26 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al.
2.1.5. Towards a Morphological Phase Diagram
From the above, cf. Eq. 16, it is seen that the Kramers’ barrier [47] determinant, Φ
R, becomes a complicated function of the main parameters of the process, such as the ones
involved in Eqs. 6–8, and the time t. Since we have built our time dynamics upon
the interfacial-region concept [41] introduced above, we might be wondering which is a morphological phase diagram of the process. While stating the diagram we may adopt a
typical rationale offered by surface-science literature, and simply look for D [Rt]Dt as
a function of βΦ
. From Eq. 16 and Eq. 17 one obtains D
[Rt] D
t = e
2 βΦ
, 18
what straight forwardly leads to Φ
≡ Φ
[Rt] = 1
2 β
× ln h
D [Rt]
D t
i .
19 This is now to say that
Φ is unambiguously determined by the global Smoluchowski-type
dynamics in the phase space as well as by the local temporal dynamics assigned to the protein velocity field in the interfacial region [39]. The morphological phase diagram can
then be built upon systematically exploring Eq. 16 in close connection with Eq. 18, that means, to have
σ
R ne
from Eq. 8 readily involved. It would enable someone to dis- criminate between different types of the nonequilibrium structures obtained, ranging from
equilibrium protein micelles in the solution, via “weakly” nonequilibrium non-Kossel SPC- A structures, to finally arrive at the highly nonequilibrium protein spherulitic cylindrolitic
fibrils-containing structures.
2.1.6. Ballistic, Hydrodynamically Stable Mode and Beyond It
When approaching the ballistic mode completely, we have to introduce the long-time su- perdiffusive scaling in a conventional way, namely
D [Rt] ∼ t
ν
, 20
where ν
≥ 0 holds for t t
o
. ν
= 0 corresponds to the standard diffusion, and can serve as a reference case, wherein
Φ = 0, and the detailed balance is assured this way in the R–space.
Otherwise, because of Φ
6= 0 it is violated and the process is out of equilibrium. The most intriguing case appears when
ν = 1, i.e. when R
2
t ∼ t
2
occurs, because now both global and local modes coincide, and the reduced variance,
κ t, given by Eq. 15 reduces,
for t t
o
, to a constant but again only if Eq. 11 under γ
→ 0 holds. Under such conditions the global R–space involving and local interfacial, v
t–engaging dynamic modes oper- ate at a fully synergistic level, greatly contributing to the overall hydrodynamics-stability
mode, presumably over all relevant dynamic scales, especially within meso- and to some extent micro-scales being mutually involved. It seems very promising from a theoretical
viewpoint but above all looks very important when seeing things from its practical, let us say, technological counterpart.
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
Kategori