Phase diagrams, finite-size scaling

In this section we study a system with purely repulsive interactions which demonstrates the importance of entropy effects on the stability of phases when the effect of the corrugation potential due to the structured surface is completely neglected. The phase diagrams are determined by finite size scaling methods, in particular the methods of Sec. IV A. 

In this section we review several studies of phase transitions in adsorbed layers. Phase transitions in adsorbed (2D) fluids and in adsorbed layers of molecules are studied with a combination of path integral Monte Carlo, Gibbs ensemble Monte Carlo (GEMC), and finite size scaling techniques. Phase diagrams of fluids with internal quantum states are analyzed. Adsorbed layers of H2 molecules at a full monolayer coverage in the /3 X /3 structure have a higher transition temperature to the disordered phase compared to the system with the heavier D2 molecules this effect is... 

FIG. 14 Phase diagram of the quantum APR model in the Q -T plane. The solid curve shows the line of continuous phase transitions from an ordered phase at low temperatures and small rotational constants to a disordered phase according to the mean-field approximation. The symbols show the transitions found by the finite-size scaling analysis of the path integral Monte Carlo data. The dashed line connecting these data is for visual help only. (Reprinted with permission from Ref. 328, Fig. 2. 1997, American Physical Society.)... 

With the availabihty of computers, the transfer matrix method [14] emerged as an alternative and powerful technique for the study of cooperative phenomena of adsorbates resulting from interactions [15-17]. Quantities are calculated exactly on a semi-infinite lattice. Coupled with finite-size scaling towards the infinite lattice, the technique has proved popular for the determination of phase diagrams and critical-point properties of adsorbates [18-23] and magnetic spin systems [24—26], and further references therein. Application to other aspects of adsorbates, e.g., the calculation of desorption rates and heats of adsorption, has been more recent [27-30]. Sufficient accuracy can usually be obtained for the latter without scaling and essentially exact results are possible. In the following, we summarize the elementary but important aspects of the method to emphasize the ease of application. Further details can be found in the above references. 

This system was modelled in terms of the lattice gas with interactions shown in Fig. Ib. The phase diagram was first calculated by the transfer matrix finite size scaling technique for various choices of the interaction parameters [Pg.122]

Fig. 18. Phase diagram of the centered rectangular lattice gas model with ==0, 3/4 2 = V3> vJ

Fig. 30. Phase diagram of a model for Si/W(110) in the temperature versus 9 plane. Experimentally determined interactions J Jj,are used. Full dots are from Monte Carlo calculations, while triangles are based on transfer matrix finite size scaling using strip widths of 8 and 12. The point labelled L indicates approximate location of Lifshitz point. The dotted line indicates the transition region between the (5 x l)and(6 x 1) phases. (From...

In order to obtain the stability diagram for the three-body Coulomb systems in the X — K)-plane, one has to calculate the transition line, Xc(k), which separates the stable phase from the unstable one. To carry out the finite-size scaling calculations, the following complete basis set was used [66] ... 

Fig. 15. Inverse maximum of the collective structure factor of composition fluctuations, N/S k 0), as a function of the incompatibility, x - Symbols correspond to Monte Carlo simulations of the bond fluctuation model, the dashed curve presents the results of a finite-size scaling analysis of simulation data in the vicinity of the critical point, and the straight, solid line indicates the prediction of the Flory-Huggins theory. The critical incompatibility, XcN = 2 predicted by the Flory-Huggins theory and that obtained from Monte Carlo simulations of the bond fluctuation model M 240, N = 64, p = 1/16 and = 25.12) are indicated by arrows. The left inset compares the phase diagram obtained from simulations with the prediction of the Flory-Huggins theory (c.f. (47)). The right inset depicts the compositions at coexistence such that the mean field theory predicts them to fall onto a straight line. Prom Muller [78]...

Of course, again these data show a dramatic size effect (Fig. 24a) and thus a finite size scaling analysis of the type shown in Fig. 21a is necessary to extract the coexistence curve in the thermodynamic limit In this way an accurate estimation of the phase diagram of polymer mixtures with NA = NB but asymmetric interactions is possible (Fig. 24a, b). 

Computer simulation is invariably conducted on a model system whose size is small on the thermodynamic scale one typically has in mind when one refers to phase diagrams. Any simulation-based study of phase behavior thus necessarily requires careful consideration of finite-size effects. The nature of these effects is significantly different according to whether one is concerned with behavior close to or remote from a critical point. The distinction reflects the relative sizes of the linear dimension L of the system—the edge of the simulation cube, and the correlation length —the distance over which the local configurational variables are correlated. By noncritical we mean a system for which L E, by critical we mean one for which L [Pg.46]

The calculated phase diagram for pentadecanoic acid using the Karaborni and Toxvaerd model has been reported in a previous publication [29]. The results can be summarized as follows (i) the model yields coexistence between a liquid and a vapor phase (ii) the liquid phase of the model monolayer is substantially denser than the LE phase of the real monolayer, and the critical point seems to be shifted to higher temperatures and (iii) the coexistence curve obtained from simulations can best be fitted with a scaling exponent of 0.32, supporting a three-dimensional character of the finite-size model system. Thus the Karaborni and Toxvaerd model yields merely qualitative agreement for the G-LE coexistence with experiments on the same systems. 

We may conclude that, in particular, the high-temperature pseudophases DE, DC/DG, AG, AE, nicely correspond to each other in both models. Noticeable qualitative deviations occur, as expected, in those regions of the pseudophase diagram where compact conformations are dominant and (unphysical) lattice effects are influential. Thus, the choice of the appropriate model depends on the question one wants to answer. Unlike temperatures are not too small and the polymer chain not too short, lattice models are perfectly suitable for the investigation of structural phases. This is particularly true for scaling analyses toward the thermodynamic limit. However, if the focus is more on finite-size effects and the behavior at low temperatures, off-lattice models should generally be preferred. 

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