1. Bare Surface Tension and Surface Fluctuations of Clusters with Long–Range Interaction D.I. Zhukhovitskii Joint Institute for High Temperatures, RAS

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

3. We define three particle types: internal and surface particles and virtual chains.

4. Average configurations yield smooth density distribution inside the transitional region:

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

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

7. 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

8. 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: at sufficiently high temperature ( T = 0.95 ), when there is no non-negative solution for  0. This difficulty is removed in proposed theory.

9. 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.

10. 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

11. Theoretical CF slice spectrum for different 

12. 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:

13. 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

14. Simulation cell: a cluster in equilibrium vapor environment

15. We isolate the surface 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.

16. CF spectral amplitudes for clusters comprising 20000 particles at  = 445, T = 0.955 : theory, simulation. BF amplitudes are shown for comparison

17. CF spectral amplitudes for clusters comprising 20000 particles at  = 10, T = 0.75 : theory, simulation. BF amplitudes are shown for comparison

18. CF spectral amplitudes for clusters comprising 20000 particles: theory, simulation. BF amplitudes are shown for comparison

19. CF spectral amplitudes for clusters comprising 20000 particles at  = –4.96, T = 0.75 : theory, simulation. BF amplitudes are shown for comparison

20. Second spectral amplitude for clusters comprising 20000 particles as a function of 

21. Deformation parameters of clusters comprising 20000 particles,  = (c/a) 2/3 – 1, at T = 0.75

22. Fission of a supercritical cluster

23. 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

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

25. 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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