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The complexity of problems of the theoreticaldescription of the crystallization process was noted bymany experts. For example, E.N. Kablov stated thefollowing in [1, p. 300] when considering the directedcrystallization process of heatresistant nickel alloys:“Description of the morphology of the phase boundary when taking into account the instabilities that ariseis very difficult even for singlephase alloys.”N.P. Lyakishev and G.S. Burkhanov [2, p. 140] adhereto the same opinion: “Until now there has been nosufficiently complete theory of crystallization ofeutectic alloys explaining the formation of variouseutectic structures.” According to [2], the first attemptat explaining the formation mechanism of lamellareutectics was undertaken by Tamman, while Fogelassumed that both phases grow simultaneously, and, inthis case, their common interfaces should be perpendicular to the interface between solid and liquidphases. Vast experimental material on the problem offormation of various structures upon crystallizationfrom solutions and melts including the description ofthe wellknown Lisegang structures and attempts atconstructing a theory of similar phenomena considered in monograph [3] of F.M. Shemyakin andP.F. Mikhalev had been accumulated already by 1938.According to [3], it was G.V. Woolf who in 1926 proposed for the first time to consider the formation of analloy crystal structure of eutectic composition as anunstable process accompanied by the alternate formation of crystallites of one component or another.Thus, this problem has been formulated over a longtime, and all available means have been used for itssolution. It is of interest to note that, for describing thealternation of rings formed during the crystallizationof a deposit from solution (Lisegang structures), Susann Weil (1931) proposed to use the formulas following from quantum mechanics, where the ring numberis an analogue of the quantum number. In this case,good agreement between the calculated and experimental results [3] was obtained.Nowadays, it is possible to consider as an established fact that the complex structures arising duringcrystallization, for example, alloys, are a consequenceof the development of certain instabilities during thephase transition. However, there are many causes forthe occurrence of instabilities and, as experimentalinvestigations show, various variants of their development can be implemented. Without claiming completeness, it is possible to list the following possibilitiesleaving the priority for the development of the phasetransition instability:(i) concentration overcooling;(ii) convective flows deforming the temperaturefield (gravitational and thermocapillary convection);(iii) phase stratification.In addition, the elastic properties of the solid phase[2] (one example is epitaxial crystallization on substrates [4, 5]) as well as the thin structure of the phaseboundary (atomically smooth or diffuse surfaces [1,p. 298]) and the adsorption phenomena [6] also give acertain contribution. The adsorption phenomenadetermine the surface energy (surface tension), wetting, and spreading and, thus, can cause thermocapillary convection [7–9]. Nevertheless, nowadays there isno general approach to description of the causes of theoccurrence of all variants of structures arising duringthe solidphase formation from melt or solution,which are a consequence of the instability of the processes accompanying the phase transition. Materialscience achieved a significant success, especially anapplied one, but the further development of this fieldof science determining the technical progress in many