Practical modern statistics

APS/Nova-Ch

Can modern statistical mechanics unravel some practical problems encountered in
model biomatter aggregations emerging in internal- & external-friction conditions?
A. Gadomski∗ and N. Kruszewska
Institute of Mathematics and Physics,
University of Technology and Life Sciences,
Kaliskiego 7, Bydgoszcz PL–85796, Poland

I. Santamaria-Holek
Facultad de Ciencias, Universidad Nacional Autonoma de Mexico,
Circuito exterior de Ciudad Universitaria, 04510 DF, Mexico

J.J. Uher
ZSMiO, Konopnickiej 6, Bielsko Biala 43-300, Poland

Z. Pawlak
Faculty of Chemical Technology and Engineering,
University of Technology and Life Sciences, Kaliskiego 7, Bydgoszcz PL–85796, Poland
Utah State Department of Health, Salt Lake City, UT 84113, USA

A. Oloyede
Queensland University of Technology, School of Engineering Systems, GPO Box 2434 Brisbane, Q 4001, Australia

E. Pechkova, C. Nicolini
Nanoworld Institute and Eminent Chair of Biophysics,
Genoa University, Corso Europa 30, 16132, Genoa, Italy,
Fondazione EL.B.A., Piazza SS. Apostoli, 66, 00187, Rome, Italy
(Dated: July 13, 2007)
Abstract
(Dis)ordered aggregations, in particular a crystal formation in biopolymer systems, appear recently
complex tasks to be undertaken by many specialists of research and technology, among which statistical physicists play their role, mainly in solving and/or elucidating the thermodynamic-kinetic
and dynamic aspects of the aggregations. Biomatter aggregations are ubiquitous in both natural
as well as laboratory systems, to mention but micelles emerging in aqueous solutions, biopolymer
spherulites or non-Kossel (single) crystals.
Although their thermodynamic behavior has mostly been studied very close to equilibrium, there
has appeared recently quite a new trend of benefiting readily from some out-of-equilibrium studies
on the amphiphilic systems listed above. It turns out natural to see an accompanying role of kinetic
effects, ranging over many time and space scales, somehow completing the way in which systems
make an attempt towards attaining, slowly or vigorously, or sometimes ”normally”, their possibly
closest-to-equilibrium states, measured along a naturally selected reaction coordinate.

The aggregations in question emerge naturally under a viscous (or, more generally, viscoelastic),
which is to say - internal-friction context. They appear because of typically asymmetric distribution(s) of structural viz hydrophobic forces throughout the amphiphilic system, thus they emerge
mainly thanks to hydrophobicity. Such aggregations show up some interesting viscoelastic properties, coming from a proper quantification of their diffusion-type non-Markovian characteristics.
The memory-involving properties mentioned are attributed to a broad survey of microrheological
effects, accompanying the aggregations of interest. They are seen at the levels of micelles as well as
of non-Kossel crystal formations. They manifest, however, under no external perturbation (load),
and the only interesting constraint one can announce is related to the degrees of freedom of the
system, pointing directly to its also structural, i.e excluded-volume effect. It is very important
from a practical viewpoint if one is capable of speeding up the process of interest, for example,
a lysozyme crystal formation. The origin of all charged particles involving formations of interest
is deeply rooted in the first law of thermodynamics for open Gibbs’ systems, here of amphiphilic
nature, complemented by the entropy-production equation and suitable (linear) flux-force relations
with Onsager’s coefficients playing their pivotal role.
The aggregations of micellar nature, in turn, may help in facilitating a process taking place in
external-friction conditions, that is, with an external perturbation (load) being applied. This can
happen to some model tribopolymerization systems, in which (e.g., reverse) micelles, emerging under
a response of the articular cartilage to a load, may easily play at least two pronounced roles: (i)

2
they can absorb and distribute some quite heavy energy inflow to a complex interlayer, named the

synovial fluid; (ii) they can facilitate the friction effect, contributing also hypothetically to a lowwear counter-effect, this time by a certain efficient roll-over sub-effect involved in the biolubrication
of two solid surfaces of the articular cartilage. The system of special interest, termed the articular
cartilage, being present in certain parts of our body, such as knees or hips, appears to be a highly
dissipative system, characteristic of an anomalous chemical reaction the dynamic behavior of which
ranges from regular to chaotic, also manifesting a certain sensitivity to a geometrical confinement,
being additionally prone to certain random-walk (load-assisted) microscopic rheological conditions of
versatile types. (At thermodynamic equilibrium, in turn, it exhibits an acid-base dynamic balance,
to be observed in the multilayer protective structure of the membranes involved.)
The solutions to versatile biomatter-aggregation tasks formulated in the internal-friction emphasized context, and in quite untypical, confinement-involving, dispersive conditions, look amenable to
a statistical-mechanics approach, mainly based on the Smoluchowski equation with time-dependent
Onsager’s coefficients, and with a Kramers’ barrier of controllable/adjustable characteristics. The
aggregations emerging within the friction-lubrication context of external-friction nature, being involved when studying the dissipative dynamics of the model articular cartilage, leading presumably
to formations of multilayers and (reverse) micelles, trapped within an interlayer between both solid
surfaces, are also manageable to the modeling in terms of a thoroughly designed autonomous dynamical system, being able to reflect a plethora of its dynamical (e.g., periodic) behaviors, manifesting
during the load-duration periods. Both main ways of modeling emphasize a practical role played
by statistical-mechanics in solving two types of particular non-ergodic problems of interdisciplinary
(bioengineering and medicine; materials and surface science) character.
PACS numbers: 05.10.Gg, 05.10.Ln, 05.40.-a, 81.10.-h, 83.50.-v, 87.15.Nn
Keywords: aggregation, crystal growth, micelles, viscosity, friction, mesoscopic system, fluctuationdissipation, biolubrication

Main abbreviations used throughout the
Chapter
IFS - internal-friction system
EFS - external-friction system
StatMech - statistical-mechanics and/or
statistical-mechanical
ES - Einstein-Smoluchowski; eES - extended
Einstein-Smoluchowski
CA - Coulomb-Amontons law
LB&S - Langmuir-Blodgett&Schaefer
AC - real articular cartilage; mAC - model articular
cartilage
SPC-A - spherical protein crystal/aggregate
aD3DS - anomalous dissipative 3D dynamical system
aRW - anomalous random walk
FP&S - Fokker-Planck & Smoluchowski
FL - friction-lubrication
N-G - nucleation-growth
MNET - mesoscopic nonequilibrium thermodynamics
AK - Avrami-Kolmogorov

RMs - reverse micelles

I.

INTRODUCTION

Biomatter aggregations, such as the ones of amphiphilic types, composed of proteins and/or lipids in a
water-based polyelectrolyte, have become recently a sub-

∗ Electronic

address: agad@utp.edu.pl

ject of intensive studies and interesting practical applications [1, 2].
As for the biomatter aggregations based on proteins in
aqueous solutions one could mention formations of protein (non-Kossel) crystals and aggregates [3–5].
As for the biomatter aggregations based mainly on
phospholipids one could invoke formations of monolayers, bilayers, reverse micelles as well as liposomes
[6, 7]. Both main types of aggregations listed above
may emerge in solutions, also those of physiological nature, showing up their basic properties in many different friction-lubrication (FL) contexts. Depending upon

whether the context is explicitly influenced by an exter−−→
nal force, Fext , causing some aggregation events in the
system to occur/change, one can divide them into two
−−→
groups:(i) internal-friction systems (IFS) when Fext ≈
−
→
0 ; (ii) external-friction systems (EFS) when clearly
−−→
−
→
Fext 6= 0 . The IFS can typically belong to microrheological systems [8], whereas EFS may generally develop
full rheological behavior [9], with shear effects and nonNewtonian characteristics as their main landmarks [10].
Both IFS and EFS are of permanent interest coming from
statistical-mechanical (StatMech) description.
As a matter of fact, the IFS are studied more intensively than the EFS[246] within the StatMech approaches
chosen to reveal their basic characteristics. It is likely
to come from the fact that IFS dynamics are typically
placed within a diffusion context, being the most ubiquitous context encountered in natural phenomena. Let us
remark that the notion of internal friction is equivalent to

the system’s viscosity, ηs , and typically, the viscosity is
related to the corresponding diffusivity Ds by means of

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the Einstein-Smoluchowski (ES) fluctuation-dissipation
formula, namely:
Ds ηs = β −1 ,

(1)

1

where β = kB T , kB - Boltzmann’s constant, T - absolute system’s temperature. Certain plausible extensions
of the ES formula, abbreviated throughout the whole
Chapter by eES, typically lead to a time-dependent (nonMarkovian) behavior that should always be assigned to
a particular system of interest [11].
In recent years, the EFS dynamics, in turn, have not
received considerable attention as far as the CoulombAmontons (CA) law,
µs =

Ff r
,
Fext

(2)

is concerned (with all quantities in the CA having
their usual meaning, e.g., µs is the static-friction coefficient). A real challenge appears, however, when frictionadhesion effects at a nanoscale molecule-size level, characteristic of pN forces, come into play. Then, the CA
context does not work effectively, and one must resort to
more microscopic description of any FL process of interest [12]. It is very important to realize that biomatter
aggregations appear as decisive factors controlling both
IFS and EFS dynamics [13]. However, due to its generality, such a statement, cannot be proved satisfactorily
in any other way than by examples. Therefore, in this
study, we have carefully selected two adequate IFS- and
EFS-involving examples that will emphasize the governing role of aggregation events, both at their dynamic and
quasi-static levels, in order to see how they could control
a specific process of interest.
Thus, as the proposed example of the IFS we would
like to offer a protein (poly)crystalline aggregation from
solution. First, we will introduce a (dis)ordered proteinaggregate formation in a space without confinement [14].

Next, we will try to reconsider the same process in
terms of a superimposed confinement, being typically of
Langmuir-Blodget & Schaefer (LB&S) type [15]. Such
confinement readily promotes incorporation of small ordered aggregates into a crystal’s microstructure [16].
This incorporation causes the speed of a protein crystal to
increase, which is an experimentally evidenced fact [17],
also confirmed by means of a computer simulation [18].
Here, we also present an analytical rationale that absorption of aggregates instead of monomers may increase the
crystal/aggregate growing pace.
As for the proposed EFS example, we have made an
attempt to embark on modeling a really complex but
extremely important system - this is the so-called articular cartilage (AC), the dynamics of which, whether
vigorous or slow, would influence the system’s behavior
substantially [19]. In the case of the AC it appears interesting to see even a slight departure from the acid-base
quasi-equilibrium of the system’s behavior, cf. [20]. A
description of such a, typically small, nonequilibrium departure leads to conjecture the dynamics of the system

as a tribopolymerization process [21]. Here, we wish to
reconsider the general tribopolymerization system [22] in
terms of a specific (first order) tribomicellization process

[23], in which a formation of lipid multilayers and (reverse) micelles [24] is believed to control the kinetics of
the AC friction-lubrication system, in which the lubrication is thought of to proceed much easier by emergence
of micelles and aggregates of various types. Their pivotal
role can be extensively discussed throughout the second
part of this study.
The Chapter is organized as follows. In the next part,
we are dealing with a model IFS which is taken for simplicity as a spherical protein crystal/aggregate (SPC-A).
Then, we are looking into main dynamic/kinetic features
of a model AC, abbreviated for the purpose of the present
study by mAC. Finally, we summarize our findings by
concluding about the usefulness of StatMech approaches
applied, and discuss some perspective of the proposal offered.

II. FIRST EXAMPLE:THE
INTERNAL-FRICTION SYSTEM (IFS)
EXPLAINED IN TERMS OF SPHERICAL
PROTEIN CRYSTAL/AGGREGATE (SPC-A)
A.

MNET-type theory of unconfined SPC-A

Brief overview
Proteins are main constituents of the living cells. They
frequently aggregate under a variety of physiological as
well as laboratory-designed conditions. Formation of ordered protein aggregates, such as non-Kossel crystals,
spherulites, fibrils , and lamellar crystals, yielding gel
phases, etc., becomes nowadays a formidable practical
task of modern science and technology [5, 25].
In case of protein crystals it also becomes a narrow bottleneck of recent crystallographic research and
structural-biology investigations towards resolving the
chemical structure of an individual protein of interest because both disciplines mentioned need high-quality crystalline material in order to carry out its systematic experimental (SAXS, AFM, etc.) investigations [15].
Amongst practitioners of protein crystal growth from
solution, constantly trying to improve the quality of the
crystals, there exists a belief that at least at low supersaturation the Burton-Cabrera-Frank (BCF) model of crystal growth by screw dislocation would suffice to describe
the formation, especially when it is completed by appropriate identification of the driving force(s), and when
having the set of main thermodynamic-kinetic and chemical parameters under sufficient control [3].
It is worth noting that even though the protein crystal phase is a minority phase, it can spread over a major
part of the system, and constitute a gel-like microstructure, mostly thanks to phase separation mediated by viscoelasticity of the solution [26]. A similar situation can

4
be expected in colloidal suspensions [27], and in both
protein- as well as colloid-containing solutions such a
mechanism is proposed to be responsible for phase ordering, although it, unfortunately, may finally induce defects into non-Kossel crystal’s structure. Such a behavior
is commonly called viscoelastic phase separation, and has
recently been established as an experimentally justified
fact, both in synthetic as well as natural polymeric systems [25–28].
Having right now permitted a historical excursion, let
us ascertain that a few methods are employed to produce
crystals. To make inorganic, e.g. semiconductor crystals,
we are privileged to choose the Czochralski [29] method,
and of course its modifications [30], or some methods
characteristic of the growth from undercooled melt or supersaturated solution. By the way, note that enormous
technological progress toward contemporary High Tech
has been gained due to an almost accidental discovery of
pulling out the single crystals from undercooled metallic
melt, produced with extreme invention about 1916 in a
German AEG lab (Berlin) by a crystallographer, chemist
and materials scientist Jan Czochralski, born in western
part of Poland (Kcynia near Poznań or Bydgoszcz), by
means of a synergistic stepwise recovery of the regular
atomic layers, constituting the needle-like single crystals
by applying a synchronized external clocking device, the
frequency of which was determined experimentally for
several common-use metals, quite of interest from electrical community of those as well as recent times [31]. Let
us underscore strongly that the method by Czochralski
looks confidential, and very similar to the truly comingfrom-nature crystal formation discovered by Burton and
coworkers (BCF).
As usual, life becomes more complicated and there is,
to authors’ best knowledge, no robust protocol developed
that can be applied to any protein crystal nucleationgrowth (N-G) phase transformations of practical interest
- it is always carefully corroborated for each individual
protein to be crystallized. The reason is due to tremendous complexity of the multi-parametric process we have
to deal with, thus, its description from the first principles
looks hopeless, and any use of approximate analytical and
numerical methods renders it possible for at most certain
specific systems of interest, e.g. that of lysozyme [4].
Thus, how can one remedy the problem that looks unsolved from the microscopic point of view? As one can
learn from the below presented rationale the mesoscopic
description might really be of help here [32].
In fact, in the fifties and sixties of the past century
J.W. Cahn made for metallurgical purposes an interesting attempt for formulating the theory of crystal growth
and interface motion in crystalline materials, also pointing to its mesoscopic character deeply rooted in irreversible thermodynamics. He mentioned certain properties of a crystal advancing toward its external phase,
such as continuous-growth property, whereby every element of the surface is capable of advancing normal to itself, whether the surface is diffuse or sharp, and whether

there exist or not, the singularities of the surface tension of the crystal. Trying then to judge whether the
continuous growth is possible to occur he introduced a
criterion of interaction of the crystallizing front with its
nearby surroundings called lattice resistance - it will be
seen in our model as some viscosity-dependent property
characteristic of the interfacial region of the spherical object of radius R that we will allow to grow in a bath of
Brownian-type particles (proteins) dispersed in the nearcrystal-surface aqueous solution.
The mesoscopic model that is offered here is qualitatively in accord with Cahn’s conception of continuoustype crystal growth [32]: It is also rooted in nonequilibrium thermodynamics at a mesoscopic level, commonly
abbreviated by MNET [33]. Its dynamics is well described by the Smoluchowski-type equation in a configurational space, with the Kramers’ barrier being involved
in the presented approach; moreover, a hydrodynamic
mode, revealed in the context of global vs. local dynamic properties of the complex system, appears to play
a unifying role towards showing up some common (hydro)dynamic features of the formation [13, 14].
Certain polycrystalline round-shaped nonequilibrium
microstructures, in turn, commonly termed spherulites,
become nowadays a real challenge both scientifically
and technologically [34]. In this study, we are going to convince the reader that the simple concept of
Avrami-Kolmogorov (AK) phase change, dealt with at
a MNET based level, is fully able to elucidate in which
kinetic and thermodynamic conditions may the spherical (three-dimensional case) or circular (two-dimensional
case) polycrystalline objects arise in an entropic milieu,
being mostly by virtue of the examples studied an aqueous protein solution.
The spherulites typically emerge during N-G phase
transformation in nonequilibrium conditions. The thermodynamic and kinetic conditions one may have in mind
are supposed to be of competitive nature. Irrespective of
the system in which the spherulites arise, whether biological (natural) or not, the competition concerns the
observation on how the main order-disorder effects manifest in the course of the spherulitic formation, see Fig.
1.
It has very recently been recognized that a reflection
of an interplay between the ordering effect on crystallization, and the disordering effect on interfacial instabilities
due to some formation of ”secondary” N-G front at the
interface, leads to switching-on a possibly second crystallization mode subject to the emergence of fibrils that
introduce polycrystalline misorientation-involving asymmetry to the system, thus spoiling the otherwise symmetric mode of the growth under consideration. It is then
envisioned that such a principle already described above
must be quite general, and to a large extent independent of the system details, namely, whether it consists of
small-molecule or large-molecule building blocks or not
[35].
We find the above observation in full agreement with

5
1.

FIG. 1: A schematic drawing of the protein spherulite: left,
simplified general view of its microstructure with the amorphous core in the center; right, the expanded part of the internal fibrillar microstructure of the spherulite, with the amorphous (blank) phase in between, for transparency shown in a
rectangular frame.

our fairly coarse-grained description of the spherulitic formation in a complex fluctuating medium of entropic character [14]. Based again on the first law of thermodynamics, and applying the well-known Gibbs’ equation for the
rate of positive entropy production in the system, we are
able, under a minimal set of necessary thermodynamic
and boundary-conditions concerning assumptions, to unambiguously derive a kinetic equation of Fokker-PlanckSmoluchowski (FP&S) type being capable of revealing
the evolution of a polynuclear system in the space of the
size of the nuclei, χ, called further the χ–space.
A careful analysis of the basic properties of such
a stochastic system, with specially emphasized phenomenological construction of its diffusion function
D(χ, t), where t is the time, virtually based upon the AK
type rationale, gives us univocally the main kinetic signature of the spherulite formation, namely, that it is a wavelike process, asymptotically arriving at a constant-speed
limit for the overall formation. In what follows we are
going to delineate the main landmarks of our argumentation on the formation of the protein-type spherulites
by the switching-on competitive mode, including formation of the spherulitic skeleton comprised of the rod-like
segments, cf. Fig. 1, the emergence of which is merely
described in terms of a fractional, typically one-half AK
exponent, νr ≈ 12 .
Anomalous transport of finite-sized particles in a viscoelastic medium is described in terms of a generalized
FP&S type equation containing memory effects. This
equation is derived from the principles of nonequilibrium StatMech and nonequilibrium thermodynamics at
the mesoscale, and is shown to account for the finite-size
of the Brownian particles. At large times, the description
leads to a non-Markovian diffusion equation for the mass
density field suitable to describe anomalous subdiffusion
and superdiffusion inside the intracellular environment.
This description is carried out in terms of the evolution of
the mean square displacement and compared with experiments. Good agreement between theoretical description
and experimental data is found.

The complex interfacial motion involving hybrid model
of crystal growth and (poly)crystalline aggregation

In what follows, the mesoscopics practically means a
deterministic kinetic description of the formation with an
account of the fluctuations of a properly chosen stochastic variable, included. The primary advantage of using
MNET for the ordered protein-spheroid formation is that
we do not arrive at a Langevin-type description, which
is a stochastic version of the Newtonian dynamics, thus,
being restricted to force-and-potential application. We
rather end up with a more flexible tool which appears to
be the Smoluchowski-type equation [36] for the probability density ρ(R, t) of forming a spheroid of radius R at
time t [28, 33, 37].
In the deterministic part, we start with a massconservation law in a form of the standard integrodifferential equation, where the integration goes over the
volume of a nucleus, V , closed by its surface area, S,
over which also the second matter-flux involving integral
is taken. The differentiation, in turn, goes over time and
is applied to the volume integral expressing a net mass
change of the system. The nucleus, assumed for simplicity in our model as being homogeneous (of constant density C) is fed by an external particle-concentration field,
c(r, t), composed of Brownian-type particles viz proteins,
and possibly their clusters, dispersed in the solution. All
sources of chemical reaction, virtually contributing to altering C and/or c(r, t) are being neglected, this way, after
applying the divergence (Gauss) theorem, giving rise to
state explicitly [38, 39]
∂
c(r, t) + div[J(r, t)] = 0,
∂t

(3)

where r denotes the position of a protein [40, 41], and
J represents the matter flux, i.e. ultimately arriving at
a local version of the above mentioned mass conservation law. After accepting the argumentation given in the
literature [39, 41], let us confine ourselves to choose a
ballistic-type case in which for a protein-type system the
matter flux is of mass-convective nature [39]
J(r, t) = c(r, t)v(r, t),

(4)

where v(r, t) represents the velocity vector field, acting
mostly in the interfacial region that can be limited to the
Stern-type macroion-depleted double layer enclosing the
growing object, see Fig. 2 [39, 42].
This influences the form of the deterministic kinetic
equation that results from applying, here without any
substantial loss of generality, Eq. (3) with Eq. (4) together, to a spherical object of radius R, with C =
const. too, assumed that now c(r, t) because of spherical symmetry finally becomes c(r, t) = ce (R). The equilibrium concentration 0 < ce (R) = 0,
K(| t − s |) = < v(t)v(s) >

f or

t 6= s,

(9)

where the time correlations K(s′ ), with the characteristic correlation time τcor ), given by means of an average
< ... >, are related to D(t) by
D(t) =

Zt

K(s′ )ds′

f or

s′ ≥ 0.

(10)

0

It is found out that the most interesting correlational
proposal to be offered here appears to be the following
[46]
 t 1−γ
τcor
D(t) ≃
×
1−γ
τcor

f or

t ≫ to ,

(11)

where γ ∈ (0, 1) is a characteristic fractional exponent,
and to is taken as an initial time. It can be anticipated
that [42], if γ → 0 is obeyed, the rounded protein-crystal
formation points to an overall ballistic, hydrodynamically stable growing asymptotic mode, since D(t) ∝ t
eventually applies. It greatly contributes to the total
hydrodynamic stability of the formation, especially in a
sufficiently mature growing stage (see, Fig. 4), since for
t >> to the late-time solutions of the stochastic
d
R = σR ne × v(t),
dt

FIG. 4: Two consecutive snapshots of the FP&S type pictures
in two different stages (A - early stage, B - late stage), showing how small randomly walking clusters, or large monomers,
are merged into some bigger ones. In the stage, B, under
inhomogeneous solution’s regime, there is a possibility of creating a ripe viz thermodynamically stable nucleus from which
the crystal formation may likely start to develop. (Note, that
the situation depicted by the stage A could possibly be seen
as a formation of a (dis)ordered cluster under diluted solution’s condition - for it the possibility of creating such nucleus
markedly diminishes in the course of time). By the envelopes
seen around each charged object the double layers, expected
to emerge as depletion zones in electrorheological solutions,
are marked [39, 42]. A liquid-liquid phase separation, enabling to form precrystalline aggregates, is also more likely to
occur in the stage B, cf. [48].

real challenge that can happen here is, however, that
Eq. (12) is fully equivalent to a Smoluchowski-type equation [36] in R–configurational space that can be solved
with suitable viz reflecting boundary conditions characteristic of a two-state dynamic process with a surmountable Kramers-type energetic barrier [47]. This barrier,
given by Φ, can be determined exactly, and both its
height and shape can be controlled by the fluctuations
[45] of the radius of the SPC-A given by the diffusion
function involved in the Smoluchowski-type dynamics,
D(R, t) = D[R(t)], defined still under t >> to as

D[R(t)] =

(12)

cf. relations (7) and (9), R ≡ R(t), are given by R ∼ t,
pointing to on-average-constant speed of the formation
dR/dt → const., which is true because σR ne typically
suppresses to a constant value, being an inverse of the
supersaturation, when t >> to , and v(t) → vmi = const.
on average as well.
There is another signature of the ballistic character of
the process, herein at the mesoscopic level. At a ”macroscopic level” of description one may see the relation given
by Eq. (4) as some signature of a ballistic character of
the process, whereas its microscopic counterpart corresponds to Eq. (11) correlation strength with γ → 0. The

R∞ 2
R ρ(R, t)dR
0

2t

,

(13)

whereby the above relation is the Einstein-type relation
in the one-dimensional phase space (R).

4.

Smoluchowski SPC-A scenario in R–space

In a picturesque way one can see the Smoluchowski dynamics as a cluster-cluster formation. Its first stage will
certainly rely on creation in a random way a single nuclei,
see Fig. 4A. Next, many nucleus can also be formed, see
Fig. 4B. The process as a whole can be described by the
FP&S formalism.

8
Thus, the global mass conservation law for the proteinspheroid object grown in the entropic milieu (e.g.,
lysozyme non-Kossel crystals or spherulites in a water
solution [39]) eventually yields the Smoluchowski-type
equation for the probability density ρ(R, t), also named
the probability distribution function (PDF)

surface, or within the interface [41], apparently in a certain conjunction with the protein-velocity correlational
field, coupled to the curvatures-involving crystal’s border.
5.

∂
∂ρ(R, t)
∂ 
D[R(t)]
ρ(R, t) =
+
∂t
∂R
∂R

∂Φ(R)
ρ(R, t) ,
+βD[R(t)]
∂R

(14)

where Φ(R) ≡ Φ[R(t)] becomes the free energy of the
thermodynamic process, ultimately contributing to the
determination of the Kramers’ barrier, and β = 1/kB T
with T the temperature. It is completely equivalent to
Eq. (12) with Eq. (9) and Eq. (10). Now, we may speak
R∞
of an ensemble-average >= Rρ(R, t)dR
0

which is a well-defined quantity. This is also the case
R∞
of >= R2 ρ(R, t)dR which stands for the

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[R(t)]/D(t) as a function of
βΦ. From Eq. (16) and Eq. (17) one obtains
D[R(t)]
= e2βΦ ,
D(t)

0

mean-squared displacement of the Φ–drifted superdiffusive process in the R–space [36]. Thus, the reduced variance, κ(t), given by
>

Φ ≡ Φ[R(t)] =

2

>

− 1,

(15)

as a measure of the fluctuations in the R–space, can be
determined too. Moreover, one is able to derive Φ(R) in
a Boltzmann-like form [39] as
1
Φ(R) = − ln[(σR ne )−1 ],
β

(16)

with (σR ne )−1 determinable from Eq. (8), this quantity
is suit to be the system’s nonequilibrium supersaturation. D[R(t)] is also determinable from the MNET-type
proposal just offered, and reads ultimately
−2

D[R(t)] = D(t)/[σR ne ]

,

(17)

where D(t) is generally given by Eq. (10) (a general timecorrelational proposal), or by Eq. (11) with the limit
of γ → 0, i.e. when the specific ballistic-type proposal pointing to the fully hydrodynamic stable mode
of the overall SPC-A process taking place within the
Stern-type double layer is envisaged. Note that by
comparing Eq. (17) with the Einstein-type definition of
D[R(t)] :=> /2t ( Eq. (13)), one has to
have for Φ(R) 6= 0, and ∆Φ(R) < 0 (thereby indicating
a natural thermodynamic course of the process), that
is a superdiffusion in our R–space. The case of Φ = 0
will particularly correspond to the standard (Einstein)
diffusion. When Φ(R) 6= 0 again but ∆Φ(R) > 0 appears to hold because of some unusual, e.g., auxetic-type
boundary elastic effects [39, 49], one may detect some
thermodynamic-kinetic anomalies either at the crystal

(18)

what straight (forwardly) leads to

2

κ(t) =

Towards a morphological phase diagram

h D[R(t)] i
1
.
× ln
2β
D(t)

(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 σR ne from Eq. (8) readily involved. It would enable someone to discriminate
between different types of the (non)equilibrium structures obtained, ranging from equilibrium protein micelles in the solution, via ”weakly” nonequilibrium nonKossel SPC-A structures, to finally arrive at the highly
nonequilibrium protein spherulitic (cylindrolitic) fibrilscontaining structures.
6.

Ballistic, hydrodynamically stable mode and beyond it

When approaching the ballistic mode completely, we
have to introduce the long-time (super)diffusive scaling
in a conventional way, namely
D[R(t)] ∼ tν ,

(20)

where ν ≥ 0 holds for t >> to . 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 >∼ t2 occurs, because now both

9
global and local modes coincide, and the reduced variance, κ(t), given by Eq. (15) reduces, for t >> to , 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 operate
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.
To sum up, we have formulated a complex-interfaceinvolving model of time (t) and temperature (T ) dependent SPC-A formations. We have shown that it is capable
of operating 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 microscale 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.

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 description 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 dynamic (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.
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 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-

+

FIG. 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 O2− 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.

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 + dχ)
at time t. One can then formulate the Gibbs entropy postulate in the form [33]
Z
ρ(χ, t)
S − Seq = −kB ρ(χ, t) ln
dχ .
(21)
ρeq (χ)
Here Seq 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 ρeq (χ) given by a Gibbs-Boltzmann type
formula


−∆W(χ)
,
(22)
ρeq (χ) ∼ exp
kB T
where ∆W(χ) is the minimum reversible work required
to establish that state, kB is Boltzmann’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.

10
In order to obtain the dynamics of the mesoscopic degrees of freedom [37] one first takes variations in Eq. (21)
Z
ρ(χ, t)
δS = −kB δρ(χ, t) ln
dχ,
(24)
ρeq (χ)
focusing only on the nonequilibrated degrees of freedom.
The probability density evolves in the χ−space along
with the continuity equation
∂ρ(χ, t) ∂J(χ, t)
+
= 0,
∂t
∂χ

(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]
Z
∂
dS
=−
JS dχ + σe ,
(26)
dt
∂χ
is the entropy flux, and
where JS = J(χ, t) ln ρρ(χ,t)
eq (χ)
σe = −kB

Z

J(χ, t)

∂
∂χ



ln

ρ(χ, t)
ρeq (χ)



dχ,

is the FP&S type equation [36] accounting readily for
the evolution of the probability density ρ in our χ-space.
This implies that the spherulitic formation of interest is
given the FP&S 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
dE = 3 and an area in dE = 2, where dE - the Euclidean
dimension of the space.
8.

Avrami-Kolmogorov (AK) phase-change model and its
two-phase modes

AK phase-change description continues to remain the
most popular method for obtaining crystallization kinetics information [51, 52]. The conceptual foundation of
this description is based on the famous combinatorial
(raindrop) problem leading to Poisson statistics. For
spherulites it can be reformulated by quantitatively determining the probability of a point being passed over
by exactly M evolving spherulites, ρsph (M ), being in the
original combinatorial description termed as the number
of the wave-fronts. ρsph (M ) takes on the standard Poissonian form

(27)

M

ρsph (M ) =

is the entropy production which is expressed in terms of
currents and conjugated thermodynamic forces defined
in the χ–space.
We will now assume a linear (meaning: small departure from the equilibrium) dependence between fluxes
and forces and establish a linear relationship between
them [33]


∂
ρ(χ, t)
J(χ, t) = −kB L[ρ(χ, t)]
ln
,
(28)
∂χ
ρeq (χ)
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)
∂ρ(χ, t)
∂
D(χ, t)ρeq (χ)
,
(29)
=
∂t
∂χ
∂χ ρeq (χ)
where we have defined the diffusion coefficient as
D(χ, t) ≡ kB L(ρ(χ,t))
. This equation, which because of
ρ(χ,t)
Eq. (22) applied together with Eq. (23) can also be
written as

e−M M
,
F(M)

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 ρsph (M =
0), eventually resulting in
ρsph (M = 0) = e−M ,

∂
∂χ

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−ωt ,

D(χ, t) ∂ρ(χ,t)
∂χ

+


∂∆W
+ D(χ,t)
kB T
∂χ ρ(χ, t) ,

(30)

(32)

since F(M = 0) = 1.
As a consequence of the above, ρsph (M = 0) also represents the points being still amorphous, 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 ρsph (M = 0) yields at
once

ν

∂ρ(χ, t)
=
∂t

(31)

(34)

where M = ωtν , resulting from a direct comparison of
the two last formulae, involves two parameters ω and

11
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 (bio)polymers the so-called
transcrystallization is a process in the course of which
nucleation does not prevail to be a rate-determining factor.
One of most solid observations on the spherulitic formation, as often as possible reported in the literature, is
that the formation occurs asymptotically in a constanttempo kinetic-thermodynamic regime [39–42, 44, 53–55].
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 appea