Snapshots of the atomic configurations of the krypton-on-graphite system at (a) low-temperature , , ordered and (b) high-temperature , , disordered states.
Planar order parameter, , as a function of temperature, , for the krypton-on-graphite system. Upon heating, the system remains commensurate almost up to the transition onset, in agreement with experimental observations. The order-to-disorder transition occurs at .
Specific heat, , as a function of temperature, , for the krypton-on-graphite system obtained from fluctuations of the total energy. The temperature is ramped at 2.4 K increments. The sharp peak centered around reveals the transition onset.
(a) Transformation of atomic coordinates to occupy reduced cell containing only four adsorption sites that fills 2D space if repeated periodically. (c) The reduced-space cell under the action of periodic boundary conditions.
PDFs of the effective-particle coordinates for the Kr-on-graphite system obtained at (a) 125 K, (b) 126 K, (c) 127 K, and (d) 130 K. In (a) and (b), the shown PDFs are centered at the -phase adsorption sites. Increasing the temperature from 126 to 127 K reveals a transition onset: the two PDFs shown in (c) and (d) indicate a finite probability for the Kr atoms to occupy all the graphitic adsorption sites, i.e., additional sites to the ones.
MSD evolution curves for various temperatures around the transition temperature, . After an initial transient, the evolution curves become straight lines with slopes that reveal two different states corresponding to temperatures below and above the transition onset. The overlap between the evolution curves at and 128 K is attributed to the proximity of the former temperature to the transition onset. The red line is used to represent the MSD at , which corresponds to the temperature that is closest to .
Computed results and linear fits of as a function of , i.e., an Arrhenius-type plot for the temperature dependence of the diffusion coefficient, . The lower branch corresponds to the ordered (commensurate) state over the temperature range from 120 to 126 K. The upper branch corresponds to a second state, characterized by higher diffusion coefficients, for temperatures of 127 K and higher. This behavior confirms that the transition onset occurs at a temperature between 126 and 127 K.
Pair correlation function profiles, , corresponding to temperatures of 40, 120, 126, 127, and 140 K. The profiles at , 126, 127, and 140 K are shifted upwards by constant values of 5, 7, 9, and 11, respectively, for visual convenience. Solid arrows pointing to the peaks that indicate coordination shells characteristic of the phase are shown in the corresponding to the lowest temperature. The braces are used to indicate temperatures below and above the transition onset, , which correspond to states that are distinguished (most importantly) by the presence or absence of the third coordination shell. Traces from the third coordination shell can still be detected in the profile at 126 K; gray arrows are used to indicate the proximity to the transition onset.
Schematic outline of the CMD procedure implemented with emphasis on the lifting scheme for studying order-to-disorder transitions in the Kr-on-graphite system. The coarse-variable space, , is mapped by implementing a MC-based sampling scheme with a quadratic penalty function (harmonic potential) that uses the current and target values of the coarse variable. Following this initialization, the constraint is released and the ensemble averaged coarse-variable evolution, , is monitored to obtain the drift velocity, , and the diffusion coefficient, , of the underlying Fokker–Planck equation.
Ensemble averaged coarse-variable evolution, , after lifting at a temperature (a) well below and (b) well above the transition temperature, . The coarse trajectories in (a) show that the system evolves to its ordered state , while those in (b) show that the coarse evolution drifts toward the disordered state . The observed crossing between some of the curves in the coarse-variable evolution may imply that a second coarse variable becomes important. This issue could be due to the small fraction of atoms promoted away from the original monolayer and is currently being investigated.
Effective free-energy landscapes for temperatures over the range . The broken dashed lines point to the bottom of the thermodynamic potential wells corresponding to the ordered and disordered states, respectively. The solid vertical lines are used to designate the regions corresponding to the ordered and disordered states, respectively. The square box shown encloses the region used to calculate the slopes that have been employed in the determination of the transition temperature, .
Temperature dependence of the slope of the effective free energy, , with respect to the coarse variable, , used for the bracketing and determination of the transition onset, . The inset corresponds to the box shown in Fig.11 and is used to highlight the slope computation.
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