Thermal Surface Fluctuations of Clusters with Long-Range Interaction D.I. Zhukhovitskii Joint Institute for High Temperatures, RAS

Liquid―vapor interface structure: smooth or stratified ? smooth or stratified ? (van der Waals) (Gibbs) Gas Liquid Gas Liquid Intermediate phase

Aim of research: 1. Working out a proper method for MD simulation of such clusters in vapor environment. 1. Working out a proper method for MD simulation of such clusters in vapor environment. 2. Calculation of slice spectra. 2. Calculation of slice spectra. 3. Estimation of fission threshold. 3. Estimation of fission threshold. 4. Development of a theory of surface fluctuations for clusters with long–range interaction. 4. Development of a theory of surface fluctuations for clusters with long–range interaction.

Cluster particles are assumed to interact via the pair additive potential where and the long–range component System under consideration

Systems with multiple length and time scales require special integrators to prevent enormous energy drift. In the force rotation approach, an artificial torque of the long – range force components F i arising from cluster rotation is removed by rotation of these forces. We impose the condition      and   are the Euler angles. They are solutions of equation set Molecular dynamics simulation

Simulation cell: a cluster in equilibrium vapor environment

Definition of a cluster: a particle belongs to the cluster if it has at least one neighbor particle at the distance less than r b, which belongs to this cluster. The problem is, how to define r b.

We define three particle types: internal and surface particles and virtual chains. Particle 1 with the radius vector r 1 that belongs to the cluster will be called internal if there exists at least one particle 2 with radius vector r 2 belonging to the same cluster that forms more than four bonds such that the conditions are satisfied.

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Average configurations yield smooth density distribution inside the transitional region:

We isolate the surface particles (pivot particles) situated between two parallel planes. The particle polar coordinates are the values of a continuous function The slice spectrum are defined as the averages both over configurations and over the Euler cluster rotation angles: The total spectrum is a sum of the capillary fluctuations (CF) and bulk fluctuations (BF) spectra.

13 Bulk fluctuations arise from randomicity of particles location. Hence, they coincide with fluctuations of the surface particles of a cluster truncated by a sphere. The total spectral density is

14 Different components of the surface fluctuations spectral amplitudes for a cluster comprising particles at  = 0 and the temperature of 0.75 interparticle potential well depth. (1), bulk fluctuations,  k = R k ; (2), capillary fluctuations,  k = Q k ; (3), the total spectral amplitude,  k = S k ; (4), the total spectral amplitude without isolation of the virtual chains.

CF spectral amplitudes for clusters comprising particles at  = 445, T = : theory, simulation. BF amplitudes are shown for comparison

CF spectral amplitudes for clusters comprising particles at  = 10, T = 0.75 : theory, simulation. BF amplitudes are shown for comparison

CF spectral amplitudes for clusters comprising particles: theory, simulation. BF amplitudes are shown for comparison

CF spectral amplitudes for clusters comprising particles at  = –4.96, T = 0.75 : theory, simulation. BF amplitudes are shown for comparison

Second spectral amplitude for clusters comprising particles as a function of 

Deformation parameters of clusters comprising particles,  = (c/a) 2/3 – 1, at T = 0.75

21 Precursor stage of a supercritical cluster

Fission of a supercritical cluster

Ratios of the second slice spectral amplitudes calculated in three reciprocally perpendicular planes, the plane of a maximum amplitude and the planes of intermediate and minimum amplitude, as a function of time for a supercritical cluster

Autocorrelation function and correlation decay time for the second slice spectral amplitude for different 

25 Gibbs and Smoluchowski treated the liquid–vapor interface as a more or less abrupt change of the density and predicted that this interface is perturbed by thermal fluctuations. Mandelstam (1913) and Buff, Lovett, and Stillinger (1965) obtained The interface width diverges due to short-wavelength fluctuations. A simple cutoff at the interparticle distance leads to the critical point paradox. A way to overcome it is introduction of the bend rigidity (Helfrich, 1973). This yields the wave vector dependent bare surface tension where is the bare surface tension. where is the bend rigidity. Unfortunately, results obtained by different researchers are inconsistent. Thus, Mecke (1999) obtained a decreasing dependence ; some derived more complicated dependences.

Probability of cluster fluctuation is defined by corresponding change in the Gibbs free energy where Assuming small fluctuation amplitudes we have derived Theory of cluster capillary fluctuations where  0 is the bare surface tension. Based on the equipartition theorem we arrive at the amplitudes of fluctuation modes

27 Limitation of the maximum surface curvature by formation of a virtual chain

Formation of virtual chains limits the local curvature of the fluctuation surface: This allows one to writeand to find the spectrum cutoff number andotherwise. If we introduced a common cutoff then we would arrive at failure of the capillary wave theory (critical point paradox): at sufficiently high temperature ( T = 0.95 ), when there is no non-negative solution for  0. This difficulty is removed in proposed theory.

By definition, the bare surface tension  0 refers to a flat (nonperturbed) interface. Due to the parachor considerations, it depends on the surface density, which is independent on the field strength (field pressure vanishes on the surface). Therefore,  0 is field independent. The quantity is also field independent by definition. Due to the relation the ordinary surface tension  proved to be field independent as well. Bulk fluctuations Bulk fluctuations are characterized by the radial distribution of surface particles and the distribution of their number

1. The case  = 0. The interface variance is reached at k max = (  0 2 /8) 1/4. Divergence of interface variance at R → ∞ is removed: and proportional interface width diverge with cluster size. In the case of gravitational attraction, the interface variance vanishes with the increase in R : 2. The case  > 0 (pseudogravitation). The maximum of spectral slice amplitude

Theoretical CF slice spectrum for different 

Surface variance  2 as a function of cluster size at 

3. The case  < 0 (Coulomb-like repuilsion). The surface variance is The maximum value  = –10 corresponds to singularity of The cluster becomes unstable with respect to fission. The classical fission threshold [Bohr and Wheeler (1939), Frenkel (1939)] supposes greater charge:

Conclusions 1.A leading order theory of surface fluctuations is proposed for clusters with a long – range particles interaction. 2.CF are damped by the attractive long – range interaction; the surface tension is independent of the field strength. 3.For the repulsive interaction, the fission threshold is defined by the bare rather than ordinary surface tension. 4.A nonlinear theory of large fluctuations is required.

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