Practical modern statistics2

Editor: Byung-Soo Kim, pp. 13-98 c 2008 Nova Science Publishers, Inc.

Chapter 1

C

AN

M

ODERN

S

TATISTICAL

M

ECHANICS

U

NRAVEL

S

OME

P

RACTICAL

PROBLEMS

E

NCOUNTERED IN

M

ODEL

B

IOMATTER

A

GGREGATIONS

E

MERGING

IN

I

NTERNAL- & EXTERNAL-F

RICTION

CONDITIONS?

A. Gadomski1∗, I. Santamaria-Holek2, N. Kruszewska1, J.J. Uher3, Z. Pawlak4,5, A. Oloyede6, E. Pechkova7,8and C. Nicolini7,8 1_{Institute of Mathematics and Physics, University of Technology and}

Life Sciences, Kaliskiego 7, Bydgoszcz PL–85796, Poland 2_{Facultad de Ciencias, Universidad Nacional Autonoma de Mexico}

Circuito exterior de Ciudad Universitaria, 04510 DF, Mexico 3_{ZSMiO, Konopnickiej 6, Bielsko Biała 43-300, Poland}

4_{Faculty of Chemical Technology and Engineering,} University of Technology and Life Sciences Kaliskiego 7, Bydgoszcz PL–85796, Poland

5_{Utah State Department of Health, Salt Lake City, UT 84113, USA} 6_{Queensland University of Technology, School of Engineering Systems}

GPO Box 2434 Brisbane, Q 4001, Australia

7_{Nanoworld Institute and Eminent Chair of Biophysics, Genoa University} Corso Europa 30, 16132, Genoa, Italy

8_{Fondazione EL.B.A., Piazza SS. Apostoli, 66, 00187, Rome, Italy}

Abstract

(Dis)ordered aggregations, in particular a crystal formation in biopolymer sys-tems, appear recently complex tasks to be undertaken by many specialists of research and technology, among which statistical physicists play their role, mainly in solving

∗_{E-mail address: agad@utp.edu.pl}

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and/or elucidating the thermodynamic-kinetic and dynamic aspects of the aggrega-tions. Biomatter aggregations are ubiquitous in both natural as well as laboratory sys-tems, 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 gener-ally, viscoelastic), which is to say - internal-friction context. They appear because of typically asymmetric distribution(s) of structural viz hydrophobic forces through-out 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 mi-celles 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 for-mation. 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 tak-ing 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) 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 low-wear 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 re-action 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, disper-sive 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 emerg-ing within the friction-lubrication context of external-friction nature, beemerg-ing involved when studying the dissipative dynamics of the model articular cartilage, leading

pre-sumably to formations of multilayers and (reverse) micelles, trapped within an inter-layer 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 pe-riods. 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.

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

1.

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 subject 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, 4, 5].

As for the biomatter aggregations based mainly on phospholipids one could invoke for-mations 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 na-ture, showing up their basic properties in many different friction-lubrication (FL) contexts. Depending upon whether the context is explicitly influenced by an external 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 non-Newtonian 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 EFS1 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 the Einstein-Smoluchowski (ES) fluctuation-dissipation formula, namely:

Dsηs=β−1, (1) whereβ=_k1

BT,kB- Boltzmann’s constant,T - absolute system’s temperature. Certain

plau-sible extensions of the ES formula, abbreviated throughout the whole Chapter by eES, typ-ically lead to a time-dependent (non-Markovian) 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 Coulomb-Amontons (CA) law,

µs=

Ff r

Fext

, (2)

is concerned (with all quantities in the CA having their usual meaning, e.g.,µsis the static-friction coefficient). A real challenge appears, however, when static-friction-adhesion effects at a nanoscale molecule-size level, characteristic of pNforces, come into play. Then, the CA context does not work effectively, and one must resort to more microscopic description of anyFLprocess 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)crys-talline aggregation from solution. First, we will introduce a (dis)ordered protein-aggregate 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 or-dered 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 be-havior 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 de-scription of such a, typically small, nonequilibrium departure leads to conjecture the dy-namics 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) tribomicel-lization 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 lu-brication 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.

2.

First Example: the Internal-Friction System (IFS) Explained

in Terms of Spherical Protein Crystal/Aggregate (SPC-A)

2.1. 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 crys-talline material in order to carry out its systematic experimental (SAXS, AFM, etc.) inves-tigations [15].

Amongst practitioners of protein crystal growth from solution, constantly trying to im-prove 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 situ-ation can 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, 26, 27, 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 so-lution. By the way, note that enormous technological progress toward contemporary High Techhas 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 Pozna´n or Bydgoszcz), by means of a syn-ergistic 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 coming-from-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 nucleation-growth (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 Rthat we will allow to grow in a bath of Brownian-type particles (proteins) dispersed in the near-crystal-surface aqueous solution.

The mesoscopic model that is offered here is qualitatively in accord with Cahn’s con-ception of continuous-type crystal growth [32]: It is also rooted in nonequilibrium ther-modynamics at a mesoscopic level, commonly abbreviated by MNET [33]. Its dynam-ics 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, com-monly termed spherulites, become nowadays a real challenge both scientifically and tech-nologically [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-(three-dimensional case) polycrystalline objects arise in an en-tropic 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 sup-posed 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.

Figure 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.

It has very recently been recognized that a reflection of an interplay between the or-dering effect on crystallization, and the disoror-dering 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 de-tails, namely, whether it consists of small-molecule or large-molecule building blocks or not [35].

We find the above observation in full agreement with our fairly coarse-grained descrip-tion of the spherulitic formadescrip-tion in a complex fluctuating medium of entropic character [14]. Based again on the first law of thermodynamics, and applying the well-known Gibbs’ equa-tion for the rate of positive entropy producequa-tion in the system, we are able, under a minimal set of necessary thermodynamic and boundary-conditions concerning assumptions, to un-ambiguously derive a kinetic equation of Fokker-Planck-Smoluchowski (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 theχ–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), wheretis the time, virtually based upon the AK type rationale, gives us univocally the main kinetic sig-nature of the spherulite formation, namely, that it is a wave-like 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 de-rived 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 in-tracellular 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.

2.1.1. 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, in-cluded. 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 radiusRat timet[33, 37, 28].

In the deterministic part, we start with a mass-conservation law in a form of the stan-dard integro-differential 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 ex-pressing 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 alter-ingC and/or c(r,t) are being neglected, this way, after applying the divergence (Gauss) theorem, giving rise to state explicitly [38, 39]

∂

∂tc(r,t) +div[J(r,t)] =0, (3)

where r denotes the position of a protein [40, 41], and J represents the matter flux, i.e.

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].

Figure 2. Cartoon of the Stern-type double layer surrounding the SPC-A. The lysozyme-type proteins, performing their random walks (two trials are depicted), would exceed by their mean-free paths (contributed to by a λB) the width of the layer, λDL, which makes mostly time-correlational contribution to the formation dynamics, cf. text for additional explanation.

This influences the form of the deterministic kinetic equation that results from ap-plying, 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)<<Ccan then be taken, like in the Mullins-Sekerka (MS) instability concept, as

the one given by Gibbs-Thomson (GT) condition ce(R) =co[1+ΓD×K1(R)](ΓD- capil-lary constant;K1(R) =2/R- the mean curvature) but now applied for low-dimensional viz non-macromolecular crystals emerging from a supersaturated solution. The above yields the kinetic equation as follows

d

dtR=σR

e_×v

mi, (5)

where the dimensionless supersaturationσRe=ce(R)/(C−ce(R)), andvmi can be consid-ered as a parameter, herein an average constant velocity of the macroion near the spheroid surface. This is by no means the case of high-dimensional or protein crystal growth, cf. Fig. 3: Here a modification of the GT boundary condition has to be expected [39]. This and the stochasticity ofv(r,t)(thus, considered in a scalar form) will be discussed below.

2.1.2. (Non)equilibrium Boundary Condition

Modification of the GT equilibrium boundary condition for the protein crystal formation results in either some modified GT condition of equilibrium type (originating from more refined consideration about the molecular nature of the surface tension, and the additional

Figure 3. Protein crystallization by LB&S thin film nanotemplate: LB&S nanotemplate, used to modify classical hanging drop method [43], trigger the specific aggregates’ forma-tion in the protein drop soluforma-tion and thereby helps to overcome the nucleaforma-tion free energy barrier, see Fig. 13.

e.g. curvature terms it may contain) or gives rise to readily nonequilibrium effects at the boundary, especially the ones coming from elastic interactions [39] between the boundary proteins and its nearby electrolyte surroundings (also, within the boundary when some addi-tional strain between the vacancies may arise), or when spherulites composed of fibrils tend to emerge, even with a tendency of lateral growth violating here somehow the continuous-growth mode invoked by Cahn which suits also our type of reasoning [41].

Thus, the overall (non)equilibrium boundary condition can be proposed as follows [39, 42]

cne(R) =co

h

1+ΓD×K1(R) +δT2×K2(R) +Σ3i=1αix(i)(R)−βk×

dR dt

i

, (6) whereδT Tolman length,K2(r) =1/R2Gaussian curvature,αithe elastic coefficients and

x(i)(R) = (Ri−Ri_o)/Ri_o(Rois the initially taken object’s radius), with the latter representing three main crystal-surface nucleation mechanisms [39]. βkstands for the kinetic coefficient, anddR/dt points to the readily nonequilibrium character of the boundary condition [42], expected to occur for spherulites, especially those of lysozyme type. Because of stating Eq. (6) in such a general form one is then able to write down a more general kinetic equation [39]

d

dtR=σR

ne_×v

mi, (7)

where

σRne=

cne(R)

C−cne(R)

(8) under a necessary restriction that all (non)equilibrium mechanisms prescribed at the bound-ary, cf. Eq. (6), do not need to operate at the same time together. They can rather be switched on/off whenever physically legitimate [44, 40, 41, 45].

2.1.3. Protein-Involving nearby Velocity Field

In the deterministic description one may notice the Frenkel-type velocity of the macroion incorporated in the resulting equation, see vmiin Eq. (7). In order to achieve a more

real-istic description of the process one is motivated to offer a correlational proposal for v(r,t). Although it can separate into its spatial (r) and temporal (t) parts, in the very vicinity of the crystal,i.e. mostly in the diffusive part of its Stern-type double layer the spatial [42] cor-relations can be postponed since they are hard to detect in such a narrow depletion region, especially when the ballistic motion of the proteins is foreseen. In turn, the temporal corre-lations inv(r,t)cannot be ruled out. They arise because the viscosity in the diffusive part of the Stern double layer decreases in time when the crystal grows. Since the crystal becomes bigger the double layer expands too, and because the external-concentration conditions are assumed unchanged, a protein-influenced viscosity change near the crystal surface results in some viscosity decrease in time. This can be described as a depletion-zone effect around the growing object. It is due to some diffusivity, D(t), effective increase in time which comes from the time correlations inv(r,t)to be inferred from the Green-Kubo formula,i.e. when the fluctuation-dissipation theorem can be applied [33]. Thus, after postponement of the spatial correlations, the correlational proposal may be a stationary Gaussian but correlated, though with a zero average velocity part

<v(t)>=0, K(|t−s|) =<v(t)v(s)> for t6=s, (9) where the time correlations K(s′), with the characteristic correlation time τcor), given by means of an average<· · ·>, are related toD(t)by

D(t) =

t

Z

0

K(s′)ds′ for s′≥0. (10) It is found out that the most interesting correlational proposal to be offered here appears to be the following [46]

D(t)≃ τcor 1−γ×

t τcor

1−γ

for t≫to, (11)

whereγ∈(0,1)is a characteristic fractional exponent, andtois taken as an initial time. It can be anticipated that [42], ifγ→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 forma-tion, especially in a sufficiently mature growing stage (see, Fig. 4), since for t>>to the late-time solutions of the stochastic

d

dtR=σR

ne_{×v(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 becauseσRnetypically suppresses to a constant value, being an inverse of the supersaturation, when t>>to, andv(t)→vmi=

const.on average as well.

There is another signature of the ballistic character of the process, herein at the meso-scopic level. At a “macromeso-scopic 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 real challenge that can happen here is, however, that Eq. (12) is fully equivalent to a Smoluchowski-type equa-tion [36] inR–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 undert>>toas

D[R(t)] = ∞

R

0

R2ρ(R,t)dR

2t , (13)

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

2.1.4. Smoluchowski SPC-A Scenario inR–space

In a picturesque way one can see the Smoluchowski dynamics as a cluster-cluster forma-tion. 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.

Thus, the global mass conservation law for the protein-spheroid 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)

∂

∂tρ(R,t) =

∂ ∂R

D[R(t)]∂ρ(R,t)

∂R +βD[R(t)]

∂Φ(R) ∂R ρ(R,t)

, (14)

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

0

Rρ(R,t)dRwhich is a well-defined quantity. This is also the case of <<R2(t)>>=R∞

0

R2ρ(R,t)dRwhich stands for the mean-squared dis-placement of theΦ–drifted superdiffusive process in the R–space [36]. Thus, the reduced variance,κ(t), given by

κ(t) = <<R

2_(t)>>

<<R(t)>>2−1, (15) as a measure of the fluctuations in the R–space, can be determined too. Moreover, one is able to deriveΦ(R)in a Boltzmann-like form [39] as

Φ(R) =−1

Figure 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 solu-tion’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 de-picted 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 dimin-ishes 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].

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

D[R(t)] =D(t)/[σRne]−2, (17) where D(t) is generally given by Eq. (10) (a general time-correlational 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)]:=<<R2(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 ofΦ=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 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.

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[R(t)]/D(t)as a function ofβΦ. From Eq. (16) and Eq. (17) one obtains

D[R(t)]

D(t) =e

2βΦ_{, (18)}

what straight (forwardly) leads to

Φ≡Φ[R(t)] = 1

2β×ln

hD[R(t)]

D(t)

i

. (19)

This is now to say thatΦis 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 σRne from Eq. (8) readily involved. It would enable someone to dis-criminate between different types of the (non)equilibrium 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-per)diffusive scaling in a conventional way, namely

D[R(t)]∼tν, (20) whereν≥0 holds fort>>to.ν=0 corresponds to the standard diffusion, and can serve as a reference case, whereinΦ=0, and the detailed balance is assured this way in theR–space. Otherwise, because ofΦ6=0 it is violated and the process is out of equilibrium. The most intriguing case appears whenν=1,i.e.when<<R2(t)>>∼t2occurs, because now both global and local modes coincide, and the reduced variance,κ(t), given by Eq. (15) reduces, fort>>to, to a constant but again only if Eq. (11) underγ→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.

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 O2−ion and twoH1+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

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 theχ–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 stateχ∈(χ,χ+dχ)at timet. One can then formulate the Gibbs entropy postulate in the form [33]

S−Seq=−kB

Z

ρ(χ,t)lnρ(χ,t)

ρeq(χ)

dχ. (21) HereSeqis the entropy of the system when the degrees of freedomχare 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

ρeq(χ)∼exp

−∆

W

₍χ₎

kBT

, (22)

where∆

W

₍χ_{)is the minimum reversible work required to establish that state, kB is}

Boltz-mann’s constant, andT 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 areaA, whereasσstands for the surface tension.

In order to obtain the dynamics of the mesoscopic degrees of freedom [37] one first takes variations in Eq. (21)

δS=−kB

Z

δρ(χ,t)lnρ(χ,t)

ρeq(χ)

dχ, (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 ∂

whereJS=J(χ,t)ln_ρeq(χ)ρ(χ,t) is the entropy flux, and

σe=−kB

Z

J(χ,t) ∂ ∂χ

lnρ(χ,t)

ρeq(χ)

dχ, (27) is the entropy production which is expressed in terms of currents and conjugated thermo-dynamic forces defined in theχ–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) =−kBL[ρ(χ,t)]

∂ ∂χ

lnρ_ρ(χ,t) eq(χ)

, (28)

where L≡L[ρ(χ,t)] is an Onsager’s coefficient, which in general depends on the state variableρ(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)ρeq(χ)

∂ ∂χ

ρ(χ,t) ρeq(χ)

, (29)

where we have defined the diffusion coefficient asD(χ,t)≡kBL(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)

kBT

∂∆

W

∂χ ρ(χ,t)

, (30)

is the FP&S type equation [36] accounting readily for the evolution of the probability den-sityρ 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 indE =2, wheredE - the Euclidean dimension of the space.

2.1.8. 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 exactlyMevolving spherulites,ρsph(M), being in the original combi-natorial description termed as the number of the wave-fronts. ρsph(M)takes on the standard Poissonian form

ρsph(M) =

e−MMM

whereMstands 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, (32)

since F(M=0) =1.

As a consequence of the above,ρsph(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−θas the crystalline fractional completion. Comparing then 1−θwithρsph(M=0)yields at once

θ=1−e−M, (33)

which provides the formula for the crystallized fraction.

The problem finally reduces to determineMas a function of geometric assumptions on the nuclei forms as well as the timetafter 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−ωtν, (34)

whereM=ωtν, resulting from a direct comparison of the two last formulae, involves two parametersωandν.ωis recognized to be dependent on the shape of the growing crystalline entities as well as on the amount and type of nucleation. νdepends 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 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 off/on the Asymmetric Growing Mode

Let us consider a situation in which the phase change takes place in a system of constant total volume,Vsph =const.[53, 54, 52]. The growth rate of the spherulitic formation can

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)

Tρ

kBT

∂ρ(χ,t)

∂χ +ρ(χ,t) ∂Φ

∂χ

. (35)

InterpretingΦ≡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 χδ, where δ= (dE−1)/dE, with dE - 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)

kBT

∂Φ

∂χρ(χ,t), (36)

whereD(χ,t)is a spherulitic diffusion coefficient, asymptotically obeying [53, 54]

D(χ,t) =D0χδtµ. (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 exponentδbeing 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 dE+1 [56] and upon the rod-like spherulitic-skeleton involving behavior, represented by the exponent νr, usually obeyingνr≈1₂[53, 54].

The latter comes from the fact that the time-dependent part of D(χ,t)from Eq. (37) originates from a smallωapproximation (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 ofθ(t)

θ(t)∼tν, (38)

wherein ν≃νr ≈ 1₂ finally applies [53, 54]. This power-law type ∼tνr 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 FP&S context[36].

Thus, the overall exponentµcan be defined, similarly as in a previous study [53, 54, 56],

i.e.by means of a simple competition-type formula, as follows

µ≡µ(dE,νr) =ν(dE)−νr, (39) whereν(dE) =dE+1 (adE–dependent part) andνrgenerally obeysνr∈[1₂; 1]with a strong preference to νr≈ 1₂, i.e. when the nucleation of rods is a-thermal but its rate manifests diffusively (kinetically), that means, not in an entirely thermodynamic way.

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, withR 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. νr=0. Thus, there is no above mentioned competition involved, and finally the overall exponent coincides with the Kolmogorov amorphization measure,

µ(dE,νr=0) =ν(dE) =dE+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(dE) =dE+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.νr6=0, one switches on the competition (asymmetry) mode, and finally arrives at

µ(dE,νr6=0) =ν(dE)−νr≈dE+1− 1

2≡dE+ 1

2, (41)

which leavesµ(dE,νr6=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 tempodR/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 (poly)crystalline formations in model protein systems, and their characteristic integer vs non-integer Kolmogorov-type measures coming from

application of the MNET-type formalism withdE=2,3, cf. text. Type of formation Geometric-exponent value

System amorphization:

dR/dt6=const. µ(dE,νr=0) =dE+1 Spherulitic formation:

dR/dt≈const. µ(dE,νr6=0)≈dE+1₂

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 ofρ(χ,t),<ρ(χ,t)}i_>,

i=1,2, where <ρ(χ,t)1>=Vsph=const.,<ρ(χ,t)0>- well-determined from the overall FP&S formalism, and<ρ(χ,t)0>

misorientation related de-amorphization of the system, finally arriving, due to the competi-tion mode mencompeti-tioned, 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

Outlook - As discussed previously, thermal fluctuations are central in the formation and behavior of soft materials. In consequence, entropic forces are determinants of the mi-crostructure formation and thus a main aspect of the (dis)order, slow dynamics and kinetics that will be the common ingredients in the systems of our interest.

Since these thermal fluctuations belong to the realm of mesosocopic world, the dynam-ics of the systems must be described by appropriate techniques. Generally speaking, one has two options for carry out this description: a) through Langevin equations andb) through FP&S type equations.

On the one hand, due to the viscoelastic nature of the heat bath in which the particles perform their Brownian motion, the use of Langevin equations requires the specification of the appropriate noise correlation [63, 50, 64] as well as the external forces. This is not clear in the general non-linear case due to the non-Markovian nature of the dynamics. As a consequence, it becomes difficult to perform the corresponding analysis in both analytical and numerical way. On the other hand, kinetic equations of the FP&S type, may incorporate memory effects through the dependence on time of the transport coefficients, or by introduc-ing memory functions [65, 66, 11, 8]. In principle, this second formulation of the problem seems to be more suitable to be generalized to the case when entropic or energetic barriers are present, in fact typically occurring in the systems under consideration, and when spatial restrictions and confinement are important. Here we will use the MNET formalism to de-rive these FP&S type kinetic equations for the PDF depending on the variables of interest [33, 42, 14, 67, 8]. The advantage of using it will become clear later.

Our first approach to the problem, still presented in Part 2.1., will be analytical, a cor-responding numerical analysis in a nonlinear case will be performed in Secs. 2.1.11., 2.2.3. and 2.3.2.. Moreover, we will first study the dynamics of passively diffusing particles in unconfined spaces. The generalization to the case when confinement and finite size effects are important will be considered later, in Part 2.2.. It is worth stressing now that in ex-periments and computer simulations confinement is frequently unavoidable. This fact does not invalidates the results of this section because they can be considered valid at short or intermediate times.

Experiments and computer simulations have been performed to characterize the (single-point) viscoelastic properties and structure of, for instance, polymer gels or actin networks [68, 69, 70]. Other processes can be present in these system, they can involve dynamical processes associated with conformation and growth of ’biomolecules’ [71, 42, 72]. Despite this complexity, the medium can be assumed as a viscoelastic matrix with a frequency dependent ’effective’ viscosityηe f f(ω), [73]. Similar conditions can also be found in, for

instance, the growth process of crystals or biomolecules in whose conformation memory, elastic and finite-size effects play an important role [42, 39].

At certain time scales, the main feature of their properties is the power-law behav-ior of the complex shear modulus [68], the creep compliance and the diffusion coefficient [69]. The viscoelasticity of these systems can be studied by means of microrheological techniques, such as the diffusing wave spectroscopy or video based methods which may characterize them in terms of the mean square displacement (MSD) of test particles that undergo subdiffusion [74, 75]. The MSD of the particle manifests a power-law dependence on time in which, in the case of small particles, the exponent can depend on the aspect ratio between the particle radius and the characteristic length of the polymer network [11] for certain values of these quantities. In the case when the linear dimension of the particle (i.e., its radiusa) and its mass are much larger than the polymers surrounding it, an apparently universal 3₄ exponent is found [74].

Mesoscopic nonequilibrium approach to viscoelasticity - Consider the motion of a testing spherical Brownian particle (macroion, spherule) of radiusathrough a complex fluid composed by other Brownian particles (macroions, spherules) or by polymer molecules. The presence of these particles introduce spatial non-homogeneities and act on the test particles through electrostatic and elastic forces [11, 8].

As we have mentioned previously, at diffusion times the relevant microscopic variable determining the state of the test particle is the position vector r. Hence, the dynamics can be described by means of the PDFρ(r,t); cf. Eq. (14). Since the PDF is normalized, it will obey the continuity equation

∂ ∂tρ=−

∂

∂r·(ρVr), (42)

where the explicit form of the probability diffusion current ρVrcan be found by assuming local equilibrium and using the rules of MNET [33, 76]. To obtain ρVr, this thermo-kinetic formalism uses an irreversibility criterion based upon the generalized Gibbs entropy postu-late [77, 33]

s(t) =−kB

Z

ρ(r,t)ln_ρρ le

dr+sle, (43) cf. Eq. (21), wheres(t)is the nonequilibrium entropy, sle the entropy at local equilibrium. In Eq. (43), the reference state is characterized through the local equilibrium PDF of Gibbs-Boltzmann form

ρle(r) =eβ[µle−∆φT], (44)

whereµleis the chemical potential at local equilibrium,∆φT(r) =φe+φB the total interac-tion potential related with external forces φe(r)and with interactions between the particle and the bathφB(r), [28]. These interactions can be considered separately from those due to external agents, because of the two well known “opposite” roles the bath plays in the dynamics of the particle: Supplying thermal energy and introducing dissipation. Note that, since energy dissipation is due to surface forces, in general it depends on the size of the particle.

Specifically, the potential∆φT(r)may in general involve energetic as well as entropic barriers [67] and, as a consequence, it could be responsible for Krames’ type dynamics of the system and thus useful to describe aggregation process, see Ref. [14] and references

therein. In fact, this local equilibrium PDF can be written in a more general and suitable form by expressing it in terms of the minimum work necessary to change the state of the system [67], see Eq. (22), exactly in the form of Eq. (23) where ∆φT(r)≡∆

W

. This ex-pression also includes, among others, the presence of activated volumes∆V(pthe pressure) and surface effects through the surface tension σ.

Now, using Eqs. (42)-(44) and (23), it is possible to derive a generalized FP&S type equation in the position space for ρ(r,t)[8, 11, 33, 14]. To this end, one may first calculate the entropy production of the system by taking the time derivative of Eq. (43), after using (42), and integrating by parts assuming that the fluxes vanish at the boundaries. Once the entropy production is obtained, linear laws can be assumed for the relation between forces and fluxes

Vr=−ζ(t)∇φT−D(t)∇ρ(r,t), (45) where ζ(t)is the time dependent Onsager coefficient entering through the linear law as-sumption [11, 33, 14]. Since ζ(t) plays the role of an effective mobility, we have in-troduced, by a fluctuation-dissipation formula, the time dependent diffusion coefficient

D(t) =β−1_{ζ(t), which constitutes a generalization of the ES formula, the eES formula.}

The substitution of Eq. (45) into (42) yields

∂

∂tρ=ζ(t)∇·[ρ∇φT] +β

−1_ζ(t)∇2_{ρ, (46)}

which is the desired generalized FP&S type equation. The time dependence of the transport coefficients introduces memory effects in the description [65, 28], whereas the first term at the right-hand side of the equation accounts for external and bath interactions with the particle, and thus is suitable to be used in both unconfined and confined motion, and in IFS and EFS conditions.

For test particles moving under IFS in an unconfined space (the physically more simple case), and with linear dimensions sufficiently large when compared with the length char-acterizing the heat bath, for instance the characteristic length of the polymer network in polymer solutions, the host complex fluid can be assumed as a continuum. In this case, at the mesoscopic level of description, the interactions between the test particle and the other components of the bath can be assimilated into the time dependence of the diffusion or ef-fective friction coefficients. Notice however that, from a microscopic point of view, such interactions are responsible for the anomalous subdiffusion performed by the particle.

Under these assumptions, it is not necessary to take into account in explicit way the interaction potential: ∆φT(r) = 0. As a consequence, the evolution equation for the test Brownian particle becomes

∂

∂tρ(r,t) =β

−1_ζ(t)∇2_{ρ(r,t). (47)}

The mesoscopic effective properties of the viscoelastic medium are often determined by analyzing the time dependence of the mean square displacement (MSD) hr2i(t)or, equiva-lently through the creep complianceχcc(t)or the complex shear modulusG′′msd(ω), withω the frequency [64, 68, 69, 11, 8]. The subindexmsd stands for the fact that this quantity is obtained by using the generalized ES relation [78, 64], i.e., by measuring the MSD of the

test particle by means, for instance, of diffusing-wave spectrometry techniques or video-based methods [69, 68, 74, 75]. As an example, it is convenient to mention that in a certain range of frequencies, it has been shown in experiments that the complex shear modulus follows the scaling behavior G′′_msd ∼ωα with α<1 [64, 74, 75]. Bear in mind that this power-law behavior is the cornerstone of microrheological viscosities.

These experimental results can be explained in the context of hydrodynamics for which it has been proven that the mobility has the general form ˆζ(ω) =ζ0τD

1+aλ−1(ω)−1 , [79, 80, 76], Here we have introduced τD as the characteristic diffusion time in order to keep the correct dimensions. Here,ζ0is the inverse Stokes friction coefficient and λis the

so called viscous penetration length λ−1=p

iω/νK, where νK is the kinematic viscosity of the host fluid [80]. When the host fluid is viscoelastic, the kinematic viscosity becomes frequency dependentνK(ω)and then one may assume the form (48). For frequencies lying in the range 1/τD<ω≪β0,i.e., times satisfyingτD>t≥β−01, the mobility coefficient can

be written as

ˆ

ζ(ω)≃ζ0τD−1(τDω)−(

1−δ

2 )_{. (48)}

The exponentδcharacterizes the subdiffusion. At this level of description, its value can be justified in terms of the characteristic dimension of diffusing particle [8, 71] or by simple comparison with the experiment. In a more detailed description, it can be justified that its value is determined by both, the elastic forces of the viscoelastic medium and the hydrody-namic interactions [11].

Here, it is essential to point out that the inverse Laplace transform of Eq. (48) yields a memory function ˜ζ(t), this shall not be confused with the time dependent mobility coeffi-cientζ(t)[65, 81]. Both quantities are related through the relation

ζ(t) =

t

Z

to

˜

ζ(z)dz. (49)

which represents a time average of the memory function.

It is interesting to notice that this interpretation also arises when considering the non-Markovian FP&S equation in the complete ordering prescription (COP), that is, in which memory effects are introduced through memory functions [65, 82]. The actual descrip-tion involving time-dependent coefficients [technically called partial ordering prescripdescrip-tion (POP)], can be related with the COP by taking into account that, in the case of slowly vary-ing fields, the integral becomes a temporal average of the memory function. The relation between both prescriptions is treated in detail in Refs. [65] and [63].

In view of these considerations, we have to accept that the inverse Laplace transform of Eq. (48) gives the memory function

˜

ζ(t)≃ζ0τ−D1 1

Γ(1−δ

2 )

t−(1+δ2 )τ−(

δ−1 2 )

D , (50)

which in turn can be integrated over time in order to obtain the effective time dependent mobility

ζ(t)≃ 2ζ0τ

−1

D

(1−δ)Γ(1−δ

2 )

t

τD

−(1+δ₂ )

This expression is useful since it can be compared with experimental results in order to obtain the value of the exponent δ for each viscoelastic medium. This can be done by examining the time dependence of the MSD, defined by hr2i(t) =R

r2ρ(r,t)dr. The evolution equation for hr2i(t)can by calculated from Eq. (47) and by using the definition above [11]. One obtains

dhr2i(t)

dt =D(t) =β

−1_{ζ(t), (52)}

where, at first, we have equality in which we have used the definition of the current diffusion coefficient [65]. From this equation one obtains that the MSD is

hr2i(t) = 4ζ0β

−1 (δ−1)(δ−3)Γ(1−δ

2 )

t

τD

3−₂δ

(53) Eq. (53) can be compared with experimental and theoretical results [64, 8, 68]. A typical behavior observed ishr2iexp≃t

3

4, from which one infers thatδ=3

2. Moreover, Eq.

(53) leads to good agreement with the large frequency experimental behavior G′′_msd ∼ω34,

reported, for instance, in Ref. [74]. The comparison can be established by using the well known relationsζ(ω)∝ η−_{e f f}1(ω)andG′′_msd(ω)≃ωηe f f.

Here, it is important to stress that, when the transported particles are smaller and lighter than the particles forming the effective medium, the expression for the exponent δdepends on the ratio between the radius of the particle to the characteristic length of, for exam-ple, the polymer network [75, 11]. This case will be studied in section corresponding to viscoelasticity in confined systems below.

0.1 1 10

ω(Hz) 0.1

1

G’’(

ω)

(arbitrary units)

τD=0.1 τ_D=0.2

τD=0.5 τ_D=1.0

Figure 6. Dependence of the complex shear modulusG′′ as a function of the frequency ω for different values of the characteristic relaxation time (see text). This behavior is typical for the cytoskeleton of eukariotic cells, among other systems.

To illustrate a characteristic behavior of the complex shear modulus, Fig. 6 shows the complex shear modulus as a function of the frequency given by the expression G′′_msd ≃

G′′₀(1+ζ−

1 4

0 ω

3

4), whereG′′

0 is a reference value and we have assumedζ0as a characteristic

time, see Ref. [8]. The value of the reference complex shear modulus wasG′′₀=0.1. Among others, this behavior of the complex shear modulus as a function of the frequency is typical for the cytoskeleton of eukariotyc cells, see, for instance, Ref. [83].

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