Surface and Bulk Fluctuations of the Lennard-Jones Clusrers D. I. Zhukhovitskii

Joint Institute for High Temperatures Russian Academy of Sciences Moscow Russia

Liquid―vapor boundary: gradual transition or layered structure? Gas Liquid Gas Liquid Intermediate phase

Investigation methods: 1. analytical; 1. analytical; 2. experimental (x-ray and neutron reflectivity 2. experimental (x-ray and neutron reflectivity measurements); measurements); 3. numerical simulation (molecular dynamics 3. numerical simulation (molecular dynamics and Monte Carlo). and Monte Carlo).

Shortcomings of the capillary wave model: 1.What is a microscopic wave surface? How should the wave overhangs be treated ? 2.The problem of capillary and bulk fluctuations separation. 3.What kind of the surface tension should be associated with the microscopic capillary fluctuations?

Cluster particles are divided in three groups: we call particle 1 with the radius vector internal if there exists at least one particle 2 with radius vector belonging to the same cluster that forms more than four bonds such that the conditions are satisfied. Here, is the number density of bulk liquid. The cluster particles that are not internal and form more than four bonds are called surface particles. The cluster particles that are not internal and form less than five bonds are assigned to the virtual chains. Proposal #1: separate the pivot particles for the wave (fluctuation) surface and virtual chains of particles loosely bound to the cluster. Thus, the maximum surface curvature is limited.

The slice snapshots of clusters formed by 1592 (1) and 2320 (2) particles. Solid circles are internal particles, hatched circles are surface particles, and open circles are virtual chains. The surface particles form a monolayer highly curved by thermal fluctuations. For the internal particles, the number of bonds is no less than 10. The particles with less than five bonds form virtual chains (overhangs).

Isolate the surface particles of a slice passing through the cluster center of mass. The polar coordinates of particles are the values of a continuous function expanded in the Fourier series: The slice spectral amplitudes were calculated by averaging over both the cluster configurations and Euler angles corresponding to the rotation of each configuration:

Slice spectral amplitudes for the clusters formed by 150 (1), 1000 (2), 3000 (3), and (4) particles at the temperature 0.75 of the interparticle potential well depth (MD simulation). Dots represent theoretical calculation for a particles cluster.

Proposal #2: The bulk fluctuations are generated primarily by the discontinuity in the spatial distribution of cluster internal particles. Therefore, such fluctuations can be simulated by the surface particles of the cluster truncated by a sphere smaller than the cluster. The total spectral density is, where

Different components of the fluctuation spectral amplitude for the surface of a cluster comprising particles at the temperature 0.75 of the interparticle potential well depth. (1), bulk,  k = R k and (2), capillary fluctuations,  k = Q k ; (3), total spectral amplitude,  k = S k ; (4), total spectral amplitude including the virtual chains.

Capillary fluctuations theory In the early study by Buff, Lovett, and Stillinger (1965), the relation Attempts to allow for the wave number dependence of the surface tension for the fluctuations result in discrepant dependences, e.g., to the increasing one (Helfrich, 1973) to the decreasing dependence (Mecke, 1999), and to more complicated dependences. was obtained, where is the coupling constant. where is the interface bending rigidity,

где According to the fluctuation theory the change of Gibbs free energy of the cluster surface is where Using the equipartition theorem we derive The condition of finiteness for the cluster excess surface

где yields the maximum value of l and the relation between the coupling constant and surface tension : Here, is a universal constant.

где The capillary fluctuation in the form of spherical harmonic Y lm where is the amplitude corresponding to, contributes to the 2D spectrum obtained in MD simulation as This contribution is defined by the Fourier coefficients of slice boundary coordinates Then the total spectral amplitude of capillary fluctuations is

Spectral amplitudes of the capillary (1, 2) and bulk (3) fluctuations for the cluster comprising particles at the temperature 0.75 of the interparticle potential well depth. Theory (1) and MD simulation (2, 3).

где The effective surface tension is defined as where is independent of m. In the “classical” theory,. The total spectral amplitude of capillary fluctuations is We consider a two-parameter (  is the Heaviside step function) and a three-parameter dependences. The parameters were determined from the best fit condition

где

Effective surface tension in the two-parameter and three-parameter approximations for the cluster formed by particles at the temperature 0.75 of the interparticle potential well depth.

Capillary fluctuations for the cluster comprising particles at T = 0.69 (1) and for the cluster comprising particles at T = 0.75 (2). Dots represent MD simulation; lines, calculation using the three-parameter effective surface tension.

Width of the liquid―vapor interface is defined by the variance Interface width diverges logarithmically as the surface area increases!

Average of the liquid―vapor interface configurations yields smoothed dependences of system characteristics in the transition region.

Small clusters and virtual chains

Small clusters with the minimum number of bonds form the virtual chains. Respective partition function is calculated analytically: Hence, the cluster vapor equation of state is

Trends of research 1.Capillary fluctuations and virtual chains near the critical point. 2.Capillary fluctuations at the surface of a liquid metal. 3.Liquid metal surface near the critical point. 4.Capillary fluctuations in strong fields. 5.Interphase boundary with a high temperature gradient.

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