**2. Spherulites' Formation in Terms of a Kinetic-Thermodynamic Model**

Herein below, let us consider in brief a model of the spherulitic growth that is based on a mass-convection conserved field instead of a diffusion field. We would like to state clearly that, considering the growth of spherulites, here from solution of a certain concentration, we propose our simple approach in which the spherulites are represent by spherical objects. Note that a spherulite is a 3D-system (there also exist some 2D objects commonly known as cylindrulites). The non-equilibrium character of the process can quantitatively be manifested by at least: (i) external concentration field feeding the growing object, (ii) internal boundary condition prescribed at the interface: spherulite-surroundings.

It is worth recalling the following experimental observation: the growth rate *v* ≡ *dR*/*dt* (*R* is the spherulite's radius; *R* ≡ *R*(*t*), where *t* is time) is mainly a parametric function of temperature *T* (the process under study is isothermal) and slightly depends on a particular system of interest. Thus, we may solely expect that asymptotically *R* ∝ *t*. Note that, especially in the long times' domain, it substantially differs from the well-known relationship *R* ∝ *t* 1/2 characteristic for purely diffusion-controlled crystal growth processes, as first uncovered by the perennially alive Mullins–Sekerka approach [13]. This approach assumes that the growth rate depends on local curvatures of the interface, the growing object vs. surroundings. From this, the square root radius vs. time relationship emerges.

An evolution equation for spherulites can be formulated as follows:


It has been shown that an evolution equation for growing objects (like polycrystals) with an ideal or perturbed spherical symmetry has the form:

$$[\mathcal{C} - \mathfrak{c}(\mathcal{R})] \dot{\mathcal{R}} = -\stackrel{\rightarrow}{\mathcal{J}} [\mathfrak{c}(\mathcal{R})] \circ \stackrel{\rightarrow}{\mathcal{H}}\_{0\prime} \tag{1}$$

where . *R* = *dR*/*dt*, *C* is the object's density, which may depend on space variables and can generally be of stochastic nature, *c*(*R*) stands for concentration of external particles at the surface, <sup>→</sup> *j* [*c*(*R*)] is the flux of particles outside the object which depends functionally on concentration and <sup>→</sup> *n*<sup>0</sup> is the outer normal to the surface of the object. Both sides of Equation (1) are given in SI units of kg/m2s.

As regarding point (ii), the concentration of the particles at the surface of the growing object is determined by thermodynamic conditions and geometry of the surface. Under assumption of local thermodynamical equilibrium near the interface, it has the form of the Gibbs–Thompson relation. In a more realistic model, the surface is far from equilibrium and its deviation from equilibrium is proportional to the growth velocity of the interface:

$$c(R) = c\_0 \left( 1 + \frac{2\Gamma}{R} - \beta \dot{R} \right) \tag{2}$$

where *c*<sup>0</sup> is the concentration field at a flat interface, Γ is the capillary constant which aims at smoothening out the surface of the growing object and is proportional to the surface tension [14], *β* is a positive kinetic coefficient, 2/*R* is twice the mean curvature of the spherical object, and the last term describes a deviation from the thermodynamic equilibrium. When *β* = 0, one gets the well known Gibbs–Thomson condition.

Let us consider point (iii). If the feed of the growing object is purely mass-convective, then:

$$
\overrightarrow{\hat{\boldsymbol{\psi}}}[\boldsymbol{c}(\boldsymbol{R})] = \boldsymbol{c}(\boldsymbol{R})\overrightarrow{\boldsymbol{\psi}}(\boldsymbol{R}),\tag{3}
$$

where <sup>→</sup> *v* (*R*) is a mass-convection velocity. (For the diffusion-limited, Mullins–Sekerka type growth, r.h.s. of Equation (3) assumes a concentration-gradient form [13]).

From the above, a basic evolution equation can be derived:

$$\left[\mathbf{C} - c\_0 \left(\mathbf{1} + \frac{2\boldsymbol{\Gamma}}{R} - \beta \dot{\boldsymbol{\mathcal{R}}}\right)\right] \dot{\boldsymbol{\mathcal{R}}} = c\_0 v\_0 \left[\mathbf{1} + \frac{2\boldsymbol{\Gamma}}{R} - \beta \dot{\boldsymbol{\mathcal{R}}}\right],\tag{4}$$

where *R*(*t* = *t*0) > 0.

The growth process described by Equation (4) is influenced by five parameters: *C*, *c*0, Γ, *β* and *v*0. However, in fact, only two parameters are physically meaningful: (i) the quantity Δ which is a measure of the saturation in the system:

$$
\Delta = \frac{\mathcal{C} - \mathfrak{c}\_0}{\mathfrak{c}\_0} = \frac{\mathcal{C}}{\mathfrak{c}\_0} - 1,\tag{5}
$$

and (ii) rescaled kinetic coefficient *β*0:

$$
\beta\_0 = \beta v\_0. \tag{6}
$$

Indeed, rescaling the bare variables *R* and *t* to dimensionless quantities *r* = *r*(*τ*) and *τ* via the relations:

$$r = \frac{\mathcal{R}}{2\Gamma'} \tag{7}$$

and

$$
\pi = \frac{v\_0}{2\Gamma} t \tag{8}
$$

is useful for carrying out a solid semi-quantitative description of the spherulite's formation equation.

From the system (4)–(8), one derives the following nonlinear differential equation

$$
\beta\_0 \left(\frac{dr}{d\tau}\right)^2 + \left(\Delta + \beta\_0 - \frac{1}{r}\right) \frac{dr}{d\tau} - \frac{1}{r} - 1 = 0. \tag{9}
$$

Equation (9) is an algebraic quadratic equation with respect to *dr*/*dτ*, where *x* = *dr*/*dτ*, which can be rewritten as:

$$
\beta\_0 \mathbf{x}^2 + \frac{(\Delta + \beta\_0)r - 1}{r} \mathbf{x} - \frac{r + 1}{r} = 0. \tag{10}
$$

The real valued roots of this equation can be determined by using the conventional method of solving quadratic equations, namely, with specifying the characteristic Δ*x*:

$$
\Delta\_{\rm X} = \frac{\left[ (\Delta + \beta\_0)r - 1 \right]^2 + 4\beta\_0 r (r+1)}{r^2} > 0,
\tag{11}
$$

and its necessary square root:

$$\sqrt{\Delta\_x} = \frac{\sqrt{\left[ (\Delta + \beta\_0)r - 1 \right]^2 + 4\beta\_0 r (r+1)}}{r} := d(r)/r,\tag{12}$$

thus, if the numerator of the fraction in Equation (12) is denoted for convenience by *d*(*r*). The roots of the quadratic equation are explicitly given by:

$$x\_{1|2} = \frac{-[(\Delta + \beta\_0)r - 1] \pm \sqrt{[(\Delta + \beta\_0)r - 1]^2 + 4\beta\_0 r(r+1)}}{2\beta\_0 r}.\tag{13}$$

One of its roots has to be ruled out. It is determined by the limiting case *β*<sup>0</sup> → 0. (This, contrary to the vanishing kinetic limit of the phenomenon, can rather be ascribed to its thermodynamic counterpart.) Finally, one gets:

$$\frac{dr}{d\tau} = x\_2 = \frac{\sqrt{\left[ (\Delta + \beta\_0)r - 1 \right]^2 + 4\beta\_0 r (r+1)}}{2\beta\_0 r} + \frac{1}{2\beta\_0 r} - \frac{\Delta + \beta\_0}{2\beta\_0},\tag{14}$$

or by employing the shorter notation with *d*(*r*)

$$\frac{dr}{d\tau} = \frac{d(r) + 1}{2\beta\_0 r} - \frac{\Delta + \beta\_0}{2\beta\_0} > 0,\tag{15}$$

where explicitly

$$d(r) = \sqrt{\left[ (\Delta + \beta\_0)r - 1 \right]^2 + 4\beta\_0 r (r+1)}.\tag{16}$$

Notice that, by presenting above the rescaled spherulite's evolution equation, that means Equation (15) with its supporting Equation (16), we have arrived at the equation on which we wish to perform a simplified structural stability analysis in order to reveal the onset of the spherulitic growth.

Numerical solutions of Equation (15) by using Euler's discretization method are presented in Figure 2. The dependence of *r*(*τ*) on the rescaled kinetic (*β*0) and thermodynamic (Δ) dimensionless parameters indicate the edge of diffusion- and mass-convection-driven growth at the onset of spherulitic formation around *β*<sup>0</sup> ≈ Δ ≈ 0.2. A square root tendency manifests earlier for the set of upper curves, thus for smaller values of *β*0.

**Figure 2.** The dependence of *r*(*τ*) on the rescaled kinetic *β*<sup>0</sup> and thermodynamic Δ dimensionless parameters. In the chosen time interval, the curves reflect a visible tendency to pass from the diffusion-like (*β*<sup>0</sup> <sup>→</sup> <sup>0</sup>) to mass-convective-type mode (*β*<sup>0</sup> - 0). Other realizations of *r*(*τ*) are presented in [12].

## **3. Results and Discussion**

First of all, let us note that, at the onset of the spherulitic growth at which the overall system is settled in front of a decision whether it will evolve either in diffusional or in non-diffusional (mass-convection-like) limit, one can derive the double root (Δ*<sup>x</sup>* = 0) of the quadratic Equation (10)

$$\mathbf{x}\_0 = \frac{1 - (\boldsymbol{\Delta} + \boldsymbol{\beta}\_0)\boldsymbol{r}}{2\beta\_0 r} = \frac{d\boldsymbol{r}}{d\boldsymbol{\tau}}\tag{17}$$

to be equivalent to

$$\frac{dr}{d\tau} = \frac{1}{2\beta\_0 r} - \frac{\Delta + \beta}{2\beta\_0} \tag{18}$$

which, after denoting the radius by *rd*, can be named the diffusion limit if

$$\frac{1}{r\_d} - \Delta + \beta\_0 > 0\tag{19}$$

or <sup>1</sup>

$$\frac{1}{r\_d} > \Delta - \beta\_0. \tag{20}$$

It implies that the critical value of the non-spherulitic or diffusional growth reads

$$r\_d = \frac{1}{\Delta - \beta\_0}.\tag{21}$$

The non-dimensional radius given by Equation (21) reflects an interplay between thermodynamic (Δ) and kinetic (*β*0) parts at the onset of the diffusion-driven but nonspherulitic growth. Cleary, one can also expect such mode of growth if one puts *d*(*r*) = 0 in Equation (15), making then use of its equivalence with Equation (18) with zeroth condition applied to Equation (16).

The non-diffusive, thus, the spherulitic onset of the evolution, arises if one assumes a proportionality of *d*(*r*) to *r*; thus, when providing a linearity thereof (but not a constancy with non-negative property), namely

$$d(r) \sim r \Rightarrow \left[ (\Delta + \beta\_0)r - 1 \right]^2 + 4\beta\_0 r (r+1) \sim r^2 \tag{22}$$

which implies that

$$(\Delta + \beta\_0)^2 r^2 - 2(\Delta + \beta\_0)r + 1 + 4\beta\_0 r^2 + 4\beta\_0 r \sim r^2 \tag{23}$$

and a simple limit of the form after postponing all in-*r* quadratic (counter-balancing) terms

$$2r(2\beta\_0 - \Delta - \beta\_0) + 1 \to 0\tag{24}$$

can ultimately be taken, which results in

$$2r(\beta\_0 - \Delta) \to -1\tag{25}$$

and eventually leads to (denote the limit by *rnd*)

$$r\_{nd} = \frac{1}{2(\Delta - \beta\_0)}.\tag{26}$$

It is if *β*<sup>0</sup> ≈ 0.2, cf. Figure 2, which is when the kinetics and thermodynamics work at the singularity-expressing onset of the spherulitic formation, with Δ ≈ 0.2, c.f. Equations (21) and (26).

Comparing Equation (26) with Equation (21), one immediately recognizes that a simple relation

$$r\_d = 2 \cdot r\_{nd} \tag{27}$$

prevails. Despite the physical fact that the factor two in proportionality relation (27) means an earlier onset (as compared to that non-spherulitic viz diffusional mode) of the spherulitic growth to manifest. The factor two can also be interpreted in graphical terms of Figure 1, namely, that in the diffusion (Mullins–Sekerka type) limit characteristic of r ∼ *t* 1/2, the only one crystalline phase builds the object (Figure 1a), while in the case of the spherulitic growth, two concurrent phases (crystalline vs. amorphous) constitute the spherulite's body, cf. Figure 1b.

Thus, this way it has been shown that, at the onset of the growth under study and presumably around the kinetic-thermodynamic singularity *β*<sup>0</sup> ∼ Δ, cf. Equations (21) and (26), the spherulitic growth will prevail earlier in terms of the spherulite's nucleus value, cf. Equation (27), if the mass-convection and nonequilibrium (Goldenfeld type [3]) boundary conditions win over those of the Mullins–Sekerka kind. This feasible singularity limit at the onset of the growth, namely, *β*<sup>0</sup> ∼ Δ, suggests that, though the model is deterministic, its extension can fairly be envisioned towards applying prospectively a stochastic approach [15], wherein the corresponding fluctuations around the *β*<sup>0</sup> ∼ Δ condition can show up in subtle or pronounced ways.

The overall spherulitic formation as viewed in the so-called entropy-production (*e*) terms [16] detected at the interface can be associated with the stochastic-fluctuational context. The scalar product <sup>→</sup> *<sup>j</sup>* [*c*(*R*)] ◦ <sup>→</sup> *n*<sup>0</sup> put at the r.h.s. of Equation (1) involves the matter flux <sup>→</sup> *j* [*c*(*R*)] of the external feeding field. The field is twofold, namely, either of mass-convective or of locally diffusional character. In the former, it does not include curvatures, whereas in the latter it receives them for granted, see Figure 1a, and it basically goes like <sup>→</sup> *<sup>j</sup>* [*c*(*R*)] ◦ <sup>→</sup> *n*<sup>0</sup> ∼ 1/*r*. For the mass-convective counterpart, one gets <sup>→</sup> *<sup>j</sup>* [*c*(*R*)] ◦ <sup>→</sup> *n*<sup>0</sup> ∼ *const*, when late times conditions readily apply. (In fact, in this time zone, the local curvatures of the Mullins–Sekerka mode also cease to grow, yielding ultimately a similar physical scenario.) The entropy production *<sup>e</sup>* <sup>=</sup> <sup>→</sup> *<sup>j</sup>* [*c*(*R*)] ◦ <sup>→</sup> *x* with <sup>→</sup> *x* representing the (Onsager type) thermodynamic force enables, while based on the same reasoning, to ascertain that the involvement of (local) curvature term and the application of the Fick's law <sup>→</sup> *<sup>x</sup>* ◦ <sup>→</sup> *n*<sup>0</sup> ∼ 1/*r* to *e* gives a bigger non-negative account to it based thoroughly on the Mullins–Sekerka [13,17] (crystal growth) mode than in the case of mass-convective, thus spherulitic mode, cf. Equation (3).

#### **4. Conclusions**

In this study, we have demonstrated, while based on the non-dimensional model (suitable for numerics), that upon mass-convection and nonequilibrium boundary criteria for the (poly)crystal's growth, such as the one of (bio)polymers addressing or that concerning biominerals (geophysical objects), realized in defects containing and condensed matter involving matrix, that a well-justifiable chance of spherulites' emergence prior to a pure diffusion-controlled crystal growth exists at the onset of the growing conditions. The argumentation line is based on the physical fact that, in spherulites (polycrystals) two phases may "synergistically" coexist, whereas in single crystals the only ordered crystalline phase has to be built in suitably, presumably at a higher energetic cost than in the former. As named by us, the unimodal crystalline Mullins–Sekerka type mode of growth, characteristic of local curvatures' presence, seems to be more entropy-productive in its emerging (structural) nature than the so-named bimodal or Goldenfeld type mode of growth in which the local curvatures do not play any crucial roles, and in which kinetics seems to win over thermodynamics. In turn, a liaison of amorphous and crystalline phases makes the system far better compromised to the thermodynamic-kinetic conditions it actually, and concurrently, follows.

The final conclusions presented are based on the peculiar evolution equation, having corroborated cooperatively mass-convective and nonequilibrium (boundary) conditions, that basically drive the growing system far from equilibrium. Interestingly, one may qualitatively predict that the entropy production [16] in such massive (poly)crystalline forms is more expressed in the unimodal non-compromised Mullins–Sekerka type mode. In this mode, a thermodynamic, close-to-equilibrium supersaturation factor prevails, unlike in its Goldenfeld-like bimodal nonequilibrium counterpart. This bimodality, i.e., a synergistic coexistence of crystalline and amorphous phases within a growing spherulite, rests on terms of internal stress–strain material conditions, and does not admit the local curvatures at the interface to prevail. The latter is not the case of any diffusion-controlled (unimodal) growth in which the so-called Mullins–Sekerka instability manifests readily. Finally, let us stress that it seems to us that the non-dimensionality of the proposed modeling suggests that the system does not depend upon experimental details, manifesting somehow a quasi-universal, that means, mainly the scaling addressing character of the performed modeling [18].

**Author Contributions:** Conceptualization, A.G. and J.S.; methodology, A.G.; software, J.S.; validation, A.G. and J.S.; formal analysis, A.G. and J.S.; investigation, A.G. and J.S.; resources, A.G. and J.S.; data curation, J.S.; writing—original draft preparation, A.G. and J.S.; writing—review and editing, A.G. and J.S.; visualization J.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by BN-WTiICh-11/2022 of the Bydgoszcz University of Science and Technology.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** On the anniversary of the 70th birthday of Jerzy Łuczka, we would like to thank him for many years of scientific cooperation and wish him good health and further success in his private and professional life.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

