Phase field theory of interfaces and crystal nucleation in a eutectic system of fcc structure: I. Transitions in the one-phase liquid region

The phase field theory (PFT) has been applied to predict equilibrium interfacial properties and nucleation barrier in the binary eutectic system Ag-Cu using double well and interpolation functions deduced from a Ginzburg-Landau expansion that considers fcc (face centered cubic) crystal symmetries. The temperature and composition dependent free energies of the liquid and solid phases are taken from CALculation of PHAse Diagrams-type calculations. The model parameters of PFT are fixed so as to recover an interface thickness of approximately 1 nm from molecular dynamics simulations and the interfacial free energies from the experimental dihedral angles available for the pure components. A nontrivial temperature and composition dependence for the equilibrium interfacial free energy is observed. Mapping the possible nucleation pathways, we find that the Ag and Cu rich critical fluctuations compete against each other in the neighborhood of the eutectic composition. The Tolman length is positive and shows a maximum as a function of undercooling. The PFT predictions for the critical undercooling are found to be consistent with experimental results. These results support the view that heterogeneous nucleation took place in the undercooling experiments available at present. We also present calculations using the classical droplet model [classical nucleation theory (CNT)] and a phenomenological diffuse interface theory (DIT). While the predictions of the CNT with a purely entropic interfacial free energy underestimate the critical undercooling, the DIT results appear to be in a reasonable agreement with the PFT predictions.     

Monte Carlo simulations of critical cluster sizes and nucleation rates of water

We have calculated the critical cluster sizes and homogeneous nucleation rates of water at temperatures and vapor densities corresponding to experiments by Wolk and Strey [J. Phys. Chem B 105, 11683 (2001)]. The calculations have been done with an expanded version of a Monte Carlo method originally developed by Vehkamaki and Ford [J. Chem. Phys. 112, 4193 (2000)]. Their method calculates the statistical growth and decay probabilities of molecular clusters. We have derived a connection between these probabilities and kinetic condensation and evaporation rates, and introduce a new way for the calculation of the work of formation of clusters. Three different interaction potential models of water have been used in the simulations. These include the unpolarizable SPC/E [J. Phys. Chem. 91, 6269 (1987)] and TIP4P [J. Chem. Phys. 79, 926 (1983)] models and a polarizable model by Guillot and Guissani [J. Chem. Phys. 114, 6720 (2001)]. We show that TIP4P produces critical cluster sizes and a temperature and vapor density dependence for the nucleation rate that agree well with the experimental data, although the magnitude of nucleation rate is constantly overestimated by a factor of 2 x 10(4). Guissani and Guillot's model is somewhat less successful, but both the TIP4P and Guillot and Guissani models are able to reproduce a much better experimental temperature dependency of the nucleation rate than the classical nucleation theory. Using SPC/E results in dramatically too small critical clusters and high nucleation rates. The water models give different average binding energies for clusters. We show that stronger binding between cluster molecules suppresses the decay probability of a cluster, while the growth probability is not affected. This explains the differences in results from different water models.     

The anisotropic hard-sphere crystal-melt interfacial free energy from fluctuations

We have calculated the interfacial free energy for the hard-sphere system, as a function of crystal interface orientation, using a method that examines the fluctuations in the height of the interface during molecular dynamics simulations. The approach is particularly sensitive for the anisotropy of the interfacial free energy. We find an average interfacial free energy of gamma=0.56+/-0.02k(B)Tsigma(-2). This value is lower than earlier results based upon direct calculations of the free energy [R. L. Davidchack and B. B. Laird, Phys. Rev. Lett. 85, 4751 (2000)]. However, both the average value and the anisotropy agree with the recent values obtained by extrapolation from direct calculations for a series of the inverse-power potentials [R. L. Davidchack and B. B. Laird, Phys. Rev. Lett. 94, 086102 (2005)].     

Argon nucleation in a cryogenic nucleation pulse chamber

Homogeneous nucleation of argon droplets has been measured with a newly designed cryogenic nucleation pulse chamber presented already in a previous paper [Fladerer and Strey, J. Chem. Phys. 124, 16 (2006)]. Here we present the first systematic nucleation onset data for argon measured in a temperature range from 42 to 58 K and for vapor pressures from 0.3 to 10 kPa. For these data we provide an analytical fit function. From the geometry of the optical detection system and the time of nucleation the experimental nucleation-rate range can be estimated. This allows a comparison of the data with the predictions of classical nucleation theory. We found 16-26 orders of magnitude difference between theory and experiment, and a too strong theoretical dependence of the nucleation rate on temperature. A comparison with the self-consistent theory of Girshick and Chiu [J. Chem. Phys. 93, 1273 (1990)] showed improved temperature dependence but still discrepancies of 11-17 orders of magnitude compared to experimental data. The thermodynamically consistent theory of Kashchiev [J. Chem. Phys. 118, 1837 (2003)] was found to agree rather well with experiment in respect to the temperature dependence and to predict rates about 5-7 orders of magnitude below the experimental ones. With the help of the Gibbs-Thomson equation we were able to evaluate the size of the critical nucleus to be 40-80 argon atoms.     

Monte Carlo simulation study of droplet nucleation

A new rigorous Monte Carlo simulation approach is employed to study nucleation barriers for droplets in Lennard-Jones fluid. Using the gauge cell method we generate the excess isotherm of critical clusters in the size range from two to six molecular diameters. The ghost field method is employed to compute the cluster free energy and the nucleation barrier with desired precision of (1-2)kT. Based on quantitative results obtained by Monte Carlo simulations, we access the limits of applicability of the capillarity approximation of the classical nucleation theory and the Tolman equation. We show that the capillarity approximation corrected for vapor nonideality and liquid compressibility provides a reasonable assessment for the size of critical clusters in Lennard-Jones fluid; however, its accuracy is not sufficient to predict the nucleation barriers for making practical estimates of the rate of nucleation. The established dependence of the droplet surface tension on the droplet size cannot be approximated by the Tolman equation for small droplets of radius less than four molecular diameters. We confirm the conclusion of ten Wolde and Frenkel [J. Chem. Phys. 109, 9901 (1998)] that integration of the normal component of the Irving-Kirkwood pressure tensor severely underestimates the nucleation barriers for small clusters.     

Isothermal crystallization kinetics and melting behavior of poly(ethylene terephthalate)/barite nanocomposites

Poly(ethylene terephthalate) (PET)/Barite nanocomposites were prepared by direct melt compounding. The effects of PET‐Barite interfacial interaction on the dynamic mechanical properties and crystallization were investigated by DMA and DSC. The results showed that Barite can act as a nucleating agent and the nucleation activity can be increased when the Barite was surface‐modified (SABarite). SABarite nanoparticles induced preferential lamellae orientation because of the strong interfacial interaction between PET chains and SABarite nanoparticles, which was not the case in Barite filled PET as determined by WAXD. For PET/Barite nanocomposites, the Avrami exponent n increased with increasing crystallization temperature. Although at the same crystallization temperature, the n value will decrease with increasing SABarite content, indicating of the enhancement of the nucleation activity. Avrami analyses suggest that the nucleation mechanism is different. The activation energy determined from Arrhenius equation reduced dramatically for PET/SABarite nanocomposite, confirming the strong interfacial interaction between PET chains and SABarite nanoparticles can reduce the crystallization free energy barrier for nucleus formation. In the DSC scan after isothermal crystallization process, double melting behavior was found. And the double endotherms could be attributed to the melting of recrystallized less perfect crystallites or the secondary lamellae produced during different crystallization processes. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 655–668, 2009     

Investigation of excess adsorption, solvation force, and plate-fluid interfacial tension for Lennard-Jones fluid confined in slit pores

The excess Helmholtz free energy functional is formulated in terms of a modified fundamental measure theory [Y. X. Yu and J. Z. Wu, J. Chem. Phys. 117, 10156 (2002)] for a short ranged repulsion and a first-order mean-spherical approximation theory [Y. P. Tang, J. Chem. Phys. 118, 4140 (2003)] for a long ranged attraction. Within the framework of the density functional theory, the density profile, excess adsorption, solvation force, and plate-fluid interfacial tension of a Lennard-Jones fluid confined in slit pores are predicted, and the results agree well with the simulation data. The phase equilibria inside the slit pores are determined according to the requirement that temperature, chemical potential, and grand potential in coexistence phases should be equal, and the plate-fluid interfacial tensions at equilibrium states are predicted consequently.     

Crystal nucleation and the solid-liquid interfacial free energy

We present the results of molecular dynamics simulation of crystal nucleation in a supercooled Lennard-Jones liquid. Temperature and baric dependences of the nucleation rate, the Zeldovich factor, nucleus size diffusion coefficient, the radius, and the pressure in a critical crystal nucleus are defined in computer simulation. The data obtained have been used in the framework of classical nucleation theory to calculate the effective surface energy of crystal nuclei γ(e). It is shown that the value of γ(e) at T = const exceeds the value of the interfacial free energy at a flat crystal-liquid interface γ(∞) and γ(e) < γ(∞) at p = const.     

Homogeneous nucleation and growth in supersaturated zinc vapor investigated by molecular dynamics simulation

Homogeneous nucleation and growth of zinc from supersaturated vapor are investigated by nonequilibrium molecular dynamics simulations in the temperature range from 400 to 800 K and for a supersaturation ranging from log S=2 to 11. Argon is added to the vapor phase as carrier gas to remove the latent heat from the forming zinc clusters. A new parametrization of the embedded atom method for zinc is employed for the interaction potential model. The simulation data are analyzed with respect to the nucleation rates and the critical cluster sizes by two different methods, namely, the threshold method of Yasuoka and Matsumoto [J. Chem. Phys. 109, 8451 (1998)] and the mean first passage time method for nucleation by Wedekind et al. [J. Chem. Phys. 126, 134103 (2007)]. The nucleation rates obtained by these methods differ approximately by one order of magnitude. Classical nucleation theory fails to describe the simulation data as well as the experimental data. The size of the critical cluster obtained by the mean first passage time method is significantly larger than that obtained from the nucleation theorem.     

Vapor-liquid interfacial properties of rigid-linear Lennard-Jones chains

We have obtained the interfacial properties of short rigid-linear chains formed from tangentially bonded Lennard-Jones monomeric units from direct simulation of the vapour-liquid interface. The full long-range tails of the potential are accounted for by means of an improved version of the inhomogeneous long-range corrections of Janec?ek [J. Phys. Chem. B 110, 6264-6269 (2006)] proposed recently by MacDowell and Blas [J. Chem. Phys. 131, 074705 (2009)] valid for spherical as well as for rigid and flexible molecular systems. Three different model systems comprising of 3, 4, and 5 monomers per molecule are considered. The simulations are performed in the canonical ensemble, and the vapor-liquid interfacial tension is evaluated using the test-area method. In addition to the surface tension, we also obtain density profiles, coexistence densities, critical temperature and density, and interfacial thickness as functions of temperature, paying particular attention to the effect of the chain length and rigidity on these properties. According to our results, the main effect of increasing the chain length (at fixed temperature) is to sharpen the vapor-liquid interface and to increase the width of the biphasic coexistence region. As a result, the interfacial thickness decreases and the surface tension increases as the molecular chains get longer. The surface tension has been scaled by critical properties and represented as a function of the difference between coexistence densities relative to the critical density.     

Nitrogen nucleation in a cryogenic supersonic nozzle

We follow the vapor-liquid phase transition of N(2) in a cryogenic supersonic nozzle apparatus using static pressure measurements. Under our operating conditions, condensation always occurs well below the triple point. Mean field kinetic nucleation theory (MKNT) does a better job of predicting the conditions corresponding to the estimated maximum nucleation rates, J(max) = 10(17±1) cm(-3) s(-1), than two variants of classical nucleation theory. Combining the current results with the nucleation pulse chamber measurements of Iland et al. [J. Chem. Phys. 130, 114508-1 (2009)], we use nucleation theorems to estimate the critical cluster properties. Both the theories overestimate the size of the critical cluster, but MKNT does a good job of estimating the excess internal energy of the clusters.     

A molecular dynamics study to determine the solid-liquid interfacial tension using test area simulation method (TASM)

Molecular dynamics (MD) studies on heat transfer from a heated nanoparticle into the surrounding fluid have indicated that the fluid next to a spherical nanoparticle can get heated well above its boiling point without observing a phase change, while a contradicting behavior was observed for a flat surface-fluid interface. Another interesting observation is that the critical heat flux was found to increase with increase in the wetting characteristics of solid. Thus, the interfacial tension or free energy of solid-liquid interface could play a pivotal role in the mechanism of heat transfer. A recent study by Gloor et al. [J. Chem. Phys. 123, 134703 (2005)] has proposed test area simulation method (TASM) for the determination of interfacial tension. The present study involves the determination and the comparison of solid-liquid interfacial tension for planar and spherical interfaces using MD based on TASM and analyze the results. A higher interfacial tension value is observed for spherical nanoparticle fluid interface compared to flat surface fluid interface. The results also indicate that the solid-liquid interfacial tension is a size and temperature dependent property. The results from this study are also expected to give better insights into the possible reasons for the observed differences in the thermal transport for spherical nanoparticle-liquid interface compared to planar-liquid interface.     

Formation mechanism of critical nucleus during nucleation process of liquid metal sodium

To deeply understand the formation mechanism of a critical nucleus during the nucleation process of liquid metal sodium, a system consisting of 10 000 Na atoms has been simulated by using molecular dynamics method. The evolutions of nuclei are traced directly, adopting the cluster-type index method. It is found that the energies of clusters and their geometrical constraints interplay to form the favorable microstructures during the nucleation process. The nucleus can be formed through many different pathways, and the critical size of the nucleus would be different for each pathway. It is also found that the critical nucleus is nonspherical and may include some metastable structures. Furthermore, the size of the cluster and its internal structure both play a crucial role in determining whether it is a critical nucleus, and this is in agreement with the simulations by computing the free energy of the Lennard-Jones system [D. Moroni, P. R. ten Wolde, and P. G. Bolhuis, Phys. Rev. Lett. 94, 235703 (2005)].     

Calculation of the crystal-melt interfacial free energy of succinonitrile from molecular simulation

The crystal-metal interfacial free energy for a six-site model of succinonitrile [N triple bond C-(CH(2))(2)-C triple bond N] has been calculated using molecular-dynamics simulation from the power spectrum of capillary fluctuations in interface position. The orientationally averaged magnitude of the interfacial free energy is determined to be (7.0+/-0.4)x10(-3) J m(-2). This value is in agreement (within the error bars) with the experimental value [(7.9+/-0.8)x10(-3) J m(-2)] of Marasli et al. [J. Cryst. Growth 247, 613 (2003)], but is about 20% lower than the earlier experimental value [(8.9+/-0.5)x10(-3) J m(-2)] obtained by Schaefer et al. [Philos. Mag. 32, 725 (1975)]. In agreement with the experiment, the calculated anisotropy of the interfacial free energy of this body-centered-cubic material is small. In addition, the Turnbull coefficient from our simulation is also in agreement with the experiment. This work demonstrates that the capillary fluctuation method of Hoyt et al. [Phys. Rev. Lett. 86, 5530 (2001)] can be successfully applied to determine the crystal-melt interfacial free energy of molecular materials.     

Hard spheres revisited: accurate calculation of the solid-liquid interfacial free energy

We revise the earlier [R. L. Davidchack and B. B. Laird, Phys. Rev. Lett. 85, 4751 (2000)] direct calculation of the hard sphere solid-liquid interfacial free energy by the cleaving walls method. The revisions of the method include modified interactions with the cleaving walls and the use of a nonequilibrium work measurements approach, which allows for a more robust control of the accuracy of the obtained results. We find that the new values are lower compared to the original ones, which is consistent with the more recent indirect estimates based on extrapolation from the soft-sphere results [R. L. Davidchack and B. B. Laird, Phys. Rev. Lett. 94, 086102 (2005)], as well as those obtained using the capillary fluctuations method [R. L. Davidchack, J. R. Morris, and B. B. Laird, J. Chem. Phys. 125, 094710 (2006)].     

Test of classical nucleation theory via molecular-dynamics simulation

A direct test of classical nucleation theory (CNT) is made using molecular-dynamics simulations. The relation between critical nucleus size and undercooling temperature is extracted and the result yields the solid-liquid interfacial energy. It is shown that the CNT, within the assumptions made for spherical nucleus in supercooled liquid, is valid in the critical regime of nucleation for a large range of undercooling and nucleus size.     

Recent developments in the kinetic theory of nucleation

A review of recent progress in the kinetics of nucleation is presented. In the conventional approach to the kinetic theory of nucleation, it is necessary to know the free energy of formation of a new-phase particle as a function of its independent variables at least for near-critical particles. Thus the conventional kinetic theory of nucleation is based on the thermodynamics of the process. The thermodynamics of nucleation can be examined by using various approaches, such as the capillarity approximation, density functional theory, and molecular simulation, each of which has its own advantages and drawbacks. Relatively recently a new approach to the kinetics of nucleation was proposed [Ruckenstein E, Nowakowski B. J Colloid Interface Sci 1990;137:583; Nowakowski B, Ruckenstein E. J Chem Phys 1991;94:8487], which is based on molecular interactions and does not employ the traditional thermodynamics, thus avoiding such a controversial notion as the surface tension of tiny clusters involved in nucleation. In the new kinetic theory the rate of emission of molecules by a new-phase particle is determined with the help of a mean first passage time analysis. This time is calculated by solving the single-molecule master equation for the probability distribution function of a surface layer molecule moving in a potential field created by the rest of the cluster. The new theory was developed for both liquid-to-solid and vapor-to-liquid phase transitions. In the former case the single-molecule master equation is the Fokker-Planck equation in the phase space which can be reduced to the Smoluchowski equation owing to the hierarchy of characteristic time scales. In the latter case, the starting master equation is a Fokker-Planck equation for the probability distribution function of a surface layer molecule with respect to both its energy and phase coordinates. Unlike the case of liquid-to-solid nucleation, this Fokker-Planck equation cannot be reduced to the Smoluchowski equation, but the hierarchy of time scales does allow one to reduce it to the Fokker-Plank equation in the energy space. The new theory provides an equation for the critical radius of a new-phase particle which in the limit of large clusters (low supersaturations) yields the Kelvin equation and hence an expression for the macroscopic surface tension. The theory was illustrated with numerical calculations for a molecular pair interaction potential combining the dispersive attraction with the hard-sphere repulsion. The results for the liquid-to-solid nucleation clearly show that at given supersaturation the nucleation rate depends on the cluster structure (for three cluster structures considered-amorphous, fcc, and icosahedral). For both the liquid-to-solid and vapor-to-liquid nucleation, the predictions of the theory are consistent with the results of classical nucleation theory (CNT) in the limit of large critical clusters (low supersaturations). For small critical clusters the new theory provides higher nucleation rates than CNT. This can be accounted for by the fact that CNT uses the macroscopic interfacial tension which presumably overpredicts the surface tension of small clusters, and hence underpredicts nucleation rates.     

氟化钾团簇的相变、成核和重结晶的分子动力学模拟

我们利用Born-Mayer-Huggins相互作用势函数对(KF)_N(N=108，256，500和864)团簇进行了分子动力学(MD)模拟。为了避免周期性边界条件对相变、成核和重结晶的干扰作用，对体系采用了自由边界。基于MD模拟结果，对团簇的熔化温度、熔化焓、扩散系数、成核速率、固液界面自由能、临界核大小等进行了计算和讨论。在对(KF)₈₆₄双晶团簇的热退火模拟中，观察到了固态的重结晶和晶粒的生长。经典的成核理论成功地解释了(KF)₈₆₄双晶团簇的重结晶MD模拟结果。     

Cavitation and crystallization in a metastable Lennard-Jones liquid at negative pressures and low temperatures

Molecular dynamics simulations have been used to investigate the kinetics of spontaneous cavitation and crystallization in a Lennard-Jones liquid at negative pressures in the temperature range where these processes compete with each other. The nucleation rate has been calculated in NVE and NpT ensembles by the method of mean lifetime and the transition interface sampling method with parallel path swapping. The data obtained have been used to determine in the framework of classical nucleation theory the value of the ratio of the solid-liquid and the liquid-void interfacial free energy for critical crystals and cavities and the values of their volumes at points where the cavitation rate of the liquid is equal to the rate of its crystallization.     

磁场影响聚合物本体结晶动力学的机理分析

在经典的热力学理论基础上,探讨了磁场对聚合物本体结晶过程的成核与生长的影响,建立了相关结晶动力学理论方程.初步认为,磁场产生的＂磁结晶效应＂可能是由于晶相与非晶相之间磁化率差异导致了两相之间磁化能的差异,也可能由于聚合物体系在结晶前会形成一种有序相,减小了体系的熵值,进而改变了结晶过程中的体系自由能,影响其成核与晶体生长,乃至整个结晶动力学方程.利用Matlab软件结合PLLA的各结晶参数值,绘制了结晶自由能与各成核临界参数之间的函数图像.结果表明,在低过冷度下,较小的自由能扰动可能导致较大的晶核临界参数变化.