Ising model phase diagram

Fig. 25 Ising model phase diagram. The reconstructed (1 x 2) and unreconstructed (1x1) regions are separated by Ising disordered phase (D).

Except for a few special cases (most notably, /inear rules (see below) and the P2 = I line for the isotropic peripheral PCA discussed above, which happens to be an endpoint of a disorder line of an exactly solved 2-dim Ising model on a triangular lattice and can be solved exactly [kinzel85b]), phase-diagrams for PCA... 

Phase diagram based on a four-state Ising-Potts model 626... 

Fig. 7. Mean-field phase diagram of the four-state Ising-Potts model (A/kB = 90 K, J lkB = 125 K). The high-spin (HS) phase, the low-spin (LS) phase, and the ferroelectric-ordered (FO) phase are shown. The arrow line corresponds to Jo/kB = — 36 K, appropriate to the [Mn(taa)] system.

Fig. 19. Monte Carlo result for the phase diagrams of an off-lattice bead rod model of a symmetric binary polymer mixture with N=20, in the plane of variables reduced temperature T =kBT/ AA and volume fraction of component A, denoted here as xx. Data are for bulk systems (full dots), and for confined films of thicknesses D=10.5a (squares) and 5a (triangles), respectively. Dashed curves represent fits to xx—xlc oc T-Tc fil, where the Ising model exponent [229,230] was chosen as Pj=l/3. From Kumar et al. [39]...

The picture presented above is not complete as it neglects non-mean field behavior of polymer blends in the temperature range close to Tc [149]. The Ising model predicts phase diagrams of thin films, which are more depressed and more flattened than those yielded by mean field approach (as marked in Fig. 31d). Both effects were shown by Monte Carlo simulations performed by Rouault et al. [150]. In principle, critical regions of phase diagrams cannot be described merely by a cross-over from a three- to two-dimensional (for very thin films) situation. In addition, a cross-over from mean field to Ising behavior should also be considered [6,150]. 

The Netherlands, 1991. Ducastelle s book is full of insights into the tools needed to compute phase diagrams, including a very detailed treatment of the Ising model and its relation to phase diagrams. I wish I had more time to spend reading this one. 

Ceder G., A Derivation of the Ising Model for the Computation of Phase Diagrams, Comp. Mat. Sci. 

Kornyshevand Vdfan [272-274] suggested the simplest way of describing the transitions between 1x1 and 1x2 states, based on an anisotropic 2D Ising model, and derived a closed expression for the temperature-surface charge phase diagram. 

The calculated phase diagram for the Karaborni and Toxvaerd force field deviates significantly from the experimental observations, but the modified model gives a more quantitative description. In both cases, the shape of the coexistence curve can be fitted using the critical exponent of the three-dimensional Ising model. The two force fields yield qualitatively different structural properties. 

Fig. 110. Mean-field phase diagram for the 3D Ising model with conq>eting interactions (Bak 1982). The dark areas indicate high-order C phases with I phases in-between (incomplete devil s staircase). P is the Lifshitz...

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