Origin of the failure of classical nucleation theory: incorrect description of the smallest clusters

We carry out molecular Monte Carlo simulations of clusters in an imperfect vapor. We show that down to very small cluster sizes, classical nucleation theory built on the liquid drop model can be used very accurately to describe the work required to add a monomer to the cluster. However, the error made in modeling the smallest of clusters as liquid drops results in an erroneous absolute value for the cluster work of formation throughout the size range. We calculate factors needed to correct the cluster formation work given by the liquid drop model. The corrected work of formation results in nucleation rates in good agreement with recent nucleation experiments on argon and water.     

WnNim(n+m=8)团簇的极性和光谱性质的理论研究

采用密度泛函理论(DFT)中的B3LYP方法,在LANL2DZ基组水平上对WnNim(n+m=8)团簇的各种可能构型进行了几何参数全优化,得到了它们的基态构型;并对基态构型的偶极距、极化率、红外光谱和拉曼光谱性质进行了分析,结果表明:团簇WnNim(n+m=8)都具有极性,富W团簇非线性光学效应强,容易被外加场极化;振动频率主要分布在0-350 cm-1范围内,团簇W4Ni4因其振动方式的特殊性,红外光谱和拉曼光谱在频率421.971 cm-1处,都有明显强峰;团簇W5Ni3因其结构的高对称性在振动光谱中出现多处共振现象.     

A monte carlo method for approximating critical cluster size in the nucleation of model systems

A formalism is presented for estimating critical cluster size as defined in classical models for nucleation phenomena. The method combines Bennett's Monte Carlo technique for determining free-energy differences for clusters containingn andn- 1 atoms with the steady state nucleation rate formalism. A simple form for the free energy of formation of then cluster [including a termA (n)n ^2/3] is used to predict critical cluster size and critical supersaturation ratio, S^*. This approach is applied to Lennard-Jones vapor clusters at 60 K. Results for free-energy differences for the 13, 18, 24, and 43 clusters predict a critical cluster size of 70 ± 5 atoms at a critical supersaturation ratio given bylnS ^*=2,45 0.15. This method is intended to provide estimates of critical cluster size for more ambitious attempts to calculate cluster free energies or for initializing conditions in microscopic simulations of nucleating systems.This material is based upon work supported by the National Science Foundation under Grant No. ATM80 15790 and the National Aeronautics and Space Administration under Grant No. NAS8-31150.     

第一性原理计算ZrnFe(n=2—13)团簇的基态结构及其磁性

从第一性原理出发，利用密度泛函理论中的广义梯度近似对Zr_nFe(n=2—13)团簇进行了结构优化、能量和频率计算.在充分考虑自旋多重度的前提下，对每一具体尺寸的团簇，得到了多个平衡构型，并根据能量高低确定了团簇的基态结构.综合团簇的结合能、二阶能量差分以及团簇的最高占据轨道和最低未占据轨道间的能隙可知Zr₅Fe，Zr₇Fe和Zr₁₂Fe团簇的稳定性相对较高，Zr₁₂Fe团簇的结构是具有I_h对称性的正二十面体，而且Zr₁₂Fe的稳定性在所有团簇中是最高的.另外，不仅Zr₅Fe，Zr₇Fe和Zr₁₂Fe团簇的稳定性相对较高，而且它们均为磁性团簇(而Zr_n团簇的磁矩在n≥5时已经发生了淬灭)，由此可知通过选择合适的掺杂元素可能得到高稳定的磁性团簇.从Mulliken布居分析结果可知，除了在Zr₁₂Fe团簇中Fe原子失去少量电荷外，其他团簇中Fe原子均从Zr原子那里得到了一定量电荷，即Fe原子在Zr_nFe(n=2—13，n≠12)团簇中是电子受体.     

Preface

This special volume of The European Physical Journal D contains papers presented at the 10th International Symposium on Small Particles and Inorganic Clusters, ISSPIC 10, which was held from October 11th to 15th, 2000, in Atlanta, Georgia, USA. The meeting was attended by over 300 scientists from all over the world, and in it 45 invited and hot-topic lectures were given, and 330 posters were presented. In keeping with the tradition of the ISSPIC meetings, the 10th anniversary symposium was devoted to a broad and balanced overview of new results, emerging trends and perspectives pertaining to the physics and chemistry of clusters. The meeting covered experimental and theoretical investigations of gas-phase and supported clusters, nanoscale cluster-based materials, and nanometer. The sessions at ISSPIC 10 were organized as mini-symposia, and topics included: electronic and structural properties of clusters, charged clusters and photoelectron spectroscopy, fast laser spectroscopy and dynamics, cluster-surface interactions and cluster deposition, helium clusters and spectroscopy, structural evolution and thermodynamics of clusters, cluster reactivity and nano-catalysis, two-dimensional quantum dots, nano-crystals and self-assembly, mechanical, electronic and transport properties of carbon nanotubes, structure and conductance properties of nanowires, formation and stability of droplets and jets, and medical applications of colloidal clusters. Most importantly, the symposium provided an interdisciplinary forum for presentation and discussion of fundamental and methodological aspects as well as technologically oriented developments. The opening session of ISSPIC 10 was dedicated to the memory of Professor Walter D. Knight, a pioneer in the field of clusters. The work of Walter and his research group at Berkeley on electronic shells in metal clusters opened new avenues in cluster science and contributed significantly to the rapid growth of this field. We would like to take this opportunity to gratefully acknowledge the financial assistance extended to the symposium by the Georgia Institute of Technology and by the United States Air Force Office of Scientific Research. We also thank the participants for contributing to the success of ISSPIC 10 through communicating their latest scientific results during the meeting and via the papers appearing in these proceedings.     

第一性原理对XMgn(X=B,Al, n=1—12)团簇的几何结构和电子性质的研究

利用密度泛函理论(DFT)的B3PW91方法,在6-311G水平上对BMg_n,AlMg_n(n=1—12)团簇进行了几何结构优化和电子性质分析. 发现随着原子个数的增加, B原子进入镁团簇的内部, 而AlMg_n和镁团簇有相似的生长模式. B,Al原子的掺杂均能使镁团簇的平均结合能增大,稳定性增强, BMg_n,AlMg_n关键词： 密度泛函理论 最低能量结构 n和AlMg_n团簇')" href="#">BMg_n和AlMg_n团簇 NBO电荷布居     

Optical second-harmonic generation by supported metal clusters: Size and shape effects

Optical Second Harmonic Generation (SHG) by metal clusters has been investigated. For this purpose clusters were generated by the deposition and nucleation of metal atoms on a LiF(100) single crystal surface under ultrahigh vacuum conditions. The size and shape of the metal particles was characterized by optical transmission spectroscopy. The SHG intensity was detected in situ as a function of cluster size during the nucleation. Fundamental wavelengths of =1064 and 532 nm were used and the SHG signal was measured for different polarization combinations of the incident and registered light. SH radiation is detectable for particles as small as approximately 1 nm. The signal grows monotonically as a function of particle size, passes a maximum and finally drops off. This behavior is discussed in terms of resonant enhancement of the signal by surface plasmon excitation and changes of ⁽²⁾ as a function of particle size and shape. In further experiments the chemisorption of oxygen on the surface of the metal particles was studied. The SH signal decreases as a function of oxygen coverage and amounts to only about 15% of the initial value upon chemisorption of one monolayer. This indicates that the SH signal originates almost exclusively from the surface of the clean clusters and that higher order bulk contributions are negligible.     

Thermodynamics of supersaturated vapor nucleation on molecular condensation nuclei

A key problem in the theory of a supersaturated vapor nucleation on molecular condensation nuclei (namely, the work of formation and the equilibrium concentrations of clusters) is considered. To calculate these quantities using the structural models of clusters, which are better suited for this purpose than the classical droplet model, we derive the equation connecting the work of transfer of a molecular condensation nucleus from the gas phase to a homogeneous cluster with a change in the number of contacts between molecules, occurring in the course of this transfer, and with the work of rupture of individual contacts.     

Experimental and Theoretical Studies on the Interaction between Isonicotinic Acid Molecules and Silver Nanoclusters

The present paper describes the studies on the adsorption behavior and charge transfer from isonicotinic acid to silver nanoparticles with experiment and theory. Compared with UV–Visible adsorption spectrum, the adsorption spectrum of Ih-Ag₁₃ cluster was quite good agreed with that of silver colloidal nanoparticle. So that one Ag₁₃ cluster as a substrate was used to simulate Raman frequencies of the adsorption configuration. Here, it is demonstrated the calculated Raman spectra are in good agreement with experimental results. The analysis of Mulliken charge was obtained by density functional theory, which indicated the charge characteristics of Ag₁₃ nanoparticle. Once isonicotinic acid molecules were adsorbed on sliver clusters, the charges transfer from isonicotinic acid to silver clusters, so that the surface charges of silver clusters are uneven.     

稳态Cu-Zr二十面体团簇电子结构的密度泛函研究

采用第一原理对以Cu为心的低能稳态Cu_nZ_(r13-n)(n=6,7,8,9)二十面体团簇的电子结构进行计算,结果表明:同一化学组分下,以Cu为心的Cu-Zr二十面体团簇中出现的同类原子聚集现象可以增强团簇的稳定性,降低费米能级(EF)上的电子数N(EF),这为低能稳态团簇拥有较小的N(EF)提供了深层次的理论解释.进一步的差分电子密度与Mulliken布居分析得知,Cu-Zr二十面体中共价键与离子键共存,成键态与反键态共存,且团簇在形成时壳层Zr与中心Cu原子是电子的提供者,壳层Cu是电子的获得者.该电荷转移方向是金属玻璃中以Cu为心的Cu-Zr二十面体团簇普遍遵循的规律,不随团簇的化学序参数及化学组分的变化而变化.计算的红外振动谱为实验上准确表征不同二十面体原子团提供了一种新的思路.     

Kinetic pathways for the isotropic-nematic phase transition in a system of colloidal hard rods: a simulation study

We study the kinetic pathways for the isotropic-to-nematic transition in a fluid of colloidal hard rods. In order to follow the formation of the nematic phase, we develop a new cluster criterion that distinguishes nematic clusters from the isotropic phase. Applying this criterion in Monte Carlo simulations, we find spinodal decomposition as well as nucleation and growth depending on the supersaturation. We determine the height of the nucleation barrier and we study the shape and structure of the cluster. More specifically, we find ellipsoidal nematic clusters with an aspect ratio of about 1.7 and a homogeneous nematic director field. Our results are consistent with theoretical predictions on the shape and director field of nematic tactoids. Classical nucleation theory gives reasonable predictions for the height of the nucleation barrier and the critical nucleus size.     

Dynamic Effects in Nucleation of Receptor Clusters

Nucleation theory has been widely applied for the interpretation of critical phenomena in nonequilibrium systems. Ligand-induced receptor clustering is a critical step of cellular activation. Receptor clusters on the cell surface are treated from the nucleation theory point of view. The authors propose that the redistribution of energy over the degrees of freedom is crucial for forming each new bond in the growing cluster. The expression for a kinetic barrier for new bond formation in a cluster was obtained. The shape of critical receptor clusters seems to be very important for the clustering on the cell surface. The von Neumann entropy of the graph of bonds is used to determine the influence of the cluster shape on the kinetic barrier. Numerical studies were carried out to assess the dependence of the barrier on the size of the cluster. The asymptotic expression, reflecting the conditions necessary for the formation of receptor clusters, was obtained. Several dynamic effects were found. A slight increase of the ligand mass has been shown to significantly accelerate the nucleation of receptor clusters. The possible meaning of the obtained results for medical applications is discussed.     

Treatment of droplike clusters by means of the classical phase integral in nucleation theory

The translation inconsistency in the theory of nucleation is discussed in historical perspective. A theory is then developed, beginning with the classical phase integral, which not only allows all approximations to be well defined, but also leads to the most natural droplike model for the cluster. The theory makes it possible to apply, in a consistent manner, the thermodynamics of curved surfaces or, alternatively, moleculardynamic numerical computation schemes to the evaluation of the partition function of the cluster. If the cluster is treated as a macroscopic drop (having the free energy of a macroscopic drop), the result for the distribution of clusters differs in only a minor way from that prescribed by the conventional theory of nucleation. It is concluded that for liquid nuclei the conventional theory is consistent, but that a replacement factor may be necessary for solid nuclei. In general, however, the major problems confronting the theory involve the precise evaluation of the work of cluster formation.     

Colloidal crystal growth at externally imposed nucleation clusters

We study the conditions under which and how an imposed cluster of fixed colloidal particles at prescribed positions triggers crystal nucleation from a metastable colloidal fluid. Dynamical density functional theory of freezing and Brownian dynamics simulations are applied to a two-dimensional colloidal system with dipolar interactions. The externally imposed nucleation clusters involve colloidal particles either on a rhombic lattice or along two linear arrays separated by a gap. Crystal growth occurs after the peaks of the nucleation cluster have first relaxed to a cutout of the stable bulk crystal.     

Nucleation on substrates from the vapour phase

Recent developments in the theory of nucleation of vapour deposits on crystalline substrates are reviewed. To facilitate comparison, the theories are formulated in a common dimensionless notation, and an examination of the major underlying assumptions reveals the basic similarity of many of them. The capture and dissociation rates are expressed in terms of the cluster geometry and the pertinent energy parameters, and from these the ‘chemical’ rate equations are set up. The following types of approximate solutions are discussed: (a) long-time asymptotic solutions, from which the conditions for saturation in the cluster concentration may be deduced, (b) a generalization of the type of approximate solution used by Logan (1969), and (c) numerical solutions employing a minimum number of simplifying assumptions. Based on a simple model, agreement between the latter two seems reasonably good. For any given set of fixed parameters (energies, geometrical constants, and arrival rate from the vapour) several temperature ranges may be distinguished. The main division is between ‘initially complete condensation’ at low temperatures and ‘initially incomplete condensation’ at high temperatures. Within each of the latter cases there are further transitions (a) between different values of the ‘critical size’ i*, and (b) from negligible growth to rapid growth of the supercritical clusters. The influence of all of these factors on the final cluster concentration is described.

The distributions of the clusters in size and spacing are discussed briefly and qualitatively, as are the types of effects that can be induced by defects or other ‘special sites’ on the substrate.

Comparisons are made with some recent experimental studies. In many of these, defects in the substrate seem to play a dominant role, and no detailed comparison with theory seems possible. One notable exception is the nucleation of rare gas crystals on graphite substrates (Venables and Ball 1970), and here, for at least two of the three gases studied, excellent quantitative agreement is obtained.     

Nonmonotonic behavior of the concentration in the kinetics of dissolution of fullerenes

A cluster model for the dissolution of C₆₀ fullerenes in a nonpolar solvent has been proposed. This model provides the explanation of a maximum experimentally observed in the time dependence of the solution concentration during dissolution. The model is based on the kinetic equations of nucleation theory and involves a balance between the flux of fullerene molecules from the solid phase and the sedimentation of large clusters from the solution. The formation of clusters is described using the drop model. Analysis of the numerical solutions of the equations reveals four qualitatively different dissolution regimes depending on the relation between the model parameters.     

Primary nucleation processes in binary oxide growth: The case of MgO

The smallest forms of stoichiometric and non-stoichiometric MgO clusters appearing on the MgO(0 0 1) surface during the growth under atomic and/or molecular deposition are investigated from first-principles and empiric potentials. The basic entities (MgO molecule and (MgO)₂ cluster) result from a very exoenergetic and spontaneous redox reaction that involves directly the deposited species (Mg and O atoms, O₂ molecule). The stoichiometric clusters, resulting from the agglomeration of MgO molecules, are very stable under non-polar forms. Their formation energy is modelized, down to very small sizes, within an independent defect model. We point out the specificity of such clusters in the framework of the classical nucleation theory. The high-energy polar isomers are associated to destabilizing macroscopic electric fields and dipoles. These forms may nevertheless be strongly stabilized by incorporating extra Mg adatoms that give part of their valence shell to the cluster and decrease the total dipole in this way, illustrating the delicate coupling between chemistry and electrostatics in growth processes of oxides. Based on these considerations, we propose a scenario describing MgO growth both in the step-flow and in the nucleation regime.     

Determination of size and concentration of critical clusters of silicon carbide in the vapour phase in homogeneous nucleation process

The concentrations of clusters of various size in the atmosphere during silicon carbide crystal growth have been calculated on the basis of fundamental ideas of homogeneous nucleation theory, taking into account the specific parameters of silicon carbide. It has been shown that the cluster concentration are sufficiently high to conclude that this is the dominant influence during the initial stages of crystal growth. In this way the assumption of the polymer theory of polytypism, namely that the polytype properties of silicon carbide can be determined from the composition of the gas phase, containing sufficiently large clusters with various polytype structures, has been confirmed.     

密度泛函理论研究ScnO(n=1—9)团簇的结构、稳定性与电子性质

采用基于密度泛函理论的BP86/CEP-121G （O原子采用6-311G^**基组）方法,对Sc_nO （n=1—9）团簇的几何结构、能量与稳定性、电子结构性质及其随团簇尺寸的变化趋势进行了研究.随着团簇原子个数的增加,O原子从位于Sc_n团簇结构的边缘转变为占据团簇的内部位置.O原子的掺入增加了Sc_n团簇的稳定性,使其能隙升高,并改变了其稳定性及电子结构性质随团簇尺寸变化的规律;含有偶数个Sc原子的氧化物团簇比其周围邻近的含有奇数个Sc原子的氧化物团簇具有相对较高的稳定性.Sc_nO团簇电离势的理论计算值与实验值符合得较好,而其电子亲和势呈现振荡交替上升的变化趋势;用最大化学硬度规律等方法表征了Sc_nO氧化物团簇的稳定性和电子结构性质. 关键词： nO团簇')" href="#">Sc_nO团簇 几何结构 电子性质 密度泛函理论