Second order front tracking algorithm for Stefan problem on a regular grid

Transcription

1 Secod order frot trackig algorithm for Stefa problem o a regular grid Robert D. Groot * Uilever Reearch Vlaardige, P.O. Box 4, 330 AC Vlaardige, The Netherlad A brief review of the Stefa problem of olidificatio from a mixture, ad it mai umerical olutio method i give. Simulatio of thi problem i D or 3D i mot practically doe o a regular grid, where a harp olid-liquid iterface move relative to the grid. For thi problem, a ew imulatio method i developed that maifetly coerve ma, ad that imulate the motio of the iterface to ecod order i the grid ize. Whe applied to a iothermal imulatio of olidificatio from olutio i D at 50% uperaturatio for oly 5 grid poit, the motio of the iterface i accurate to 5.5%; ad for 0 poit the reult i accurate to.5%. The method hould be applicable to D or 3D with relative eae. Thi ope the door to large cale imulatio with modet computer power. Keyword: crytallizatio; umerical imulatio; phae field; level et. Itroductio Meltig ad freezig from olutio or from a pure liquid are example of movig boudary problem. Thee problem are decribed by time-depedet differetial equatio that eed to atify boudary coditio at a iterface, where the poitio of thi iterface i ot kow at forehad but ha to be determied a part of the olutio []. Thi cla of problem i kow a Stefa problem, after Joef Stefa ( ) who tudied the meltig of polar ice i 889 []. Although the problem i over a cetury old it till draw a lot of attetio, firtly becaue of the large umber of practical applicatio, ad ecodly becaue olutio are difficult to obtai. Applicatio are i the field of biology, where e.g. ice formatio ca iduce cryo-ijury i freezig cell [3, 4]. Other applicatio are i food ytem, i particular i the dedritic growth of ice i ugar olutio [5, 6], ad further applicatio are i metal weldig [7, 8, 9]. A authoritative review of the field ha bee give by Lager [0], a more recet review i by Ata et al. []. Whe crytal grow i coditio far from equilibrium, growth ca become utable, givig rie to patter formatio [0]. Material for which the olid-fluid iterface are rough at the molecular level but moothly rouded o a macrocopic cale, growth i rapid. For uch ubtace, growth i cotrolled by the diffuio field i the eighbourhood of the olidificatio frot oly. Subtace i thi category iclude mot metal ad alloy ad ome orgaic crytal. Ice i a itermediate cae, i which lowly growig facet occur parallel to the baal plae, but urface are rouded ad grow rapidly i the hexagoal directio [0]. Whe uch a olid grow ito a upercooled melt or olutio, a flat iterface i geerally utable. Udulatio form at oe particular wavelegth, leadig to dedritic or cellular growth. The early tage of uch itability have bee oberved for ice growig from a alt olutio by Körber ad Rau [3]. Dedritic ice growth for pure water wa tudied by Shibkov et al. [], ad from ugar olutio by Butler [5,6]. The oet of uch growth itabilitie wa firt tudied theoretically by Mulli ad Sekerka for a flat iterface [3], ad for a pherical crytal growig ito a uperaturated olutio [4]. Trivedi ad Kurz [5] reaalyed the tability problem ad exteded the growth itability criterio to large thermal Péclet umber, or growth velocitie. A major tep forward i the udertadig of capillarity effect ad the electio of the tip radiu ad velocity ha come from Lager ad Müller-Krumbhaar [6, 7], who predicted a uiveral growth law for dedritic growth rate, which wa teted experimetally [8]. Thee reult demotrate detailed udertadig of the problem of crytal growth itability ad reultig morphology for certai cae of idealied geometry. However, whe it come to olid morphology i cae where everal dedrite grow imultaeouly, or form a three-dimeioal etwork, aalytic theory caot be ued. To aalye thoe cae, we eed to reort to umerical method. Several method are ued i the literature. Broadly peakig, thee are the phae-field method [9], frot trackig method [0] ad the level et method []. Each method ha their merit ad diadvatage. The phae-field method i defied o a regular grid. The olid-liquid iterface i meared out over everal grid poit, which elimiate igularitie i the equatio of motio, but it require may grid poit ad hece large computig power. The level et ad frot trackig method imulate a harp iterface, which elimiate uphyical artefact aociated with a wide iterface. Moreover, boudary coditio at (movig) boudarie ca be implemeted aturally. The dow ide i that without precautio, volume or ma are ot coerved [], which prompt the ue of very mall grid cell, or adaptive grid refiemet. Thi make the * Correpodig author. addre: rob.groot@uilever.com (R.D. Groo.

2 level et method either more computatioally iteive tha phae-field, or more complicated. Both i level et ad i frot trackig method boudary coditio for the Stefa problem are eforced i a phyical way, by determiig the field gradiet at the iterface. The reultig fiite differece cheme may be globally of ecod order i grid pacig ad time tep [0], but i curret implemetatio the update equatio for the phae frot i of firt order i the grid pacig. Thi limit the applicatio to rather hallow cocetratio or temperature field ear the iterface, which i tur force the ue of relatively fie grid. Here we et out to track the root caue why frot updatig i oly of firt order i the grid ize, ad how thi i liked to volume o-coervatio. To thi ed we cocetrate o a D ytem oly, but it i aticipated that the reultig cheme ca be ued i higher dimeio a well. With a ecod order cheme to update the phae frot a coarer grid ca be ued with the ame accuracy a i preet firt order cheme. Thi i tur ope the way to large cale imulatio with modet computig power. Thi article i orgaized a follow. I the ext ectio, we give a brief ummary of the Stefa problem, ad a hort review of the olutio method i give i Sectio 3. Sectio 4 preet a error aalyi of variou method to determie the firt ad ecod derivative of a cocetratio or temperature field at or ear a iterface, ad the ma coervatio problem i aalyed i Sectio 5. Baed o thi, a ecod order ma coervig update cheme i developed i Sectio 6, ad cocluio are ummarized i Sectio 7. The Stefa problem The geeric problem i depicted i Fig.. To make the preetatio more cocrete, we will refer to the olid a ice ad to the liquid phae a a cocetrated olutio, or matrix, although the ame relatio hold for other material, like metal. Geerally, temperature gradiet i the olid phae are differet from thoe i the liquid. Without olute (cocetratio C) the urface temperature T equal the meltig temperature of pure water, T = T 0. I that cae the iterface temperature i fixed, ad the growth or meltig rate of the ice phae follow from a heat balace. Fig.. Temperature ditributio, heat flow ad olute ditributio ear a ice-matrix iterface. The heat traport per uit of area i the matrix phae i q m = mt m, where m i the thermal coductivity of the matrix phae. Similarly, heat traport i the ice phae i give by q ice = icet ice. Thu, the et amout of heat extracted from the iterface per uit of area ad time i give by q m q ice= icet ice mt m. Freezig water produce latet heat L per uit of ma. For ormal growth rate v N, the heat produced per uit of time ad per uit of area i L icev N, where ice i the deity if ice. Produced heat mut balace the extracted heat, hece the growth rate mut atify L v T T () ice N ice Thi i geerally kow a the (fir Stefa coditio. Ofte the approximatio i made that the deity of the olid phae equal that of the pure olvet. Whe thi approximatio i made, all traport i diffuive. I a biary olutio, there i aother boudary coditio to be atified. Geerally, the olute doe ot, or to a very limited amout, partitio ito the olid phae. Therefore, the olid ca oly move whe the olute diffue away. Aalogouly to the thermal diffuio a i Eq. (), thi coditio i N ice ice ice ( k) C v D C D C D C () Here, k i the olute partitio coefficiet, C i the olute cocetratio at the liquid ide of the urface, C ice i the olute cocetratio at the olid ide of the urface, ad D ice ad D m are the Fick diffuio coefficiet of the olute i the ice phae ad i the matrix phae repectively. For mot practical applicatio, whe the olid i water ice, we ca put k = C ice = D ice = 0. A third boudary coditio pecifie the temperature (or olute cocetratio) at a curved olidliquid iterface. Due to the olid-liquid urface eergy, the Laplace preure lead to a hift i the meltig poit at the iterface, which i kow a the Gibb-Thomo boudary coditio. Thi i m m T T ( C ) / R v (3) where T m(c ) i the meltig temperature for a flat iterface at the local olute cocetratio C ; i the Gibb-Thomo parameter; ad R i the geometric mea radiu of curvature (poitive for curvature toward the olid, egative toward the liquid). For pure water = wit 0/ icel K m, where wi = 0.09 J m i the water-ice urface eergy, T 0 = 73.5 K, ice = 97 kg m 3 ad L = 333 kj kg. See e.g. Va Wete ad Groot for detailed thermodyamic parameter of the water-ucroe ytem [3]. Parameter i the kietic coefficiet, which decribe a velocitydepedecy of the iterface temperature [9]. I mot phyical cae of diffuio-limited growth vaihe. The three equatio Eq. ( 3) fix the urface velocity, the urface temperature ad the urface cocetratio for a give et of temperature ad cocetratio gradiet. Further to the boudary coditio, the Stefa problem i determied by the traport of heat ad ma i the liquid ad olid phae. Whe the olid deity equal the deity of the pure olvet, all traport i diffuive, ad the traport equatio are m T c p T (4) t m N m

3 C t DC Thee equatio apply to both the olid ad the liquid phae. Whe there i a deity differece betwee olid ad pure olvet, olidificatio will iduce flow i the liquid phae. Due to thi flow, heat ad ma traport i ot oly diffuive, but are alo covective. To decribe the effect of covectio, oe hould replace the partial time derivative i the liquid phae by the covective derivative [4, 5] D u t t where u i the velocity field i the liquid. Thi hould be obtaied from the Navier-Stoke equatio, ubject to appropriate boudary coditio at the olid-fluid iterface. For a icompreible, Newtoia fluid thee are [4, 5, 6, 7] u 0 u u u P u t where i the liquid deity, P i the (iotropic) preure ad i the dyamic vicoity. Here, we cocetrate o the D cae, hece hydrodyamic will ot be icluded. A importat parameter i the Stefa problem i the Lewi umber, which i defied by the ratio of thermal diffuivity ad olute diffuio: (5) (6) (7) Le (8) c p D D If Le thermal diffuio i much fater tha olute diffuio. I that cae, the temperature profile will be much wider tha the cocetratio profile. For freezig aqueou olutio, thi i ofte the cae. I the limit Le, the ytem i iothermal, ad Eq. () ad (4) become irrelevat. I the limit Le 0, Eq. () ad (5) become irrelevat, ad ice growth i determied by thermal diffuio oly. I practice, thermal diffuivity ad olute diffuivity deped o both compoitio ad temperature. Therefore, the traport equatio are coupled ad o-liear. 3 Solutio method Very few aalytical olutio are available i cloed form. Thee are maily oe-dimeioal cae of a ifiite or emi-ifiite regio with imple iitial ad boudary coditio, ad cotat thermal propertie. Thee olutio take o the form of fuctio of a igle variable x/t (or r/t i pherical ymmetry) ad are kow a imilarity olutio []. Thee olutio are particularly ueful to check the validity of umerical olutio method. A example for the iothermal cae i a half pace i give i the Appedix. 3. Movig grid method from a pure melt. I either cae oly oe traport equatio eed to be coidered, but i geeral we eed to olve both the temperature ad the olute profile. Oe method to olve thee umerically i to ue a movig grid []. I thi method, the grid i deformed to match the poitio of the (harp) olid-fluid iterface. Sice after each time tep the iterface alway coicide with a grid poit, the gradiet at the iterface ca be determied traight-forwardly from umerical differetiatio o a regular grid. I the iothermal cae the cocetratio at the iterface i fixed at it thermodyamic equilibrium value, hece the iterface poitio at the ext time tep follow from the boudary coditio Eq. () a 3Ceq 4C C ( t ( td (9) m C h Here, ( i the iterface poitio, C eq i the equilibrium olute cocetratio, ad C ad C are the cocetratio at ditace h ad h from the iterface. I thi approximatio, the cocetratio gradiet at the iterface i obtaied from a parabola fit through the kow cocetratio value at the iterface ad the ext two poit. After each time tep the grid i re-adjuted to maitai a uiform grid tartig at (t+. A the grid i adjuted, the cocetratio field at the ew grid poitio i iterpolated from the old cocetratio field at the old poitio. While the cocetratio at each poit decreae i time, the iterface ha to move to maitai the ma balace. Ufortuately, a ytematic error of order O(h ) i made i determiig the cocetratio gradiet at the iterface. Error thu accumulate ad diturb the ma balace, ule precautio are take. A movig grid method wa alo ued by Wollhöver et al. [4] to calculate the coupled temperature ad cocetratio profile i freezig alt water. They gue a poitio (t+ at the ext time tep ad olve the cocetratio profile from the diffuio equatio uig a decompoitio method by Vichevetky [8]. The cocetratio at the iterface defie the local temperature, which i the ued a a boudary coditio to olve the temperature profile. I geeral, thi olutio will ot atify the firt coditio Stefa coditio Eq. (). Hece a ew poitio are tried util both boudary coditio Eq. () ad () are atified. 3. Implicit ad explicit update cheme A movig grid method i practical i oe dimeioal ytem, but difficult to apply i higher dimeio. I D or 3D, the grid would eed to deform with the olid, which make it more difficult to olve the diffuio equatio efficietly. There are two way i geeral to olve the diffuio equatio umerically, explicitly ad implicitly. I a explicit olver, the profile at the ew time tep i writte explicitly a fuctio of the previou olutio, e.g. if temperature T i defied at grid poitio x = h, the temperature i the ext tep i obtaied a eq I umerical work oe ofte coider either iothermal (olute diffuio-limited) growth, or growth 3

4 T ( t T ( t T ( T ( T ( h / (0) Thi olve the heat equatio explicitly, but ufortuately the olutio i table oly for time tep t h () where i a cotat that deped o the problem. Thig get wore whe we olve a diffuio equatio i combiatio with movig boudary coditio with curvature-drive growth [0]. It ha bee how by Hou et al. [9] that the preece of capillarity term i the iterface evolutio equatio ca lead to a evere umerical tability cotrait if the iterface i updated explicitly. I that cae, the above criterio take the form 3 t h () I Vichevetky method metioed above, the diffuio equatio at the ext time tep i olved implicitly from the olutio at the curret time tep. Thi i imilar to the Crak-Nicolo cheme, which predate it by ome 0 year [30]. Strictly peakig, both method apply to liear differetial equatio oly. The poit of thee update algorithm i that they are table becaue they combie the (utable) forward olutio ad the (table) backward olutio. Coequetly, oe ca take big time tep without umerical itabilitie. For the movig boudary problem thi i a large advatage. The Crak-Nicolo cheme ca be applied i D, D ad 3D ytem, ad ivolve the olutio of a large (NN) matrix equatio. The D update cheme i baed o Tx ( t Tx ( T ( T ( t t ) x x (3) t Hece the mea time derivative betwee time t ad t+t i related to the average quare gradiet at the old ad ew time tep. Thu, the temperature at the ext time tep i defied implicitly ad hould be olved from T ( t x t T ( t x T ( x t T ( x (4) At thi poit the ew temperature i expreed i the previou temperature field, but we till eed to olve a ordiary differetial equatio. Whe the temperature field i repreeted o a regular grid, we obtai a matrix equatio for a N-dimeioal vector: apart from thoe i a mall bad aroud the mai diagoal, the amout of work to olve thi equatio i of order O(N). I the example give i Eq. (5), where the matrix i tridiagoal, thi ca be doe efficietly with the Thoma algorithm. With a tridiagoal matrix we ca accout for earet eighbour iteractio o the grid. To obtai more accurate reult ear boudarie we may ue a petadiagoal olver [3], o that we ca iclude ecod eighbour iteractio. To olve the et of equatio i Eq. (5) it i importat that the matrix ha limited bad width. Thi ca be guarateed oly if the temperature field i defied o a regular lattice. If i a D or 3D imulatio the grid poit are irregular, a i a co-movig grid, matrix iverio become iefficiet. For thi reao, it i highly deirable to olve the diffuio ad heat equatio o a regular Carteia grid. To eable geeraliatio to D ad 3D imulatio we ue a fixed grid, ad allow the iterface to move acro the grid. 3.3 Fixed grid method 3.3. Phae-field method There are two mai way to olve movig boudary problem o a fixed grid: the phae-field method ad harp iterface method. The mai igrediet of the phae-field method coit i ditiguihig betwee phae with a o-coerved order parameter, or phae field, which i cotat withi each phae. Thi field varie moothly acro a patially diffue iterfacial regio of fiite width W. It dyamic i the coupled to that of the temperature field i uch a way that the equatio for the two field reduce to Eq. (), (3) ad (4) i the o-called harp iterface limit of the model, itroduced by Karma ad Rappel [9, 3], where the iterface i curved o a legth cale much larger tha W. Parameter mut be tued i a particular way by aalyig the phae-field aymptotic, o that the Gibb-Thomo coditio, Eq. (3), i correctly atified. Otherwie a uphyical velocity-depedet term arie i the iterface temperature, i.e. 0 i Eq. (3), wherea it hould vaih for diffuio-limited growth. AijTj ( t Fi ( (5) j where the diagoal elemet of matrix A are A ii = +t/h ; the elemet above ad below the diagoal are A i,i± = t/h ; ad where F( i a kow vector followig from Eq. (0) ad (3). For iulated boudary coditio the firt ad lat diagoal elemet are A = A NN = +t/h.thi equatio ca be olved by traightforward matrix iverio. For the particular cae that we have here, where all matrix elemet are zero Fig.. Solid-liquid iterface i phae field model for width W = ad grid ize h = 0.4. A typical example of a phae-field i [9, 3] = tah(x/w), for W = ad grid ize h = 0.4, a how i Fig.. The quare dot i Fig. deote the grid poit i the phae-field model. Although the width 4

5 parameter i W =, the iterface i actually meared out over a wide area. For itace, the ditace betwee poit A ad B i Fig. i.8, ad cover about 7 grid poit. If we check from the graph where the phae-field ru horizotally, we may ue poit A' ad B' i Fig.. The ditace A'B' cover about grid poit. Thu, eve though the harp iterface limit i a major tep forward compared to older formulatio, it till require may grid poit to decribe jut oe iterface. The coditio of mall curvature (W R) aggravate the problem. I two ad three dimeio thi require vat computatioal ytem. Aother problem with the phae-field method arie for olidificatio from olutio, whe we deal with large Lewi umber. I that cae, the olute diffuio layer i very thi while the temperature profile i exteded. Udaykumar ad Mao [33] report that i their calculatio of ice growig from alt olutio the olute boudary layer are extremely thi a compared to the width of the temperature field. Thu, ule very fie mehe are ued, the width of the diffue iterface which i pread over a few meh cell, ca be comparable to the olute boudary layer thicke. Thi i clearly uphyical. Therefore, a harp treatmet of the iterface i highly deirable i calculatig the olidificatio of impure material. Fially, phae-field aymptotic for uequal diffuivitie i the olid ad liquid phae ca be problematic [34]. Correctio term that are icoitet with the harp-iterface equatio are geerated, ad omootoic behaviour i required i the iterfacial regio. Thi require extra grid reolutio ad hece lower computatioal performace. Geeralizatio of the phae-field approach to hadle dicotiuou material propertie require a better udertadig of the mappig betwee the phae-field model ad the harp-iterface formulatio to avoid problem Frot trackig method Sharp iterface ca be imulated i variou way. Oe obviou way i to track the poitio of the iterface explicitly. Example are the immered boudary method [0, 35, 36] ad the immered iterface method [37]. I the immered boudary method, itroduced by Juric ad Tryggvao [35, 38] the effect of the iterface i tramitted to the field equatio olver uig moothed delta fuctio ad Heaviide fuctio, which how up a ource term i the traport equatio. I thi ee, the effect of the iterface i till pread out over ome grid cell. I the immered iterface method, the iterface i mathematically harp, ad it effect i accouted for by icludig the phyical boudary coditio. Trackig the iterface poitio i atural i a D ytem. Thi method wa ued by Wollhöver et al. [4] to calculate the coupled temperature ad cocetratio profile i freezig aqueou alt olutio. They ued a movig grid to olve the traport equatio. Udaykumar ad Mao [33] reproduced their reult with a fixed grid, uig a ecod order dicretizatio for the Poio equatio o irregular domai to impoe the Gibb-Thomo boudary coditio [39]. Thi latter method ca alo be ued i D ad 3D ytem [38]. I two dimeio the ice frot i a curve, repreeted by iterfacial poit. Thee poit are ditributed at a ditace of roughly the grid ize. At each iterfacial poit, curvature follow from the relative poitio of the eighbourig poit. Together with the Gibb-Thomo boudary coditio (Eq. (3), with = 0) thi determie the temperature depreio at each iterfacial poit. To fid the iterfacial temperature gradiet, a ormal to the iterface i cotructed. Normal probe poit ad are choe at oe ad two grid ditace away from the iterface. The temperature at thee poit i determied by biliear iterpolatio. Fially, the temperature gradiet at the iterface follow from a parabola fit through the temperature at the iterface marker ad odal probe ad [0]: T 3T it 4T T h it (6) Note the imilarity with Eq. (9), where the ame approach wa ued i D. The ame method i ued to determie the olute cocetratio gradiet. I a fiite differece cheme to olve the diffuio equatio, we eed to obtai the quare gradiet of the cocetratio ad temperature at each grid poit. Whe a olid-liquid boudary cut acro the grid, a complicatio arie for the poit that are cloe to the boudary. I thoe cae the field value at the iterface are ued to etimate the ecod derivative i the adjacet field poit. From thi poit oward, the method i imilar to the method by Wollhöver et al. [4]. The ormal velocity i olved from oe boudary coditio, Eq. (), ad the dicretied temperature ad compoitio field are olved uig the Crak-Nicolo cheme. The the iterface poitio i advaced i time to obtai a trial olutio for the ext time tep, ad the actual diplacemet of each iterfacial poit, a well a their temperature ad olute cocetratio are olved iteratively [33]. The mai differece betwee the method of Tryggvao et al. [35, 38] ad Udaykumar et al. [0, 33, 36] i the foreaid mearig of delta-fuctio at the iterface i the former method. I the latter method, o meared delta-fuctio are itroduced, but the field value at the iterface are ued a boudary coditio for the adjacet field. Effectively thi mea that cell which are cut through by the iterface are divided up, ad each part i merged with the ext cell. The iterface thu become a cell boudary. The trick i the to fid the correct field update equatio for thee compoed cell. I thi paper, we cloely follow the latter method by Udaykumar et al., ad will focu o thi update algorithm Level et method Aother way to decribe a harp iterface i the level et method [, 34]. Thi method, firt itroduced by Oher ad Sethia [] i imilar to a phae-field model i that the olid-liquid iterface i repreeted a the zero cotour of a level et fuctio, φ(r,, which ha it ow equatio of motio. The movemet of the iterface i take care of implicitly through a advectio equatio 5

6 for φ(r,, which ha the phyical iterpretatio of a (iged) ditace to the iterface. Ulike the phaefield model, there i o arbitrary iterface width itroduced i the level et method; the harp-iterface equatio ca be olved directly ad, a a reult, o aymptotic aalyi i required. Dicotiuou material propertie ca alo be dealt with i a imple maer. Level et method are particularly deiged for problem i multiple pace dimeio i which the topology of the evolvig iterface chage durig the coure of evet, ad for problem i which harp corer ad cup are preet [40]. Thi method wa firt applied to the Stefa problem o a regular grid by Gibou et al. [4]. The level et method ha bee compared to the phae-field method for dedritic growth [4, 43], ad lead to the ame reult. However, the level et method i computatioally more demadig tha phae-field. Thi ca be repaired by coiderig oly a arrow bad aroud the iterface where the level et fuctio i defied [40]. Moreover, a drawback of the level et method i that it i ot volume preervig [] ad thu prompt the ue of adaptive grid refiemet [44, 45], or pecial eforcemet of coervatio law [46]. Aother importat reao to apply adaptive grid refiemet, i to capture mall cale that would otherwie ot be take ito accout. Thi method wa firt applied to the Stefa problem by Che et al. [47]. Yag ad Udaykumar [48] decribe a eay implemetatio of the level et method. I thi repect, the immered boudary method ad the level et method are imilar they both imulate a mathematically harp iterface. 4 Error aalyi of umerical derivative A metioed above, the update cheme of Eq. (9) lead to umerical error: the total olute ma i ot coerved. It i hypotheized that the problem i caued by a error i either the (iterpolated) urface cocetratio gradiet, or the quare gradiet cloe to the iterface. If the gradiet ad hece the calculated growth velocity doe ot exactly match the cocetratio reductio aroud the olid phae, the iterface will ot be diplaced correctly ad Eq. () will ot be atified. I particular, thi problem will occur whe the grid i coare a compared to the gradiet that occur ear the urface. For thi reao, we firt tudy how umerical error i derivative i D deped o the grid ize. Coider a Carteia grid of pacig h, where the olid-liquid iterface i located at poitio h (ee Fig. 3). The poitive directio i poitig toward the liquid phae, ad the egative directio toward the olid. Without lo of geerality we ca chooe the origi of x-axi at the firt grid poit ahead of the iterface. Fig. 3. Solid-liquid iterface i harp iterface at grid poitio h. Liear iterpolatio are idicated by red dotted lie. I the method by Crak [], a parabola i fitted through the poit x = h, 0 ad h, with fuctio value f, f ad f repectively. Thi lead to the followig etimate (deoted by C) for the firt derivative i x = h ad for the ecod i x = 0: f [( ) f [ f f 0 ( ) f ( ) h ( ) ( ) h 0 f ] f 0 f ] ( C) (7) We tet the accuracy of umerical differetiatio with tet fuctio f(x) = exp( (x+h)). The exact derivative (F exac at the iterface ad the ext grid poit are f ' = ad f '' = exp( h) repectively. The umerical value (F um) are obtaied from Eq. (7). Fig. 4. Relative error i firt (black) ad ecod (red) umerical derivative uig Crak method (C) for h = 0.5 (lef ad for = 0.5 (righ, ad UMS method for f ' (gree). The error i Crak method for f ' i O(h ); other error are O(h). The relative error, F um/f exact, are how i Fig. 4. For a fixed value of h = 0.5 the error i the firt derivative at the iterface icreae from 0 to 6% a icreae from 0 to (black curve); ad the error i the ecod derivative at the ext grid poit decreae from +5% to % (red curve). The calig of the relative 6

7 error with grid ize i how i the right-had graph: the error i f ' icreae h, ad the error i f '' icreae h. The O(h) error i the quare gradiet at the firt grid poit may be a problem. Sharp cocetratio gradiet will be a rule rather tha the exceptio, ad the power of the imulatio method will be determied by it ability to repreet large value of h. Hece, we would like to have a method that i maifetly of order O(h ). Udaykumar et al. [0, 33] ued Eq. (6) to etimate the urface gradiet f ' (deoted by UMS), where the value T ad T were defied at the probe poitio ad. Thi would lead to ecod order accuracy if the temperature value at probe poitio were exact. However, for a biliear iterpolatio to obtai the value T ad T, the error i f ' i proportioal to h. Thi ca be ee if we check thi method i oe dimeio. I that cae, the probe poitio i grid ( 6 3 ) ( 3 f h f ( )( ) 6 f0 h f ( )( ) ( 3) f 0 ) coordiate are =, ad =. Iterpolatig the fuctio value liearly (ee Fig. 3), the firt derivative at the urface become f [ 3 f 4f 0 (5 4) f ( ) f] ( UMS ) (8) h The method ued by Udaykumar to determie f '' i idetical to that of Crak, Eq. (7), o thi too ha a accuracy of O(h). Reult for the error i f ', uig the UMS method, are alo how i Fig. 4. Sice the ecod derivative determie the rate of chage i the firt grid poit, ad ice the larget variatio i a cocetratio profile will be ear the iterface, it i importat to have thi up to d order i h. To thi ed we fit a 3 rd order polyomial through the fuctio value f.. f. Straightforward calculatio ow give f 0 (4 ) ( ) ( ) ( ) ( ) f f ( ) ( ) f f ( ) (9) The expreio for f '' at the firt grid poit ear the iterface, give i Eq. (9), i ideed accurate to order O(h ), ee Fig. 5. A a by-product of thi aalyi we fid a expreio for f ' to O(h 3 ). A ecod order determiatio of the quare gradiet ear the iterface require iformatio from the ecod ad third grid poit. Thu, if we wih to implemet a Crak-Nicolo cheme, we eed to ivert a petadiagoal matrix, rather tha tridiagoal. A efficiet code to olve a petadiagoal ytem of equatio i give i Ref. [3]. iterpolatio for the fuctio at the probe poit i Eq. (6) by quadratic iterpolatio. Thi limit the method to rather hallow gradiet. h Relative error [%] f' (C) f' (UMS) f' (pree f'' (C) f'' (pree Table. Relative percetage error of variou method to determie f ' ad f '' 5 The ma coervatio problem Fig. 5. Relative error i firt ad ecod umerical derivative for Crak method ad preet method for = 0.5. The preet error i f ' i O(h 3 ) ad for f '' it i O(h ). The preet method for f ' i valid up to h =. At that poit, the error i oly 3.4%; it icreae to about 4% at h =. The ecod derivative at the firt grid poit i more forgivig: the preet method how a maximum error of +.8% at h =.3 ad drop to 5% at h = 3. For compario, ome reult are collected i Table. Whe the decay rate i of the order of the grid pacig, h ~, the error i the UMS method to calculate the urface gradiet i ome 5%. Surpriigly, the error oly reduce to 0% if we replace the liear To tet the practical efficacy of the fixed grid method to imulate the Stefa problem we tudy the (impler) iothermal cae ad check for ma coervatio. For the ake of thi tet we et the diffuivity at D = (idepedet of olute cocetratio); ad we cale the olute cocetratio o that the equilibrium cocetratio i C eq =. Grid poit are choe at x = (+½)h, where h =, 0 N, ad N i the umber of grid poit. At t = 0 the iterface i located at (0) = 0, ad the cocetratio at all grid poit i give by C(x ) = C = C 0. At the right-had boudary, a Neuma boudary coditio i impoed, fixig the cocetratio gradiet at the value of C'() = 0. Thi i impoed by mirrorig the lat poit i the right-had boudary to calculate the ecod derivative. To ecod order i h we thu have: C''(x N ) = (C N C N +C N )/h = (C N C N )/h. At the lefthad boudary we impoe the Stefa coditio d/dt = C'(, = C '(, together with the equilibrium coditio C(, =. For each value of we obtai the miimum grid umber k for which the diffuio equatio applie, 7

8 ad we obtai the ditace betwee ad x k i grid uit, a k it / h k / h Thu, we arrive at the followig et of equatio: ( t ( t C ( ) ( ) / t C t t C Ck ( t t Ck ( t Ck ( t Ck ( () C ( t t C ( ) ( ) ( ) / ( ) ( ) t t C t t C t t h C t t C t CN ( t t C ( t C ( t / h C ( t C ( N N N N I thi et of equatio k < < N. It hould be oted that k i iitially k = 0. However, whe ( croe poit x k betwee t ad t+t, we eed to icreae k by ad olve the et of equatio agai for fewer grid poit. Moreover, hidde i the firt two lie of Eq. () i a implicit depedece o (t+, a the derivative metioed there deped i a o-liear way upo ( ad (t+ (ee Eq. (7) ad Eq. (9)). For thee reao the ytem of equatio eed to be olved i a iterative way, alteratig betwee a Crak-Nicolo olutio for the cocetratio field at t+t with a fixed trial value for the iterface poitio, ad a olutio method to obtai (t+ for a fixed cocetratio field. It i importat to ote that C k''(, a appearig at the right-had ide of Eq. (), ca be either a pecial ecod derivative, a give by Eq. (9), or a ordiary ecod derivative o a regular grid. If (t+ pae a grid poit, the k-value i icreaed, hece grid poit k of the old cocetratio field C k( i o loger adjacet to the ice frot. Therefore, the ecod derivative of the old cocetratio field i tored eparately before we eter the iterative cheme to determie (t+. Note further that C k''(t+ deped ot oly o (t+, but alo o the cocetratio at the iterface (ee Eq. (9). Thi i a (fixed) iput term, ad i therefore take to the right-had ide of the equatio for the olutio procedure. To fid (t+ we ue the followig cheme: Step : etimate ew frot poitio from 0 ( t ( t C ( C (0) / () ad et couter j =. Step : determie k ad for the ew frot poitio from Eq. (0). Step 3: olve C(,t+ from the lat three lie of Eq. () for a fixed value of (t+ Step 4: determie ew frot poitio from j C ( C ( t C t ( j / (3) ( t ad icreae couter j = j+. Step 5: retur to Step if j 4. Step 5 could be replaced by a covergece criterio, but it i foud that four iteratio uffice to obtai ix decimal place accuracy i the ew poitio, eve for a time large time tep like t = 0.5h /D. For implicity a fixed umber of iteratio i take. After four iteratio the ew cocetratio field i copied oto the old field, ad time i icreaed by t. The quare gradiet of the cocetratio field i the tored for iput to the ext time tep, ad the ext time tep i take. Fig. 6. Frot poitio ad ma for iothermal imulatio uig Crak equatio for the urface gradiet (black ad red) ad uig Eq. (9) for the urface gradiet (gree ad blue). To tet the accuracy of thi cheme, imulatio are doe for mall ytem of N = 0 grid poit. A error i ma coervatio will be more proouced i mall ytem the i large ytem becaue the error i a urface effect. I the preetatio below we recale the ytem ize to legth L =. We tart at each grid poit with the iitial cocetratio C 0 = 0.5, ad we maitai the equilibrium cocetratio C =. I thee uit, the diffuio cotat i D = h = /N, hece we recale time to t* = Dt/L. Furthermore, the total ma i the ytem i M = C 0dx = 0.5, hece the frot poitio hould coverge to ( = 0.5 for t*. The frot poitio ad total ma i the ytem are how i Fig. 6a. The gree ad black curve give the frot poitio, calculated from two approximatio for the cocetratio gradiet at the ice frot. The black 8

9 curve i for Crak approximatio (Eq. (7)), ad the gree curve i for the preet 3 rd order approximatio (Eq. (9)). The red ad blue curve repectively give the total ma for the two imulatio. The ma i calculated by itegratig piecewie parabola iterpolatio. Note that we tart with too high ma, becaue the firt grid poit at x = h/ ha cocetratio C = C 0 = 0.5, but at the iterface at x = 0 we have C = C =. Hece the ma itegral i roughly M h/ = Fig. 6b how that thi value i ot coerved, although the maximum deviatio from the mea value i of the order of 0.5%, ad the variatio betwee maximum ad miimum of the curve i about.5%. Surpriigly, uig a third order polyomial to etimate the urface gradiet (blue curve i Fig. 6b) ha hardly ay advatage over a parabola. Probably thi i becaue the error made i the ecod derivative i of order h, which will domiate the error i the urface gradiet ( h 3 ). To tabilize the total olute ma i the imulatio, a correctio method i ought for. Sice the gradiet i ot determied exactly, thi i a likely ource of error. Ad if the frot poitio i ot updated i balace with the cocetratio icreae the itegrated ma will tart to hift i the coure of time. Therefore, the poitio update cheme i corrected to ( t ( Dft C / (4) C where D i the diffuivity, C i the mea cocetratio gradiet over the time iterval t a determied by the iterpolatio procedure, ad f i a gradiet correctio factor. Let m be the ma determied by umerical itegratio. Uig piecewie liear iterpolatio (for vaihig lope at the right wall) we obtai N / h N k C Ck C C C m (5) Now let M be the exact ma, which hould be coerved. If i a previou time tep ma m deviate from M we eed to hift the frot poitio by a amout uch that m C = M, or m M C D( f ) t (6) C C The gradiet correctio factor to force ma coervatio i thu foud a M m f (7) DtC Note that the gradiet correctio factor i dimeiole, hece the ame equatio applie whether we ue the grid ize a uit of legth, or ay other choice for the uit of legth. Fig. 7. Frot poitio ad ma for iothermal imulatio uig Crak equatio for the urface gradiet (black ad red) ad uig Eq. (9) for the urface gradiet (gree ad blue), impoig correctio factor f (Eq. (7)) at every time tep. Whe the gradiet correctio factor i calculated from the field gradiet i the lat time tep (Step, Eq. ()) ad ubequetly applied i Step 4 above (Eq. (3)) to obtai the ext frot poitio, we fid the reult how i Fig. 7. Wherea i Fig. 6 a mall differece ca be ee betwee the frot poitio obtaied from the reult from d ad 3 rd order polyomial to calculate the urface gradiet, i Fig. 7 the two curve uperimpoe exactly. Moreover, they coverge exactly to the correct value ( = 0.5 for t* ; ad throughout the imulatio the total olute ma equal m = 0.5. Deviatio typically occur i the ixth decimal place. Oe might expect the cheme to ocillate, but it doe ot. A elargemet of the total olute ma i how i Fig. 7b. thi how the tability of the correctio cheme, oly durig the firt few time tep igificat deviatio occur becaue we tart with the wrog ma m 0.506, becaue of the fiite grid ize. To tet the efficacy of thi cheme, the grid ize wa varied i a fiite ize calig aalyi from h = /60 to /5. We fid the imulatio reult how i Fig. 8. The left picture how the imulatio without correctio factor, ad the right-had graph how the reult with correctio factor. The correctio factor doe correct the total ma cotet, ad hece the fial grid poitio, but the dyamic with correctio clearly how a depedece o grid ize. Without correctio, all curve uperimpoe up to t* 0. ad the tart to deviate, but with correctio factor a maximum deviatio occur ear t* 0.6, ad the the curve coverge agai. Hece, although the correctio factor olve the ma coervatio problem, it itroduce aother problem. 9

10 Fig. 8. Simulatio for variou grid ize without correctio factor (lef ad with correctio (righ. Without correctio ma i ot coerved, but with correctio dyamic deped o grid ize. Fig. 9. Frot poitio a fuctio of grid ize at t* = Dt/L = without correctio factor (lef, ad at t* = Dt/L = 0.6 with correctio (righ. I both cae the error i liear i the grid ize. If we plot the frot poitio a fuctio of grid ize, we fid that both with ad without correctio a error proportioal to the grid ize h i made, ee Fig. 9. Thi i ufortuate, a it implie that we would eed a very fie meh for accurate imulatio. Thi i tur mea that geeraliatio to D or 3D will be computatioally iteive. To ummarize, eve though we have icluded expreio for the firt ad ecod order derivative ear the iterface that are accurate to ecod order, we till fid a firt order error i the update cheme. Hece, we coclude that the iitial hypothei, that error are caued by iaccurate umerical derivative, i icorrect. Thu, we eed to earch for a ew update cheme that i iheretly ma coervig, ad i accurate to order h. Thi i the ubject of the ext ectio. 6 A ma coervig ecod order update cheme To coerve ma, we make a choice for the deity ditributio fuctio. We maitai grid poit at halfiteger lattice poitio, x /h = (+½), ad we defie the deity ditributio at itermediate poit by liear iterpolatio. A example i how i Fig. 0. Each cell ru from < x/h < +; ad the ma flux from cell to cell, J, i located at x/h =, ee Fig. 0. At the right-had boudary x = L = Nh we impoe a iulatig boudary coditio. Thi ca be impoed by mirrorig the cocetratio field i x = L, which implie that the cocetratio field ru horizotal betwee L h/ ad L. For thi cocetratio field the total ma i lattice uit i exactly give by m N C C C C C k k N k C ( ) C k C N (8) Thi approximate the ma of a arbitrary cocetratio profile to order h. Fig. 0. Sharp olid-liquid iterface at grid poitio k. Liear iterpolatio are idicated by red dotted lie. Cocetratio fluxe ad diffuivitie are defied at cell boudarie. Uig Fick law the ma fluxe betwee ucceive cell i (uig grid uit, h = ) J J k D C D C C ( k) (9) where D i the local diffuivity, which i defied at the locatio of J, ad may deped o the temperature ad olute cocetratio of the eighbourig cell: D = D ((C +C )/,(T +T )/). D i the diffuivity at the growig urface. No ma i paig through the 0

11 right boudary, therefore the rate of chage of the cocetratio i the lat bi i give by C J D ( C C ) (30) N N N N N For all cell that are ot adjacet to the ice frot we have Therefore we have C J J ( k N ) (3) N C J N ( J N J N) (3) k... ( J k J k ) J k Hece the time derivative of the total ma i m J ) C (33) k ( C Ck ) C ( Uig the Stefa coditio ( J k / C ) ad impoig ma coervatio we obtai the time derivative of C k a J k ( Ck / C ) J k C C k (34) ( ) k It hould be oted that the right-had ide of Eq. (34) i ot equal to the ecod derivative at the firt grid poit; there are correctio to O(h). Thi i the reao why the update cheme of the previou ectio (Eq. ()) i oly correct to O(h). Reverely, if we bae the time derivative o a quare gradiet that i correct to O(h ), ma coervatio mut be violated to O(h). Thi explai the violatio of ma coervatio how i Fig. 8a ad Fig. 9a. If o the other had, we ue a O(h ) quare gradiet for C k ad force ma coervatio by chagig the motio of the frot, thi motio mut deviate from the correct Stefa coditio, reultig i a O(h) error i the frot poitio. Thi explai the O(h) deviatio from the correct frot evolutio, how i Fig. 8b ad Fig. 9b. I geeral, we ca ue ay approximatio for the flux J k at the urface, a log a the ame approximatio i ued for the growth rate of the cocetratio i the firt lattice poit C k (Eq. (34)) ad for the motio of the iterface through the Stefa coditio. Two approximatio for the gradiet at the urface a give i Eq. (7) ad Eq. (9) are reproduced here: ( ) ( ) C C Ck Ck ( ) ( ) ( 6 3 ) ( 3 ) C C Ck ( )( ) ( ) C ( ) k ( ) C ( ) k [ O( h [ O( h 3 )] )] (35) Thu, we may write the urface gradiet i geeral a C = ac +b 0C k+b C k++b C k+; the coefficiet ca be read off from Eq. (35). To arrive at a forward itegratio cheme for fiite time tep we tart with the ma balace N C k C ( ) C k ' C' N k' C' ( ') C' k' (36) where the primed quatitie are for the ew time tep. For implicity, we coider a explicit update cheme; geeraliatio to a Crak-Nicolo cheme i traightforward but tediou i programmig. Firt, we ote that the um over C' at the right-had ide follow from Eq. (3) a N N C' C J k ' t (37) k' k' The idex k' deote the umber of the grid poit adjacet to the frot, i the ext time tep. Now two cae may occur, where k' = k or k' = k+. I the former cae the frot doe ot cro a grid poit, i the latter cae i doe. We tart with the firt cae, where k' = k. I a forward itegratio cheme the ditace from frot to the firt grid poit evolve accordig to the Stefa coditio a ' t J k / C (38) where J k = D C, ad D i the diffuivity at the urface. Subtitutio of Eq. (37 38) ito Eq. (36), ad uig k' = k, we fid the firt layer cocetratio i the ext time tep a C' k ( ) Ck ( C C' ) ( J kc' / C J k ( ') J k ( Ck / C ) J k ' C Ck t ( ') ) t (39) Note that i the limit of cotiuou time thi i idetical to the time derivative obtaied i Eq. (34), but if we accout for a fiite time tep we eed to replace by ' at the right-had ide to obtai exact ma coervatio. The time derivative of the urface cocetratio ued here i the forward time derivative, Ċ = (C (t+ C ()/t. We ow coider the ecod cae, where k' = k+. I thi cae, the ice frot croe a grid poit, hece the ditace from frot to the ew firt grid poit chage to ' t J k / C (40) Becaue the um over C at the left-had ide of Eq. (36) tart at k+, ad at the right-had ide it tart at k'+ = k+, a term C k+ i left over at the left-had ide which doe ot cacel out. Moreover, whe ' i ubtituted ito Eq. (36), the + term i Eq. (40) lead to a extra C' term at the right-had ide. Collectig term, we fid

12 C' k Ck C' ( ) Ck ( C C' ) ( J kc' / C J k) t ( ') J k ( Ck / C ) J k ' C ( ') Ck ' Ck C Ck t ( ') ( ') (4) Thu, we fid the ame term a time derivative for C k+ a for C k for the cae where o grid poit wa paed, albeit that a J k+ term i replaced by J k+. Remarkably, a extra term appear that caot be idetified eaily a a time derivative, but it ca be iterpreted a a cocetratio gradiet. To geeralie thi evolutio algorithm to a Crak-Nicolo cheme we hould replace Eq. (38) ad Eq. (40) by ' = ½t J k/c ½t J' k/c' ad ' = ½t J k/c ½t J' k+/c' + repectively, ad olve for the primed variable. Ma coervatio wa checked for ytem of N = L/h = 5 to 80 grid poit, uig a fixed time tep of Dt/h = The ytem were itegrated to time Dt/L =. The iitial ad fial ma itegral varied at mot by a relative error of 0 4, i.e. it i cotat to the machie preciio. Thi i far uperior to the reult how i Fig. 6 ad Fig. 7. To check the calig to grid ize, the grid pacig wa varied from h = 0. dow to h = To acertai that all ytem had equal ma from the tart, the iitial value of the frot poitio wa varied. Solvig from ½C +½(+)C 0 = C 0 (the ma i the firt grid cell), we fid the iitial frot poitio a C C 0 0 (4) C C0 Oe may argue that imulatio for differet grid ize tart at a differet time t 0, a coarer grid have a head tart. A the frot poitio iitially evolve a ( = ( ½ (ee Appedix), the time offet i obtaied a t 0 = 0 /, where i obtaied a the lope i a plot of a fuctio of t. After correctig for thi offet we fid the reult how i Fig.. Fig.. Frot poitio for variou grid ize after recalig the iitial frot poitio (lef for grid ize h = 0., 0., 0.05, 0.05 ad 0.05; ad the iitial lope of a fuctio of grid ize (righ. Dotted lie idicate aalytic limitig law. Fig. a how excellet calig. Oly the data for the larget grid ize (black lie) deviate viibly from the data obtaied from maller grid ize. The black dotted lie give the aalytical iitial behaviour = t, where i obtaied from ( C 0 / C ) erfc exp (43) 4 A derivatio i give i the Appedix. For = 0.5 we fid exact = To obtai a meaure of the umerical error, the iitial lope of the curve i Fig. a have bee determied for 0.05 < Dt/L < 0.. The reult are how a fuctio of grid ize i Fig. b. The black dotted lie agai give the aalytical reult exact = The umerical reult deviate from the exact reult a um exact.h 0.8h 3, i.e. the error i ideed of ecod order i the grid ize. Note that the relative error i quite mall, oly.5% for h = 0. (te grid poi, ad 0.35% for h = Eve for 5 grid poit i the imulated pace we have a modet error of 5.5%. Thi ope the door to large cale imulatio with modet computer power. 7 Summary ad cocluio A brief review of the Stefa problem ad the mai umerical olutio method i give. We cocetrate o the problem of olidificatio from olutio. For thi problem, calculatio i D or 3D are mot practically doe o a regular grid, where the olid frot move relative to the grid. I the phae-field method the iterface i pread out over everal grid poit. Moreover, for mixture the width of the iterface i grid uit hould be maller tha the rage of the olute diffuio field. Thi implie that may grid poit are eeded for imulatio of realitic complexity, which i computatioally iteive. Alterative method are level et ad explicit frot trackig method, of which the latter are coceptually eaier. Phyical boudary coditio at the movig iterface ca be applied traightforwardly. The motio of the iterface deped o the temperature ad olute gradiet at the iterface, ad the evolutio of the cocetratio ad temperature field away from the iterface are determied by quare gradiet. I curret implemetatio, thee are determied to a error that i liear i the grid ize. Thi may be a problem.

13 Sharp cocetratio gradiet will be a rule rather tha the exceptio, ad the power of a imulatio method i determied by it ability to repreet large field variatio with few grid poit. Therefore, higher order iterpolatio are ued to obtai the gradiet at, ad quare gradiet ear the iterface, to ecod order i the grid ize. It i urpriigly foud that, eve with uch ecod order preciio i the field derivative, ma i coerved oly up to firt order i the grid ize. Thi i a problem, becaue it implie that till may grid poit are eeded i D ad 3D calculatio to obtai acceptable accuracy, leadig to large computatioal cot. Therefore, a ew method i developed from the leadig priciple of local ma coervatio. I thi method, the motio of the iterface i accurate to ecod order i the grid ize. I thi method, the time derivative for the field poit adjacet to a iterface i ot equal to the quare gradiet i that poit, but cotai correctio to firt order i the grid ize. A update cheme for fiite time tep i derived. Whe the iterface croe a grid poit durig a time tep, the update cheme for thi firt grid poit cotai a additioal term which ca be iterpreted a a gradiet term, rather tha a quare gradiet. Whe applied to D imulatio, we fid maifet ma coervatio to 4 decimal place. To tet the method, we tudy a iothermal problem with a uperaturatio of 50%. Whe oly 5 grid poit are ued i the iterval, we fid the growth velocity of the iterface accurate to 5.5% a compared to the exact reult, ad for 0 poit the reult i accurate to.5%. It i aticipated that the method ca be geeralized to D ad 3D imulatio with relative eae. Thi ope the door to large cale imulatio of the Stefa problem with modet computer power. Appedix - Iitial growth law for iothermal coditio We coider a half pace where the ucroe cocetratio at t = 0 i give by C 0 for x > 0, ad ha a fixed value C at the urface. The cocetratio profile follow C/ t = C/ x, ad the frot move a d/dt = C/C. We ow ue a calig aatz [] where the cocetratio profile oly deped o the combiatio = x/t, hece C(x, = u(x/ = u(). Thi plaar cae i imilar to the claical Frak phere olutio [], ad i give here for completee. Uig / x = t / / ad / t = ½ t 3/ /, the diffuio equatio become u'' + ½u' = 0, which i olved directly a u' = exp( ¼ + cota. Itegratig agai give the geeral olutio C(x, = u(x/ = A erf(½ xt / )+B. Applyig boudary coditio C(() = C ad C( ) = C 0, we olve A ad B from the ytem C = A erf(½ t / )+B ad C 0 = A+B. Becaue C i a cotat, ( mut be proportioal to t / to cacel the factor t / i the error fuctio of the geeral olutio. Thu, we oly have a time-idepedet boudary coditio at the movig iterface whe the iterface move with the calig law ( / where i a cotat. I other word, the timeidepedece of the boudary coditio at the urface force the urface motio to a quare root time behaviour. Solvig for A ad B we obtai the olutio C / ( x, C0 ( C C0)erfc( xt )/erfc( Now we apply the Stefa coditio to fid, ad obtai C(, ( C C0) C C exp( t erfc( 4 / / / ) where the fial tep follow from the time derivative of the above quare root time behaviour of the movig frot. We fially obtai the implicit relatio betwee the rate cotat ad the uperaturatio: ( 0 4 ) t C / C ) erfc( ) exp( ) (A) To obtai the ivere fuctio, we firt tudy the limit for mall ad large value of. For mall we obviouly have 4 /, ad i the limit of large we fid the aymptotic expaio /. Hece a reaoable firt approximatio for i: 4 / ( 4/ ) (A) ( ) Thi approximatio atifie the limit for mall ad large, hece it form a good tartig poit for a umerical olutio for the ivere of Eq. (A). Uig the Newto-Rapho method, a ext approximatio to i obtaied a 4( 0) (A3) ( / ) where i the fuctio give i Eq. (A), ad i the deired value of the uperaturatio for which we wat to calculate the correpodig lope. Applyig Eq. (A3) iteratively, a olutio to 4 decimal place i typically obtaied i three iteratio, whe the etimate of Eq. (A) i ued a a tartig poit. 4 Referece [] J. Crak, The Mathematic of Diffuio, ecod ed., Claredo pre, Oxford, 975, pp [] J. Stefa, Ueber die Theorie der Eibildug, ibeodere über die Eibildug im Polarmeere, Aale der Phyik 78 (89) [3] C. Körber ad G. Rau, Ice Crytal Growth i Aqueou Solutio, i: D.E. Pegg, A.M. Karow Jr., (ed.), The Biophyic of Orga Cryopreervatio, Pleum Pre, New York, 987, pp [4] K. Wollhöver. Ch. Körber, M.W. Scheiwe ad U. Hartma, Uidirectioal freezig of biary aqueou olutio: a aalyi of traiet diffuio of heat ad ma, It. J. Heat Ma Trafer 8 (985) [5] M.F. Butler, Itability Formatio ad Directioal Dedritic Growth of Ice Studied by Optical Iterferometry, Cryt. Growth De. (00) 3-3. [6] M.F. Butler, Growth of Solutal Ice Dedrite Studied by Optical Iterferometry, Cryt. Growth De. (00)