MNET Applied to the Spherulitic Formation

Can Modern Statistical Mechanics Unravel Some Practical Problems . . . 27 To sum up, we have formulated a complex-interface-involving model of time t and temperature T dependent SPC-A formations. We have shown that it is capable of operat- ing under fully ballistic conditions that might univocally point to the total hydrodynamic stability of the process manifested over certain meso-to-microscale ranges. At the mi- croscale it implies that the mean free path of the macroion exceeds the width of the partially disordered double layer, while at the mesoscale the ballistic character mentioned above may contribute even more readily, merely via the constant-value-approaching growing mode, dR dt → const., as well as by means of the ES dynamics in the R–space. Since the main assumptions of the model have been verified on some experimental data on lysozyme [14, 39, 42], it seems that it looks also worth noting from a practical viewpoint, especially when recalling the crystal-formation protocol invoked above.

2.1.7. MNET Applied to the Spherulitic Formation

As appropriately mentioned by someone: “Biology is wet and dynamic”, therefore, any biological process should be examined for a description in terms of wetness and dynamics. In our case, the term ’wet’ should rather be assigned to the below invoked MNET descrip- tion because it is specially suited to any entropic liquid-involving milieu, such as the one characteristic of water-containing phase, thus being wet by definition see, Fig. 5. The phase changes, such as liquid-liquid phase separations or solute-solvent segregations, and the likes, are typically recognized as very dynamic processes in which the slowest dy- namic stochastic mode is going to determine the pace of the change. Thus, there should be no doubt - the phase change under study is a very dynamic process, see below. + Figure 5. A macroion viz model protein, or a fairly idealized spheroidal lysozyme protein cluster immersed in water, presented here for simplicity as a big circle with a plus sign in the middle. It is typically surrounded by two kinds of sheath irrotational and intermediate viz partly rotational, marked by angular arms with big dots. The ends of each water molecule in the sheaths are equipped with: one O 2 − ion and two H 1 + ions, marked by light and dark big dots, respectively. As an outer region, next to the macroion or, charged cluster and the sheaths, a bulk water phase can be seen, here as a hexagonal sub-matrix. The solute- solvent type interactions described by a Flory-Huggins-type mixing parameter, typically can propagate from inner to outer water sheaths, being the strongest upon an electrostatic contact of the first partly rotational water sheath and the macroion or, charged cluster surfaces. As was already mentioned, MNET provides a suitable framework based on which one can study the behavior of systems defined at the mesoscale. The formulation of any MNET- type theory heavily rests upon the fact that a reduction of the observational time and length 28 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al. scales of a system usually entails an increase in the number of degrees of freedom which have not yet equilibrated - they therefore exert a certain influence on the total dynamics of the system. Those degrees of freedom, χ –s the number of which can generally be reduced by the adiabatic-elimination procedure, thereby emphasizing the role of the slowest dynamic mode, with χ - herein the volumes of the growing nuclei, may represent the stochastic reaction coordinate of the system, properly defining the state of the system under N-G phase change in the χ –space. The characterization of the state of the system essentially relies on the knowledge of ρ ≡ ρ χ , t, the probability density of finding the system at the state χ ∈ χ , χ + d χ at time t. One can then formulate the Gibbs entropy postulate in the form [33] S − S eq = −k B Z ρ χ , t ln ρ χ , t ρ eq χ d χ . 21 Here S eq is the entropy of the system when the degrees of freedom χ are at local equilibrium. If they are out of equilibrium, the contribution to the entropy arises from deviations of the probability density ρ χ , t from its equilibrium value ρ eq χ given by a Gibbs-Boltzmann type formula ρ eq χ ∼ exp − ∆ W χ k B T , 22 where ∆ W χ is the minimum reversible work required to establish that state, k B is Boltz- mann’s constant, and T is the temperature of the bath. Variations of the minimum work for an open thermodynamic system are typically given by [33] ∆ W = ∆ E − T ∆ S + p ∆ V − µ ∆ M + σ∆ A + · · · , 23 where, after using a standard notation, extensive quantities refer to the system and intensive to the heat bath. The last term represents the work performed on the system to modify its surface area A, whereas σ stands for the surface tension. In order to obtain the dynamics of the mesoscopic degrees of freedom [37] one first takes variations in Eq. 21 δ S = −k B Z δρ χ , t ln ρ χ , t ρ eq χ d χ , 24 focusing only on the nonequilibrated degrees of freedom. The probability density evolves in the χ −space along with the continuity equation ∂ρ χ , t ∂ t + ∂ J χ , t ∂χ = 0, 25 where J χ , t is an unknown probability current [50]. In order to obtain its value, one proceeds to derive the expression of the entropy production dS dt which results from the continuity equation 25 and the Gibbs’ equation 24. After a partial integration, one then provides [33] dS dt = − Z ∂ ∂χ J S d χ + σ e , 26 Can Modern Statistical Mechanics Unravel Some Practical Problems . . . 29 where J S = J χ , t ln ρ χ ,t ρ eq χ is the entropy flux, and σ e = −k B Z J χ , t ∂ ∂χ ln ρ χ , t ρ eq χ d χ , 27 is the entropy production which is expressed in terms of currents and conjugated thermo- dynamic forces defined in the χ –space. We will now assume a linear meaning: small departure from the equilibrium depen- dence between fluxes and forces and establish a linear relationship between them [33] J χ , t = −k B L [ ρ χ , t] ∂ ∂χ ln ρ χ , t ρ eq χ , 28 where L ≡ L[ ρ χ , t] is an Onsager’s coefficient, which in general depends on the state variable ρ χ , t, in particular on the reaction coordinate χ [33]. To derive this expression, locality in χ −space has to be assured, for which only fluxes and forces with the same tensorial characteristics become mutually coupled [37]. Then, the resulting kinetic equation follows by inserting Eq. 28 to the continuity equation 25 ∂ρ χ ,t ∂ t = ∂ ∂χ D χ , t ρ eq χ ∂ ∂χ ρ χ , t ρ eq χ , 29 where we have defined the diffusion coefficient as D χ , t ≡ k B L ρ χ ,t ρ χ ,t . This equation, which because of Eq. 22 applied together with Eq. 23 can also be written as ∂ρ χ ,t ∂ t = ∂ ∂χ D χ , t ∂ρ χ , t ∂χ + D χ , t k B T ∂∆ W ∂χ ρ χ , t , 30 is the FPS type equation [36] accounting readily for the evolution of the probability den- sity ρ in our χ -space. This implies that the spherulitic formation of interest is given the FPS dynamics [33, 37], where the dynamics are realized as drifted diffusion in the phase space of the mesoscopic reaction coordinate, χ , which is the volume of a single spherulite: A ’real’ volume in the space of d E = 3 and an area in d E = 2, where d E - the Euclidean dimension of the space.

2.1.8. Avrami-Kolmogorov AK Phase-Change Model and Its Two-Phase Modes