Viscoelasticity in Confined Systems by Computer Experiment

44 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al. Figure 11 shows the MSD as a function of time for different sizes of the particle. At short times, the MSD follows a power-law characterizing subdiffusion, see Eq. 53. As shown in this section the value of the exponent seems to be controlled by HI. At large times, it saturates to a constant value due to the confinement imposed by the boundaries. Figure 12 shows the effective running diffusion coefficient calculated by taking the time derivative of Eq. 62, [65]. Qualitatively, the behavior of D t coincides with that reported from experiments, see Ref. [69]. The analysis performed in this section may be extended to the case when the forces applied by the boundaries have non-linear character see, for example, Ref. [8]. Those forces may have different origin. One important case however, is that of ionic channels in which the entropic forces as those analyzed in Sec. 2.1.10. seem to control the subdiffusion presented by the particles [90], in an entirely similar way as the one presented in this section.

2.2.3. Viscoelasticity in Confined Systems by Computer Experiment

The boundary conditions, whether designed for the confined system, case A, or just for its “infinite” viz unconfined counterpart, case B, do not change drastically the RW exponents, ν d –s involved in the scaling law 54, see Fig. 8. On the contrary, they do change it slightly just by a few percent at most. From the small-scale computer simulations presented in Sec. 2.1.11. it follows that because of the interaction-driven RW that we actually apply to the HP viscoelastic matrix, cf. Ref. [8], a chance of having on average a more vigorous RW appears if: i many weak-interactive paths, composed of P residues are distributed over the matrix in such a way that, at least, one percolation track is firmly assured; ii certain, long enough, glides of the walkers, equipped with seemingly realistic H − P flipping properties are allowed to succeed: these on average control the RW process, and enable one to get the desired dynamics thereof. Thus, even when the percolation threshold, picking somewhat between fifty and sixty percent, depending on whether the percolation under study is site- or bond- percolation the average threshold should be about ca. 55 [91] is reached, it suffices to perform few lattice constants lasting flights just to arrive at ν d ≈ 1 - the microscopic speed characteristic of amphiphile crystal growth [14]. Altogether, the confinement for such quite small systems, even if the MC trials are repeated many times, does not change visibly the value of ν d . It is also well known that such interaction-controlled RW, the behavior of which is modified by hydrophobicstructural forces [6, 13], can be employed to get elastic, or in more general terms, viscoelastic [92], properties of the 2 D amphiphilic material under study. From [64] and [8], cf. the references therein, it follows that when a truly asymptotic temporal behavior is attained, the response function of the material, termed the creep com- pliance [1], designated by χ cc , would obey a very similar squared scaling law, namely χ cc ∼ t 2 ν d . 63 Therefore, the complex shear modulus G ′′ [78, 83, 8], see Fig. 6, can be represented by another, this time inverse, scaling law G ′′ ∼ t −2 ν d , 64 Can Modern Statistical Mechanics Unravel Some Practical Problems . . . 45 where t - as above in Eq. 54; note that χ cc ∝ G ′′−1 holds finally. It is interesting to note that when ν d ≈ 3 8 , one gets the most common temporal behavior of G ′′ , namely G ′′ ∼ t − 3 4 , that represents linearly elastic harmonic response of the medium, e.g. that of the cytoskeleton [8]. This temporal property translated to frequency domain gives G ′′ ∼ ω i 3 4 , the most characteristic case of material’s linear viscoelastic response; bear in mind that ν d = 3 8 is sometimes called a de Gennes’ slightly-constrained diffusional exponent, cf. [93], or refer to original literature [94]. Let us point out, that the small-scale simulations presented show, that a non-Kossel amphiphilic crystal emerges when ν d 3 8 , distinctly, attaining exactly the limit of ν d = 1, just a ballistic-growth exponent [14]. From the above it can be concluded that, for example, an amphiphilic aggregate grows as a crystal in a terraces-involving way, when on average the creep compliance, χ cc , of the representative terrace responds in a quadratic fashion, or “nonlinearly”, in time. In other words, the complex modulus of the ’average’ terrace should behave with the interaction frequency, ω i , as G ′′ ∼ ω i 2 , cf. Fig. 6. Otherwise, when G ′′ ∼ ω i ε , with ε 6= 2 holds, the terrace’s material is either too soft ε 2 or too stiff ε 2 , thus, not responding properly in a crystal-like manner, which is ultimately to conclude, that an unorderly ag- gregate is presumably being formed, cf. the b.cs. of the analytic model [14], and refer to b.cs. given by Eq. 6. Note that the viscoelastic limit of growingyielding protein or, in general, amphiphilic crystals is still immersed within the soft-response range of ε 2. It is, however, worth noting that only large-scale computer simulation [95] can verify thor- oughly the values of the exponents derived. If they would have been successful, a combined analytic-simulational viz hybrid model could have become a powerful tool.

2.2.4. Nanocrystallography: New Theoretical Approach