[Paper Review] First-order phase transition and the equation of state in a 2D granular fluid

Mark D. Shattuck|arXiv (Cornell University)|Oct 30, 2006

Gas Dynamics and Kinetic Theory1 references3 citations

TL;DR

This study presents experimental evidence of a first-order phase transition in a driven 2D granular fluid under isobaric conditions, where a discontinuous transition occurs between a gas and crystal phase with concurrent jumps in density and temperature. The transition is suppressed when particle count is incommensurate with system size, exhibits rate-dependent hysteresis, and obeys the Lindemann criterion, demonstrating that thermodynamic concepts like phase coexistence can emerge in nonequilibrium steady states despite energy dissipation.

ABSTRACT

We present experimental evidence for a first-order freezing/melting phase transition in a nonequilibrium system -- an oscillated two-dimensional isobaric granular fluid. The steady-state transition occurs between a gas and a crystal and is characterized by a discontinuous change in both density and temperature. It is suppressed if the number of particles is incommensurate with the cell size, shows rate-dependent hysteresis, and obeys the Lindemann criterion for melting. Further, the measured equation of state both above and below the phase transition compares well with theory.

Motivation & Objective

• To investigate whether first-order phase transitions can occur in a nonequilibrium steady state (NESS) granular system.
• To examine the equation of state of a 2D granular fluid under isobaric conditions and compare it with kinetic theory predictions.
• To determine whether thermodynamic concepts like free energy and phase coexistence can be meaningfully applied to driven, dissipative systems.
• To explore the role of system size, particle number, and driving rate in stabilizing or suppressing the phase transition.
• To validate the Lindemann criterion for melting in a granular crystal formed under NESS conditions.

Proposed method

• Experiments were conducted using 26–85 stainless steel ball bearings (D = 3.175 mm) in a 2D container (17.5D × 20D) with a freely floating weight to maintain constant pressure.
• Particles were driven from below via a sinusoidally oscillating plunger, with the driving strength quantified by the nondimensional parameter Γ = A(2πf)²/g.
• High-speed digital imaging (840 Hz) tracked particle positions and velocities with 6 μm accuracy, enabling measurement of granular temperature, density, and radial distribution functions g(r).
• Temperatures were calculated in the frame of the floating weight to avoid spurious contributions from bulk motion, using time and particle-averaged kinetic energies.
• The compressibility factor χ was measured experimentally and compared to theoretical predictions G_T from kinetic theory, with α = χ / G_T used to infer inelasticity.
• Radial distribution functions were used to extract G_exp and compare with G_T, enabling independent estimation of the inelasticity parameter e.

Experimental results

Research questions

• RQ1 Can a first-order phase transition occur in a driven, dissipative granular system under isobaric conditions, despite the absence of energy conservation?
• RQ2 Does the observed phase transition exhibit characteristics such as hysteresis, coexistence of phases, and discontinuous changes in both density and temperature?
• RQ3 To what extent does the equation of state of the granular fluid agree with kinetic theory predictions in both the gas and crystal phases?
• RQ4 How does the system's behavior depend on particle number and system size, particularly when the number of rows R is integer versus half-integer?
• RQ5 Is the Lindemann criterion for melting valid in a granular crystal formed under nonequilibrium steady-state conditions?

Key findings

• A first-order phase transition was experimentally observed in a 2D granular fluid under isobaric conditions, marked by a discontinuous jump in both density and temperature upon crossing the transition point.
• The transition is suppressed when the number of rows R is incommensurate with the system size, indicating a strong dependence on lattice commensurability.
• The system exhibits rate-dependent hysteresis, with distinct paths for increasing and decreasing driving strength, confirming the presence of metastable states.
• The compressibility factor χ measured experimentally agrees well with theoretical predictions G_T from kinetic theory up to high densities, validating the use of kinetic models in the dilute and dense gas regimes.
• The Lindemann criterion for melting was satisfied, as the crystal phase exhibited a critical amplitude of atomic vibrations consistent with melting.
• The inelasticity parameter e, inferred from particle tracking and radial distribution functions, ranged from 0.3 to 1.3, with lower values observed at higher collision velocities, and values >1.0 possible due to unmeasured particle spin.