Amorphization vs Polycrystallization: t + div[J t] = 0,

30 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al. where M stands for the average number of the evolving spherulites passing through a point; F M denotes the factorial of M. The probability of any point not being passed over by a spherulite is given by value of ρ sph M = 0, eventually resulting in ρ sph M = 0 = e −M , 32 since F M = 0 = 1. As a consequence of the above, ρ sph M = 0 also represents the points being still amor- phous, i.e. not been passed over by the spherulites, and thus, it is equal to amorphous fractional completion, θ , which, in turn, results in having 1 − θ as the crystalline fractional completion. Comparing then 1 − θ with ρ sph M = 0 yields at once θ = 1 − e −M , 33 which provides the formula for the crystallized fraction. The problem finally reduces to determine M as a function of geometric assumptions on the nuclei forms as well as the time t after which the children spherulitic phase is born from the parent phase of apparently amorphous overall character. It then leads to the general solution θ ≡ θ t = 1 − e − ω t ν , 34 where M = ω t ν , resulting from a direct comparison of the two last formulae, involves two parameters ω and ν . ω is recognized to be dependent on the shape of the growing crystalline entities as well as on the amount and type of nucleation. ν depends upon the nucleation type and growth geometry but not upon the amount of nucleation [52]. For example, in biopolymers the so-called transcrystallization is a process in the course of which nucle- ation does not prevail to be a rate-determining factor. One of most solid observations on the spherulitic formation, as often as possible re- ported in the literature, is that the formation occurs asymptotically in a constant-tempo kinetic-thermodynamic regime [44, 40, 41, 39, 42, 53, 54, 55]. This is due to the fact that the droplets-involving amorphization kinetics, usually pronounced in a vigorous manner, is successfully balanced in a dynamic way during the phase change by a counter-effect which appears to be the fibrillization. It then leads in a common way to the typically undesired formation of the the polycrystals termed spherulites, see Fig. 1 [51]. The effect is fully manifested in the late-time zone, and because the fibrillization leads ultimately to a surface-clustering sub-effect caused by the pieces rodes of the crystalline skeleton, it somehow acts in its final stage as the surface tension in case of equilibrium systems, i.e. as if the system was, at the moment, in equilibrium with its outer thermostat phase. Yet, the system unavoidably departs from this local quasi-equilibrium if there is still an ample place, left for the pieces to enter the as-yet unoccupied space within the growing object. If there is actually no ample place left, the overall evolution terminates, showing up a characteristic cessation-to-growth stage, frequently reported by experimenters [56].

2.1.9. Amorphization vs Polycrystallization:

Switching offon the Asymmetric Growing Mode Let us consider a situation in which the phase change takes place in a system of constant total volume, V sph = const. [53, 54, 52]. The growth rate of the spherulitic formation can Can Modern Statistical Mechanics Unravel Some Practical Problems . . . 31 be unambiguously determined by the MNET formalism outlined above, with the volume χ playing its pivotal role. Proceeding as indicated previously one yields the growth rate [42] J χ , t = − L χ T ρ k B T ∂ρ χ ,t ∂χ + ρ χ , t ∂Φ ∂χ . 35 Interpreting Φ ≡ Φ χ as an entropic Gibbs’ potential the free energy suitable for the spherulitic formation, with Φ χ ∼ ln χ asymptotically, and assuming naturally that the volume-dependent Onsager’s coefficient L χ [33] follows a power law of the type χ δ , where δ = d E − 1d E , with d E - the Euclidean dimension of the system, one provides the expression of the Smoluchowski-type [36] probability current J χ , t = −D χ , t ∂ρ χ ,t ∂χ − D χ , t k B T ∂Φ ∂χ ρ χ , t, 36 where D χ , t is a spherulitic diffusion coefficient, asymptotically obeying [53, 54] D χ , t = D χ δ t µ . 37 Note that Φ χ may again fully participate in constituting the so-called Kramers’ barrier, characteristic of the two-state amorphization-spherulitization picture that we would like to convey. In Eq. 37 one sees that D χ , t is postulated to be factorized into two parts: a χ – dependent part, with the geometrical exponent δ being involved it pinpoints to the spheru- lite-surface prevailing behavior, e.g. the one in which surface tension may thoroughly be involved, as well as some time-dependent part, in which the spherulite-formation exponent µ depends upon the Kolmogorov-type amorphization measure d E + 1 [56] and upon the rod-like spherulitic-skeleton involving behavior, represented by the exponent ν r , usually obeying ν r ≈ 1 2 [53, 54]. The latter comes from the fact that the time-dependent part of D χ , t from Eq. 37 originates from a small ω approximation the approximation applied to AK-parameter from Eq. 34 [53], which typically holds for highly viscous systems as ours, and which arrives at an algebraic asymptotic behavior of θ t θ t ∼ t ν , 38 wherein ν ≃ ν r ≈ 1 2 finally applies [53, 54]. This power-law type ∼ t ν r contribution is assumed to enter then the diffusion function D χ , t leading ultimately to a certain time- rescaling of the observables arising from the whole FPS context[36]. Thus, the overall exponent µ can be defined, similarly as in a previous study [53, 54, 56], i.e. by means of a simple competition-type formula, as follows µ ≡ µd E , ν r = ν d E − ν r , 39 where ν d E = d E + 1 a d E –dependent part and ν r generally obeys ν r ∈ [ 1 2 ; 1 ] with a strong preference to ν r ≈ 1 2 , i.e. when the nucleation of rods is a-thermal but its rate manifests diffusively kinetically, that means, not in an entirely thermodynamic way. 32 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al. For the amorphization kinetic-thermodynamic formalism that does not lead at all to the typical spherulitic formation since it over-estimates its average tempo dR dt, with R the average spherulite radius 2 which has to conform asymptotically to constancy of dR dt, one has to conjecture the rods-involving contribution [57, 58] to be rejected, i.e. ν r = 0. Thus, there is no above mentioned competition involved, and finally the overall exponent coincides with the Kolmogorov amorphization measure, µ d E , ν r = 0 = ν d E = d E + 1, 40 which leads to a strongly super-diffusive hydrodynamic overall behavior in χ –space, ulti- mately resulting in a non-constant behavior of dR dt, leaving it as an increasing function of time - a hydrodynamically unstable mode [59]. This way, the asymmetric rods-involving crystalline mode is switched off, and we finally end up with a fluctuating randomly close- packed system ν d E = d E + 1 is also a generic measure of random close-packing, very characteristic of amorphous systems containing the randomly placed and oriented crys- talline drops [56, 57, 58]. When one is able to conjecture [51, 57, 58] that the rods-involving contribution is ulti- mately not being rejected, i.e. ν r 6= 0, one switches on the competition asymmetry mode, and finally arrives at µ d E , ν r 6= 0 = ν d E − ν r ≈ d E + 1 − 1 2 ≡ d E + 1 2 , 41 which leaves µ d E , ν r 6= 0 to be a non-integer competition-type exponent, pointing readily to a symmetry breaking within the system. If it is involved for both spherulitic formations on either athermal or thermal nucleation seeds, it always properly yields the constant average tempo dR dt of the overall formation [53, 54, 51, 60, 61] a stable hydrodynamic mode, i.e. when being stabilized by the counter-effect considered, which is also featured as a kinetic-thermodynamic signature of the spherulitic growth by the presented rationale. The present study can be summarized concisely in a tabular form, cf. Table 1. Table 1. Types of possible polycrystalline formations in model protein systems, and their characteristic integer vs non-integer Kolmogorov-type measures coming from application of the MNET-type formalism with d E = 2, 3, cf. text. Type of formation Geometric-exponent value System amorphization: dR dt 6= const. µ d E , ν r = 0 = d E + 1 Spherulitic formation: dR dt ≈ const. µ d E , ν r 6= 0 ≈ d E + 1 2 To summarize in part, it can be ascertained that by switching on the asymmetric rod-like crystalline-skeleton based mode we are able to make somehow a specific, likely crystalline 2 It can be evaluated after calculating the two first central moments of ρ χ ,t, ρ χ ,t i , i = 1, 2, where ρ χ ,t 1 = V sph = const., ρ χ ,t - well-determined from the overall FPS formalism, and ρ χ ,t R d E ∝ V sph = const. holds. Can Modern Statistical Mechanics Unravel Some Practical Problems . . . 33 misorientation related de-amorphization of the system, finally arriving, due to the competi- tion mode mentioned, at a constant-speed characteristic behavior of the protein spherulites SPC-A, readily emerging from the offered MNET-type description. Such a description can also be viewed as an interesting practical study on a passage between nano- the fibrils as parts of the spherulites and micro-structures the spherulites for themselves emerging in a complex viscoelastic system [62], contributing this way to modern concepts [51] of emerging science called often nanobiology.

2.1.10. MNET Approach to Viscoelasticity in Unconfined Systems