40 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al.
path of 4 sites length of the flight is equal to 4, the ν
d
has been carefully obtained as ∼ 1,
see Fig. 9. Such realizations have already been tested experimentally by means of diffusion wave
spectroscopy [64, 83]. They confirmed a fully viscoelastic behavior of the tested matrix characteristic of the exponents such as those coming from our simulations, see Fig. 8. But
above all they confirm that the HP “flicker” walker quite typically performs a superdiffusive RW along a crystal’s terrace of the same physical nature. The RW exponent of such
realization equals one, as in case of analytic studies [14].
2.2. Theory and Experiment for Confined SPC-A and Protein
CrystalsAggregates
2.2.1. Non-Kossel Crystals Spherulites Seen in Terms of Aggregates’ Incorpora-
tion into Crystals Computer simulations [18] as well as certain analytic studies [55] are indicative of the fact
that when a special-purpose confinement is superimposed on the ’infinite’ system, being a growing polycrystal, the aggregation sub-events have to be taken seriously into account.
It is mainly due to the fact that when a space-limiting nanotemplate [17] is introduced to the system, the system yields a certain additional number of aggregates that are eventually ab-
sorbed by the growing crystal, presumably prompting an increase of the crystal-formation speed [43]. Such an observation must then be formally taken into account. Following a ra-
tionale presented in [18], and establishing the space limitation by noticing that there exists, close to a steady state, a finite radius in the SPC-A model [55] just conveyed above, one may
look for modification of the crystal’s speed, Eqs. 7-8 and 12, by means of proposing a reliable aggregates’ count in favor of the evolving system. This count is supposed to modify
directly the equivalent of the interfacial supersaturation,
σ
R ne
in such a way that it must be multiplied, cf. [18] and [55], by a factor that sets properly the aggregation influence. Such a
factor, denoted by δ
aggr m
takes into account which is an excess of aggregates of size m 1
when confronting it to m = 1. The following rationale can be developed. Based on [85] in
the outer part of the electrostatic Double Layer eDL, the number of let us say, monomers, m
= 1 and tetramers m = 4 appears to be twice so big than the corresponding number of the same units of most frequent appearance [43, 18] in the inner part of the eDL. It is so
because the osmotic pressure [86] outside but close to the crystal’s surface and inside at the crystal’s surface are roughly related with one another by a factor of two. But, because
of many obvious reasons [87], the crystal is assumed to absorb rather more tetramers than monomers. Thus, outside the crystal the described situation is ruled by a factor
δ
aggr out
= 4 always because on average a net mass excess of tetramers is always four times so big than
its monomeric counterpart. Inside the crystal, in turn, δ
aggr ins
= 4 − 1 = 3 since statistically a difference between the mass of m
= 4 unit and its m = 1 counterpart always prevails. It ultimately results in the experimentally well-evidenced [43] increase of the lysozyme
crystal’s speed, since in Eq. 7 deterministic picture or in Eq. 12 stochastic picture the asymptotic counterpart of
σ
R ne
, known as c
o
C − c
o
, to be detected in the late stage of the growing process, had to be multiplied finally by a
δ
aggr m
= δ
aggr out
δ
aggr ins
, which reads for the case presented as
δ
aggr m
=
4 3
≈ 1.33.... It is in excellent accord with [43], and has also been reliably confirmed by the computer simulation [18].
Can Modern Statistical Mechanics Unravel Some Practical Problems . . .
41
2.2.2. Viscoelasticity in Confined Systems
