Free energy of cluster formation and a new scaling relation for the nucleation rate
Kyoko K. Tanaka, Jürg Diemand, Raymond Angélil, Hidekazu Tanaka
aa r X i v : . [ phy s i c s . a t m - c l u s ] M a y Free energy of cluster formation and a new scaling for nucleation
Free energy of cluster formation and a new scaling relation for the nucleationrate
Kyoko K. Tanaka, J¨urg Diemand, Raymond Ang´elil, and Hidekazu Tanaka Institute of Low Temperature Science, Hokkaido University, Sapporo 060-0819,Japan Institute for Computational Science, University of Z¨urich, 8057 Z¨urich, Switzerland (Dated: 17 September 2018)
Recent very large molecular dynamics simulations of homogeneous nucleation with (1 − · Lennard-Jonesatoms [Diemand et al. J. Chem. Phys. , 074309 (2013)] allow us to accurately determine the formationfree energy of clusters over a wide range of cluster sizes. This is now possible because such large simulationsallow for very precise measurements of the cluster size distribution in the steady state nucleation regime. Thepeaks of the free energy curves give critical cluster sizes, which agree well with independent estimates basedon the nucleation theorem. Using these results, we derive an analytical formula and a new scaling relationfor nucleation rates: ln J ′ /η is scaled by ln S/η , where the supersaturation ratio is S , η is the dimensionlesssurface energy, and J ′ is a dimensionless nucleation rate. This relation can be derived using the free energyof cluster formation at equilibrium which corresponds to the surface energy required to form the vapor-liquidinterface. At low temperatures (below the triple point), we find that the surface energy divided by that of theclassical nucleation theory does not depend on temperature, which leads to the scaling relation and implies aconstant, positive Tolman length equal to half of the mean inter-particle separation in the liquid phase.PACS numbers: 05.10.-a, 05.70.Np, 05.70.Fh, 64.60.QbKeywords: molecular dynamics simulation, nucleation, phase transitions, scaling relation I. INTRODUCTION
The nucleation process of supersaturated vapors intoliquids (or solids) has been studied for a long time, how-ever, there is still a serious gap in our understanding.The classical nucleation theory (CNT) is a very widelyused model for describing nucleation and provides thenucleation rates as a function of temperature, supersat-uration ratio, and macroscopic surface tension of a con-densed phase. However, several studies have found thatthe CNT fails to explain the nucleation rates observedin experiments − . For example, the error is the orderof 10 − for argon . In addition to laboratory ex-periments, numerical simulations of molecular dynamics(MD) or Monte Carlo (MC) simulations showed that thenucleation rates obtained by numerical simulations aresignificantly different from predictions by the CNT − .Until now several modifications to the CNT were pro-posed. It was also noted that several nucleation rate datasets exhibited empirical temperature scalings . Al-though there have been significant advances in the theo-retical models, a quantitatively reliable theoretical modeldoes not yet exist.Recently, Diemand et al. presented large-scale molec-ular dynamics (MD) simulations of homogeneous vapor-to-liquid nucleation of (1 − × Lennard-Jones atoms,covering up to 1.2 µ s (5 . × steps). The simulationscover a wide range of temperatures and supersaturationratios. This study measured various quantities such asnucleation rates, critical cluster sizes, and sticking proba-bilities of vapor molecules, and it was successful in quan-titatively reproducing argon nucleation rates at the samepressures, supersaturations and temperatures as in the SSN (Supersonic Nozzle Nucleation) argon experiment .Here we use these MD results to determine the free ener-gies of cluster formation (Sec. III) and their scaling (Sec.IV), which is expected to be of use in the construction ofa high-precision nucleation model. II. EMPIRICAL SCALING RELATIONS
Hale and Thomason suggested that the nucleationrate J obtained by MC simulations using LJ moleculeswas scaled by ln S/ ( T c /T − . over a range of J =(10 − )cm − s − which corresponds to (10 − − − ) σ − τ − , where T , T c , σ , and τ are the tempera-ture, critical temperature, a parameter of length (= 3 . .
16 ps). Figure 1 shows that nu-cleation rates obtained by the MD and MC simulationsfor LJ molecules and experimental results for argon as afunction of ln S/ ( T c /T − . and ln S/ ( T c /T − . . Thescaling by ln S/ ( T c /T − . works for MC simulationsover a limited range, however, the nucleation rates ob-tained by all MD simulations and some experiments arerather scaled by ln S/ ( T c /T − . . The fitting function islog J = 17 . S/ ( T c/T − . −
51. This linear, empiri-cal scaling relation seems to work well over a surprisinglywide range of nucleation rates, J = (10 − − − ) σ − τ − for the MD data and the NPC (Nucleation Pulse Cham-ber) experiment , but not for the MC simulations. In-terestingly, a different scaling relation, ln S/ ( T c/T − / has been found from experimental nucleation rates forseveral different substances such as water , toluene ,and nonane . Our results suggests the scaling relationdepends on the substance type.ree energy of cluster formation and a new scaling for nucleation 2 −30 −20 −10 −30 −20 −10 J [ σ − τ − ] ln(S)/[T c /T−1] J [ σ − τ − ] ln(S)/[T c /T−1] ab * * J [ c m − s − ] J [ c m − s − ] FIG. 1. Nucleation rates obtained by the MD simulationswith LJ molecules and the experimental results for argon as afunction of (a) ln S/ ( T c /T − . and (b) ln S/ ( T c /T − . .The results for various supersaturation ratios S and temper-atures T ∗ (= kT /ǫ in the Boltzmann constant k and thedepth of the LJ potential ǫ ) obtained by the large-scale MDsimulations and the previous ones are shown by thefilled circles and the crosses, respectively. The results forMC simulations are shown with square markers, where thetemperatures are T ∗ = 0 . , . , . , and 0.335. The trian-gles show the experimental results for argon . We adopt T c = 1 . ǫ/k (or 151 K) in the simulations (or experiments).In (b) the fitting function (solid line) for J [ σ − τ − ] is givenby log J = 17 . S/ ( T c/T − . − However, linear empirical scaling relations contradictone of the most basic, general expectations from nucle-ation theory: according to the nucleation theorem, thesize of the critical cluster i ∗ is determined by the deriva-tive d (ln J ) /d (ln S ) . These empirical scalings there-fore imply a constant critical cluster size i ∗ at each tem-perature over a wide range in J . The corresponding freeenergy functions would need to peak at exactly the samesize over a wide range in S and J , which seems impossi-ble to achieve with any reasonably smooth surface energyfunction. Instead of a linear relation, one would insteadexpect some downward curvature in Fig. 1, which is con-sistent with the MD data points alone, but not in com-bination with the NPC experiment. III. RECONSTRUCTING THE FORMATION FREEENERGY FROM MD SIMULATIONS
We now derive the free energies of cluster formationdirectly from MD results and compare them with predic-tions from three widely used models: In the (modified)classical nucleation theory CNT (or MCNT) and in the semi-phenomenological (SP) model , the free energies∆ G i are respectively∆ G i, CNT kT = − i ln S + ηi / , (1)∆ G i, MCNT kT = − ( i −
1) ln S + η ( i / − , and (2)∆ G i, SP kT = − ( i −
1) ln S + η ( i / −
1) + ξ ( i / − , (3)where S = P /P e is the supersaturation ratio ofmonomers using the saturated vapor pressure P e andthe partial pressure of monomers P , η and ξ aretemperature-dependent quantities which can be fixedfrom the condensed phase surface tension, bulk densityand the second virial coefficient . Note that the CNTassumes large cluster sizes, it is not expected to work forsmall clusters and its ∆ G i does not vanish at i = 1, i.e.,for monomers.The formation free energy of a cluster is directly re-lated to the equilibrium size distribution n e ( i ):∆ G i kT = ln (cid:18) n (1) n e ( i ) (cid:19) , (4)where n (1) is the number density of the monomers .For small subcritical clusters ( i < ∼ i ∗ ), the steady statesize distribution n ( i ), which can be measured in MDsimulations, agrees very well with the equilibrium sizedistribution n e ( i ) , which lets us obtain ∆ G i for smallclusters . Obtaining the full free energy land-scape, including the crucial region around the criticalsizes, requires a more sophisticated method, which takesthe difference between steady state and equilibrium sizedistributions into account. A first procedure of this kindwas proposed by Wedekind and Reguera based on meanfirst passage time (MFPT) method. In principle it allowsa full reconstruction based on a large number of smallsimulations, each one is run until it produces one nucle-ation event. However, the observation of one event doesnot demonstrate that the simulations are really samplingthe assumed steady state nucleation regime, the passagetimes might include some initial lag time and a significanttransient nucleation phase, which precedes the steadystate regime . Both time-scales become quite large forLJ vapor-to-liquid nucleation at low temperatures .Our recent, very large scale nucleation simulations al-low very precise measurements of the cluster size distri-bution during a clearly resolved steady state nucleationregime and under realistic constant external conditions .Here we present a new method to obtain the full free en-ergy landscape from these steady state size distributions:The nucleation rate is the net number of the transitionfrom i -mers to i + 1-mers and given by J = R + ( i ) n ( i ) − R − ( i + 1) n ( i + 1) , (5)where R + ( i ) is the transition rate from a cluster of i molecules, i -mer, to ( i +1)-mer per unit time, i.e., theaccretion rate, and R − ( i ) is the transition rate from i -mer to ( i -1)-mer per unit time, i.e., the evaporation rateree energy of cluster formation and a new scaling for nucleation 3of i -mer. R + ( i ) is given by R + ( i ) = αn (1) v th (4 πr i / ),where α is the sticking probability, v th is the thermalvelocity, p kT / πm , and r is the radius of a monomer,(3 m/ πρ m ) / where m is the mass of a molecule and ρ m is the bulk density. The evaporation rate is obtained fromthe principle of detailed balance in thermal equilibrium: R − ( i + 1) n e ( i + 1) = R + ( i ) n e ( i ) . (6)From Eqs.(5) and (6), the nucleation rate is given by J = " ∞ X i=1 R + ( i ) n e ( i ) − ≃ R + ( i ∗ ) n e ( i ∗ ) Z, (7)with the Zeldovich factor, Z .From Eqs.(5) and (6), we obtain n e ( i ) n ( i ) = n e ( i − n ( i − (cid:18) − JR + ( i − n ( i − (cid:19) − . (8)Equation (8) is a recurrence relation and enables us toobtain n e ( i ) if J, n ( i ) and n e ( i −
1) are known . Fig. 2shows n e ( i ), n ( i ), and ∆ G i ( S ) derived by Eq. (8) for atypical example ( T ∗ = kT /ǫ = 0 . S = 16 . ). ∆ G i ( S = 1) is a surface term corresponding to thework required to form the vapor-liquid interface. FromEq. (8), we obtain ∆ G i ( S = 1):∆ G i ( S = 1) = ∆ G i ( S ) + ( i −
1) ln S, (9)using the dependence of the supersaturation in the the-ories except the CNT. Fig. 2 also shows ∆ G i ( S = 1).The surface terms of free energy ∆ G i ( S = 1) at varioustemperatures and supersaturation ratios obtained by MDsimulations are shown in Figure 3, where we evaluated R + ( i ) using α obtained by the MD simulations (Table IIIin Diemand et al.). From Figure 3, we confirm ∆ G i ( S =1) depends only on temperature, which implies that thevolume term in Eqs.(2) and (3) works very well.The peaks of the free energy curves give critical clustersizes which agree very well with those from the nucleationtheorem (see Fig. 7). Since the nucleation rates, whichenter into the nucleation theorem, do not depend on thedetailed cluster definition, this good agreement providesa robust confirmation, that the simple Stillinger crite-rion used here gives realistic cluster size estimates.An earlier study found that critical sizes based on theStillinger definition are up to a factor 2 larger than in-dependent estimates from the nucleation theorem. Thiscontradiction can be resolved by a detailed comparisonwith other MD simulations at very similar conditions :Using the initial supersaturations S in the nucleationtheorem (as in ) instead of the actual supersaturation S during the simulation , leads one to underestimate thecritical sizes by up to a factor of 1.8, which eliminates thediscrepancy reported in . −10 −5 T* =0.6 n e ( i ) , n ( i ) i ∆ G i ( S ) / k T S =17 S =1 n e (i)n(i) S =17 ab FIG. 2. (a) ∆ G i ( S ) / ( kT ) as a function of i , where T ∗ = kT /ǫ = 0 . S = 16 . ). The dashed line shows ∆ G i ( S ) / ( kT ) at S = 1. (b) Theequilibrium number density of i -mers n e ( i ) [ σ − ] (solid curve)and the steady number density obtained by the simulation n ( i ) [ σ − ] (circles). T * =1.0T * =0.8T * =0.6 ∆ G i ( S = ) / ( k T ) i T * =0.5T * =0.4T * =0.3 FIG. 3. ∆ G i ( S = 1) as a function of i for various tempera-tures. At each temperature, we show ∆ G i ( S = 1) obtainedby the different values of the supersaturation ratio. The cir-cles show the critical clusters derived by the maximum of∆ G i ( S ) for various supersaturation ratios S . We can confirm∆ G i ( S = 1) depends on only T . IV. A NEW SCALING FOR NUCLEATION RATES
Fig. 4 shows the surface energy ∆ G i ( S = 1) dividedby that of the CNT, ∆ G i ( S = 1) / ( ηi / kT ), as a func-tion of i − / . The theoretical evaluations are also shownin Fig. 4. The simulation results agree with the SPmodel at 0 . < ∼ i − / <
1, but deviate from the modelree energy of cluster formation and a new scaling for nucleation 4 i −1/3 SPMCNTT * ∆ G i ( S = ) / ( η i / k T ) CNTT * ab Eq.(10)
FIG. 4. (a) ∆ G i ( S = 1) / ( ηi / kT ) as a function of i − / at kT /ǫ ≤ .
6. Results obtained from 11 MD simula-tions are plotted with symbols: different symbols indicatethe MD results starting from different supersaturation ra-tios. We find that they are universal, which implies that∆ G i ( S =1) / ( ηi / kT ) is independent of temperature for ( T ≤ . G i ( S = 1) / ( ηi / kT ) =1 . − i − / ) (the dotted-dashed line). The results by theSP (dotted lines) and MCNT (dashed line) are also shown.∆ G i ( S = 1) / ( ηi / kT ) = 1 in the CNT. (b) The same as (a)but for all temperatures. −4−202 i [ ∆ G i ( S = ) − ∆ G i , CN T ( S = ) ]/ ( η k T ) T * [ ∆ G i − ∆ G i , M CN T ]/ ( η k T ) −202 T * =0.60.81.0 FIG. 5. The difference in ∆ G i ( S =1) between MD resultsand the CNT divided by ηkT , [∆ G i ( S = 1) − ∆ G i, CNT ( S =1)] / ( ηkT ) as a function of i at various temperatures. Differentsymbols indicate the MD results starting from different super-saturation ratios. The results of McGraw and Laaksonen are also shown by dotted lines for T ∗ = 0 . , . G i − ∆ G i, MCNT ] / ( ηkT ),which is valid for any value of S . for larger clusters of i − / < .
5. Surprisingly, ∆ G i ( S =1) / ( ηi / kT ) is almost the same for all results obtainedby 11 MD simulations for temperatures below the triplepoint. This indicates that ∆ G i ( S = 1) / ( ηi / kT ) is afunction of i and independent of the temperature. Fromthe fitting of the results, we obtain∆ G i ( S = 1) ηi / kT = f ( i ) = A (1 − i − / ) , (10)where A = 1 .
28. The fitting function is also shownby the dotted-dashed line in Fig. 4. Equation (10) im-plies a constant, positive Tolman length of δ = 0 . r and the constant A sets an effective normalisation factorfor the planar surface energy (or the surface area), if weinterpret ∆ G i ( S = 1) / ( ηi / kT ) = a i γ i / (4 πr γ ), where γ i = γ [1 − δ/ ( r i / )] and a i are the surface tension andsurface area of the cluster and γ is the planar surface ten-sion. Equation (10) could be a promising candidate foran accurate nucleation theory, in which A is temperatureindependent below the triple point. Our result indicatesthat at low temperatures the Tolman relation is valideven for very small clusters including 2-30 atoms.McGraw and Laaksonen obtained ∆ G i of largeclusters ( i > ∼
50) with density functional calculations.They found that the deviation of ∆ G i from the CNTis temperature dependent, but independent of the clustersize. Figure 5 shows the difference of ∆ G i ( S = 1) betweenMD results and the CNT, i.e., ∆ G i ( S = 1) − ∆ G i, CNT ( S =1) as a function of i . We find these differences arenearly constant around i ∼
10 for each temperature.But they increase with the size for i >
20. Accordingto McGraw and Laaksonen (1997) , on the other hand,[∆ G i ( S = 1) − ∆ G i, CNT ( S = 1)] / ( ηkT ) are calculated tobe -2.46, -3.26, and -4.88 for T ∗ = 0 . , .
8, and 1.0, re-spectively.Using Eq. (10), the critical cluster i ∗ is obtained by i ∗ = (cid:18) Aη S (cid:19) s − A ln Sη ! , (11)from the following relation − ln Sη + 23 i − / ∗ f ( i ∗ ) + i / ∗ f ′ ( i ∗ ) = 0 , (12)where we have assumed that the molecular volume is farsmaller in the liquid phase than in the gas phase. Thedetailed derivation is given in the Appendix.We also derive the analytical formula for the nucleationrate: ln J ′ = ln[ αZi / ∗ ] + ( i ∗ + 1) ln S − i / ∗ ηf ( i ∗ ) , (13)where J ′ is a dimensionless nucleation rate defined by J ′ = J/ (4 πr n v th ) with the saturated number densityof monomers n sat (= n (1) /S ) and the Zeldvich factor isgiven by Z = 13 i − / ∗ r Aηπ (1 − i − / ∗ ) . (14)ree energy of cluster formation and a new scaling for nucleation 5 −30 −20 −10 J [ σ − τ − ] ln S * J [ c m − s − ] FIG. 6. The nucleation rate as a function of the supersatura-tion ratio. The analytical formula for the nucleation rates areshown by solid lines. The results for various temperature andsupersaturation ratios by the large-scale MD simulations and the previous ones are shown by the filled circles andthe crosses, respectively. The results for MC simulations are shown by the squares, where the temperature is T ∗ = 0 . . Fig. 6 shows the nucleation rate as a function of ln S ob-tained by the MD simulations and the analytical formula.We find good agreements between the analyses and thesimulations for the various temperatures and supersatu-ration ratios.Our finding that ∆ G i ( S = 1) / ( ηi / kT ) is independentof the temperature leads to a scaling relation. Equa-tion (12) indicates that i ∗ is a function of only ln S/η .Thus from Eq. (13) ln J ′ /η is determined only by ln S/η ,neglecting a term including Zeldovich factor which issmaller than the other terms. Fig. 7 shows the size ofcritical clusters and ln J ′ /η obtained by MD and MCsimulations and experiments as a function of ln S/η . Weconfirm that ln J ′ /η is scaled by ln S/η almost perfectlyfor MD simulations, at T ∗ ≤ .
6. At high temperatures( T ∗ > . J ′ /η deviates from the scaling relation.This would come from the deviation of f ( i ), i.e., f ( i ) de-pends on T at T ∗ > . T ∗ = 0 . T ∗ = 0 . V. SUMMARY AND CONCLUSIONS
We derived for the first time the formation free energyof a cluster over a wide range of cluster sizes and tem-peratures from recent very large-scale MD simulations.The peaks of the free energy curves give critical clustersizes, which agree well with independent estimates basedon the nucleation theorem. This implies that the simpleStillinger criterion used here gives realistic cluster sizeestimates.At low temperatures the free energies show a univer-sal deviation from the CNT, which allows us to derive a l n J ’ / η ln S / η * i* experiment ab MC previous FIG. 7. We propose that (a) the size of critical clus-ter and (b) ln J ′ /η are determined only by ln S/η , where J ′ = J/ (4 πr n sat v th ). The analytical formula obtained byour model are shown by the solid lines. Panel (a) shows,that the critical clusters sizes derived from the maximum of∆ G i ( S ) (filled circles) and from the nucleation theorem (opencircles, i NT in Diemand et al. ) agree very well with eachother and also with our analytical model. In (b), the resultsfor various temperature and supersaturation ratios by thelarge-scale MD simulations and the previous ones areshown by the filled circles and the crosses, respectively. Theresults for MC simulations are shown with square markers. The triangles show the experimental results for argon . new scaling relation for nucleation: ln J ′ /η is scaled byln S/η . This scaling relation predicts the critical clus-ter size very well. The relation can be explained froma surface energy required to form the vapor-liquid inter-face and implies a constant, positive Tolman length of δ = 0 . r . Generally, ∆ G i ( S = 1) is written as the sur-face energy multiplied by the surface area, a i γ i . In thetheory, the cluster is always assumed to be spherical andhas the same density as the bulk liquid. However, ouranalyses of cluster properties show larger surface areas(Ang´elil et al. ). The higher normalisation ( A ≃ . G i ( S ) relative to the models might becaused by these larger surface areas. The scaling relationand the relation between the cluster properties and ∆ G i should be investigated in more detail for various materi-als. VI. ACKNOWLEDGMENTS
We thank the anonymous reviewers for their valuablesuggestions which have improved the quality of the pa-per. This work was supported by the Japan Society forthe Promotion of Science (JSPS). J.D. and R.A. acknowl-edge support from the Swiss National Science Foundation(SNSF).ree energy of cluster formation and a new scaling for nucleation 6
VII. APPENDIX
The general expression for the minimum work ∆ G ( r )required to form a cluster of radius r , is given by ∆ G ( r ) = V l v l [ µ l ( P l ) − µ g ( P g )] − ( P l − P g ) V l + a i γ i , (15)where µ l and µ g are the chemical potentials of liquid andgas, P l and P g are the pressures of metastable liquid andgas, and v l and V l (= iv l = 4 πr /
3) are the molecular vol-umes of liquid and the volumes of a cluster respectively.Using µ l ( P e ) = µ g ( P e ) and µ l ( P l ) − µ l ( P e ) = v l ( P l − P e ),we obtain∆ G ( r ) = V l v l [ µ g ( P e ) + v l ( P l − P e ) − µ g ( P g )] − ( P l − P g ) V l + a i γ i , = V l v l [ µ g ( P e ) − µ g ( P g )] + ( P g − P e ) V l + a i γ i , = V l v l [ µ g ( P e ) − µ g ( P g )] + i ( S − kT v l v g + a i γ i , (16)where v g is the molecular volume in the gas phase. Forthe case v l << v g , the second term on the right handside of Eq. (16) is negligible. Assuming a i γ i = 4 πr γ , weobtain the formula for the critical radius r cr called theKelvin relation from ∂ ∆ G ( r ) /∂r = 0: r cr = 2 γv l ∆ µ , (17)where ∆ µ = µ ( P g ) − µ g ( P e ) = kT ln S .From Eq. (10), the result from MD simulations shows a i γ i = 4 πr Aγ (1 − r /r ) , (18)thus we obtain the following relation at r = r cr : − πr ∆ µv l + 8 πr cr Aγ − πr Aγ = 0 . (19)From Eq. (19), r cr is given by r cr = Aγv l ∆ µ s − ∆ µr v l Aγ ! , (20)which corresponds to Eq. (11). M. Volmer and A. Weber, Z. Phys. Chem. , 277 (1926). V. R. Becker and W. D¨oring, Ann. Phys. , 719 (1935). J. Feder, K. C. Russell, J. Lothe, and G. M. Pound, Adv. Phys. , 111 (1966). R. J. Anderson, R. C. Miller, J. L. Kassner, and D. E. Hagen, J.Atmos. Sci. , 2508 (1980). J. L. Schmitt, G. W. Adams, and R. A. Zalabsky, J. Chem. Phys. , 2089 (1982). J. L. Schmitt, R. A. Zalabsky, and G. W. Adams, J. Chem. Phys. , 4496 (1983). G. W. Adams, J. L. Schmitt, and R. A. Zalabsky, J. Chem. Phys. , 5074 (1984). A. Dillman and G. E. A. Meier, J. Chem. Phys. , 3872 (1991). D. W. Oxtoby, J. Phys.: Condens. Matter , 7627 (1992). Y. Viisanen, R. Strey, and H. Reiss, J. Chem. Phys. , 4680(1993). A. Laaksonen, I. J. Ford, and M. Kulmala, Phys. Rev. E ,5517 (1994). Y. Viisanen and R. Strey, J. Chem. Phys. , 7835 (1994). K. H¨ameri and M. Kulmala, J. Chem. Phys. , 7696 (1996). K. Iland, J. W¨olk, R. Strey, and D. Kashchiev, J. Chem. Phys. , 154506 (2007). S. Sinha, A. Bhabhe, H. Laksmono, J. W¨olk, R. Strey, and B.Wyslouzil, J. Chem. Phys. , 064304 (2010). K. Yasuoka and M. Matsumoto, J. Chem. Phys. , 8451(1998). K. Yasuoka and M. Matsumoto, J. Chem. Phys. , 8463(1998). P. R. ten Wolde and D. Frenkel, J. Chem. Phys. , 9901 (1998). K. J. Oh and X. C. Zeng, J. Chem. Phys. , 4471 (1999). B. Senger, P. Schaaf, D. S. Corti, R. Bowles, D. Pointu, J.-C.Voegel, and H. Reiss, J. Chem. Phys. , 6438 (1999). P. R. ten Wolde, M. J. Ruiz-Montero, and D. Frenkel, J. Chem.Phys. , 1591 (1999). K. Laasonen, S.Wonczak, R. Strey, and A. Laaksonen, J. Chem.Phys. , 9741 (2000). K. J. Oh and X. C. Zeng, J. Chem. Phys. , 294 (2000). H. Vehkam¨aki and I. J. Ford, J. Chem. Phys. , 4193 (2000). B. Chen, J. I. Siepmann, K. J. Oh, and M. L. Klein, J. Chem.Phys. , 10903 (2001). P. Schaaf, B. Senger, J.-C. Voegel, R. K. Bowles, and H. Reiss,J. Chem. Phys. , 8091 (2001). J. W¨old and R. Strey, J. Phys. Chem. B , 11683, (2001). J. Merikanto, H. Vehk¨amaki, and E. Zapadinsky, J. Chem. Phys. , 914 (2004). K. K. Tanaka, K. Kawamura, H. Tanaka, and K. Nakazawa, J.Chem. Phys. , 184514 (2005). H. Matsubara, T. Koishi, T. Ebisuzaki, and K. Yasuoka, J.Chem. Phys. , 214507 (2007). J. Wedekind, J. W¨olk, D. Reguera, and R. Strey, J. Chem. Phys. , 154515 (2007). M. Horsch, J. Vrabec, and H. Hasse, Phys. Rev. E , 011603(2008). M. Horsch, and J. Vrabec, J. Chem. Phys. , 184104 (2009). J. Wedekind, G. Chkonia, J. W¨olk, R. Strey, and D. Reguera, J.Chem. Phys. , 114506 (2009). I. Napari, J. Julin, and H. Vehkam¨aki, J. Chem. Phys. ,154503 (2010). K. K. Tanaka, H. Tanaka, T. Yamamoto, and K. Kawamura, J.Chem. Phys. , 204313, (2011). J. Diemand, R. Ang´elil, K. K. Tanaka, and H. Tanaka, J. Chem.Phys. , 074309. (2013). K. K. Tanaka, A. Kawano, and H. Tanaka, J. Chem. Phys. ,114302 (2014). B. N. Hale, Phys. Rev. A , 4156 (1986). B. N. Hale, Metall. Trans. A , 1863 (1992). B. N. Hale, J. Chem. Phys. , 204509 (2005). B. N. Hale and M. Thomason, Phys. Rev. Let. , 046101(2010). V. I. Kalikmanov,
Nucleation theory , Lecture notes in Physics(Springer, Dordrecht, 2013) , Vol. 860. J. Wedekind and D. Reguera, J. Phys. Chem. B , 11060(2008). V. Shneidman, K. Jackson, and K. Beatty, Phys. Rev. B , 3579(1999) R. McGraw and A. Laaksonen, Phys. Rev. Lett. , 2754 (1996). R. McGraw and A. Laaksonen, J. Chem. Phys. , 5284 (1997). R. Ang´elil, J. Diemand, K. K. Tanaka, and H. Tanaka, J. Chem.Phys. , 074303 (2014). L. D. Landau and E. M. Lifshitz,
Stastistical Physics (Pergamonpress, Oxford, 1980), section 162. A. Laaksonen and R. McGraw, Europhys. Lett.35