**1. Introduction**

Defects in condensed matter phase are ubiquitous in their appearances and types. As for the point defects, one can mention atomic vacancies and interstitial positions for the abundant atoms covering the corresponding material's locations.

As for extended defects, in turn, it is appropriate to list grain boundaries, two-dimensional defects, very characteristic of polycrystals, whether composed of metallic or of (bio)polymeric material. Stepping up one dimension higher, one can address (screw) dislocations and disclinations, to mention but two [1].

Spherulites are complex defects emerging in condensed phase. They bear something from the notion of extended defects, but they mean something distinctly more than this. They are typically recognized as an imperfect crystalline phase comprised from radially distributed polycrystal's needles, crossed at certain non-crystallographic angles with each other, but intermingled with a not-yet-crystallized amorphous phase in between, cf. Figure 1. The most known technique to visualize the spherulites as defects is polarized light microscopy, which yields the famous Maltese cross. Especially, the amorphous, non-radial phase of the object is named a "band defect" the expression of which depends on peculiarities of the system studied [2]. It seems as if it was a process of competition between crystal and amorphous emerging phases that tend to evolve in a diffusional manner, *r*(*t*) ∼ *t* 1/2 (*r*—arbitrary spherulite's radius in diffusion-controlled mode; *t*—time),

**Citation:** Siódmiak, J.; Gadomski, A. Spherulites: How Do They Emerge at an Onset of Nonequilibrium Kinetic-Thermodynamic and Structural Singularity Addressing Conditions? *Entropy* **2022**, *24*, 663. https://doi.org/10.3390/e24050663

Academic Editor: Antonio M. Scarfone

Received: 12 April 2022 Accepted: 6 May 2022 Published: 9 May 2022

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and then, they change their growing mode toward its mass-convection-like (long times) counterpart, *r*(*t*) ∼ *t*. This implies that the evolution goes with a constant speed [3–5] and proceeds with certain massive steps in absorbing the amorphous matter of the system. It looks as if it was that the emerging (poly)crystalline phase imposes a confinement on the diffusion space, yielding an accelerated absorption of the diffusing material onto the crystal phase, with a remnant non-absorbed amorphous phase still available for diffusion to occur. The method of revealing this phenomenon turns out to be the DSC (Differential Scanning Calorimetry), and the underlying process is coalescence of the spherulitic material, also resulting in a structural impingement of the spherulites [6].

**Figure 1.** Mullins–Sekerka (M–S) type and Goldenfeld (G) type: (**a**) M–S, local curvatures are indicated by blue dashed circles; (**b**) G, mean curvature is signified by blue dashed circle.

An intriguing question that appears sounds: Why the spherulitic evolution changes its mode from the diffusion-like to that mass-convection-like? Amongst many answers to this question, there is at least a pronounced streamline of arguments proclaiming that the growing system of interest is evolving in nonequilibrium thermodynamic boundary conditions [3,7]. Being motivated by the aforepresented and not-answered-in-full experimental observations, in what follows, we are attempting to provide a simple theoretical rationalization that it is convincingly seen in terms of our type of modeling.

To achieve our goal, we shall employ in a natural way a spherical approximation to a conserved-mass deposition, mimicking the spherulitic growth. It seems to be really natural here, because the complex defects called spherulites assume ultimately a spherelike form. The spherical approximation to the mass conservation law has been presented elsewhere [8–10]. However, the essential novelty applied to it is going to rely on a special boundary condition of nonequilibrium character [11]. Another relevant precondition applied to the spherulite's modeling, especially important for (bio)macromolecular realizations, is that the mass-convective instead of purely diffusive mass transportation conditions are employed decisively to create the spherulite's evolution. However, the diffusion limit is not completely ruled out, but it is also discussed when considering the onset of spherulitic formations [8–11].

In the subsequent sections, we shall present the model of spherulitic formation, capturing both diffusional and non-diffusional/mass-convection-like competitive modes, and bearing a signature of nonlinear ordinary differential equation (ode), solvable when resorting to its numerical solutions [12]. The presentation of the spherulite-formation model in terms of nonlinear ode with the corresponding initial conditions gives also a chance of its qualitative analysis. It leads to determining spherulite's characteristic linear dimension valid for both regimes recalled, albeit the radius' value for the immature spherulite (rather, its prerequisite) at the onset of the spherulitic growth is half the size as it would be for its purely diffusive counterpart. The proposed semi-quantitative model looks fairly manageable to solicit firm-basis addressing conclusions toward spherulites' formation. It is because this is to a major extent presented in terms of rescaled variables [12], both independent and dependent (the key parameters as well). Thus, all quantities of interest are nondimensional, and the number of the key governing parameters is reduced from five to two, see Section 2. It univocally allows to conclude on the principal features of the proposed modeling, showing up basic signatures of the spherulitic growth [3,4,7], and its, most importantly, inherent passage from diffusional to mass-convection-like limit, a type of structural, nonequilibrium phase transition [6].

The article is organized as follows. In Section 2, a kinetic-thermodynamic nonequilibrium model of spherulitic growth is presented, whereas in Section 3 its main results toward the onset of the spherulitic growth are disclosed, and its properties, also certain proposals for legitimate and/or useful extensions, are discussed. Section 4 serves for the main conclusions.
