Metastable liquid-liquid coexistence region in the phase diagram of the Ag–Cu system, computed using the free energy functions from CALPHAD-type calculations of Ref. 19. Note that coexistence line (thin solid), the spinodal line (dashed solid), and the bifurcation line (heavy gray line). Regions I and II may show different nucleation properties. For explanation, see the text.
Free energy density surface at in the phase field theory, counted relative to a homogeneous liquid of the critical composition. Note that at this temperature, besides having two stable solid compositions in equilibrium (at ), two metastable liquid compositions can also be in equilibrium with each other (at ).
Properties of the metastable planar liquid-liquid interface as a function of temperature: (a) surface tension; (b) 10%–90% interface thickness. Convergence of the former to zero and the divergence of the latter happen with the appropriate mean field critical exponents.
Properties of liquid-liquid nuclei as a function of initial liquid composition at : (a) Concentration profiles (black lines: Cu rich liquid nuclei forming in initial liquid compositions falling between the coexistence and spinodal lines to the left of the critical composition; gray lines; Ag rich liquid nuclei forming on the opposite side of the phase diagram). (b) Nucleation barrier for liquid-liquid nuclei as a function of the initial liquid composition (the vertical solid and dashed lines indicate the coexistence and spinodal compositions, respectively). (c) Critical radii of liquid-liquid nuclei defined via the Gibbs surface for the concentration as a function of the initial liquid concentration.
Variation of the nucleation barrier for the Cu and Ag rich crystal nuclei along the constant driving force lines that cross the critical point. (a) The corresponding trajectories (dashed and dash-dot lines, respectively). For comparison the metastable liquid-liquid coexistence line is also shown (solid line). (b) Barrier height vs reduced temperature for Cu rich nuclei. (c) Barrier height vs reduced temperature for the Ag rich nuclei.
Structural order parameter vs composition trajectories for points along the constant driving force line for the Cu rich solution. From left to right: , 0.975, 1.0, 1.025, and 1.05. Note the gradual transition from crystal-like nuclei into liquidlike, when approaching the metastable critical point from both sides.
Free energy for various nucleation pathways as a function of temperature (a) on the left and (b) on the right of the critical composition. [Notation: and stand for normal solutions, while and for composite solutions that are rich in the component sown in the subscript, respectively. Solid and dashed lines: for the Ag and Cu rich normal solutions, respectively, observed when descending in the phase diagram at the critical composition, and following the coexistence line below (trajectories denoted by heavy black lines and arrows in Fig. 1). Dash-dot lines: for the composite solutions along the bifurcation line (the trajectories denoted by heavy gray lines and arrows in Fig. 1)].
Structural order parameter vs composition trajectories for points along the coexistence line for the Cu rich solution. From left to right: , 0.95, 1.0, 0.95, and 0.85. Note the gradual transition from crystal-like nuclei into liquidlike, when approaching the metastable critical point from both sides.
Properties of Cu rich normal and composite nuclei forming at : Shown are the radial phase field (solid) and the composition profiles (dashed) for (a) the Ag rich normal, (b) the Cu rich normal, and (c) the Cu rich composite nuclei. The solutions at the bifurcation point are denoted by heavy lines. Note the well-defined liquid layer around the crystalline core forming at small supersaturations and the convergence of the two types to each other at the bifurcation composition. (d) The respective nucleation barriers. ( and stand for normal solutions, while for the composite solution that are rich in the component in the subscript.)
Structural order parameter vs composition trajectories for the three types of solutions shown in Fig. 9, at the initial liquid composition of . Note the long horizontal line for the composite solution that represents the liquid layer around the solid core.
Nucleation barrier vs initial liquid composition for the three types of solutions existing on the left and right of the critical composition at (a) , (b) , and (c) . The coexistence and spinodal compositions are denoted by vertical solid and dashed lines, respectively.
Radial phase field (light lines) and composition (heavy lines) profiles for the three types of solutions existing on the left of the critical composition at (a) , (b) , and (c) .
Lines representing equal nucleation barrier heights for the Ag and Cu rich normal solutions [ in the text] and for the Ag rich normal solution and the Cu rich composite solutions [ in the text]. Note that they terminate in a common point falling on the bifurcation line.
Phase selection in the Ag–Cu system, according to the minimum of the nucleation barrier at (upmost curve), (central curve), and (bottom curve). Note the complex behavior below the critical point. ( and stand for normal solutions that are rich in the component denoted by the subscript, while denotes the normal solution forming on the Cu rich branch of the coexistence line.)
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