document: the phase - field method has become the method of choice for simulating microstructure formation during solidification . 
   it owes its popularity mainly to its algorithmic simplicity : the cumbersome problem of tracking moving solid - liquid interfaces or grain boundaries is avoided by describing the geometry in terms of one or several phase fields . 
   the phase fields obey simple partial differential equations that can be easily coded by standard numerical methods . 
   the foundations of the phase - field method and its application to solidification have been the subject of several recent review articles @xcite , and it seems of little use to repeat similar information here . instead , in this paper 
   several topics are discussed where robust phase - field modelling tools are not yet available because some fundamental questions remain open . in sec . 
   [ sec2 ] , the thin - interface limit of two - sided phase - field models is examined , and it is shown that the currently available approaches can not in general eliminate all effects linked to the finite interface thickness . in sec . 
   [ sec3 ] , orientation - field models for polycrystalline solidification are discussed , and it is shown that the standard equation of motion usually written down for the orientation field is not appropriate for the evolution of coherent crystalline matter . 
   finally , in sec .  [ sec4 ] , the inclusion of microscopic fluctuations in the phase - field equations is reviewed , and it is shown that the standard approach can not be used in a straightforward way to investigate the process of nucleation .    the common point of these topics is that they pose challenges or limitations for straightforward computations . 
   indeed , a characteristic feature of the phase - field method is that its equations can often be written down following simple rules or intuition , but that their detailed properties ( which have to be known if quantitative simulations are desired ) become only apparent through a mathematical analysis that can be quite involved . 
   therefore , it is not always easy to perceive the limits of applicability of the method . 
   it is hoped that the present contribution will be helpful to point out some pitfalls and to stimulate further discussions that will facilitate the solution of these issues . 
   the precision and performance of phase - field models have been greatly enhanced in the last decade by a detailed control of their properties . 
   phase - field models are rooted in the mean - field description of spatially diffuse interfaces by order parameters . however , to be useful for simulating microstructure formation in solidification , phase - field models need to bridge the scale gap between the thickness of the physical solid - liquid interfaces and the typical scale of the microstructures . 
   this is achieved by increasing the interface width in the model , sometimes by several orders of magnitude . 
   obviously , this procedure magnifies any physical effect that is due to the diffuseness of the interface . 
   therefore , to guarantee precise simulations , all these effects have to be controlled and , if possible , eliminated . the privileged tool to achieve 
   this is the so - called _ thin - interface limit _ : 
   the equations of the phase - field model are analysed under the assumption that the interface thickness is much smaller than any other physical length scale present in the problem , but otherwise arbitrary . 
   the procedure of matched asymptotic expansions then yields the effective boundary conditions valid at the macroscale , which contain all effects of the finite interface thickness up to the order to which the expansions are carried out . 
   this procedure was pioneered by karma and rappel , who analysed the symmetric model of solidification ( equal diffusion constants in the solid and the liquid ) and obtained a thin - interface correction to the expression of the kinetic coefficient @xcite . 
   the use of this result has made it possible to carry out quantitative simulations of free dendritic growth of a pure substance , both at high and low undercoolings @xcite . 
   it turned out , however , that the generalisation of this method to a model with arbitrary diffusivities is far from trivial @xcite , since several new thin - interface effects appear , which can not all be eliminated simultaneously . 
   a solution to this problem was found later for the case of the one - sided model ( zero diffusivity in the solid ) with the introduction of the so - called antitrapping current @xcite , and it was shown that quantitative simulations of alloy solidification are possible with this model @xcite , including multi - phase @xcite and multi - component alloys @xcite . 
   recently , several extensions of the antitrapping current were put forward to generalise the approach to the case of finite diffusivity in the solid @xcite , and simulations were presented which show that the approach works well for the instability of a steady - state planar interface @xcite and for free dendritic growth @xcite . however , as will be shown below , this is only a partial solution to the problem of developing a general quantitative model , since there is a second , independent thin - interface effect that can not be removed by an antitrapping current , namely , the kapitza resistance .    for the sake of concreteness , consider the standard phase - field model for the solidification of a pure substance as discussed in refs . 
   the evolution equation for the phase field reads @xmath0 where @xmath1 is the phase field , with @xmath2 and @xmath3 corresponding to solid and liquid , respectively , @xmath4 is the relaxation time of the phase field , @xmath5 is the interface thickness , and @xmath6 is a dimensionless coupling constant . the field @xmath7 is a dimensionless temperature defined by @xmath8 , where @xmath9 , @xmath10 and @xmath11 are the melting temperature , latent heat , and specific heat , respectively . 
   it is assumed for simplicity that @xmath11 is the same in both phases . 
   the temperature is governed by a diffusion equation with a source term , @xmath12     + \frac 12 \partial_t h(\phi ) . 
   \label{diffusion}\ ] ] here , @xmath13 , which satisfies @xmath14 , is a function that describes the release or consumption of latent heat during the phase transition , and @xmath15 interpolates between the thermal diffusivities of the liquid and the solid , @xmath16 and @xmath17 , @xmath18 where the interpolation function @xmath19 satisfies @xmath20 and @xmath21 . for simplicity , 
   crystalline anisotropy has not been included in the above model because it is not necessary for the present discussion . 
   furthermore , the equations have been stated in the language of a two - sided thermal model , but with some modifications ( as detailed in refs . 
   @xcite ) , they also apply to the isothermal solidification of a binary alloy . in this case , @xmath7 is a dimensionless chemical potential ( conjugate to the concentration of one of the alloy components ) , and @xmath15 is the chemical diffusivity .    in the following 
   , two simple one - dimensional solutions of these equations will be analysed . 
   the first is a steady - state planar front that propagates with constant velocity @xmath22 in the positive @xmath23 direction into a liquid of undercooling @xmath24 ( @xmath25 for @xmath26 ) , and leaves behind a constant temperature . 
   this solution only exists if the liquid is undercooled beyond the hypercooling limit , that is , @xmath27 . 
   the sharp - interface solution to this problem is readily obtained and reads @xmath28 for an interface located at @xmath29 ( in the frame moving with the interface ) . here , 
   @xmath30 and @xmath31 are the limit values of the temperature when the interface is approached from the solid and the liquid side , respectively . in the standard formulation of the free boundary problem of solidification 
   , it is assumed that the temperature is the same on the two sides of the interface , @xmath32 . then , the use of the two boundary conditions @xmath33 , where @xmath34 is the linear kinetic coefficient , and @xmath35 ( the stefan boundary condition ) determines the solution , @xmath36 ( a simple consequence of heat conservation ) , and @xmath37 . 
   the phase - field equations can be analysed and related to this sharp - interface solution by the method of matched asymptotic expansions in the limit where the interface thickness @xmath5 is much smaller than the diffusion length @xmath38 . 
   this calculation has been presented in detail in refs . 
   @xcite and will not be repeated here . 
   the essential outcome is that , in general , the two asymptotes of the bulk phases do _ not _ correspond to the same temperature . the difference is given , to the lowest order , by @xmath39 , \label{tjump1}\ ] ] where @xmath40 is the equilibrium profile of the phase field . 
   the physical interpretation of this temperature jump is _ trapping _ : when the diffusivity decreases upon solidification , the heat generated at the rear of the interface gets trapped . in the alloy version of the model , 
   this is nothing but the well - known solute trapping effect . indeed , in sharp - interface models of alloy solidification the chemical potential exhibits a jump at the interface when solute trapping occurs . in the phase - field model , 
   the temperature profile through the interface is determined by the interplay between the rejection of latent heat and the diffusion away from the interface ; therefore , it is natural that the heat source function @xmath13 and the diffusivity function @xmath15 appear in eq . 
   ( [ tjump1 ] ) . 
   whereas , thus , this discontinuity is physically correct , it generates problems for simulations . to see this , is is sufficient to rewrite eq . 
   ( [ tjump1 ] ) in order to make the relevant scales apparent . since the only length scale in eq . 
   ( [ pf ] ) is the interface thickness @xmath5 , the equilibrium solution @xmath41 is a function only of the reduced variable @xmath42 . using this together with the interpolation of @xmath15 given by eq . 
   ( [ ddef ] ) , eq . 
   ( [ tjump1 ] ) becomes @xmath43 @xmath44\ ; 
   d\eta\qquad { \rm and } \label{fdef}\ ] ] @xmath45 the temperature jump is thus proportional to the velocity , the interface thickness , and the difference of the two integrals ; the latter depends only on the choice of the interpolation functions . if @xmath5 is the physical interface thickness ( a few angstroms ) , this effect is negligibly small , but if the interface thickness is increased by a large factor to make simulations feasible , this leads to potentially large errors in the simulations .    as discussed in detail in refs . 
   @xcite , it is not possible to eliminate this macroscopic discontinuity simply by the choice of appropriate interpolation functions , due to other constraints not discussed here . 
   the solution put forward in ref . 
   @xcite and further developed in ref . 
   @xcite is the introduction of an antitrapping current : eq . 
   ( [ diffusion ] ) is replaced by @xmath46 where the antitrapping current @xmath47 is given by @xmath48 where @xmath49 is a shorthand for the time derivative @xmath50 , @xmath51 is the unit normal vector to the interface , and @xmath52 is a new interpolation function . 
   this term induces a current which is directed from the solid to the liquid , and proportional to the interface velocity ( through the factor @xmath49 ) . 
   it thus `` pushes '' heat from the solid to the liquid side of the interface when the interface moves , and can be used to adjust the temperature jump at the interface . for the one - sided model ( @xmath53 ) with the standard choices @xmath54 and @xmath55 , it was shown that a constant @xmath56 leads to a vanishing jump in @xmath7 , because it modifies the function @xmath57 in eq . 
   ( [ fdef ] ) such that @xmath58 . 
   thus , continuity of the temperature between the two sides of the interface ( local equilibrium ) is restored for arbitrary @xmath5 and @xmath22 , as long as the asymptotic analysis remains valid . 
   recently , several authors have put forward generalisations of this approach @xcite for arbitrary ratio of the diffusivities . for the case 
   analysed above ( that is , the current far inside the solid vanishes ) , they reduce to the simple prescription that the same expression for the antitrapping current can be used , but with an additional prefactor that can be written as @xmath59 , @xmath60 indeed , the asymptotic analysis shows @xcite that in this way the temperature jump can be eliminated . 
   however , this is not the only thin - interface effect that can arise in the two - sided case . to see this , consider now a different situation , namely an immobile interface in a temperature gradient . 
   such an interface can be easily obtained in experiments by maintaining a pure substance between two walls which are held below and above the melting temperature , respectively . when the interface is stationary , @xmath61 by definition , and eq . 
   ( [ diffusion ] ) implies that the system is crossed by a constant heat current flowing from the liquid into the solid , @xmath62 with @xmath63 a positive constant . as before , 
   the centre of the interface is located at @xmath29 , and the solid is located in the domain @xmath64 . 
   this situation can be analysed without performing a perturbation expansion , since it is sufficient to integrate eq . 
   ( [ current ] ) to obtain a solution for @xmath7 , @xmath65 where @xmath66 is the temperature at @xmath29 . 
   the sharp - interface solution for this case is simply given by @xmath67 matching the asymptotes of the phase - field and sharp - interface expressions , it is straightforward to show that there is again a temperature jump given by @xmath68,\ ] ] this time proportional to the _ 
   current_. if the phase - field profile is replaced by its equilibrium shape , this can be rewritten as @xmath69 with @xmath70    this temperature jump corresponds to a surface thermal resistance , also called kapitza resistance , first found for an interface between liquid helium and metal @xcite . 
   indeed , in a sharp - interface picture it is generally necessary to assign a surface resistance to an interface for a complete description of heat transfer , because transport through an interface can be decomposed into three elementary steps : transport in one bulk phase , crossing of the interface , and transport in the other phase . 
   the surface resistance describes the kinetics associated with the crossing of the interface ( its inverse is sometimes referred to as the interfacial transfer coefficient ) . 
   it is characterised either by the value of the resistance , @xmath71 , or by a length that is obtained by dividing this resistance by the conductivity of the liquid phase . here , this characteristic length is simply @xmath72 , which is of the order of the interface thickness . since 
   this quantity is actually an interface excess of the inverse diffusivity ( in complete analogy to the interface excesses for equilibrium quantities obtained by the well - known gibbs construction ) , it can also be negative  this does not violate the laws of thermodynamics because the _ local _ transport coefficients are strictly positive . 
   if the surface resistance is finite , the temperature in the sharp - interface model is _ not _ continuous at the interface , but exhibits a jump that is proportional to the current crossing the interface . in the alloy version of the model 
   , this corresponds to a jump in chemical potential that is proportional to the solute flux @xcite . 
   such discontinuities have been thoroughly investigated @xcite , and can be measured in experiments @xcite and detected in molecular dynamics simulations @xcite for solid - liquid interfaces . 
   thus , like the trapping effect , the surface resistance is a natural effect that is proportional to the interface thickness . 
   if the interface thickness is to be upscaled , it should therefore also be eliminated . 
   however , is is immediately clear that this effect can not be eliminated by any antitrapping current proportional to @xmath49 as given by eq . 
   ( [ antinew ] ) : since the interface does not move , @xmath73 and the antitrapping current vanishes , independently of the current @xmath63 that crosses the interface . 
   the authors of both refs . 
   @xcite have recognised the importance of the current @xmath63 . 
   they have developed generalised expressions for the antitrapping current with coefficients that depend on the value of @xmath63 . 
   as long as the interface velocity remains non - zero , the formal asymptotic analysis shows that it is still possible to eliminate the temperature jump . 
   however , for a fixed current @xmath63 , the expressions of the coefficients diverge when @xmath22 tends to zero , such that the asymptotic analysis is not valid in this limit . 
   thus , it seems unlikely that this approach can be used as a robust method for simulations .    in summary , there exist two independent thin - interface effects , one proportional to @xmath22 , and one proportional to @xmath63 . on a very fundamental level , this is just the consequence of the fact that the interface motion is driven by a diffusion equation , which has two independent boundary conditions . 
   the corresponding physical quantities are the currents on the two sides of the interface , or one current and the velocity . a general solution to eliminate both thin - interface effects ( which are linearly independent ) does not seem to exist at this moment , but the above considerations can at least be used to obtain simple criteria when the prescription of eq . 
   ( [ antinew ] ) can be used . 
   indeed , eqs . 
   ( [ tjump1a ] ) and ( [ tjump2 ] ) show that if @xmath74 ( note that , since @xmath7 is dimensionless , @xmath63 has the dimension of a velocity ) , the kapitza effect is much smaller than the trapping effect , and can thus be neglected . 
   this is generally the case for equiaxed dendritic growth , in which the gradients outside the growing dendrite , which determine the growth speed , are much larger than the gradients inside the solid . 
   indeed , it was shown in ref .  @xcite that eq . 
   ( [ antinew ] ) works well in this case . 
   however , problems might arise in the case of alloy solidification in a temperature gradient or for multicomponent alloys with widely different solute diffusivities , since in this case large currents of heat or certain solutes may cross an interface whose velocity is controlled by a different diffusion field . 
   such cases have to be critically examined before simulation results can be trusted . 
   the size and shape of the crystalline grains formed upon solidification is one of the most important factors that determine materials properties . 
   therefore , phase - field models that are to be helpful for materials design must be capable of dealing with the evolution of polycrystals , both during solidification of individual columnar or equiaxed grains from the melt and during the subsequent evolution of the grain structure after impingement . 
   this can be achieved using the multi - phase - field approach @xcite , in which each grain is represented by a different phase field , even if they are of the same thermodynamic phase . 
   the properties of each individual grain boundary or interface can then be specified separately @xcite , and it has been demonstrated that good quantitative control of the grain boundary properties can be achieved @xcite . 
   the problem of handling several hundreds or even thousands of phase fields simultaneously can be solved by recognising that only a small number of fields are important at any given point of space ( see for example @xcite ) . 
   an alternative approach is the orientation - field method . 
   its starting point is the remark that it would be desirable , both for efficiency and simplicity , to formulate a model that works only with a small number of field variables . 
   indeed , the orientation of a crystal can be described by one scalar quantity ( an angle ) in two dimensions , and three scalars in three dimension ( for instance , the euler angles ) . 
   orientation - field models for pure substances in two dimensions that work with a single phase field , an orientation field ( the local angle of the crystalline structure with respect to a fixed coordinate system ) , and the temperature field were put forward in refs . 
   @xcite , and generalised for alloy solidification @xcite and to three dimensions @xcite . 
   while these models are elegant and simple in their formulation and therefore hugely appealing , it is pointed out here that the evolution equation of the angle field , which takes the form of a simple relaxation equation , does not correctly describe the microscopic evolution of the orientation field since it does not take into account the connectivity of matter and the resulting geometrical conservation laws .    for simplicity , anisotropy and crystallographic effects 
   will again be neglected , and it is sufficient to consider a two - dimensional system . 
   the dimensionless free energy of the orientation - field model is @xcite @xmath75 d\vec r , \label{modeli}\ ] ] where now @xmath76 and @xmath2 in the liquid and the solid , respectively , @xmath77 and @xmath78 are positive constants , @xmath79 and @xmath80 are monotonous functions that satisfy @xmath81 and @xmath82 , and @xmath83 is the local free energy density , with @xmath7 the same dimensionless temperature field as previously ; the standard choice is @xmath84 . 
   recently , an alternative model was developed @xcite , @xmath85 d\vec r , \label{modelii}\ ] ] where @xmath86 is a constant . in the following 
   , these models will be called model i and model ii . 
   they both have some features that distinguish them from standard phase - field models . 
   model i contains a term proportional to @xmath87 , which has a singular derivative at @xmath88 . 
   model ii has only a regular square gradient term in @xmath89 , but it is multiplied by a singular function of the phase field @xmath1 , which diverges in the limit @xmath90 ( the solid ) . 
   these singular features are needed to create stable grain boundary solutions , that is , localised spatial regions where the phase field departs from its solid value and the angle field exhibits rapid variations . 
   both models have a variational structure for the dynamics of the phase field and the angle field , that is @xmath91 @xmath92 which means that both @xmath1 and @xmath93 evolve such as to follow the gradient of the free energy , with @xmath94 and @xmath95 being the corresponding mobilities ( which may be functions of the fields ) . in the following 
   , it will be shown that eq . 
   ( [ thetaeq ] ) is incorrect for coherent crystalline matter .        to illustrate the problems with this equation of motion , 
   it is again useful to analyse a simple one - dimensional situation , which is a tricrystal . 
   a slab of crystalline orientation @xmath96 is sandwiched between two crystals of identical orientation @xmath97 , as shown in the left side of fig . 
   the two crystals on the sides of the system are assumed to be clamped to a substrate , that is , @xmath97 for all times . in both models 
   , this initial condition evolves with time : the orientation of the central slab remains homogeneous , but changes with time to approach the orientation of the outer crystals . 
   the final state is a uniform solid of orientation @xmath97 : the central slab has disappeared . 
   of course , this process can take place since it corresponds to a minimisation of the free energy : the two grain boundaries with their positive grain boundary energy are eliminated . 
   however , the pathway of this dynamics is not appropriate for the evolution of a coherent crystal . 
   in fact , eq . 
   ( [ thetaeq ] ) corresponds to the dynamics of matter which has orientational , but no positional order , such as a liquid crystal . 
   indeed , if in model i the term proportional to @xmath87 is omitted or in model ii the singular coupling function is replaced by a regular one , the resulting model can be mapped to the standard landau - de gennes model for nematic liquid crystals in two dimensions @xcite . 
   the free energies in eqs . 
   ( [ modeli ] ) and ( [ modelii ] ) have been designed to stabilise grain boundaries , which do not exist in a nematic liquid crystal . 
   the energetics of the models are thus quite different from liquid crystals . in contrast 
   , the type of the dynamics has stayed the same .    to understand where is the difference in dynamics between liquid crystals and crystals , consider the elongated molecules of a nematic liquid crystal characterised by a director field of a certain orientation @xmath96 . since the molecules have no bonds , 
   it is possible to change the local orientation while keeping the centres of mass fixed , by just making each molecule rotate around its centre of mass ( of course , in a dense liquid crystal , this exact procedure is not possible because of steric exclusion , but the director can still be changed with only short - range displacements of the centres of the molecules ) . 
   the system is thus free to _ locally _ change orientation in order to lower its free energy , and thus follows eq . 
   ( [ thetaeq ] ) . 
   this is obviously not the case in crystalline matter : it is not possible to rotate a unit cell without displacing the surrounding neighbours , because bonds ( or , more generally , the positional ordering of elements ) define a connectivity . 
   it is easy to grasp that the evolution depicted in fig.[fig1 ] is impossible if the connectivity of the central slab is preserved . 
   thus , a consistent evolution equation for @xmath93 has to take into account this connectivity , or , in other words , the evolution of the positions . 
   this is , in general , a complicated undertaking . 
   two elementary situations where it easy to obtain an equation are ( i ) rigid body rotation , in which case the ( advected ) time derivative of the local angle is given by the curl of the local velocity field , or ( ii ) purely elastic deformations of the solid , in which case the orientation is not an independent quantity but can be deduced from the elastic displacement field .    here 
   , a third possibility will be briefly discussed , namely , plastic deformation . 
   this corresponds precisely to a change in the connectivity of matter . 
   if the matter in question can be considered reasonably crystalline ( as opposed to , for example , an amorphous material ) , its geometry can formally always be described by a density of dislocations , which are singularities of the displacement field if a perfect crystal is taken as the reference state . 
   if , furthermore , grain boundaries remain coherent ( that is , no grain boundary sliding takes place ) , the evolution of the local orientation can be linked to the motion of dislocations . 
   a complete description is far outside of the scope of this article ; the interested reader is referred to ref . 
   @xcite for a detailed introduction to the continuum theory of defects . here 
   , only two simple examples will be qualitatively treated for illustration . 
   consider again the tricrystal configuration . in the sketch shown in fig . 
   [ fig2 ] , only one set of crystal planes is shown for clarity , and the central slab has a small misorientation with respect to the outer crystals . 
   in this situation , the two low - angle grain boundaries consist of individual edge dislocations . the inner crystal can now rotate by an elementary process : take one of the edge dislocations of the left grain boundary ( marked by a circle ) and make it glide towards the other grain boundary . 
   this process involves only local reconnection events . 
   when the dislocation arrives at the right grain boundary , it can annihilate with a dislocation of the opposite sign . as a result 
   , one dislocation has disappeared from each grain boundary . 
   of course , this process can repeat itself until no dislocation is left , and the grain boundaries have disappeared . 
   it should be stressed that this pathway for rearrangement exhibits large energy barriers , since the elastic energy of a single dislocation is much higher in the centre of the slab than at its original position within the grain boundary . 
   therefore , if only thermal fluctuations are driving this process ( no external strains ) , it will be extremely slow .    on a more quantitative level 
   , the misorientation through a grain boundary is linked to the density of dislocations by simple geometrical arguments . 
   therefore , it is natural that the misorientation is lowered when the dislocation density in the grain boundaries decreases . 
   furthermore , it is obvious that the rotation rate of the central slab is proportional to the current of dislocations crossing the crystal . 
   thus , a consistent equation of motion for the orientation should be based on the evolution of the dislocation density . 
   however , the development of such an equation is a difficult task , because the motion of dislocations is determined by their complicated elastic interactions , as well as by external strain and interactions with other defects . despite intense activity on the phase - field modelling of defects , elasticity , and plasticity ( see @xcite for a recent overview ) , such an equation seems at present out of reach . 
   let us now come back to the outcome of the simulations for the tricrystal configuration . 
   the functional derivative of the gradient term in eq . 
   ( [ modeli ] ) of model i generates a non - local diffusion equation for the angle field , which has to be regularised as described in ref . 
   @xcite . for a constant mobility 
   , the nonlocal interaction between the grain boundaries leads to a rotation rate that is almost independent of the distance between the grain boundaries . in model ii , the rotation rate of the central crystal decreases exponentially with the distance between the grain boundaries @xcite . in both cases , 
   the central slab eventually disappears . 
   while , quantitatively , neither of these evolutions is likely to be accurate , qualitatively the result is the same as the one achieved by dislocation motion . 
   to see that there can be qualitative differences between the two dynamics , consider now a circular grain of orientation @xmath98 inserted in an infinite monocrystal of orientation @xmath97 . 
   suppose that the misorientation ( which is equal to @xmath98 ) is small , such that the grain boundary is made of individual dislocations separated by a typical distance @xmath99 which is much larger than the lattice spacing . 
   furthermore , suppose that the grain radius @xmath100 is large , @xmath101 , such that on the scale of the grain the boundary can still be described as a continuous line . for simplicity , 
   disregard any anisotropy in the grain boundary energy or mobility . 
   then , the grain will shrink by standard motion by curvature , and the dislocations will simply move towards the centre of the grain . 
   note that the motion of the dislocations might not be strictly radial due to their coupling to the crystal structure ; however , this does not change the present discussion , as long as no annihilation of dislocations takes place . indeed , in this case , the total number of dislocations is conserved , and the dislocation density is simply proportional to @xmath102 , which _ increases _ with time as the grain shrinks . 
   this means that the misorientation also increases with time , and if the outer crystal is fixed , the circular inner grain has to perform a rigid body rotation away from the orientation of the outer crystal . 
   this seems surprising at first , since for low - angle grain boundaries the grain boundary energy is an increasing function of the misorientation . 
   however , this process is perfectly possible if it leads to a decrease of the total energy of the grain boundary , which is given by @xmath103 , with @xmath104 the misorientation - dependent grain boundary energy . 
   its time derivative is @xmath105,\ ] ] where @xmath106 is the derivative of @xmath107 with respect to the misorientation . 
   the evolution can thus take place if the first term , which is always negative since @xmath108 , is large enough to outweigh the second one , which is positive . in that case 
   , the geometrical constraints thus predict an increase of @xmath98 with time . 
   the orientation - field models make exactly the opposite prediction : since the angle field evolves _ locally _ such as to lower the energy , the misorientation of the inner grain should _ decrease _ with time . 
   recently , this situation was investigated by numerical simulations @xcite using the phase - field crystal model @xcite , which gives a faithful microscopic picture of dislocations . 
   an increase of the misorientation with time was observed , consistent with the geometrical constraints . 
   a previous study that had compared phase - field and molecular dynamics simulations @xcite and had reached different conclusions was limited to high misorientations , such that the above hypotheses were not satisfied .    in conclusion , the simple relaxation equation for the angle field , eq . 
   ( [ thetaeq ] ) is not consistent with the coherent crystalline structure of matter , and can sometimes lead to predictions that are even qualitatively wrong . for practical purposes , the quantitative importance of the committed errors might be small when the evolution of a large - scale grain structure is considered , but this has to be confirmed for each case at hand . 
   it is worth mentioning that orientation - field models have been used to investigate the interplay between the positional and orientational degrees of freedom during the solidification of spherulites @xcite or in the presence of foreign - phase particles @xcite . 
   these studies were performed with a vanishing orientational mobility @xmath95 in the solid , and are thus not affected by the problem pointed out here . indeed , in the interfacial region where the structure of the solid in not yet fully established , the concept of a rotational mobility is valid . 
   many phase - field simulations include fluctuations , which are often introduced in a purely qualitative way to trigger instabilities or to create some disorder in the geometry of the microstructures . 
   the role of fluctuations has been investigated more quantitatively in connection with the formation of sidebranches in free dendritic growth @xcite . 
   the standard approach is to include fluctuations as langevin terms in the field equations , with coefficients deduced from the fluctuation - dissipation theorem . before proceeding further , 
   this procedure will be summarised . 
   after inclusion of noise , eqs . 
   ( [ pf ] ) and ( [ diffusion ] ) for the solidification of a pure substance become ( see ref . 
   @xcite for details ) @xmath109 @xmath110 where @xmath111 is assumed ( symmetric model ) , and lengths and times have been scaled by the interface thickness @xmath5 and the phase - field relaxation time @xmath4 , respectively . here , @xmath112 and @xmath113 are random fluctuations of the phase field and random microscopic heat currents , respectively . 
   they are assumed to be @xmath114-correlated in space and time , @xmath115 @xmath116 with dimensionless amplitudes @xmath117 and @xmath118 given by @xmath119 @xmath120 where @xmath99 is the spatial dimension , and the quantity @xmath121 is determined by materials properties only , @xmath122 where @xmath123 , @xmath9 , @xmath11 , @xmath10 , and @xmath124 are boltzmann s constant , the melting temperature , the specific heat , the latent heat , and the capillary length , respectively . 
   the latter is given by @xmath125 , where @xmath107 is the surface free energy . 
   with the help of this expression for the capillary length , @xmath121 can be rewritten as @xmath126 , which makes its physical meaning more transparent : it is the ratio of the thermal energy and a capillary energy scale , and can thus be seen as a non - dimensional temperature .    in a finite - difference discretization of timestep @xmath127 and grid spacing @xmath128 , the noise terms are implemented by drawing , at each grid point @xmath129 and for each time step @xmath130 , independent gaussian random variables of correlation @xmath131 where @xmath132 and @xmath133 are now kronecker symbols , and similarly for @xmath134 . 
   this procedure was shown to yield the correct interface fluctuations at equilibrium in numerical simulations @xcite . 
   an obvious question then arises , namely , can this method also be used to explore nucleation ? 
   phase - field methods have been used recently to investigate homogeneous and heterogeneous nucleation , both in single - phase and multi - phase systems ( see , for example , @xcite ) . in particular , it was found that for high undercoolings , diffuse - interface models yield better agreement with experiments than classical nucleation theory , since the size of the nuclei is not much larger than the thickness of the diffuse interfaces ; therefore , the free energy barriers calculated in phase - field models can differ significantly from classical nucleation theory . 
   is it sufficient , then , to add thermal noise as described above to obtain quantitative simulations of nucleation processes ? 
   the answer to this question is negative . 
   the reason is that , for strong noise , field equations like the phase - field model are renormalized by the fluctuations . 
   this is a well - known fact in statistical field theory , but its implications do not yet seem to have been fully appreciated in the phase - field community . 
   therefore , it is useful to briefly sketch a few calculations that can be found in textbooks ( see , for example , @xcite ) . 
   they are , therefore , neither new nor complete ; however , they will prepare the ground for understanding the conclusions on the phase - field method at the end of this section .    instead of the full phase - field model , 
   consider a single equation for a scalar field @xmath1 that reads @xmath135 where @xmath136 is a non - conserved noise that is @xmath114-correlated , @xmath137 with @xmath138 a suitably non - dimensionalized temperature ( such as @xmath121 , see the discussion after eq .  ( [ fexpt ] ) ) , and the deterministic part of the equation derives from the functional @xmath139d\vec r , \label{hamil}\ ] ] where @xmath140 is a local potential of the field @xmath1 ( lengths , times , and energies are dimensionless ) . 
   it is important to stress that @xmath141 is _ not _ a free energy functional , but the hamiltonian of the field theory . 
   ( [ langevin ] ) generates an evolution in which each microscopic field configuration appears with probability @xmath142 in the limit of infinite evolution time . 
   here , @xmath143 is the partition function , @xmath144 and @xmath145 denotes a functional integration over the field @xmath1 . 
   the free energy is then obtained by the standard formula @xmath146 . 
   the free energy can be calculated exactly for the case of a quadratic potential , @xmath147 , where @xmath148 is a constant . to carry out the calculations , 
   it is useful to consider a discrete version of the model . for simplicity , consider as the domain of integration @xmath22 a @xmath99-dimensional torus of size @xmath149 with periodic boundary conditions . 
   when this system is discretized with the usual finite difference formulae using @xmath150 grid points in each direction and hence a grid spacing @xmath151 , the integral in eq . 
   ( [ hamil ] ) becomes a sum over a finite number of variables . in one dimension , @xmath152 , \label{hamilex}\ ] ] with the convention that @xmath153 . 
   for the discretized system , the functional integration in eq . 
   ( [ funcint ] ) is replaced by a simple integration over the field variables at each grid point , @xmath154 since the hamiltonian of eq . 
   ( [ hamilex ] ) is a quadratic form in the @xmath155 s , this is a @xmath150-dimensional gaussian integral which can be evaluated using standard formulae . 
   the most convenient way is to use a discrete fourier transform to find the eigenvalues of the quadratic form . 
   the final result for the free energy is ( up to a constant that can be dropped ) @xmath156 for dimensions @xmath157 , the same calculation can be repeated without difficulties , and the result is @xmath158 where the sum is now over an independent index @xmath159 for each dimension ( @xmath160 ) , and is normally taken over the first brillouin zone , @xmath161 .    for an arbitrary potential @xmath140 , an exact calculation is generally impossible . 
   statistical field theory has developed sophisticated approximation methods , in particular perturbation expansions . 
   formally , every potential can be written as a perturbation of a quadratic potential . the perturbation expansion ( where the expansion parameter is the temperature , which sets the fluctuation strength ) is cumbersome and usually visualised in terms of diagrams @xcite . 
   fortunately , the first order result can be understood in a relatively simple manner if we are interested in homogeneous systems . 
   more precisely , consider the spatial average of the field , @xmath162 which is a fluctuating quantity . 
   the probability distribution of @xmath163 can be written as @xmath164 where @xmath165 is the free energy density . to first order in the perturbation expansion , @xmath166 where the correction to the original ( `` bare '' ) potential @xmath167 is identical to the exact result for the quadratic potential , with the constant @xmath168 replaced by the second derivative of the bare potential , taken at @xmath163 . 
   this results from a quadratic approximation ( second - order taylor expansion ) of the bare potential around @xmath163 . 
   the result @xmath165 is a renormalized potential for @xmath163 .    ) and from numerical simulations , for @xmath169 , @xmath170 , @xmath171 . 
   only the part close to one of the potential wells is shown . 
   the zero of @xmath172 was chosen at the minimum of the renormalized potential . 
   the bin size for the histograms was @xmath173 . ] 
   these calculations can be readily verified numerically . as an example 
   , the standard double - well potential was used , @xmath174 ( usually called @xmath175-potential in the field - theory literature ) , and simulated in a two - dimensional system of size @xmath176 with a grid spacing of @xmath170 and @xmath169 , using the standard discretization method described above with a timestep @xmath177 , and an initial condition @xmath178 . in time intervals of @xmath179 , @xmath163 was calculated , and in total @xmath180 points were sampled . then , the free energy can be obtained by making a histogram of the values of @xmath163 , and taking the logarithm of the counts ( the normalisation contributes only a constant to @xmath172 and can be disregarded ) . 
   the comparison between the simulation and the prediction of eq . 
   ( [ theory ] ) in fig . 
   [ fig3 ] shows excellent agreement . 
   it can be seen that the minimum of the free energy density is shifted with respect to its `` bare '' value @xmath181 . 
   this can be understood intuitively by the following reasoning . 
   the system starts in the well of the `` bare '' potential , at @xmath181 . 
   the random fluctuations push the system in both directions with equal probability , but since the potential is asymmetric , the restoring force is larger for fluctuations towards @xmath182 than towards @xmath183 ; therefore , smaller values are more likely to occur . in the example chosen here , the shift is small ( the minimum is close to @xmath184 ) , but for increasing temperature , the correction becomes larger and larger ( for an example of such simulations , see @xcite ) , and eventually a phase transition occurs ( the double well disappears ) ; in this regime , of course the first - order perturbation result is inaccurate . 
   the correction also depends on the discretization . 
   this is physically sound : a finer discretization introduces more degrees of freedom per unit volume in the discretized system , and hence allows for more fluctuation modes that contribute to the free energy . with a slight change of perspective , 
   this can also be seen as the natural result of a coarse - graining procedure . 
   indeed , if the free energy is calculated from a given microscopic model by coarse - graining ( averaging ) over cells with a certain size @xmath128 larger than the size of the microscopic elements , both the free energy density and the amplitude of the fluctuations that remain after the averaging ( which thus have a wavelength larger than @xmath128 ) depend on the choice of @xmath128 , as was recently demonstrated explicitly for a simple lattice gas model @xcite .    however , a problem arises in the continuum picture : it is easy to verify that , when the grid spacing @xmath128 tends to zero , the sum in eq . 
   ( [ theory ] ) diverges for @xmath185 . 
   this is a classical example of an ultraviolet divergence . 
   thus , eq .  ( [ langevin ] ) has no continuum limit , and if it is written down in continuum language , it is implicitly understood that an ultraviolet cutoff must be specified . 
   a reasonable physical value for a cutoff in condensed - matter systems is the size of an atom . 
   let us now discuss the implications of these facts for phase - field modelling . 
   even though the above calculation have not been carried out for the full model ( @xmath1 and @xmath7 ) , it is clear that renormalization occurs . if a phase - field model is seen as a simulation tool for a problem that is defined in terms of macroscopic parameters , the relevant quantities that need to be adjusted in the model are the renormalized ones . 
   for instance , thermophysical properties are usually interpolated assuming that the phase field takes fixed values in the bulk phases ( @xmath186 ) . if , on average , this is no longer the case , such as in the example of fig . 
   [ fig3 ] , these interpolations become incorrect . 
   an obvious idea to cure this problem is to choose the `` bare '' potential such that the renormalized potential has the desired properties . for the @xmath175-potential , which is renormalizable , one may choose @xmath187 and determine the constants @xmath188 and @xmath189 by the two conditions @xmath190 and @xmath191 using eq . 
   ( [ theory ] ) . 
   for the example shown above , the values @xmath192 and @xmath193 indeed restore the correct bulk properties . 
   however , in a quantitative phase - field model , the macroscopic properties not only of the bulk phases , but also of the interfaces need to be controlled . 
   it is far from obvious that the above procedure , designed for homogeneous systems , will work . 
   this is even more so for the critical nucleus needed to evaluate the nucleation barrier . 
   it is instructive to examine some orders of magnitude . in nickel , 
   the value of @xmath121 is @xmath194 @xcite , of order unity ; it can be expected that this value is of similar order of magnitude for other substances with microscopically rough interfaces . 
   an inspection of eqs . 
   ( [ fnoisedef][fudef ] ) reveals that _ if _ phase - field simulations are carried out with the `` natural '' interface thickness , which is of the order of the capillary length @xmath124 , the fluctuations are of order unity ( recall that @xmath117 and @xmath118 are equivalent to @xmath138 in the numerical example ) , and renormalization can not be neglected . 
   this is a natural consequence of the fact that real solid - liquid interfaces do indeed exhibit very strong fluctuations , as evidenced from molecular dynamics simulations @xcite ; therefore , a mean - field approximation ( such as the phase - field model without noise ) is not accurate . 
   in contrast , if ( as in refs . 
   @xcite ) a much larger interface thickness is used , the fluctuation strength is greatly reduced , and the difference between `` bare '' and renormalized free energy is small . 
   note , however , that even in this limit a sufficient refinement of the grid would create noticeable fluctuation corrections . 
   we are thus faced with the conclusion ( opposite to the usual point of view in phase - field modelling ) that the use of the simple prescription of ref . 
   @xcite is more precise for larger interface thickness and coarser grids . 
   it is noted in passing that the concept of the sharp - interface limit , central for the asymptotic analysis in the deterministic case , has to be reexamined because a new length scale ( the microscopic cutoff for the fluctuations ) has been introduced .    in conclusion 
   , it is clear that the use of the phase - field method with fluctuations is subject to caution , at least on small length scales . to gain a better understanding , the fluctuation effects on the couplings of the phase - field variables need to be investigated 
   furthermore , a good control of the discretization effects needs to be achieved ; the introduction of a simple cutoff will most likely be insufficient , since the renormalized free energy of eq . 
   ( [ theory ] ) also depends on the grid structure . 
   while a large body of results on these topics can certainly be found in the field - theory literature , the development of quantitative models for specific materials remains a challenging task . 
   in this paper , some open questions concerning various aspects of phase - field modelling of solidification have been discussed , and potential future directions of research have been outlined . 
   the selection of topics is necessarily incomplete , both concerning the problems and the potential solutions . 
   for instance , the rapid development of the phase - field crystal approach @xcite and related methods currently opens up interesting new perspectives for the modelling of polycrystals , which are not discussed further here . 
   the common point of the topics treated here is that they illustrate the dual nature of the phase - field method . on the one hand 
   , it is a genuine representation of condensed - matter systems and their evolution in terms of order parameters on a mesoscopic scale . on the other hand , with the help of mathematical analysis , 
   it can be turned into an efficient simulation tool for the solution of free boundary problems . as in the past , the development of more efficient and robust models for materials modelling will most likely benefit from the pursuit and confrontation of _ both _ of these two complementary viewpoints . 
   therefore , the further development of the phase - field method remains an exciting research topic at the frontiers of physics , mathematics , and materials science . 
   i thank jean - marc debierre , tristan ducousso , alphonse finel , lszl grnsy , herv henry , alain karma , yann le bouar , jesper mellenthin , tams pusztai , and james warren for stimulating discussions on these and many other topics . 
   t. ducousso , _ tude de la solidification dirige par la mthode du champ de phase  : comparaison thorie  exprience pour un alliage binaire dilu _ , ph.d . 
   thesis , universit paul czanne , marseille , france , 2009 .
summary: three different topics in phase - field modelling of solidification are discussed , with particular emphasis on the limitations of the currently available modelling approaches . 
   first , thin - interface limits of two - sided phase - field models are examined , and it is shown that the antitrapping current is in general not sufficient to remove all thin - interface effects . 
   second , orientation - field models for polycrystalline solidification are analysed , and it is shown that the standard relaxational equation of motion for the orientation field is incorrect in coherent polycrystalline matter . 
   third , it is pointed out that the standard procedure to incorporate fluctuations into the phase - field approach can not be used in a straightforward way for a quantitative description of nucleation .