6500 J. Phys. Chem. B 2005, 109, 6500-6504 Anisotropic Interfcil Free Energies of the Hrd-Sphere Crystl-Melt Interfces Yn Mu, Andrew Houk, nd Xueyu Song* Deprtment of Chemistry, Iow Stte UniVersity, Ames, Iow 50011 ReceiVed: August 17, 2004; In Finl Form: December 28, 2004 We present relible method to define the interfcil prticles for determining the crystl-melt interfce position, which is the key step for the crystl-melt interfcil free energy clcultions using cpillry wve pproch. Using this method, we hve clculted the free energies γ of the fcc crystl-melt interfces for the hrd-sphere system s function of crystl orienttions by exmining the height fluctutions of the interfce using Monte Crlo simultions. We find tht the verge interfcil free energy γ 0 ) 0.62 ( 0.02k B T/σ 2 nd the nisotropy of the interfcil free energies re wek, γ 100 ) 0.64 ( 0.02, γ 110 ) 0.62 ( 0.02, γ 111 ) 0.61 ( 0.02k B T/σ 2. The results re in good greement with previous simultion results bsed on the clcultions of the reversible work required to crete the interfces (Dvidchck nd Lird, Phys. ReV. Lett. 2000, 85, 4571). In ddition, our results indicte γ 100 > γ 110 > γ 111 for the hrd-sphere system, similr to the results of the Lennrd-Jones system. I. Introduction The interfce between crystl nd its melt is one of the most importnt subjects with considerble studies in physics, chemistry, nd mteril sciences. In prticulr, there hs been persistent interest focusing on the clcultion of the interfcil free energy γ, due to both its fundmentl physics vlue nd the prcticl pplictions for determining mteril properties. As mtter of fct, lthough mny theoreticl nd experimentl studies hd been done, fundmentl understnding nd detiled microstructurl description of the interfce between crystl nd its melt re still in its erly stge. 1,2 Theoreticlly, the primry pproch to study the structure nd thermodynmics of the crystl-melt interfce hs been densityfunctionl theory (DFT). 3-6 However, for the hrd-sphere system considered in this pper, the vlue of the interfcil free energy γ obtined depends very much on the pproximtions used nd the density profile prmetriztions employed in the DFT studies, rnging from 0.25k B T/σ 2 to 4.00k B T/σ 2. 7 The DFT studies lso disgree drmticlly in the degree of orienttion dependence of the interfcil free energies. Experimentlly, the interfcil free energy γ cn be estimted from mesurements of crystl nucletion rtes interpreted through clssicl nucletion theory. After the studies of γ for number of mterils, Turnbull 8 found strong empiricl correltion between the interfcil free energy nd the ltent het of fusion per unit re given by the reltion γ C T f HF 2/3 s, where F s is the number density of the solid phse nd C T is the Turnbull coefficient tking 0.45 for most metls nd 0.32 for other mostly nonmetllic mterils. An interesting physicl rgument hs been proposed by Lird to explin the Turnbull rule. 9 Recently, some experiments 10,11 of the crystlliztion kinetics of colloidl systems tht closely pproximte the hrdsphere system hve been interpreted within clssicl nucletion theory to provide n experimentl estimte for the interfcil free energy of the hrd-sphere system of γ (0.55 * To whom correspondence should be ddressed. E-mil: xsong@istte.edu. Prt of the specil issue Dvid Chndler Festschrift. ( 0.02)k B T/σ 2. 12 This result is in good greement with tht predicted using Turnbull s empiricl reltionship bove. 7 However, the method of mesuring crystl nucletion rtes cnnot be used to determine the nisotropy of the interfcil free energy, nd the vlues obtined re not very ccurte due to the pproximtions inherent in clssicl nucletion theory. More ccurte nd direct techniques hve been developed, exmining the shpe of the interfce where it intersects with grin boundry. Unfortuntely, such experiments re difficult, nd so fr only few mterils hve been studied. For exmple, experimentlly determined γ of bismuth, 61.3 10-3 J/m 2,is reltively independent of crystl orienttion nd for Al the nisotropy is bout 2%, 13,14 but direct comprison with hrdsphere system results is not strightforwrd becuse of the longrnge ttrction interctions. The determintion of the interfcil free energy γ my lso be obtined using computer simultions. In recent yers, the clcultions of crystl-melt interfcil free energies hve been performed for Lennrd-Jones nd hrd-sphere systems by using number of techniques of computer simultions. For the Lennrd-Jones system, Broughton nd Gilmer 15 used cleving potentil pproch. In tht work, series of fictitious externl cleving potentil re used to crete interfces in bulk solid nd liquid phses nd then to bring the interfces into contct in nerly reversible mnner. The virtul work required to crete the crystl-melt interfce is then directly relted to the interfcil free energy. However, this erly work ws unble to ccurtely determine the nisotropy of the interfcil free energies. Recently, Dvidchck nd Lird pplied vrition of this technique to clculte the interfcil free energy of the hrd-sphere nd Lennrd-Jones systems using moleculr dynmic (MD) simultions nd provided sufficient ccurcies of nisotropy of the interfcil free energies. 16,17 An lternte pproch vi simultions, cpillry wve method, 2,18-20 hs been used nd pplied to number of model systems of metls in the pst few yers. 21-26 The pproch is bsed on the fct tht most crystl-melt interfces of interest re flt nd homogeneous over mcroscopic length scles but 10.1021/jp046289e CCC: $30.25 2005 Americn Chemicl Society Published on Web 02/12/2005
Free Energies of Hrd-Sphere Crystl-Melt Interfces J. Phys. Chem. B, Vol. 109, No. 14, 2005 6501 re rough nd inhomogeneous over microscopic length scles. Therefore, the mgnitude of the height fluctutions of the interfce depends on the stiffness of the interfces which is relted to the interfcil free energy. With this method, Morris nd Song 26 clculted the interfcil free energies for the Lennrd-Jones system using moleculr dynmic simultions, nd the results re in good greement with clcultions of the interfcil free energies bsed upon the cleving potentil pproch. 17 But the order of orienttion-dependent interfcil free energies is different from the result of hrd-sphere system. 16 In this work, we used this pproch to clculte the interfcil free energies for hrd-sphere system vi Monte Crlo simultions. It is found tht the orienttion-dependent interfcil free energies hve the sme order s those in the Lennrd-Jones system, but overll the greement between these two methods is good. Furthermore, we hve developed relible wy to determine the interfce position which is the key step for the cpillry wve method. The pper is orgnized s the following. In section II, we briefly outline the cpillry wve pproch nd the rtionle of our determintion of the interfce position. In section III, simultion results re presented with discussions relted to previous works. Some conclusions re drwn in section IV bsed upon our results. II. Cpillry Wve Approch Consider two-dimensionl crystl-melt interfce which is divided by grids with grid sizes y nd z ; the devition of the interfce from its verge position cn be defined by height function h(r ij ), where r ij ) i y ŷ + j z ẑ denotes the position on the interfce. As is well-known, it is the essentil ide of cpillry wve theory tht the free energy cost of interfcil height fluctutions is proportionl to the increse in the interfcil re cused by the fluctutions. 2 Assuming tht the grdients h/ y nd h/ z re very smll, this cn be written s F CW ) γ (θ) 2 [( h y) 2 + ( h z) 2 ] dydz (1) where γj(θ) is the interfcil stiffness governing fluctutions in the height of the crystl-melt interfce nd θ gives the orienttion of the interfce. The Fourier trnsform of interfcil height function h(r ij ) my be defined by y z h q ) A ij h ij exp(iq r ij ) (2) where A is the re of the interfce. In most of simultions using the cpillry wve method, the y-direction is very nrrow so tht the z-direction is long enough to probe the long wvelength fluctutions, which is essentil to extrct out the stiffness within resonble computtionl time. Thus, the simultion system is qusi-one-dimensionl system. With this in mind, the interfcil free energy in the Fourier spce then becomes F CW ) γ (θ) q 2 h 2 q 2 (3) q From the equiprtition theorem, one cn obtin 2 h q 2-1 ) γ (θ)q2 k B T The interfcil stiffness γj(θ) is relted to the interfcil free (4) TABLE 1: Summry of the Interfces Simulted, Including the Short Direction (the y Direction in the Text) in the Simultions interfce short direction energy γ(θ) by interfcil free energy interfcil stiffness (100) [001] γ 0(1 + 2 / 5ɛ 1 + 4 / 7ɛ 2) γ 0(1-18 / 5ɛ 1-80 / 7ɛ 2) (110) [001] γ 0(1-1 / 10ɛ 1-13 / 14ɛ 2) γ 0(1 + 39 / 10ɛ 1 + 155 / 14ɛ 2) (110) [1h10] γ 0(1-1 / 10ɛ 1-13 / 14ɛ 2) γ 0(1-21 / 10ɛ 1 + 365 / 14ɛ 2) (111) [1h10] γ 0(1-4 / 15ɛ 1 + 64 / 63ɛ 2) γ 0(1 + 12 / 5ɛ 1-1280 / 63ɛ 2) In this pper, we choose tht the x direction is norml to the interfce, the y nd z directions re the short nd long directions in the interfce, respectively. The equtions for interfcil stiffness in terms of γ 0 nd the nisotropy prmeters defined by eq 6 re shown. γ (θ) ) γ(θ) + γ (θ) (5) For n isotropic system, the interfcil stiffness reduces bck to the interfcil free energy. For n nisotropic system, in terms of the norml vector n ) (n 1, n 2, n 3 ) to the crystl-melt interfce, the interfcil free energy my be written s 26 γ(n) ) γ 0[ 1 + ɛ 1( n 4 i - 3 i 5) + ɛ 2( 3 n 4 i + 66n 2 1 n 2 2 n 2 3-17 i 7 )] (6) From this eqution, we cn derive equtions relting the interfcil stiffnesses nd free energies; the derived equtions re given in Tble 1. From Tble 1, it cn be clerly seen tht the prefctors of the nisotropy prmeters ɛ 1 nd ɛ 2 re much lrger for the stiffness thn the free energy; tht is, the interfcil stiffness is more nisotropic. 21-26 Through exmining the more nisotropic interfcil stiffness, we cn determine the vlues of the nisotropy prmeters ɛ 1 nd ɛ 2 more ccurtely. Thus, this is sensitive method to determine the nisotropy of the interfcil free energies. The key step of the cpillry wve method is how to clculte the height function of the crystl-melt interfces. In this pper, we developed relible method to define the interfcil prticle to determine the position of the crystl-melt interfce. To get height function derived from tomic configurtions, we first define locl order prmeter for ech prticle. Following the method in ref 25, we choose set of N q wve vectors qb i stisfying exp(iqb rb) ) 1 for ny vector rb connecting the nerest neighbors in perfect fcc lttice. We omit one of ech pir of ntiprllel wve vectors nd thus N q ) 6. Then we define the locl order prmeter s φ ) 1 1 N q Z nn exp(iqb rb) 2 (7) rb qb where the sum on rb runs over ech of Z nn nerest neighbors between the first- nd second-neighbor shells in perfect lttice. This order prmeter will be one for perfect fcc lttice nd less thn one otherwise. To reduce the effects of moleculr vibrtions due to therml fluctutions nd to improve the discrimintion between the liquid nd solid phses, n verged locl order prmeter φh is defined for ech prticle: 25 φj i ) 1 Z nn + 1 (φ i + φ j ) (8) j where j runs over ll Z nn nerest neighbors of prticle i. This
6502 J. Phys. Chem. B, Vol. 109, No. 14, 2005 Mu et l. Figure 1. A snpshot of the system with one (111) interfce generted from Monte Crlo simultions detiled in section III. The solid circles represent the liquid prticles, while the open circles correspond to the solid prticles. The interfcil prticles re indicted by open circles with cross. Figure 2. () Order prmeter φh vs position for ech prticle in n instntneous configurtion of two (111) interfces (shown in Figure 1). The center region, where the order prmeter is smll, is the liquid region, while the regions where φh is lrge correspond to the crystl region. (b) Order prmeter Z nns vs position for ech prticle in the sme configurtion. The prticles with the order prmeter Z nns ) 0 correspond to the liquid prticles, while the prticles with Z nns g Z s () 7) correspond to the solid prticles. The prticles tht hve the vlues of the order prmeter 0 < Z nns < Z s correspond to the interfcil prticles. procedure helps to eliminte the difference of the locl order prmeters between the prticle nd its environment due to therml fluctutions. Figure 1 shows snpshot of the system with one (111) interfce, nd Figure 2 shows the corresponding locl order prmeter φh i of ech prticle in the snpshot. From Figure 2, it cn be seen tht the vlue of the locl order prmeter of the liquid phse is smll, while the solid phse hs the locl order prmeter with lrge vlues. However, only with the order prmeter φh i, it is difficult to identify the interfcil prticles relibly. As the determintion of interfcil prticles leds to the interfce position nd the stiffness is sensitive to the position of the interfce (cf Figure 4), relible wy to determine these interfcil prticles is crucil for the ccurte extrction of the stiffness. To define the interfcil prticles, we first tke certin vlue φh s s threshold vlue of the order prmeter for the solid phse; tht is, the prticles with the order prmeter φh g φh s belong to TABLE 2: System Geometries nd Number of Prticles in Simultions nd Resultnt Interfcil Stiffnesses γj long with the Fitted Interfcil Stiffnesses from the Anisotropy Prmeters E 1 nd E 2 interfce geometry number of prticles interfcil stiffness from simultion fitted interfcil stiffness with ɛ 1, ɛ 2 (100)[001] 66.28 6.26 109.58 44 800 0.55 0.55 (110)[001] 70.20 6.26 88.58 38 400 0.71 0.71 (110)[1h10] 70.21 6.64 93.96 43 200 0.49 0.43 (111)[1h10] 65.81 6.64 95.87 41 400 0.80 0.80 The geometries re shown with ll lengths in units of σ, while ll interfcil stiffnesses re in units of k BT/σ 2. the solid phse. According to this bsic criterion, we cn clculte the number Z nns of the nerest neighboring solid prticles for ech prticle. Figure 2b shows the number Z nns of the nerest neighboring solid prticles of ech prticle in the sme configurtion s tht in Figure 2. And now we cn define the other threshold vlue Z s s new criterion to identify the solid prticles; tht is, the prticles with Z nns g Z s re solid prticles. In this pper, we choose Z s ) 7. For perfect fcc crystl, ech prticle hs 12 nerest neighbors. Considering the effects of instntneous moleculr vibrtions due to therml fluctutions, the prticles which re surrounded by more thn hlf of the 12 nerest neighbors re tken s solid prticles. Thus, we choose Z s ) 7 here. In fct, the results re quite robust with respect to other resonble choices of Z s. For exmple, with the choice of φh s ) 0.15 for the (111) interfce, s in Figure 4 the stiffness chnges re within 4% if Z s vries from 4 to 7. With these two criteri, we cn define the prticles with the locl order prmeter φh i < φh s nd t the sme time the number of the nerest neighboring solid prticles 0 < Z nns < Z s s the interfcil prticles. Tht is, the prticles tht hve smll locl order prmeter nd t the sme time re next to solid prticles re defined s interfcil prticles. Thus, the intrinsic interfce between the solid nd liquid phses cn be determined relibly by those interfcil prticles. It cn be esily understood tht the vlue of Z nns for ech prticle depends on the criterion φh s. Thus, the criticl prmeter φh s is the only bsic nd crucil prmeter in this method. A relible method to determine this crucil prmeter φh s is presented in section III. With this method, we evlute the height function h(x) of the crystl-melt interfces nd clculte the nisotropic interfcil free energies for the hrd-sphere system. Our results clerly indicte tht the nisotropy is wek but cn be ccurtely resolved using this pproch due to the sensitivity of the height fluctutions of the interfce on the nisotropy. III. Simultion Results nd Discussion In this pper, we studied the fcc crystl-melt interfces of the hrd-sphere system using Monte Crlo simultions with NPT ensemble. First, pure crystl nd liquid phses re simulted seprtely with periodic boundry conditions, which hve the density pproprite for the bulk phse t the melting pressure P ) 11.57. 27 Subsequently, the crystl nd liquid systems re joined together to crete two crystl-melt interfces. Four different crystl-melt interfce simultions re performed; these geometries re summrized in Tble 2. The simultion boxes re chosen to be nrrow in the [001] direction (4 0 ) or the [1h10] direction (3 2 0 ). The choice of this qusi-one-dimensionl geometry ensures tht the height fluctutions of the interfce re essentilly functions of only one direction, which mkes the nlysis esier. In this pper, we choose tht the x direction
Free Energies of Hrd-Sphere Crystl-Melt Interfces J. Phys. Chem. B, Vol. 109, No. 14, 2005 6503 Figure 3. Inverse of the verge squred mplitude of the height fluctutions, hq 2-1 vs q 2, for different crystl-melt interfce orienttions. Ech orienttion includes two interfces. The vlues of hq 2-1 re clculted for the cse of φh s ) 0.15 s result of procedure shown in Figure 4. In the insets, the solid lines re the fit to smll q portions (long wvelength modes), nd the liner behvior indictes the vlidity of the cpillry wve method. The error brs re of the order of the size of the symbols. is norml to the interfce nd the y nd z directions re the short nd long directions in the interfce, respectively. The system is run for 200 000 MC steps t the melting pressure to be equilibrted nd run for 500 000 MC steps for dt collection. The configurtions of the system re stored every 100 MC steps during the run for dt collection. Using the bove method of defining the interfcil prticles, we cn determine the discrete height function of the crystlmelt interfce by simply clculting the verge position of ll the interfcil prticles in ech grid. To obtin good sptil resolution nd t the sme time to ensure tht there re sufficient interfcil prticles in ech grid to define the height function, we choose y, z 0 ( 0 is the lttice constnt of fcc crystl). The vlues of h q 2-1 re clculted for ech of the geometries for the cse of φh s ) 0.15, nd the results re shown in Figure 3. As cn be seen in Figure 3, lthough the globl curves re not stright lines s nticipted in eq 4, in the smll q region (the long wvelength modes) the simultion results follow the liner behvior, which indictes the roughness of the interfces with Gussin sttistics. In fct, becuse the simultion system is qusi-one-dimensionl geometry structure (L z. L y ), the height fluctutions in the short direction (y direction) re neglected, which suggests tht the height fluctutions within the equl length region in the long direction (z direction) should lso be neglected; tht is, only the smll q modes of q < 2π/L y re vlid (2π/L y 1 in our cse). The devition for lrge q depends on the detils of the method of clculting the height function of the interfce nd the inherent nonliner behvior of prticle s motion t short length scle. In ddition, for the very long wvelength modes which cn be comprble with the length of the long direction of simultion system, the devition is due to the fct tht these modes need very long relxtion nd smpling time to be converged. Thus, by fitting the smll q (long wvelength) portion with best linerity to the form given in eq 4 for ech of the different interfce orienttions, we obtined the interfcil stiffness from the slope of the fitted stright line. However, the vlues for the stiffnesses found from the fittings depend on the choice of the prmeters to define Figure 4. Dependence of the interfcil stiffness on the crucil prmeter φh s for different crystl-melt interfce orienttions. the interfce position. We hve found the crucil prmeter is φh s in our cse, nd relible procedure for the choice of φh s is given below. As discussed bove, the criticl prmeter φh s is the only bsic nd crucil prmeter to define the interfcil prticles, in turn, the interfce position. However, the choice of the vlue of φh s is not relible if it is chosen directly from Figure 2 by visul inspection s done previously. On one hnd, if the vlue of the criticl prmeter φh s is chosen to be too smll, it will occur inevitbly tht number of prticles which essentilly belong to liquid phse t equilibrium re tken s the interfcil prticles; on the other hnd, if the vlue of φh s is too lrge, the prticles essentilly belonging to solid phse re regrded s the interfcil prticles. In both cses, the profile of the interfce is chnged drmticlly by those pseudo interfcil prticles, nd ccordingly, the height fluctutions of the interfce become stronger due to greter noise, which results in the fct tht the clculted results of the interfcil stiffnesses re smller thn the true vlue. Thus, there should exist n intermedite vlue for the criticl prmeter φh s by which the position of the interfce cn be determined reltively more ccurtely, nd the corresponding clculted results of the interfcil stiffnesses should hve mximum. To determine the vlue of the prmeter φh s, we clculted the interfcil stiffnesses for different interfce orienttions with different φh s. The dependences of the interfcil stiffness on the crucil prmeter φh s for different interfces re shown in Figure 4. From Figure 4, it cn be clerly seen tht the interfcil stiffnesses of ll interfces depend very much on the criticl prmeter φh s nd hve distinct peks round φh s ) 0.15, ll indicting tht the pproprite vlue of this criticl prmeter is φh s ) 0.15. Therefore, the corresponding pek vlues re the best estimte of the equilibrium interfcil stiffnesses. Results bsed on the choice of φh s ) 0.15 re listed in Tble 2. All interfcil stiffnesses from simultions should be used with equtions given in Tble 1 to clculte the vlues of the prmeters γ 0, ɛ 1, nd ɛ 2 by lest-squres fitting. But, here, we clculted the prmeters using the three stiffnesses of (100)[001], (110)[001], nd (111)[1h10] interfces due to the fct tht their h q 2-1 curves hve better linerity in the smll q region. This procedure yields γ 0 ) 0.62, ɛ 1 ) 0.056, nd ɛ 2 )-0.0073. The fitted interfcil stiffnesses of different crystl-melt interfces with these prmeters re given in Tble 2. We lso clculted the interfcil free energies of different interfces with vrious orienttions using these prmeters ccording to the equtions given in Tble 1. These results cn be compred
6504 J. Phys. Chem. B, Vol. 109, No. 14, 2005 Mu et l. TABLE 3: Comprison with the Results of Dvidchck nd Lrid for Interfcil Free Energies of Different Crystl-Melt Interfce Orienttions (All Interfcil Free Energies Are in Units of k B T/σ 2 ) current work Dvidchck nd Lrid γ 0 0.62 ( 0.02 0.61 ( 0.01 γ 100 0.64 ( 0.02 0.62 ( 0.01 γ 110 0.62 ( 0.02 0.64 ( 0.01 γ 111 0.61 ( 0.02 0.58 ( 0.01 with the results of Dvidchck nd Lrid 16 bsed on the cleving potentil method, which is shown in Tble 3. As cn be seen in Tble 3, the verge interfcil free energy γ 0 ) 0.62k B T/σ 2 of the current work is in very good greement with 0.61k B T/σ 2 from Dvidchck nd Lrid using the cleving potentil method vi simultion nd 0.616k B T/σ 2 from the clssicl nucletion theory estimte. 28 However, there re some differences in the interfcil free energies of different interfce orienttions between our results nd tht of Dvidchck nd Lrid. Both clcultions show tht the interfcil free energies re slightly nisotropic nd the (111) interfce hs the lowest interfcil free energy, but for the (100) nd (110) interfces, our results indicte γ 100 > γ 110, in contrst to γ 100 < γ 110 of the result from Dvidchck nd Lrid. Our results indicte tht for the hrd-sphere system the order of the interfcil free energies of different crystl-melt interfce orienttions is γ 100 > γ 110 > γ 111, which is similr to the results of the Lennrd- Jones system. 17,26,29 IV. Conclusion In summry, we developed relible method to define the interfcil prticles to determine the position of the intrinsic crystl-melt interfce nd hve computed the height fluctutions of the interfce from different crystl-melt orienttions using the Monte Crlo technique. From the interfcil fluctutions, we hve clculted the nisotropic interfcil stiffnesses with different criticl prmeter φh s nd extrcted the equilibrium interfcil stiffnesses of those crystl-melt interfces. Fitting the interfcil stiffnesses to the equtions given in Tble 1, we clculted the verge interfcil free energy γ 0 ) 0.62 ( 0.02k B T/σ 2 nd the nisotropic prmeters ɛ 1 ) 0.056 nd ɛ 2 ) -0.0073. We find tht the nisotropy of the interfcil free energy is wek nd the interfcil free energies of different crystl-melt interfce orienttions re γ 100 ) 0.64 ( 0.02, γ 110 ) 0.62 ( 0.02, nd γ 111 ) 0.61 ( 0.02k B T/σ 2. Our results re in good greement with the results of Dvidchck nd Lrid, 29 bsed on the clcultions of the reversible work required to crete the interfces. The nisotropic order is γ 100 > γ 110 > γ 111, which is similr to the results of the Lennrd-Jones system. However, we should point out tht the results in current work re bsed upon prticulr wy to define the interfce with moderte system size nd cn be refined by more extensive clcultions bsed on lrger simultion systems in future work. Combined with the good greement of Lennrd-Jones system between two different methods to determine the crystl-melt interfcil free energies, 17,26 it is resonble to stte tht the cpillry wve pproch cn be relible nd efficient technique to clculte crystl-melt interfcil free energies. Acknowledgment. The uthors re grteful for the finncil support by NSF Grnt CHE0303758. A.H. is n REU student supported by NSF Grnt CHE0139152. We re indebted to Jmie Morris for mny helpful discussions. References nd Notes (1) Pimpinelli, A.; Villin, J. Physics of Crystl Growth; Cmbridge University Press: Cmbridge, 1998. (2) Hoyt, J. J.; Ast, M.; Krm, A. Mter. Sci. Eng. 2003, R41, 121. (3) Curtin, W. A. Phys. ReV. Lett. 1987, 59, 1228. (4) Curtin, W. A. Phys. ReV. B: Condens. Mtter 1989, 39, 6775. (5) Mrr, D. W.; Gst, A. P. Phys. ReV. E: Stt. Phys., Plsms, Fluids, Relt. Interdiscip. Top. 1993, 47, 1212. (6) Ohnesorge, R.; Lowen, H.; Wgner, H. Phys. ReV. E: Stt. Phys., Plsms, Fluids, Relt. Interdiscip. Top. 1995, 50, 4801. (7) Mrr, D. W. J. Chem. Phys. 1995, 102, 8283. (8) Turnbull, D. J. Appl. Phys. 1950, 21, 1022. (9) Lird, B. B. J. Chem. Phys. 2001, 115, 2887. (10) Smits, C.; vn Duijneveldt, J. S.; Dhont, J. K. G.; Lekkerkerker, H. N. W.; Briels, W. J. Phse Trnsitions 1990, 21, 157. (11) Underwood, S. M.; Tylor, J. R.; vn Megen, W. Lngmuir 1994, 10, 3550. (12) Mrr, D. W.; Gst, A. P. Lngmuir 1994, 10, 1348. (13) Glicksmn, M. E.; Vold, C. Act Metll. 1969, 17, 1. (14) Npolitno, R. E.; Liu, S.; Trivedi, R. Interfce Sci. 2002, 10, 217. (15) Broughton, J. Q.; Gilmer, G. H. J. Chem. Phys. 1986, 84, 5759. (16) Dvidchck, R. L.; Lird, B. B. Phys. ReV. Lett. 2000, 85, 4751 nd references therein. (17) Dvidchck, R. L.; Lird, B. B. J. Chem. Phys. 2003, 118, 7651. (18) Lcsse, M. D.; Grest, G. S.; Levine, A. J. Phys. ReV. Lett. 1998, 80, 309. (19) Sides, S. W.; Grest, G. S.; Lcsse, M. D. Phys. ReV. E: Stt. Phys., Plsms, Fluids, Relt. Interdiscip. Top. 1999, 60, 6708. (20) Chcón, E.; Trzon, P. Phys. ReV. Lett. 2003, 91, 166103. (21) Hoyt, J. J.; Ast, M.; Krm, A. Phys. ReV. Lett. 2001, 86, 5530. (22) Morris, J. R.; Lu, Z. Y.; Ye, Y. Y.; Ho, K. M. Interfce Sci. 2002, 10, 143. (23) Hoyt, J. J.; Ast, M. Phys. ReV. B: Condens. Mtter 2002, 65, 214106. (24) Ast, M.; Hoyt, J. J.; Krm, A. Phys. ReV. B: Condens. Mtter 2002, 66, 100101(R). (25) Morris, J. R. Phys. ReV. B: Condens. Mtter 2002, 66, 144104. (26) Morris, J. R.; Song, Xueyu J. Chem. Phys. 2003, 119, 3920. (27) Frenkel, D.; Smit, B. Understnding Moleculr Simultion: from lgorithms to pplictions; Acdemic Press: Sn Diego, CA, 1996. (28) Ccciuto, A.; Auer, S.; Frenkel, D. J. Chem. Phys. 2003, 119, 7467. (29) Lird, B. B. Privte communiction, revised simultions bsed on the cleving method give the sme order γ 100(0.592) > γ 110(0.571) > γ 111(0.558) with somewht lower numbers.