Common tangent rule

The common tangent rule above is the thermodynamic condition for the equilibrium between A and B phases. The temperature dependence of the concentrations at A and B states outlines the phase coexistence curve, which is called the binodal line. 

IPB proceeds at equal rates. Owing to symmetry, at a-fi-co triple junction, the B component concentration at IPB (from the )-phase side) must be equal to = 1/2. At planar regions of IPB, equilibrium concentrations are found with the help of the common tangent rule and are equal to Caa, = c — ACa,/2 between a and a> phases ... 

In the present section, we introduce a thermodynamic model of phase separation in a nanoparticle and analyze how the depletion effects can be taken into account The last part is devoted to the analysis of the influence of size and depletion on phase diagrams of regular solutions. In the Appendix, we present the rule of parallel tangents construction (not to confuse with common tangent rule) for the extreme points of the phase transition. In the following, the starting single phase is called old or parent phase and the newly formed phase is called new one. 

As a first step, we will look at the equilibrium compositions in the parent phase Cjjoo = Cp R —> 00) and in the new phase C oo = Cn R —> 00), corresponding to the full separation in the infinite bulk material (at every fixed temperature T). The conditions for optimal concentration C oo and Cp co and solubihty limits can be found according to the common tangent rule ... 

The common tangent to the minima in the free-energy curve determines the composition of the two phases in the glass and the proportions of each are determined by the lever rule. 

Furthermore, the values of the derivatives with respect to composition, m2 = 9gn/9c cpj and mi = dgn/dclc, are fixed by the phase equilibrium according to the rule of common tangents. So, we have... 

The equihbrium concentrations in the new phase and in the ambient parent phase are determined by the rule of the parallel (not common) tangents drawn from lines of concentration dependencies of Gibbs free energy densities on concentrations for new and old phases [61]. 

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