Numerical Tests of Nucleation Theories for the Ising Models. Abstract

Transcription

1 to be submitted to Physial Review E Numerial Tests of Nuleation Theories for the Ising Models Seunghwa Ryu 1 and Wei Cai 2 1 Department of Physis, Stanford University, Stanford, California Department of Mehanial Engineering, Stanford University, Stanford, California (Dated: January 5, 2010) Abstrat The lassial nuleation theory (CNT) is tested systematially by omputer simulations of the 2D and 3D Ising models. While previous studies suggested potential problems with CNT, our numerial results show that the fundamental assumption of CNT is orret. In partiular, the Beker-Doring theory aurately predits the nuleation rate if the orret droplet free energy funtion is provided as input. This validates the oarse graining of the system into a 1D Markov hain with the largest droplet size as the reation oordinate. Furthermore, the droplet free energy predited by CNT mathes numerial results very well, after a logarithmi orretion term from Langer s field theory and a onstant orretion term are added. PACS numbers: qe, My, q 1

2 I. INTRODUCTION Nuleations are ubiquitously observed in many different systems inluding superooled fluids 1 3, nano-materials 4, polymerization proesses 5 and eletro-weak phase transitions 6. The standard theory used to desribe the nuleation phenomena is the lassial nuleation theory (CNT) CNT onsiders the droplets of the stable phase spontaneously formed in the bakground of the meta-stable phase. The maximum in the droplet free energy as a funtion of droplet size is the free energy barrier of nuleation, and is the dominant fator in the determination of the nuleation rate. A widely used form of CNT is the Beker-Doring theory 7 that predits the nuleation rate from a steady-state solution of a one-dimensional Markov hain model. While CNT suessfully aptures many qualitative features of nuleation events, the predition of the nuleation rate based on CNT annot be ompared quantitatively with experiments, given the gross approximations made in the theory. During the past 50 years, many modifiations and extensions of CNT have been developed. For example, Lothe and Pound 11 onsidered the ontributions from extra degrees of freedom of a luster (in addition to its size) to its Gibbs free energy of formation. Langer 12,13 developed a field theory to extend the Beker-Doring steady-state solution to inlude the effet of other mirosopi degrees of freedom of a luster. Zeng and Oxtoby 14 improved the temperature dependene of the nuleation rate predited by CNT by expressing the droplet free energy as a funtional of the radial density profile ρ(r). To date, many nuleation theories have been developed, but it is very diffiult to verify them experimentally, due to the diffiulties in measuring nuleation rates aurately. While for a theory it is more onvenient to study homogeneous nuleations in a single-omponent system, suh onditions are diffiult to ahieve in experiments 15. Instead, experimental measurements are usually influened by surfae strutures and impurities that are diffiult to ontrol. Computer simulations have the opportunity to probe nuleation proesses in great detail and to quantitatively hek the individual omponents of the nuleation theories. The inrease of omputational power and the development of advaned sampling algorithms allow the alulation of nuleation rates for model systems over a wide range of onditions A prototypial nuleation problem is the deay of the magnetization in the 2D or 3D Ising model, whih has been studied by omputer simulations for several deades Both 2

3 agreement 22,28,29 and disagreement 20,21,25,30 between numerial results and CNT preditions have been reported. When the CNT preditions of nuleation rate do not agree with numerial results, several potential problems of CNT were usually disussed. For example, a suspet is the appliation of surfae tension of marosopi, flat, interfaes to a small droplet 21. The validity of oarse-graining the many-spin system into a one-dimensional Markov hain was also questioned 21,25. Nuleation theories usually express the rate in the Arrhenius form, with a free energy barrier and a pre-exponential fator. Usually both terms are not omputed in the same study. Hene, we often annot onlude whih one auses the disrepany between CNT and numerial simulations, and how the theory should be improved. Only rarely did numerial results lead to lear onlusions on the validity of the fundamental assumptions made in CNT 27. In addition, most existing works used brute fore Monte Carlo simulations to ompute the nuleation rate, limiting the studies to high temperature or large magneti field onditions, or both. The aim of this paper is to investigate several nuleation theories systematially by testing their assumptions separately, using omputer simulations of 2D and 3D Ising models. We ompute the nuleation rate by the forward flux sampling (FFS) method 31, whih allows the rate to be alulated over a muh wider range of external field and temperature onditions than that possible by brute fore Monte Carlo simulations. To test the individual omponents of CNT (Beker-Doring theory), the free energy F (n) of the droplet as a funtion of droplet size n is omputed using the umbrella sampling method 18. The kineti prefator of the ritial luster, f +, whih is part of the Beker-Doring theory, is omputed independently from Monte Carlo simulations starting from the ensemble of ritial lusters. The nuleation rate predited by the Beker-Doring theory, using the so omputed F (n) and f + as inputs, is ompared with the nuleation rate diretly omputed from the FFS method. We find that, provided with the orret droplet free energy F (n), the Beker-Doring theory predits the nuleation rate very aurately. This onfirms that the oarse-graining of the Ising model as a one-dimensional Markov hain, as invoked in CNT, is a very good approximation, whih was also noted earlier 24,27. Disrepanies between the droplet free energy F (n) predited by CNT and numerial results have been reported earlier 24,25. Here we show that if a logarithmi orretion term and a onstant orretion term are added, the theoretial predition of droplet free energy agrees very well with the numerial result. The 3

4 logarithmi orretion term was first derived from Langer s field theory, but was ustomarily put as a orretion to the kineti prefator. Our analysis shows that this orretion term should be plaed in the free energy funtion F (n) in order to orretly predit the size of the ritial nuleus. The logarithmi orretion term has an analyti expression in 2D, and hene ontaining no fitting parameters. It ontains one fitting parameter in 3D due to the existing ontroversies on its temperature dependene. In 2D, the onstant orretion term an be determined from analyti expressions for the free energies of very small droplets and hene ontains no fitting parameters. In 3D, the onstant orretion term is treated as a fitting parameter in this work. Our analysis resolved some of the previously reported disrepanies between numerial simulations and CNT. For the 2D Ising model, the logarithmi orretion term to the droplet free energy was often negleted 21,24. Beause the logarithmi orretion term is positive and substantial in 2D, the omission of this term would ause CNT to grossly overestimate the nuleation rate. For the 3D Ising model, the logarithmi orretion term is muh smaller relative to the other terms. However, the temperature dependene of the surfae free energy was sometimes ignored 25,30. While the surfae free energy an be approximated as a onstant at very low temperatures 28, it dereases appreiably with temperature above a quarter of the ritial temperature. Overestimating the surfae free energy would lead to an overestimate of the nuleation barrier and an underestimate of the nuleation rate. The paper is organized as follows. Setion II summarizes a number of nuleation theories and their appliations to the 2D and 3D Ising model. Setion III presents the numerial methods we employ to test these theories. The numerial results are ompared with the nuleation theories in Setion IV. A brief summary is given in Setion V. II. NUCLEATION THEORIES A. Brief Review of Nuleation Theories In 1926, Volmer and Weber first introdued the onept of ritial droplet and estimated the nuleation rate in a supersaturated vapor by the following equation, I f + ( exp F ) k B T (1) 4

5 where F is the formation free energy of the ritial droplet, and f + is the attahment rate of moleules to the ritial droplet. F is the maximum of the droplet free energy as a funtion of droplet size n. The Volmer-Weber theory also gives the droplet free energy funtion in the following form, F (n) = σ S δµ n (2) where σ is the effetive surfae tension and S is surfae area. δµ is the bulk hemial potential differene per moleule between the liquid and the vapor phases. n is the total number of moleules in the droplet. The onepts of ritial droplet, its free energy, and the attahment rate of moleules, still remain important to date for our understanding of the nuleation proess. Other dynamial fators, suh as multiple rerossing of the free energy barrier, originally ignored in the Volmer-Weber theory, was reognized later. In 1935, Beker and Doring 7 modelled the time evolution of the droplet population using a one-dimensional Markov hain model 10, and obtained a steady-state solution for the nuleation rate. This solution finally pinpoints the kineti prefator 49 in the nuleation rate, whih is expressed as I = f + ( Γ exp F ) k B T where Γ is known as the Zeldovih fator 8,9 defined by Γ ( ) η 1/2 2 F (n), η = 2πk B T n 2 (4) n=n The flatter is the free energy urve near the ritial size n, the smaller is the Zeldovih fator. Hene the Zeldovih fator aptures the multiple rerossing of the free energy barrier, whih lowers the nuleation rate. There are two fundamental assumptions in CNT that are independent of eah other. First, the time evolution of the droplet population an be desribed by a 1D Markov hain model. Seond, the free energy of a droplet an be written as Eq. (2), where σ is the surfae tension of marosopi interfaes. We an test the first assumption if we an ompute the nuleation rate using a numerial method that does not rely on the Markovian assumption. We an test the seond assumption by omputing the free energy funtion by umbrella sampling. In 1967, Langer 13 developed a field theoretial approah to take into aount all degrees of freedom of a droplet when alulating the steady-state solution for the nuleation rate. (3) 5

6 This is a generalization of the Beker-Doring theory to inorporate mirosopi (flutuation) degrees of freedom of the droplet. Langer s field theory was later used to derive a orretion term to the nuleation rate in the droplet model In the literature, the field theory orretion is usually expressed as an extra term in the pre-exponential fator in Eq. (3). But it an also be expressed as a modifiation to the free energy funtion in Eq. (2), hanging it to F (n) = σ S δµ n + τk B T ln n (5) While both approahes an give rise to similar preditions to the nuleation rate, we will show later that it is more self-onsistent to inlude the orretion term in the free energy. The field theory predits that, for an isotropi medium, the oeffiient of logarithmi orretion term is τ = 5 in 4 2D37 and τ = 1 in 9 3D38. However, it was later predited that the shape flutuation of a 3D droplet should be suppressed below the roughening temperature 36, whih leads to τ = 2. Our numerial results onfirm the τ = 5 predition 3 4 for the 2D Ising model, under a wide range of temperatures. This ontradits an earlier numerial study 21 whih suggests that τ is lose to zero at low temperatures and only rise to 5 4 at high temperatures. On the other hand, our numerial results are not onsistent with the above preditions for τ for the 3D Ising model. This problem may be related to the finding by Zia and Wallae 39, that the exitation spetrum around a 3D droplet is affeted by the anisotropy of the medium, but that around a 2D droplet is not affeted by anisotropy. Beause the Ising model is fundamentally anisotropi (e.g. with ubi symmetry), the field theoreti predition based on isotropi medium may not apply for the 3D Ising model. All the nuleation theories mentioned above share several fundamental assumptions: (1) only isolated droplets are onsidered and the interation between droplets is negleted; (2) a droplet is assumed to be ompat with a well-defined surfae; (3) the surfae energy expression derived from a marosopially planar surfae an be applied to the surfae of a very small droplet. The first two assumptions are valid at temperatures muh lower than the ritial temperature and at small magneti field. Under these onditions, the density of droplets is very small and eah droplet tends to be ompat. We will not test the two assumptions in this study. In other words, our numerial simulations will be limited to the low temperature and small field onditions where these assumptions should be valid. Models that aount for droplet interations exist in the literature 40 but will not be disussed in this paper. 6

7 Even with the logarithmi orretion term, Eq. (5) still deviates from the numerially omputed droplet free energy (from umbrella sampling) by a onstant. This onstant term is likely to be aused by the third assumption listed above. We found that in the 2D Ising model this disrepany an be removed by adding a onstant term to the droplet free energy F (n) = σ S δµ n + τk B T ln n + D(T ) (6) In 2D, the onstant term D an be determined without any fitting to the numerial results. At eah temperature T, D an be obtained by mathing F (n) with analyti expressions of droplet free energy that are available for very small droplets (Appendix A). Unfortunately, in 3D even Eq. (6) does not desribe the free energy well enough for small droplets. This prevents the use of the analyti expressions of small droplet free energies to determine D. Hene in 3D we need to treat D as a fitting parameter. B. Nuleation Theories Applied to the Ising Model The Ising model is desribed by the following Hamiltonian H = J s i s j h i,j i s i (7) where J > 0 is the oupling onstant and h is the external magneti field. The spin variable s i at site i an be either +1 (up) or 1 (down), and the sum i,j is over nearest neighbors of the spin lattie. For onveniene, we set J = 1 in the following disussions. In our simulations, we study the relaxation of the magnetization starting from an initial state magnetized opposite to the applied field h. To be speifi, we will let h > 0 and the initial state has s i = 1 for most of the spins. The dynamis follows the Metropolis single-spin-flip Monte Carlo (MC) algorithm with random hoie of trial spin. The simulation time step is measured in units of MC step per site (MCSS). The 2D model onsists of a square lattie and the 3D model onsists of a simple ubi lattie, with periodi boundary onditions (PBC) applied to all diretions. 1. Beker-Doring Theory To ompute the nuleation rate using Eq. (2) and Eq. (3), the surfae free energy σ and bulk hemial potential differene δµ must be known for the Ising model. The hemial 7

8 potential differene is simply δµ = 2h, whih is a good approximation not only at low temperature but also near the ritial temperature 41. On the other hand, the surfae free energy σ is more diffiult to obtain. This is beause the Ising model is anisotropi at the mirosopi sale and the free energy of a surfae depends on its orientation. Therefore, the input to the Beker-Doring theory should be an effetive surfae free energy σ eff, whih is an average over all possible orientations given the equilibrium shape of the droplet. σ eff is a funtion of temperature T not only beause the surfae free energy of a given orientation depends on temperature, but also beause the equilibrium shape of the droplet hanges with temperature σ eff σ (10) σ eff k T B FIG. 1: (olor online) Effetive surfae free energy σ eff as a funtion of temperature for the 2D Ising model from analyti expression 21. The free energy of the surfae parallel to the sides of the squares, σ (10), is also plotted for omparison. We follow the definition of Shneidman 21, whih gives the analyti expression of σ eff (T ) for the 2D Ising model, as shown in Fig. 1. The free energy of a droplet an be written as, 50 F 2D (n) = 2 πn σ eff (T ) 2 h n (8) where n is the total number of up-spins in the droplet. From the maximum of F 2D (n), we obtain the ritial droplet size of the 2D Ising model, n 2D = π σ2 eff(t ) 4 h 2 (9) as well as the free energy barrier F 2D = π σ2 eff(t ) 2h 8 (10)

9 We also obtain the Zeldovih fator defined in Eq. (4) Γ 2D = 2 k B T h 3/2 π σ eff (T ) (11) Assuming the ritial droplet has a irular shape, the attahment rate an be written as f + = 2β 0 π n 2D (12) where β 0 is the average spin-flip frequeny at the boundary of the droplet. As an approximation, β 0 exp( σ eff (T )/k B T ). Combining all, we obtain the nuleation rate predited by the Beker-Doring theory I 2D BD(h, T ) = β 0 [ 2 h k B T exp π ] σ2 eff(t ) 2 h k B T (13) Given the analyti expressions for σ eff (T ) in 2D, the preditions of the Beker-Doring theory an be omputed expliitly. For example, at k B T = 1.5 and h = 0.05, we have n 2D = 463, F 2D = 46.3, Γ 2D = , f + = 34.0, and I 2D BD = MCSS 1. The numerial results (in Setion III B) under the same ondition are n = 496, F = 61.3, Γ = , f + = 39.2, and I = MCSS 1. As disussed further below, the four orders of magnitude disrepany in the nuleation rate mainly omes from the underestimate of F 2D disrepany. by Eq. (8). The logarithmi and onstant terms in Eq. (6) are needed to remove this For the 3D Ising model in a simple ubi lattie, there is no analyti expression for surfae free energy for arbitrary surfae orientations. A parametri expression exists only for the {001} surfae 43. Therefore, the equilibrium shape and the equivalent surfae free energy of the 3D droplet is not known. Similar to Eq. (8), the free energy of a 3D droplet an be written as, F 3D (n) = σ eff (T ) α n 2/3 2 h n (14) where α = (36π) 1/3 is a geometri fator expressing the surfae area of a sphere with unit volume. Contrary to the ase of 2D Ising model, the analyti expression of σ eff (T ) is not known in the 3D Ising model, and it will be used as a fitting parameter in our study. Following the same proedures as above, we obtain the ritial nuleus, free energy barrier and Zeldovih fator for the 3D Ising model, n 3D = α3 σ 3 eff(t ) 27 h 3 (15) 9

10 F 3D = α3 σeff(t 3 ) (16) 27 h 2 9 Γ 3D h 2 = πk B T α 3 σeff 3 (T ) (17) Finally, the nuleation rate predited by the Beker-Doring theory is [ IBD(h, 3D α3 σ eff (T ) T ) = β 0 9π k B T exp α3 σeff(t 3 ] ) 27 h 2 k B T (18) Given that σ eff (T ) is unknown and has to be treated as a fitting parameter, it is more diffiult to test Eq. (18) quantitatively. 2. Langer s Field Theory Langer s field theory predits a orretion term to the droplet free energy, as in Eq. (5). In 2D Ising model, τ = 5, and this orretion term not only inreases the free energy barrier, 4 but also inreases the size of the ritial droplet. The ritial droplet size predited by the field theory is, n 2D/F T = πσeff + πσeff 2 + 8τk 2 BT h 4 h This equation is to be ompared with Eq. (9) predited by the Beker-Doring theory. We will see (in Table I) that Eq. (19) agrees muh better with numerial results than Eq. (9), indiating that the field theory orretion should be put in the free energy funtion instead of the kineti prefator. The τk B T ln n orretion term also modifies the ritial nuleus size in the 3D Ising model. The analyti expression for n 3D (19) given by the field theory an be obtained by solving a third order polynomial equation. The expression is omitted here to save spae. We also note that in the 3D Ising model, there have been preditions that τ depends on temperature: τ = 1 9 above the roughening temperature T R 38 and τ = 2 3 below T R 36. But our numerial results do not support these preditions. 10

11 III. NUMERICAL METHODS A. Forward Flux Sampling Here we give a brief overview of the forward flux sampling (FFS) method used in this study. The full mathematial details an be found in the literature 31. To ompute the transition rate from the initial state A to the final state B, FFS uses a series of interfaes in the phase spae defined through an order parameter λ. State A is defined as the phase spae region in whih λ < λ A, while state B orresponds to λ > λ n. The interfaes between A and B are defined as hyperplanes in the phase spae where λ = λ i, i = 0, 1, 2,, n 1, λ A < λ 0 < < λ n. In priniple, the hoie of the order parameter λ should not affet the alulated rate onstant, whih means λ need not be the true reation oordinate. In the FFS method, the nuleation rate I from A to B is expressed as a multipliation of two terms I = I 0 P (λ n λ 0 ) (20) where I 0 is the average flux aross the interfae λ = λ 0 (i.e. leaving state A), and P (λ n λ 0 ) is the probability that a trajetory leaving state A will reah state B before returning to state A again. Beause it is impossible to reah interfae λ = λ i+1 without reahing interfae λ = λ i first, the probability P (λ n λ 0 ) an be deomposed into a series multipliation, P (λ n λ 0 ) = n 1 i=0 P (λ i+1 λ i ) (21) where P (λ i+1 λ i ) is the probability that a trajetory reahing λ i, having ome from A, will reah λ i+1 before returning to A again. In this work, we set λ to the size of the largest droplet n in the simulation ell. The rate I 0 is obtained by running a brute fore Monte Carlo simulation, during whih we ount how frequently droplets with size larger than λ 0 are formed. An ensemble of onfigurations at interfae λ = λ 0 (for trajetories oming from A) is stored from this MC simulation. The next step is to run MC simulations with initial onfigurations taken from the ensemble at interfae λ = λ 0. A fration of the trajetories reahes interfae λ = λ 1 before returning to state A. From these simulations the probability P (λ 1 λ 0 ) is omputed and an ensemble of onfigurations at interfae λ = λ 1 is reated. The proess is repeated to ompute the probability P (λ i+1 λ i ) for eah i = 1,, n 1. 11

12 As an example, Fig. 2 plots the probability P (λ i λ 0 ) P (λ 1 λ 0 )P (λ 2 λ 1 ) P (λ i λ i 1 ) for the 2D Ising model at k B T = 1.5 and h = In this test ase, we find I 0 = MCSS 1 with λ 0 = 24 from a brute fore Monte Carlo simulation with 10 7 MCSS. 15, 000 onfigurations are then olleted at eah interfae, whih allows the nuleation rate to be determined within 5%. The probability of reahing interfae λ = λ n from interfae λ = λ 0 is P (λ n λ 0 ) = with λ n = Following Eq. (20), the nuleation rate under this ondition is I FFS = MCSS P (λ i λ 0 ) λ i FIG. 2: (olor online) The probability P (λ i λ 0 ) of reahing interfae λ i from λ 0 without oming bak to A at k B T = 1.5 and h = 0.05 in the 2D Ising model. Short horizontal lines indiate error bars. It is important to note that FFS does not require the transitions between different interfaes to be Markovian. Neither does it require the transitions to satisfy detailed balane, unlike other sampling methods 19,44,45. Therefore, it an be used to test the fundamental assumption of the Beker-Doring theory, whih states that the nuleation proess an be oarse-grained into a one-dimensional Markov hain. FFS would fail if there is no separation of time sale, i.e. if the time spent on a reation path is omparable to (instead of muh shorter than) the dwell time in state A or state B. B. Computing Rate from Beker-Doring Theory Having omputed the nuleation rate from FFS, we will use it as a benhmark and ompare it with the nuleation rate predited by the Beker-Doring theory. To gain more insight from this omparison, we will split the Beker-Doring theory into two parts and test 12

13 them separately. Part I is summarized in Eq. (3), whih expresses the nuleation rate in terms of the attahment rate f +, Zeldovih fator Γ, and the free energy barrier F. Part II is the predition of the droplet free energy F (n), whih was disussed in Setion II.B. We will ompute the droplet free energy F (n) numerially by umbrella sampling (US) 18. The result then allows us to speifially test Part II of the Beker-Doring theory. To test Part I of the Beker-Doring theory, we will ompute F and Γ from the free energy funtion F (n) obtained by US, and plug them into Eq. (3). The attahment rate f + an also be omputed separately, as explained below. (a) F = 61.3 n = 496 (b) F (n) n n 2 (t) t (MCSS) FIG. 3: (olor online) (a) Droplet free energy F (n) obtained by US at k B T = 1.5 and h = 0.05 in the 2D Ising model. (b) Flutuation of droplet size n 2 (t) as funtion of time. As an example, Fig. 3 (a) shows the droplet free energy F (n) omputed from US at k B T = 1.5 and h = The size of the largest droplet is the order parameter, and a paraboli bias funtion is used, following Auer and Frenkel 18. The maximum of this urve indiates that the ritial droplet size is n = 496 and the free energy barrier is F = The Zeldovih fator an be alulated from the seond derivative of this urve at n, whih gives Γ = We then ollet many onfigurations from the US simulation, when the bias potential is entered at the ritial droplet size. Using eah onfiguration as an initial ondition, we run Monte Carlo simulations and obtain the effetive attahment rate from the following equation, f + = n2 (t) 2 t, (22) 13

14 where n 2 (t) is the mean square flutuation of the droplet size. n(t) n(t) n(t = 0), n(t) is the droplet size at time t, and represents ensemble average from these Monte Carlo simulations. The Monte Carlo simulations are stopped when n(t) exeeds 4. The result for the means square flutuation n 2 (t) at k B T = 1.5 and h = 0.05 is plotted in Fig. 3 (b), whih shows a linear funtion of time. From the slope of this urve, we obtain f + = 39.1 MCSS 1. A similar approah was used by Brendel et al. 24 to ompute the interfae diffusion oeffiient. Combining the values of f +, Γ, F and plug them into Eq. (3), we find that the Beker- Doring theory would predit the nuleation rate to be I BD = MCSS 1, if the orret free energy funtion F (n) is used. This is very lose to the FFS result I FFS = MCSS 1 given in the previous setion. The agreement onfirms the validity of Part I of the Beker-Doring theory. Comparisons over a wider range of onditions are presented in the following setion. IV. RESULT A. Comparison of Nuleation Rate We have omputed the nuleation rates using two different methods over a wide range of onditions: h = , T = T for 2D and h = , T = T for 3D, where T is the ritial temperature at zero field (k B T = in 2D and in 3D). In the first method, the nuleation rate is diretly omputed by FFS. In the seond method, the nuleation rate is omputed from the Beker-Doring Eq. (3), but using the free energy urve obtained from US, as desribed in Setion III.B. The pre-exponential fator, f + Γ, is found to have a weak dependene on T and h, and varies by about a fator of 2 in the entire range of T and h onsidered in this study. The alulations are performed on a 3 GHz Linux luster. Eah FFS alulation for a given (T, h) ondition takes about 50 CPU-hours for the 2D Ising model and 200 CPU-hours for the 3D Ising model. Eah US alulation takes a similar amount of time as an FFS alulation. As shown in Fig. 4, the nuleation rate over these onditions spans more than 20 orders of magnitude. Yet, most of the rates predited by the two methods are within 50% of eah other. This is a strong onfirmation of Part I of the Beker-Doring theory, i.e. Eq. (3). It 14

15 (a) k B T = (b) I (MCSS 1 ) I FFS / I BD h h () (d) 2 I (MCSS 1 ) k B T = I FFS / I BD h h FIG. 4: (olor online) (a) The nuleation rate I omputed by FFS (open symbols) and Beker- Doring theory with US free energies (filled symbols) in the (a) 2D and () 3D Ising models. (b) The ratio between nuleation rates obtained by FFS and Beker-Doring theory at different temperatures in the () 2D and (d) 3D Ising models. The symbols in (b) and (d) math those defined in (a) and (), respetively. onfirms that for the purpose of omputing nuleation rate, it is valid to oarse grain the Ising model to a one-dimensional Markov hain, with the size of the largest droplet being the reation oordinate. Detailed balane 27 between neighboring states along the Markov hain, as is assumed by the Beker-Doring theory, is also onfirmed 51. This means that the Beker-Doring theory an predit the nuleation rate of the 2D and 3D Ising models aurately, provided that the orret free energy funtion F (n) is used. This is onsistent with an earlier report by Brendel et al

16 TABLE I: Size of ritial droplet at k B T = 1.5 in the (a) 2D and (b) 3D Ising models. n omm the size at whih the average ommittor probability P B = 0.5. n F/US obtained by US simulation reahes maximum. n F/BD expression, Eq. (2), reahes maximum. n F/FT the free energy, Eq. (5), reahes maximum. is is the size at whih F (n) is the size at whih the lassial free energy is the size at whih the field theoreti expression for (a) h n omm n F/US n F/BD n F/FT (b) h n omm n F/US B. Comparison of Critial Droplet Size There are two ommon definitions of the ritial droplets. In the first definition, a droplet is of ritial size if its probability to grow and over the entire system is 50%. In other words, a ritial droplet has a ommittor probability of 50%. In the seond definition, a droplet is of ritial size if it orresponds to the maximum of the free energy urve F (n). It is of interest to verify whether these two definitions are equivalent. After eah FFS simulation under a given (T, h) ondition, an ensemble of 15,000 spin onfigurations are saved at eah interfae λ i. It is straightforward to ompute the ommittor probability for eah of the saved spin onfigurations and the average ommittor probability P B (n) for eah n = λ i. By fitting the data of P B (n) to a smooth urve, we an extrat the ritial value n for whih P B = 0.5. Some of the ritial nuleus size obtained this way are listed in Table I under the heading of n omm. The droplet sizes that orrespond to the maximum of the free energy urve obtained by US are listed in Table I under the heading of n F/US. For both 2D and 3D Ising models, n omm and n F/US agree with eah other within 2%. This onfirms that the two definitions for the ritial nuleus are equivalent, provided that the orret free energy urves F (n) are 16

17 D P(P B ) 0.15 P(P B ) P B P B FIG. 5: (olor online) (a) Histogram of ommittor probability in an ensemble of spin onfigurations with n = 496 for the 2D Ising model at k B T = 1.5 and h = Representative droplets are also shown, with blak and white squares orresponding to +1 and 1 spins, respetively. (b) Histogram of ommittor probability in an ensemble of spin onfigurations with n = 524 for the 3D Ising model at k B T = 2.20 and h = used. It also proves that the size of the largest droplet is a good reation oordinate. Fig. 5(a) plots the histogram of the ommittor probability for an ensemble of spin onfigurations with n = 496 for the 2D Ising model at k B T = 1.5 and h = The average ommittor probability of this ensemble is 49.4%. About 94% of the spin onfigurations in this ensemble have ommittor probabilities within the range of 49 ± 5%. This further onfirms that the size of the largest luster, n, is a very good reation oordinate of the nuleation proess. Fig. 5(b) plots the histogram of the ommittor probability within an ensemble of spin onfigurations with n = 524 for the 3D Ising model at k B T = 2.20 and h = The average ommittor probability of this ensemble is 50%. About 80% of the spin onfigurations in this ensemble have ommittor probabilities within the range of 50 ± 5%. The spread of the ommittor probability distribution is wider than the 2D ase, and is onsistent with an earlier report

18 C. Droplet Free Energy of 2D Ising Model The previous setions show that the Beker-Doring theory performs well as long as the orret droplet free energy F (n) is provided. We now ompare the theoretial preditions of F (n) with numerial results by US. We will fous on 2D Ising model in this setion and will disuss F (n) in the 3D Ising model in the next setion. (a) (b) F (n) US BD Langer τ k B T ln n F(n) Exat US BD Langer τ k B T ln n n n FIG. 6: (olor online) (a) Droplet free energy urve F (n) of the 2D Ising model at k B T = 1.5 and h = 0.05 obtained by US (irles) is ompared with lassial expression (dashed line) and field theory expression (solid line). Logarithmi orretion term 5 4 k BT ln n is also drawn (dotted urve) for omparison. (b) Magnified view of (a) near n = 0, together with the results from analyti solutions (squares) available for n 17 (see Appendix A). Fig. 6 plots the F (n) urves for k B T = 1.5 and h = Numerial results from US and preditions from the Beker-Doring theory and Langer s field theory are plotted together. It is lear that the logarithmi orretion term τk B T ln n from the field theory is substantial. The field theory predition, whih ontains this orretion term, agrees very well with numerial US results (after the onstant term is added, see Appendix A). The free energy used in CNT, Eq. (8), whih laks this orretion term, is signifiantly lower. Obviously, if this free energy urve is used, the Beker-Doring theory will overestimate the nuleation rate by several orders of magnitude. Our result also shows that, the field theory preditions, though derived under the assumption of infinitesimal h, are still valid at finite h in the range of field onsidered in this study. Our results shows that the marosopi surfae free energy (at zero h) an be safely 18

19 applied to a droplet (at finite h) 29, provided that the onstant orretion term is added (see Appendix A). Brendel et al. 24 reported that the effetive surfae free energy exeeds that of the marosopi surfae free energy by 20%. But this was aused by the neglet of the logarithmi orretion term in that study. Our results ontradit the previous report 21 that τ is lose to zero at low temperatures (T = 0.59 T and 0.71 T ) and only goes to 5 near T = 0.84 T 4. In the previous study 21, only small lusters (n < 60) are sampled, but not using the umbrella sampling tehnique. We suspet this approah is suseptible to the error aused by the lak of statistis at low temperatures (espeially for lusters with n > 30). Beause the field theoreti orretion term τk B T ln n beomes smaller at low T, it ould be masked by the statistial error. To support our finding, the free energy urves for luster sizes up to n = 1800 at a wide temperature range (from 0.44 T to 0.84 T ) are attahed in a supplementary doument 46. τ = 5 4 is neessary in the entire temperature range to aurately desribe the droplet free energy. In the literature, the field theory orretion is usually expressed as an extra preexponential fator inserted into the Beker-Doring formula of the nuleation rate. Both a pre-exponential fator and a hange to the free energy urve an modify the nuleation rate. So it may appear impossible (or irrelevant) to deide whih approah is more orret. However, a loser inspetion shows that it is indeed possible to tell whether the orretion should be interpreted as a free energy hange, or a kineti pre-fator. This is beause selfonsisteny requires that the maximum of the free energy urve F (n) should math the droplet size n whose ommittor probability is 50%, as disussed in Setion IV.B. Table I(a) lists the ritial droplet sizes n F/BD, whih orrespond to the maximum of the free energy urves F (n) predited by the Beker-Doring theory, Eq. (8). They are signifiantly smaller than the ritial droplet sizes n omm that orresponds to a 50% ommittor probability. With the field theory orretion term in the free energy, the ritial droplet sizes n F/FT agree muh better with n omm. This result learly shows that the field theory orretion should be plaed in the free energy funtion F (n), instead of being a kineti prefator. It is of interest to ompare the various free energy expressions disussed so far with the analyti (exat) expressions 47 for F (n) that are available for 0 n 17. It is somewhat surprising that the field theory predition of F (n) (after orreted by a onstant term, see 19

20 Appendix A) agrees very well with both the numerial (US) data from US and the analyti expressions, for suh small values of n. This is another onfirmation for the field theory predition of the free energy urve, Eq. (6). Shneidman et al. 21 also observed the effet of the logarithmi orretion term, but expressed it in terms of size-dependent prefator, and suspeted that it is aused by oagulation of droplets. Our results show that this is not a oagulation (many-droplet) effet, beause the logarithmi orretion term is derived by onsidering the shape flutuation of a single droplet. To summarize, the free energy expression from CNT must be modified by two terms, i.e. a logarithmi orretion term τk B T ln n from field theory and a onstant term to math the free energy of very small droplets. In 2D, both terms an be determined ompletely from existing theories and ontain no fitting parameters. D. Droplet Free Energy of 3D Ising model In the following, we will examine a number of funtional forms to see whih one best desribes the numerial data of the droplet free energy F (n) in the 3D Ising model. Beause there is no analyti solution to the effetive surfae free energy in 3D, σ eff (T ) must be treated as a fitting parameter in our analysis, whih reates more unertainty in our onlusions. For example, we annot unambiguously determine the oeffiient τ in the logarithmi orretion term (in Eq. (6)) from the numerial results. Another diffiulty in determining τ is that in 3D the logarithmi orretion term is muh smaller ompared with the first two terms in Eq. (6). To redue the omplexity from finite h, we omputed droplet free energy at zero field for a range of temperature k B T = 2.0,..., 2.8 by US. Fig. 7 plots the results at k B T = 2.40 and h = 0. First, we fit the data to the original Beker-Doring form, Eq. (14), plus a onstant orretion term, i.e., F (n) = σ eff (T ) α n 2/3 2 h n + D(T ) (23) where σ eff (T ) is a free fitting parameter at eah temperature. We find that Eq. (23) annot desribe the droplet free energy well in the entire range of n. Sine we expet it to be more aurate in the ontinuum limit of large n, we fit the US data to Eq. (23) only in the range of n > 50. The resulting term D(T ) is in the range of 1.1 (at k B T = 2.0) to 2.4 (at 20

21 (a) F (n) n US BD Langer τ k B T ln n (b) F(n) Exat US BD Langer τ k B T ln n B exp( Cn/B) n FIG. 7: (olor online) (a) Droplet free energy F (n) of the 3D Ising model at k B T = 2.40 and h = 0 obtained by umbrella sampling (irles) is ompared with lassial expression (dots) and field theory expression (solid line). Logarithmi term τk B T ln n is also plotted (dashed line). The differene in preditions by lassial expression and field theory are very small ompared to F (n) itself and annot be observed at this sale. (b) Magnified view of (a) near n = 0, together with the exat solution of small droplets (squares, see Appendix A) and the exponential orretion term (rosses). [ ] /2 k B T = 2.8). The error in the fit is defined as R (F (i) F fit (i)) 2, where F (i) 700 i=50 is the numerial data from US, and F fit (i) is the value given by Eq. (23). The resulting R is in the range of and inreases with inreasing temperature. Signifiant disrepany between the US data and the fit is observed in the range of n < 50, whih will be further disussed below. The next funtion to be onsidered for the fit inludes the logarithmi orretion term, F (n) = σ eff (T ) α n 2/3 + τ(t ) k B T ln n 2 h n + D(T ) (24) in whih τ is a free parameter for eah temperature T. The error of the fit is now redued to about R 0.01 for all temperatures and is now independent of temperature. This means that the logarithmi term improves the desription of the temperature dependene of the free energy of large droplets (n > 50). But the disrepany in the range of n < 50 still remains. This is different from the 2D Ising model, where Eq. (6) desribes the droplet free energy very well even without any fitting parameters. 21

22 Perini et al 38 used the following funtional form to fit their free energy data, F (n) = σ eff (T ) α n 2/3 + K(T ) n 1/3 + τ k B T ln n 2 h n + D(T ) (25) where τ = 1 9 is onstrained to be a onstant52. The parameter K orresponds to the extra energy of ledges that appear on 3D droplets. We find that the quality of the fit using Eq. (25) is similar to that using Eq. (24). 53 However, the resulting K(T ) is an inreasing funtion of temperature. This is ounter-intuitive beause the ontinuum droplet approximation is expeted to be better at higher temperatures where equilibrium droplet shape beome more spherial. Hene we would expet K(T ) to derease with inreasing temperature. Therefore, we believe Eq. (24) is a more appropriate funtional form than Eq. (25). In the following, we will not inlude the ledge energy term, and will treat τ as a funtion of temperature during the fitting. We find that the fit in the range of n < 50 an be signifiantly improved by adding an exponential term. The US data at all T and in the range of 4 < n < 750 an be well fitted by the following funtion, 54 [ F (n) = σ eff (T ) α n 2/3 + τ(t ) k B T ln n 2 h n + A + B(T ) exp C n ] B(T ) where A and C are onstants independent of T, and B(T ), σ eff (T ), and τ(t ) are funtions of T. The fitted parameters are: A = 0.06 and C = It turns out that B(T ) an be well desribed by a linear funtion: B(T ) = 9.12 k B T The ontribution of the exponential term is plotted in Fig. 7(b). In the range of T and h onsidered in this study, the size of the ritial nuleus is larger than 100. Hene the nuleation rate predited by CNT under these onditions is only affeted by the auray of F (n) in the range of n > 100. When Eq. (26) is used, the numerial values of the logarithmi term is in the range of 5 to 0.5, for 2.0 k B T 2.8 and 100 < n < 750. It is the major orretion term to the lassial expression of the droplet free energy, Eq. (14). In omparison, the onstant term is A = 0.06 and the magnitude of the exponential term is less than 0.02 for n > 100. Fig. 8(a) shows the fitted values of σ eff in the temperature range of T. σ eff dereases with T, as expeted. In the limit of large T, the differene between the free energies of (100) and (110) surfaes diminishes, the droplet beomes spherial, and σ eff onverges to the free energy of (100) surfaes. In the limit of T 0, we expet σ eff to onverge to (6/π) 1/3 times the surfae tension of the (100) surfae. This is beause as T 0, the shape 22 (26)

23 (a) 3 (b) σ eff T R σ (100) 0.1 T R σ eff 1.5 τ k B T k B T FIG. 8: (olor online) (a) Surfae free energies of the 3D Ising model as funtions of temperature. Cirles are fitted values of σ eff from Eq. (26), dashed line is the expeted behavior of σ eff over a wider range of temperature, and solid line is the free energy of the (100) surfae 43. Numerially fitted values of σ eff from Heermann et al. 29 are plotted as +. (b) τ values that give the best fit to the free energy data from US. τ an be roughly desribed by a linear funtion of T shown as a straight line. No abrupt hange is observed near the roughening temperature T R. of the droplet beomes ubial 28, and (6/π) 1/3 is the surfae area ratio between a sphere and a ube, both having unit volume. The expeted shape of σ eff (T ) over this temperature range is plotted as a dashed line, whih is similar to the ase of 2D Ising model shown in Fig. 1. In summary, we expet σ eff to derease from to 0 as temperature inreases. For example, at k B T = 2.71, σ eff = 1.6. This may explain the disrepany reported by Pan et al. 25, in whih σ eff = 2 is assumed at k B T = In Vehkamaki et al. 30, the nuleation rate predited by CNT was reported to have a weaker temperature dependene than the numerial results. This is probably aused by the use of the same surfae free energy (at T = 0.59 T ) in the entire temperature range (0.54T to 0.70 T ). Fig. 8(b) shows the fitted values of τ as a funtion of temperature. Over the range of temperature onsidered here, τ an be approximated by a linear funtion of T, τ = 0.26 k B T The fat that τ < 0 in 3D is onsistent with theoretial preditions. But τ is found to derease with temperature, and no disontinuity at the roughening temperature is observed. This is ontrary to the theoretial preditions of τ = 2 at T < T 3 R and τ = 1 9 at T > T R. The hange of τ with T may be the onsequene of a gradual hange of anisotropy effets as temperature hanges 39. More investigation is needed to resolve the ontroversy of 23

24 τ in the 3D Ising model. The differene between 2D (where τ = 5/4 remains a onstant) and 3D Ising model on the behavior of τ remains intriguing. V. SUMMARY In this paper, we have used two independent methods to alulate the nuleation rate of Ising model in 2D and 3D, in order to hek independently the different assumptions of the nuleation theories. The Markov hain assumption with the largest droplet size as the reation oordinate is found to be aurate enough to predit nuleation rate spanning more than 20 orders of magnitude, provided that the orret droplet free energy funtion is used. The logarithmi orretion term is found to be essential to droplet free energy in 2D. Our numerial results verified the field theory predition that τ = 5/4 in 2D. However, for the 3D Ising model, our numerial results are not onsistent with existing theories on the oeffiient τ of the logarithmi orretion term, suggesting that some important physis may still be missing in the existing theories. An exponential funtion seems to be neessary to desribe the free energy of small 3D droplets, but it is not needed for the 2D droplets. A promising diretion for future researh is to numerially ompute the surfae free energy of different orientations in 3D and to build the effetive surfae free energy σ eff from Wulff onstrution. This would eliminate σ eff as a fitting parameter and would enable a more stringent test of CNT for the 3D Ising model. APPENDIX A: THE CONSTANT TERM IN DROPLET FREE ENERGY In this appendix, we disuss how to obtain the onstant orretion term in the droplet free energy funtion, Eq. (6), for the 2D Ising model by onsidering the exat free energy expressions of small lusters. Shneidman et al. 21 used a similar approah to improve the preditions of droplet distributions. A related problem was disussed by Wilemski 48. We will also list the free energy expressions of small 3D lusters. Even though they annot be used to determine the onstant orretion term, they are useful for omparison purposes, as in Fig. 7(b). Beause the free energy expression, Eq. (5), is based on a ontinuum droplet model, we expet it to be inaurate for very small droplets, where the disreteness of the lattie 24

25 beomes appreiable. On the other hand, the ontinuum approximation should work better for large lusters, i.e. in the ontinuum limit. Therefore, we expet that Eq. (5) an be used to aurately predit the free energy differene between two large droplets, F (m) F (n), if both m 1 and n 1. This justifies the addition of a onstant term in Eq. (6). The value of the onstant term an be determined by mathing Eq. (6) with the exat values of F (n) for small n. Fortunately, for small enough n, the exat expression of the droplet free energy an be written down by enumerating all possible shapes of the droplet with size n and summing up their ontributions to the partition funtion. For simpliity, we will onsider the ase of h = 0. For example, a droplet of n = 1 is simply an isolated spin +1 surrounded by spins 1. The partition funtion of this droplet in the 2D Ising model is, Ω 2D 1 = e 8βJ (A1) where β 1/(k B T ). Similarly, the partition funtion of droplets of size 2, 3 and 4 are, Ω 2D 2 = 2 e 12βJ (A2) Ω 2D 3 = 6 e 16βJ (A3) Ω 2D 4 = e 16βJ + 18 e 20βJ (A4) The number in front of the exponential term orresponds to the multipliity of lusters of a given shape. Analyti expressions for the partition funtions of 2D droplets have been obtained up to n = 17 with omputer assistane 47. We have obtained similar expressions for the droplet partition funtions in the 3D Ising model for n from 1 to 7. Ω 3D 1 = e 12βJ (A5) Ω 3D 2 = 3 e 20βJ (A6) Ω 3D 3 = 15 e 28βJ (A7) Ω 3D 4 = 3 e 32βJ + 83 e 36βJ (A8) Ω 3D 5 = 48 e 40βJ e 44βJ (A9) Ω 3D 6 = 18 e 44βJ e 48βJ e 52βJ (A10) Ω 3D 7 = 8 e 48βJ e 52βJ e 56βJ e 60βJ (A11) 25

26 Given the droplet partition funtions, the droplet free energy F (n) defined in this paper an be obtained from the following equation, e βf (n) = Ω n 1 + i=1 Ω i (A12) Numerially, the summation in the denominator onverges very quikly after summing over 2 to 3 terms. As an approximation, we may write F (n) k B T ln Ω n. But this approximation is not invoked in Setion IV. The droplet free energy omputed from Eq. (A12) is used to determine the onstant term D in Eq. (6), by requiring that F (n) from the two equations mathes at a given n = n 0. In this work, we have always used n 0 = 1. Setting n 0 to larger values (as long as the analyti expression exists) does not hange the numerial results appreiably. For example, onsider the 2D Ising model at k B T = 1.5, h = 0 and J = 1. The free energy of a droplet of n = 1 is F (1) 8, whereas Eq. (8) predits that F (1) = 2 πσ eff 4.3. This means that a onstant orretion term D 3.7 is needed. 1 J. E. MDonald, Am. J. Phys. 30, 870 (1962). 2 F. F. Abraham, Homogeneous Nuleation Theory, Aademi, New York, (1974). 3 R. J. Speedy and C. A. Angell, J. Chem. Phys. 65, 851 (1976); P. H. Poole, F. Siortino, U. Essmann, and H. E. Stanley, Nature 360, 324 (1992). 4 E. M-V. and R. Bowles, Phys. Rev. Lett. 98, (2007). 5 R. J. Young, Introdution to Polymers, CRC Press, NY, (1981). 6 M. Gleiser and E. Kolb, Int. J. Mod. Phys. C 3, 773 (1992). 7 R. Beker and W. Doring, Ann. Phys. (N.Y.) 24, 719 (1935). 8 Ya. B. Zeldovih, Ann. Phys. (N.Y.), 18, 1 (1943). 9 J. Frenkel, Kineti Theory of Liquids, Oxford University Press, Oxford, (1946). 10 L. Farkas, Z. Phys. Chem. (Munih) 125, 236 (1927). 11 J. Lothe and G. Pound, J. Chem. Phys. 36:2080 (1962). 12 J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 31, 688 (1959). 13 J. S. Langer, Ann. Phys. (N.Y.) 41, 108 (1967); J. S. Langer, Phys. Rev. Letter. 21, 973 (1968); J. S. Langer, Ann. Phys. (N.Y.) 54, 258 (1969). 14 X. C. Zeng and D. W. Oxtoby, J. Chem. Phys. 94, 4472 (1991). 26