Can Modern Statistical Mechanics Unravel Some Practical Problems . . .
33 misorientation related de-amorphization of the system, finally arriving, due to the competi-
tion mode mentioned, 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 disorder, 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 and b through FPS 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 FPS 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 FPS 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’ viscosity
η
e f f
ω , [73]. Similar conditions can also be found in, for
34 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al.
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 radius a and its mass are much larger than the polymers surrounding it, an apparently
universal
3 4
exponent is found [74].
Mesoscopic nonequilibrium approach to viscoelasticity - Consider the motion of a
testing spherical Brownian particle macroion, spherule of radius a through 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 PDF
ρ
r, t; cf. Eq. 14. Since the PDF is normalized, it will
