Nucleation and growth of ellipsoidal crystals in a supercooled binary melt

Abstract

When considering the evolution of particulate assemblages in supercooled and supersaturated liquids, the shape of crystals often differs from spherical. Existing experiments show that evolved crystals often are ellipsoidal. Motivated by important applications in materials physics, chemistry, and biophysics, we develop here a growth theory of a polydisperse ensemble of ellipsoidal crystals in a binary supercooled melt. An integro-differential model of kinetic and balance equations supplemented by initial and boundary conditions that describe desupercooling dynamics and particle-volume distribution function with allowance for the ellipsoidal shape of growing crystals is formulated and solved analytically using the saddle-point technique for a Laplace-type integral. The distribution function increases up to the maximal volume of particles and shifts to larger crystal volumes with time. The solute concentration substantially changes the evolution of a particulate assemblage. The melt supercooling decays faster and the particle-volume distribution function is lower with increasing the initial solute concentration. The theory under consideration generalizes previously developed growth theories for spherical crystals in a binary melt (Alexandrov 2014 {\it J. Phys. A: Math. Theor.} {\bf 47} 125102) and ellipsoidal crystals in a single-component melt (Nikishina and Alexandrov 2021 {\it Phil. Trans. R. Soc. A} {\bf 379} 20200306).

Analysis of an integro‐differential model for bulk continuous crystallization with account of impurity feeding, dissolution/growth of nuclei and removal of product crystals

“…Let us especially emphasize that there are many parameters and external processes having a considerable impact on the bulk crystallization phenomenon. These include, for example, (i) crystal growth/dissolution rate [12][13][14][15][16], (ii) crystal geometry and structure [17][18][19][20][21][22], (iii) mass and heat exchange with the environment [23][24][25][26], (iv) crystallizer design Eugenya V. Makoveeva and Alexander A. Ivanov are equally contributing authors. [27,28], (v) mixing process of supersaturated/supercooled liquid [29][30][31], (vi) withdrawal of product crystals [32,33], and (vii) interaction between crystals (their coalescence, agglomeration, and fragmentation) [34][35][36][37][38].…”

“…Let us especially emphasize that there are many parameters and external processes having a considerable impact on the bulk crystallization phenomenon. These include, for example, (i) crystal growth/dissolution rate [12–16], (ii) crystal geometry and structure [17–22], (iii) mass and heat exchange with the environment [23–26], (iv) crystallizer design [27, 28], (v) mixing process of supersaturated/supercooled liquid [29–31], (vi) withdrawal of product crystals [32, 33], and (vii) interaction between crystals (their coalescence, agglomeration, and fragmentation) [34–38]. A mathematical model that takes into account all of the above processes is extremely difficult both to formulate and to analyze.…”

On the Theory of Unsteady-State Operation of Bulk Continuous Crystallization

“…In addition to the nucleation kinetics, the bulk-phase-transformation process is significantly influenced by a number of other factors. These include, for example, the crystal growth rate law (quasi-stationary, non-stationary, empirical) [9][10][11][12][13][14]; the shape hypothesis of growing crystals (spherical and ellipsoidal or irregular shape approximations) [15][16][17][18][19][20]; external mass inflows or heat dissipation through system boundaries [21][22][23][24][25]; the type and shape of the crystallizer [26,27]; the presence or absence of stirring in the bath liquid [28,29], crystal discharge from the crystallizer [30,31]; the presence or absence of crystal fragmentation, attrition, coagulation and Ostwald ripening processes [32][33][34][35][36][37][38]; simultaneous occurrence of nucleation and directional solidification [39,40]; and other factors. Simultaneous consideration of all these characteristics in a single mathematical model of the crystallizer is hardly possible due to its extreme complexity.…”