Solution landscapes of the simplified Ericksen–Leslie model and its comparisonwith the reduced Landau–deGennes model

We investigate the solution landscapes of a simplified Ericksen–Leslie (sEL) vector model for nematic liquid crystals, confined in a two-dimensional square domain with tangent boundary conditions. An efficient numerical algorithm is developed to construct the solution landscapes by utilizing the symmetry properties of the model and the domain. Since the sEL model and the reduced Landau–de Gennes (rLdG) models can be viewed as Ginzburg–Landau functionals, we systematically compute the solution landscapes of the sEL model, for different domain sizes, and compare them with the solution landscapes of the corresponding rLdG model. There are many similarities, including the stable diagonal and rotated states, bifurcation behaviours and sub-solution landscapes with low-index saddle solutions. Significant disparities also exist between the two models. The sEL vector model exhibits the stable solution C ± with interior defects, high-index ‘fake defect’ solutions, novel tessellating solutions and certain types of distinctive dynamical pathways. The solution landscape approach provides a comprehensive and efficient way for model comparison and is applicable to a wide range of mathematical models in physics.

The Na+,K+/CN-,I- ternary reciprocal system

The DTA method was used to obtain the phase diagram of the Na+, K+/CN-, I- ternary reciprocal system. This system forms completely miscible solid solutions with two ternary minima with the parameters xNa+ = 0.50, xI- = 0.57, θ = 412°C, and xNa+ = 0.75, xI- = 0.20, θ = 428°C. The NaI-KCN diagonal is the stable diagonal.

Miscibility Gaps in Fused Salts. Note XII. Systems Formed with Thallium (I) Halides and Lithium or Sodium Halides: the Ternary Li, Tl/Cl, I

Demixing was investigated in the molten salt families formed with thallium (I) halides and lithium or sodium halides. Measurements were carried out on the twelve stable diagonals of the corresponding reciprocal ternary mixtures. Liquid-liquid equilibria were found in the three systems TlCl+LiF, TlBr+LiF and TlI+LiCl while in the two systems formed with Til and LiF or NaF measurements could not be carried out owing to the thermal instability of Til.The reciprocal ternary Li, Tl/Cl, I exhibits demixing in a temperature range which is convenient for investigation: therefore the whole composition square was studied. The upper critical solution point of the gap lies along the stable diagonal at 734°C and xLiCl = 0.58. The projection of the stratification lens occupies 38.8% of the composition square. Moreover an “a priori” prediction of the liquid-liquid equilibria was carried out by means of the conformal ionic solutiontheory.

Miscibility Gaps in Fused Salts. Note X. The Reciprocal Ternary System K, Li/Br, F

Solid-liquid and liquid-liquid equilibria in the reciprocal ternary system K, Li/Br, F were fully measured.The projection of the miscibility gap occupies 20.8% of the composition square; the upper critical point of the gap is along the stable diagonal at 953°C and xLIF = 0.70.Attempts to predict the liquid-liquid equilibria by means of the conformal ionic solution theory using temperature independent interaction parameters lead to an incorrect symmetry of the gap. The symmetry can be improved if the temperature dependence of the interaction parameters of the binary mixtures is taken into account.

Miscibility Gaps in Fused Salts

An analysis was made of the demixing phenomena in the reciprocal ternaries Li, Me/Br, F (Me = Zn, Pb, Cd). It was found that the system Li, Zn/Br, F shows only a "tendency" to demix; the system Li, Pb/Br, F exhibits a miscibility gap (MG) which could be completely detected in the whole composition square; finally the system Cd, Li/Br, F shows a MG which was only partially measured along the stable diagonal. The trend in demixing of these systems agrees with the values of the standard enthalpy variations of the metathetical reactions.