-field parameters in terms of physical and substance parameters, which are given by the equations ␭=I1 I2d0 ,共4.6兲 ␣ = ␤ d0 ,共4.7兲 ␴ u 2=KBTM 2c L2ld,共4.8兲 ␴ ␾ 2=I1KBTM 2c ␤ d0 2L2ld.共4.9兲 In the last relations, the two first equations 共4.6兲and 共4.7兲are the usual expressions determined by the standard asymptotic procedure, whereas Eq. 共4.8兲reflects the identification between the conserved stochastic currents of both phase-field model and sharp-interface projection. In this sense, a major result of our approach has been the derivation of an expression for the noise strength of the phase field, that is Eq. 共4.9兲, from the above calculations. With this election of the model parameters, the phase-field simulations will present the correct equilibrium statistics in the limit of small interface thickness ␧→0. Therefore, the nonvariational phase-field formulation of Eqs. 共2.1兲and 共2.2兲 can be used to quantitatively account for thermodynamical fluctuations in solidification processes. V. TEST OF THE APPROACH In order to test the validity of our approach, we have performed two-dimensional 共2D兲phase-field simulations to obtain the power spectrum of the interfacial fluctuations of a solid-liquid stationary flat interface. Introducing the Fourier transform of the interface position ␰ 共y,t兲as ␰ k共t兲 =兰dy ␰ 共y,t兲e−iky, the power spectrum of a stationary planar front in scaled variables is given by S共k兲=具 ␰ k ␰ −k典= 冕 dk⬘ 2 ␲ 具 ␰ k ␰ k⬘典=KBTM ␥ 1 k2,共5.1兲 where ␥ =ldL2d0/TMcis the scaled interfacial surface energy. In the simulations, space and times have been scaled using length and time scales of l=10−8 m and ␯ =9⫻10−10 s, respectively. The functions h,ghave been chosen to be h共 ␾ 兲 = ␾ and g⬘共 ␾ 兲=共1− ␾ 2兲2so that the model does not have a variational structure. The substance parameters used in the simulations correspond to the values of the pure SCN in the three-dimensional 共3D兲case, and are given by d0=0.2817, ␤ =3.0331 关26,27兴, and ␴ u 2=0.001 432. For this choice, and using Eqs. 共4.6兲–共4.9兲, the phase-field parameters take the values ␭=3.13, ␣ =10.76, and ␴ ␾ 2=0.051 58. The interface thickness has been taken to be ␧=0.3. The simulations have been implemented with a finite differences scheme on a 50⫻512 lattice with ⌬x=⌬y=0.2 and ⌬t=0.005. We have used the initial conditions ␾ 共x,y,0兲 =−tanh共x/␧冑2兲,u共x,y,0兲=0. Nonflux and periodic boundary conditions have been imposed in the xand ydirections, respectively. The numerical implementation of the stochastic terms has been carried out by generating Gaussian-distributed random numbers at each of the lattice sites. The correlations of these numbers can be determined by discretizing the time and spatial delta functions in Eqs. 共2.4兲and 共2.5兲by substituting ␦ 共x−x⬘兲→ ␦ ii⬘/⌬xand ␦ 共t−t⬘兲→ ␦ nn⬘/⌬t. The divergence of the stochastic current in Eq. 共2.2兲has been discretized by using a forward differences scheme ⵱·兩q共r,t兲兩i,j=关qx共i +1,j兲−qx共i,j兲兴/⌬x+关qy共i,j+1兲−qy共i,j兲兴/⌬y. The power spectrum statistics has been obtained as a time average among the last 3⫻106time steps in a long-term simulation of 3.5⫻106steps, and is represented by a dashed line in Fig. 1. The solid line in Fig. 1 depicts the theoretical prediction given by Eq. 共5.1兲and, as it can be seen, an excellent agreement is found between theoretical and numerical results. The vertical dashed line in the figure represents the wavelength associated with the effective thickness of the interface, and determines the expected breakdown of the phase-field description. VI. DISCUSSION AND CONCLUSIONS To summarize, we have obtained an asymptotic projection of the fluctuating phase-field equations 共2.1兲and 共2.2兲to a SHARP-INTERFACE PROJECTION OF A FLUCTUATING…PHYSICAL REVIEW E 71, 061603 共2005兲 061603-7 sharp interface description. This has been worked out by means of a hybrid asymptotic procedure, combining sharp interface and small noise limits. As a result, the projected equations adopt the form of a moving boundary problem with a conserved stochastic force qin the equations for the diffusion field Eqs. 共3.16兲and 共3.70兲, and an interfacial noise z共s,t兲in the Gibbs-Thompson condition Eq. 共3.67兲. Other authors have previously introduced fluctuations in phasefield models 关20–22兴, but their approaches applied only for the case of variational formulations and were restricted to noises from a thermodynamical origin. In this context, it has been claimed 关20兴that the presence of a nonconserved noise such as ␩ in the equation for the phase field is not relevant for the dynamics of the phase-field model, and thus could be omitted in simulations. In order to check the importance of the nonconserved phase-field noise, we have carried out a numerical test with the same parameters reported in Sec. V but taking ␴ ␾ =0. In this case, the power spectrum is plotted as a dotted line in Fig. 1. The clear disagreement with both the theoretical prediction and the simulations of the complete model indicates that the phasefield noise is indeed necessary in order to obtain quantitative results. This can be explained by noting that Eq. 共3.67兲establishes a direct relation between the nonconserved phasefield noise ␩ and the interfacial fluctuations appearing in the Gibbs-Thompson equation, usually associated to kinetic attachment effects 关13兴. Thus, the small significance of the ␩ noise reported in Ref. 关20兴is probably due to the fact that the stationary power spectrum was calculated in the limit of vanishing kinetics ␤ =0. Therefore, we conclude that, in the presence of kinetic effects, the phase-field noise is relevant for a quantitative description of the solidification process. It is interesting to discuss the scaling of the noise terms as proposed on the one hand in Eqs. 共3.1兲and 共3.2兲and on the other hand in the ␧3/2 factor explicitly appearing in the equation for the phase field, Eq. 共2.1兲. As it has already been commented, the assumption of the order relations Eqs. 共3.1兲 and 共3.2兲has permitted to manage a double expansion 共sharp interface and small noise兲by formally using a single small parameter. The specific powers of ␧appearing in these relations have been chosen for maintaining fluctuations as small perturbations for the dynamics of both inner and outer equations. On the contrary, the multiplicative factor of the noise term of the equation for the phase field, Eq. 共2.1兲, has a different motivation. It is well known that there are problems in the formulation of stochastic field equations when the noise terms are delta-correlated in space. In such cases, some kind of regularization is required. In Ref. 关24兴, for instance, this regularization was provided by the correlation length of the noise, in such a way that the results did depend on that parameter. In the present case, the regularization is provided by the interface width ␧. The projection of the bulk noise into the interface gives a fluctuation term that in principle should diverge as ⑀ goes to zero. The ⑀ 3/2 factor of the noise term in Eq. 共2.1兲exactly cancels out this divergence, and has been introduced in the formulation of the model precisely to make results independent of ⑀ , specifically regarding the interfacial noise term zin Eqs. 共3.67兲and 共3.68兲. In conclusion, we have proposed an asymptotic procedure to obtain the sharp-interface projection of a generic 共not necessarily variational兲phase-field model with fluctuations. We have tested the validity of our approach by comparing the