Viscoelasticity in Confined Systems

Can Modern Statistical Mechanics Unravel Some Practical Problems . . . 41

2.2.2. Viscoelasticity in Confined Systems

When addressing the dynamics and aggregation of SPC-A’s, it is very important to take into account the effects of the boundaries because they can modify the subdiffusion of the particles under study. In this case, the general approach offered in Sec. 2.1.10. can still be used if one introduces the effects of the boundaries on the averaged system’s behaviors. The presence of solid boundaries introduces two main effects on the dynamics of the particles: a hydrodynamic interactions particleboundary and b confinement. Hydrodynamic interactions HI particleboundary - From a microscopic point of view, these effects must be taken into account by considering the Oseen tensor of hydro- dynamics [88, 36]. It is clear that this approach introduces mathematical complications in the description that make it difficult to be manageable in an analytical way. However, ana- lytical progress is possible if we introduce in average form the stresses that the boundaries introduce through HI. That is, these stresses propagate and modify the state of motion of the fluid near the particle. As a consequence, the calculation of the force the host fluid exerts over the surface of the particle as the average stresses over the surface, is modified in such a way that the mobility or friction coefficient of the particle in the medium becomes rescaled by a factor depending upon the radius of the particle to its distance to the wall [88, 11]. Near to a solid wall, the first order correction to the mobility can be expressed in the form [88, 89] ζ t ≃ ζ 1 − 0.625 a h ζ ∗ t, 55 where h is the distance of the particle to the wall when moving parallel the boundary. The physical situation is shown in schematical form in Fig. 10. In writing 55, we have used the fact that the effective mobility coefficient is a function of time, thus being ζ ∗ a dimensionless function accounting for memory effects. Comparison between Eqs. 55 and 51 yields ζ ∗ t ∼ t τ D − 1 + δ 2 . 56 When the particle is moving near a “flexible” wall or in a arbitrary direction with respect to the boundary normal, at first order one may assume that the coefficient that modifies the mobility will change in the form ζ t ≃ ζ 1 +C a d ζ ∗ t, 57 where for simplicity we have considered that the particle is moving into a sphere of radius d, see Fig. 10. Thus, the combination C d −1 is a way to introduce in average form the effects of HI due to the boundaries. Confinement - A simple way to introduce the confinement due to the boundaries, i.e., finite-size effects imposed by the heat bath, is by means of a force entering into the evolution equation 46. The simplest although general model accounting for these effects is the linear force − ∇φ T = F w r = − κ w r , where the confinement constant κ w can be related with the characteristic linear size L of the heat bath by considering the relation κ w ∼ φ w L. Here, φ w is an energy characterizing the interaction with the wall. It is clear that the wall introduces a “passive” force, in the sense that it does not depend, for example, on the charge of the particles and its electrostatics interactions with an external potential, see Sec. 2.1.2.. 42 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al. Figure 10. Schematic situation of a particle diffusing through a viscoelastic medium in a heat bath imposing confinement. ~v represents the velocity of the particle. By taking into account these ingredients, the corresponding generalized FPS type equation 46 becomes ∂ ∂ t ρ = β −1 ζ t ∇ 2 ρ + ζ t κ w ∇ · [ ρ r ] . 58 In the case of a small particle moving through a viscoelastic medium made up by a polymer solution, an equation similar to 58 has been used in Ref. [11] to derive a parameter- dependent expression for the exponent, 1 + δ 2, entering into the power-law 53. In view of this, we may obtain a similar time functionality for the mobility ζ t by analyzing the behavior of the relaxation function χ r t = R xx ρ dx [81]. The evolution equation for χ r is d d ˜t χ r ˜t = − ακ w ζ 2 ζ ∗ ˜t χ r ˜t, 59 where for simplicity in notation we have introduced α = 1 + C a d , and the dimensionless variables ˜t = ζ −1 t and ˜r = d −1 r , see also Fig. 10. From this expression it is clear that the time dependence of the effective friction coefficient ζ ∗ t is determined by the behavior of χ r ˜t: ζ ∗ ˜t = 1 ακ w ζ 2 d dt ln χ r ˜t. 60 This equation can be used to obtain [11, 8] ζ ∗ t ∝ t ζ 1 ακ w ζ 2 −1 , 61 which is consistent with a stretched exponential form of the relaxation function typical of viscoelastic systems: χ r ∼ χ r0 exp h − αζ 2 C 1 ˜t 1 ακ w ζ 2 i . Here C 1 is a constant characterizing the relaxation. Can Modern Statistical Mechanics Unravel Some Practical Problems . . . 43 Using these results and Eq. 58, one may calculate the following explicit expression for the MSD hr 2 it ≃ 3d 2 κ 2 w ζ 4 [1 − 2C 1 a d ] × h 1 + coth 2 κ w ζ 2 C 1 1 − 2C 1 a d h t ζ i [1−2C 1 a d ] κ w ζ 2 i −1 , 62 where we have assumed a d ≪ 1 and expanded α −2 up to first order. Interesting, this expres- sion introduces both confinement and finite-size effects due to HI. Both are reflected in the explicit form of MSD as a function of time and on the exponent characterizing subdiffu- sion. It is convenient to mention that in describing anomalous subdiffusion, the exponent of the power law entering Eqs. 61 and 62 must satisfy the relation ακ w ζ 2 −1 ≤ 1, which imposes restrictions to the values of the parameters. 0.0001 0.001 0.01 0.1 1 t seconds 0.01 0.1 1 MSD arbitrary units a=0.001 a=0.05 a=0.08 a=0.1 Figure 11. Mean square displacement as a function of time as given by Eq. 62 for different radius of the particle. Confinement and finite-size effects are reflected in the slope of the MSD at short times and the saturation of the curve at large times. This result shows the correct behavior when compared with the experimental results of Ref. [69]. The values of the parameters used are: d = 1.2, κ = 1, ζ = 1, C 1 = 1. 0.001 0.01 0.1 1 ts 0.001 0.01 0.1 1 10 Dt arbitrary units a=0.001 a=0.05 a=0.2 a=0.1 Figure 12. Time dependent diffusion coefficient D t as a function of time. Dt has been calculated by taking the time derivative of the MSD given in Eq. 62. This result shows the correct behavior when compared with the experimental results of Ref. [69]. The values of the parameters used are: d = 1.2, κ = 2.0, ζ = 0.8, C 1 = 1. 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