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Europhysics Letters (EPL)

Finite-size scaling behavior of the tracer surface diffusion coefficient near a second-order phase transition

F. Nieto, Alexander Tarasenko, C. Uebing

Published Sep 1, 1998• 14 citations
Publisher

Abstract

We investigate the finite-size scaling behavior of the tracer surface diffusion coefficient in the vicinity of a second-order phase transition. For this purpose, we use a lattice gas model of repulsively interacting particles on a square lattice. For all lattice sizes L studied, the measured tracer surface diffusion coefficient, D t , is a smooth function, having an inflexion point at critical temperature. Its derivative, ∂ D t /∂(1/ k B T ), exhibits cusp-like maximum which is a) sharply pronounced and b) converged to T c (∞) for large lattice sizes. We have analyzed the finite-size behavior of D t and obtained its critical exponent σ t = 0.665 ± 0.003.

DOI
10.1209/epl/i1998-00397-8

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We investigate the finite-size scaling behavior of the tracer surface diffusion coefficient in the vicinity of a second-order phase transition. For this purpose, we use a lattice gas model of repulsively interacting particles on a square lattice. For all lattice sizes L studied, the measured tracer surface diffusion coefficient, D t , is a smooth function, having an inflexion point at critical temperature. Its derivative, ∂ D t /∂(1/ k B T ), exhibits cusp-like maximum which is a) sharply pronounced and b) converged to T c (∞) for large lattice sizes. We have analyzed the finite-size behavior of D t and obtained its critical exponent σ t = 0.665 ± 0.003.
We present a finite-size-scaling study of a two-dimensional square-lattice-gas model with nearest-neighbor repulsion and next-nearest-neighbor attraction. This system may be used as a simplified model of adsorbates on transition-metal (100) surfaces. An accurate determination of the tricritical temperature is obtained by the utilization of the degeneracy of the three largest eigenvalues of the transfer matrix at a tricritical point. We obtain the phase diagram and critical and tricritical exponents, all quite consistent with previously known results. We also find that a recent conjecture relating the critical exponent $\ensuremath{\eta}$ to the correlation-length amplitude seems to work well even at the tricritical point.
We investigate a hard-square lattice gas on the square lattice by means of transfer-matrix and Monte Carlo methods. The size of the hard squares is equal to two lattice constants, so the simultaneous occupation of nearest-neighbor sites as well as of next-to-nearest-neighbor sites is excluded. Near saturation of the particle density, this system is known to undergo a phase transition to one out of four partially ordered phases. We find that this transition displays strong finite-size corrections to scaling and that the correlation functions deviate from isotropy to rather large distances. In contrast with an earlier study, we find that the critical temperature exponent of the transition is not Ising-like.
The second-order phase transition in the two-dimensional (2D) classical Coulomb gas of half-integer charges on a square lattice is investigated by using Monte Carlo simulations. Based on the finite-size scaling analysis, we estimate the second-order phase transition temperature Tc and the static critical exponents β and ν with a new numerical analysis method. More precise critical temperature Tc = 0.1311(2) and critical exponents β/ν = 0.1152(12) and ν = 0.857(15) are obtained. The estimated value of ν indicates that the charge lattice melting transition is different from the pure 2D Ising transition.
The spectrum of the transfer matrix of a semi-infinite two-dimensional Isin model with marginally inhomogeneous couplings has been determined exactly in the finite-size-scaling limit at the bulk critical point. Depending on the local temperature at the boundary the surface phase transition of this system is either of first order or of second order with nonuniversal critical exponents. For a second-order transition, the excitation energies of the transfer matrix are inversely proportional to the linear size (N) of the system, but the levels are not equidistantly spaced. On the other hand, for a first-order transition the lowest gap vanishes algegraically, but faster than 1/N.
A two-dimensional lattice-gas model with honeycomb symmetry is investigated by using the real-space renormalization group (RSRG) approach with a number of RSRG transformations based on clusters of different symmetries and sizes (up to 42 sites). The accuracy of calculations is found to be a nonmonotonic function of the size of a cluster, and to depend strongly on its symmetry. The highest obtained accuracy with respect to the determination of the critical value of the pair interaction parameter is 0.38%. The phase diagram of the Ising antiferromagnetic spin model and of the corresponding lattice-gas model is constructed with this accuracy. The ordered phase in the lattice system is shown to appear in the very narrow density interval, 0.447<n<0.553.0.447<n<0.553.0.447<n<0.553. In addition, the coverage dependence of the chemical diffusion coefficient and mean-square density fluctuations are studied at temperatures below the critical one. Both quantities are demonstrated to exhibit singularities at the critical coverages. The type of singularities (in particular, the critical slowdown of diffusion) is in agreement with the predictions of the scaling theory and also with recent results obtained for a model treating the effect of surface reconstruction on diffusion.