Nanobubbles in a Lennard-Jones Fluid

1. Nanobubbles in a Lennard-Jones Fluid D.I. Zhukhovitskii Institute for High Energy Densities JIHT

2. Introduction 1. Kinetics of the first-order transitions is based on statistics of nanoobjects (embryos of a new phase). 2. For the vapor–liquid <—> liquid–vapor transitions, a cluster (nanobubble) is far from being a drop (bubble). 3. The particles that form an embryo are divided into the surface and internal ones (effective phases); then, an interpolation between small and large sizes is possible. 4. The layer of surface particles is the same for a cluster and nanobubble.

3. 5. Clusters (nanobubbles) are no casual formations! Constituent particles are strongly correlated.

4. Arbitrary size cluster in the ‘effective phases’ model The size distribution of clusters:

5. Effective surface tension vs. cluster size

6. Nucleation rate vs. barrier height of the critical size cluster for mercury (red) and water (green) vapors

7. Cluster surface energy vs. cluster radius (~ g1/3) for LJ system. MD simulation (red) and theory water (green)

8. Arbitrary size nanobubble in the ‘effective phases’ model Here, S is the same as for a cluster but the relation between g and gs is different. The size distribution of nanobubbles:

9. Here, The critical size satisfies the condition The equation of spinodal is For small bubble sizes (gb = 0),

10. Energy of bubble formation at spinodal (1) and in its vicinity (2) vs. bubble size

11. A model of asymmetric fluctuations in a liquid High density (bond-rich particles) Low density (bond-deficient particles) Density fluctuations are asymmetric. Bond-deficient particles must have the maximum possible number of bonds but the system must find itself below the percolation threshold! Such particles clusterize to form bond-deficient clusters (BDC’s). We assume that BDC’s coincide exactly with low-density regions (no stochastic fluctuations). The number density of BDC’s comprising kbond-deficient particles is fk.. The equimolecular size of these BDC’s is g. The number of bubbles with sizes from g to g + dg in the volume V is Ngdg. Obviously, ng = <Ng>/V = fk(dg/dk)–1.

12. The number density of voidusters with the equimolar sizes from g to g + dg is The pre-exponential factor Cis determined from the relation between microscopic and macroscopic fluctuations of the number of particles N in volume V :

13. On the other hand, thermodynamic relation reads therefore, We use the van der Waals equation with parameters defined by the saturation line rather than critical point to estimate the compressibility: Here, is a solution of the equation with a single parameter w=nvs/nls Eventually we find and for the Lennard-Jones fluid at T = 0.75, C = 0.0114.

14. Eventually, we can write the size distribution of nanovoids in the form This makes it possible to calculate the void fraction in a liquid

15. MD simulation Particles are assumed to interact via the pair additive potential where and is the long–range component.

16. Simulation cell: a droplet in an equilibrium vapor environment

17. Definition of a cluster: a particle belongs to the cluster if it has at least one neighbor particle at the distance less than rb, which belongs to the same cluster. Bond-deficient cluster (BDC) is a group of bond-deficient particles. Definition of BDC: a particle belongs to the BDC if it has less than bmax bonds and at least one neighbor bond-deficient particle at the distance less than rb, which belongs to the same BDC.

18. Typical nanobubble with a transitional size

19. Typical BDC’s in a liquid and clusters in dense gas

20. BDC distribution over the number of bond-deficient particles for p = 0

21. BDC equimolar size vs. the number of bond-deficient particles for p = 0

22. Nanobubble distribution over the equimolar size of nanobubbles for p = 0

24. Compressibility factor of BDC’s as a function of their volume fraction at p = 0

25. Bulk and surface particle distributions over the number of bonds

26. Bond deficiency in nanobubbles (theory) At , the bond deficiency is With and ,we have Borrowing we arrive at (1) If we have (2) Equimolar size and bond deficiency of nanobubbles (simulation) where Rcm is the largest distance between voiduster center of mass and bond-deficient particle.

27. Bond deficiency as a function of the nanobubble equimolar size

28. Nanobubble distribution over the equimolar size for p = 0.8

29. Bubble in a liquid in the vicinity of a spinodal in the center of the drop. MD simulation yields the spall threshold p = –0.65, while a theoretical estimation is p = –0.62.

30. Conclusions • A nanobubble in a liquid is a region of bond-deficient particles . • In the MD simulation, the maximum number of bonds for bond-deficient particles is dictated by the non-percolation condition. • Size distribution of nanobubbles over the equimolar size correlates with the ‘effective phases’ theory. Bond deficiency is the most important quantity for thermodynamics of nanoobjects. • Pre-exponential factor in the size distribution of nanobubbles can be determined from the relation between microscopic and macroscopic fluctuations of the number of particles in a liquid.

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