Phase Coexistence and Phase Changes in Clusters

1
Phase Coexistence and Phase Changes in Clusters

• By
• Joseph H. Noroski
• April 17, 2003

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Two Common Phases

• Solid Particles exist in well-defined sites,
  oscillating around their equilibrium positions in
  small amplitude motions. The potential well is
  deep and steep, yielding a low density of states.
• Liquid Particles exhibit diffusive motion with
  large mean square displacements, have soft
  vibrational modes, and can permute (exchange
  positions) among themselves. The potential wells
  are less deep and steep, yielding a high density
  of states.

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The Bulk World

• The freezing point (Tf) and the melting point
  (Tm) are identical for a bulk sample a typical
  first order transition.
• Phase equilibria can be described entirely by
  macroscopic thermodynamic variables and, as such,
  are STATIC equilibria.

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The Cluster World

• The freezing and melting points are not the same.
  There exists an equilibrium between phases over
  a range of T.
• Phase equilibria must be considered in dynamic
  terms that is, consider ENSEMBLES of clusters.
• The individual clusters freely pass from one
  phase to another.

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Other Types of Phases in Clusters

• Surface-melted solid core, liquid surface, and a
  few floaters above the surface The surface
  becomes liquid-like when about 1 atom in 30
  floats above.
• Soft solid clusters can pass between a limited
  number of solid-like potential minima with
  relatively large amplitude motions no permuting
  of identical atoms among non-equivalent sites
• Slush short duration of various phase-like forms

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Examining Clusters

• The majority of cluster research is done via
  computer simulation. The experimental
  difficulties, of course, result from synthesizing
  something with 10 1000 atoms in a reproducible
  way.
• Probe atoms inserted into the clusters possess
  different spectra in one phase or another. These
  spectra are used to identify different phases.

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Solid-Liquid Equilibrium in Clusters

• There is only one minimum in the potential well
  below Tf and only one minimum above Tm.
• For clusters there exists a RANGE of T (at a
  single pressure) for which the equilibrium (s) ?
  (l) exists. So, Keq liq/sol
  exp(-?F/kBT). ?F is the free energy difference
  in the minima for the solid and liquid.

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Conditions for Coexistence

• S-bends in the probability distributions of, for
  example, the canonical ensemble, P(E), or the
  grand canonical ensemble, P(N).
• For example, an S-bend in the microcanonical
  caloric curve results when ln P(E) for the
  corresponding canonical distribution has two
  inflections.

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A Triple Point is Now a Triple Range????

• The Gibbs Phase rule says that three phases of
  one component can exist at only one T, the
  so-called triple point, T3.
• Simulations show that surface-melted clusters can
  coexist with solid and liquid clusters over a
  range of T.
• For clusters each phase is as much a component as
  it is a phase.
• The Gibbs Phase rule has no meaning for ensembles
  of small systems.

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More on Solid/Liquid/Surface-Melted Equilibrium

• The grand canonical partition function is of the
  form Zc(mc)Zs(ms)Zf(mf)Zint(mc, ms, mf). c
  core s surface f floater int
  interaction. mxs are defects. For example, ms
  are floater-vacancy pairs.
• Conditions for coexistence (?ln Zc/?mc)T 0 and
  (?ln ZsZf /?mf)T 0.
• Number of analytical solutions indicates the
  number of phases present.
• New types of phase diagrams are needed.

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A Model for Phase Changes

• Vekhter and Berry used simulations to examine
  Ar55 and (KCl)32 clusters.
• Over a T range of 730-760 K two phases observed
  for KCl crystal-like and liquid-like. Over a T
  range near 40 K three phases observed for Ar55
  solid, liquid, and surface-melted.
• The third phase of Ar55 is due to a third peak in
  the density of states not present for (KCl)32.

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A Model for Phase Changes (cont.)

• We can explain (KCl)32 by a two-level model with
  a nondegenerate ground state (where the solid
  is) and an N1-fold degenerate excited state
  (where the amorphous liquid states are). The
  difference in energy is d.
• The population of the excited level is n1(T)
  N1exp(-d/kT)/1 N1exp(-d/kT). It is key to
  note that, as N1 grows, the solid to liquid
  transition temperature range gets smaller.

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A Model for Phase Changes (cont.)

• Ar55 requires an intermediate energy level, dsm.
• s solid m surface-melted l liquid
• Two consecutive transitions, (s) to (m), followed
  by (m) to (l), occur if Tsl Tsm, where kBTsm
  dsm/ln(Nsm) and kBTsl (d1 - dsm) /ln(N1/Nsm).
• If Tsm Tsl, no surface-melted phase seen.
• If Tsm Tsl, 3 phase coexistence is seen.

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Other efforts with clusters

• Nigra, et. al. have used parallel tempering Monte
  Carlo and multihistogram methods to study phase
  changes of (H2O)8.
• There is a sharp Tm at 178.5 K.
• However, a solid to solid phase change is found
  at 12 K. Here, the (H2O)8 clusters change
  symmetry from strictly D2d to a mixture of D2d
  and S4.
• Others cluster research involves efforts to store
  data in condensed form using small clusters.

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References

• R. S. Berry, in Large Clusters of Atoms and
  Molecules, edited by T.P. Martin (Kluwer,
  Dordrecht, 1996), p. 281.
• R. S. Berry, in Scientific American, (August,
  1990), p. 68.
• P. Nigra, M. A. Carignano, and S. Kais, J. Chem.
  Phys. 115, 2621 (2001).
• Benjamin Vekhter and R. S. Berry, J. Chem. Phys.,
  106, 6456 (1997).