Screening of hydrodynamic interactions

Theoretical Outline — Collective Diffusion and Screening of Hydrodynamic Interactions... 

A model that can take these findings into account is based on the idea that the screening of hydrodynamic interactions is incomplete and that a residual part is still active on distances r > H(c) [40,117]. As a consequence the solvent viscosity r s in the Oseen tensor is replaced by an effective... 

Fig. 59. Incomplete screening of hydrodynamic interactions in semi-dilute polymer solutions. Presentation of different regimes which are passed with increasing concentration. A,C Unscreened and screened Zimm relaxation, respectively, B enhanced Rouse relaxation. (Reprinted with permission from [12]. Copyright 1987 Vieweg and Sohn Verlagsgemeinschaft, Wiesbaden)...

With respect to the screening of hydrodynamic interactions, one is confronted with the occurrence of a multiple-transition behavior. Instead of the expected crossover from ordinary (unscreened) Zimm to enhanced Rouse relaxation, one observes, at increasing concentrations, additional transitions from enhanced Rouse to screened Zimm and from screened Zimm to enhanced Rouse relaxation. This sequence of crossover effects are highly indicative of an incomplete screening of hydrodynamic interactions. 

Borsali, R., Vilgis, T. A., and Benmonna, M., Viscosity of weakly-charged polyelectrolyte Solutions the screening of hydrodynamic interactions, Macromol. Theory SimuL, 3, 73-77 (1994). 

In concentrated suspensions of charged colloidal particles, the strong Coulomb interaction prevents the particles from moving freely and leads to an effective screening of hydrodynamic interaction [66]. Such a screening may be particularly... 

Riese DO, Wegdam GH, Vos WL, Sprik R, Fenistein D, Bongaerts JH, Griibel G (2000) Effective screening of hydrodynamic interactions in charged colloidal suspensions. Phys Rev Lett 85(25) 5460-5464... 

Muthukumar, M. and Edwards, S.F., 1983. Screening of hydrodynamic interaction in a solution of rodlike macromolecules. Macromolecules, 16,1475-1478. 

The coupled Brownian motion of polymer segments can be resolved into a linear combination of the motion of Rouse modes, essentially a Fourier transformation in position along the chain. The Rouse modes behave as independent, overdamped, harmonic oscillators. Hydrodynamic interactions arise from the flow field accompanying one segment s motion creating flow at other segments. Increase of concentration leads to a screening of hydrodynamic interactions. 

In both poor and good solvents, one expects the screening of hydrodynamic interactions to increase the net friction felt by the chain, causing the mean cyclization rate to decrease. In good solvents, this retardation is compensated by excluded volume screening. From the net increase in Ie/Im for I in THF, it appears that the latter effects predominate. In contrast to the results for Py-polystyrene Py, here the cyclization rate is quite sensitive to the molecular weight of matrix chains. In addition, there seems to be no simple explanation at this time for the more complicated behavior of I + PVAc in a poor solvent. 

We shall consider first the screening of hydrodynamic interactions, which turns out to be of major importance in a semi-dilute solution. We will then introduce the scaling laws of collective modes in semi-dilute solutions, emphasizing here the role of universal behavior as a function of the concentration variable. The experimental results are then discussed within the framework of this universal behavior. 

The dynamics of concentration fluctuations expressed by eqn [161] (or eqn [34]) of the Omstdn-Zemike type exhibits the diflrrsion coeflrdent D =feT/(6nj 0), eqn [72], which can also be derived from fluctuations of gel-like networks with screening of hydrodynamic interactions. Thus, it is often rderred to as the gd mode. A decrease in mesh size increases the rdaxation rate of the gel mode. Experimentally, DLS for semidilute solutions rrsually exhibits another mode of motions with a very slow decay rate. This slow mode may be assodated with some inhomogeneity depending on solvent quality and sample preparation, or may be rdated to the translational... 

A comparison with Burchard s first cumulant calculations shows qualitative agreement, in particular with respect to the position of the minimum. Quantitatively, however, important differences are obvious. Both the sharpness as well as the amplitude of the phenomenon are underestimated. These deviations may originate from an overestimation of the hydrodynamic interaction between segments. Since a star of high f internally compromises a semi-dilute solution, the back-flow field of solvent molecules will be partly screened [40,117]. Thus, the effects of hydrodynamic interaction, which in general eases the renormalization effects owing to S(Q) [152], are expected to be weaker than assumed in the cumulant calculations and thus the minimum should be more pronounced than calculated. Furthermore, since for Gaussian chains the relaxation rate decreases... 

In order to resolve these challenges, it is essential to account for chain connectivity, hydrodynamic interactions, electrostatic interactions, and distribution of counterions and their dynamics. It is possible to identify three distinct scenarios (a) polyelectrolyte solutions with high concentrations of added salt, (b) dilute polyelectrolyte solutions without added salt, and (c) polyelectrolyte solutions above overlap concentration and without added salt. If the salt concentration is high and if there is no macrophase separation, the polyelectrolyte solution behaves as a solution of neutral polymers in a good solvent, due to the screening of electrostatic interaction. Therefore for scenario... 

Necessarily for any number of particles more than two, eqn. (211) cannot be solved exactly, even if v° = 0 and U = 0. When there are more than two particles, the motion of one particle, say j, causes both k and / to move. Now because k and / are perturbed by j, then the perturbation to the motion of k is felt by /. The motion of j affects / directly and also indirectly through k. These indirect effects are not usually very important, especially in chemical kinetics, because the particles most likely to react are those which are closest together. Under such circumstances, the direct effect is stronger than the transmitted and reflected components. These effects have been considered by Adelman [481], Freed and Muthukumar [482] and Allison et al. [483]. Adelman draws an interesting parallel between the screening of hydrodynamic repulsion and the electrolyte screening of a coulomb interaction [481]. 

Unfortunately, there is no accurate theoretical estimate of the strength of hydrodynamic interaction or the extent of hydrodynamic screening of polydisperse branched polymers. Hydrodynamic screening usually correlates well with excluded volume screening. As was demonstrated... 

In concentrated solutions, with the increase of the polymer concentration, the screen effect of hydrodynamic interactions is enhanced due to the interpenetration of polymer chains. We can assume that the hydrodynamic screening length is close to the screening length of volume exclusion of monomers as given by... 

Neglect of Hydrodynamic Interactions.—The coupling of hydrodynamic flow exerts a major influence on the dynamics of colloidal dispersions.In certain special cases, however, it has proved reasonable or expedient to neglect the hydrodynamic interactions. One such instance is the very dilute, electrostatically-stabilized dispersion in which particles interact via a screened Coulomb potential, that is, equation (2) with ku 1. 

In a later article [59] the importance of hydrodynamic interactions in the formation of certain microphases was demonstrated by close comparison of simulations using DPD and Brownian dynamics (BD). Whilst both simulation methods describe the same conservative force field, and hence should return the same equilibrium structure, they differ in the evolution algorithms. As mentioned above, DPD correctly models hydrodynamic interactions, whereas these are effectively screened in BD. Distinctly different equilibrium structures were obtained using the two techniques in long simulations of a quenched 2400 A3B7 polymer system, as shown in Fig. 1. Whilst DPD ordered efficiently into the expected state of hexagonal tubes, BD remained trapped in a structure of interconnected tubes. By way of contrast, both DPD and BD reproduced the expected lamellar equilibrium structure of A5B5 on similar time scales, see Fig. 2. 

In the second of his two well-known papers (of 1976) devoted to the dynamics of entangled polymer solutions, de Gennes used a "two-fluid model" to determine the effects of hydrodynamic interactions. Among the many results reported in this paper are two of particular interest to us here, viz., (1) that these interactions are not screened in the static, long wavelength limit and (2) that static screening does occur provided that the wavelength is less than the radius of the polymer chain. 

Relaxation of Single and of Entangled Macromolecules. In the absence of hydrodynamic interactions (HI) the normal modes of a polymer are Rouse modes, which act as overdamped harmonic oscillators. With HI the Rouse modes are still nearly normal modes, but the relaxation spectrum is modified. The HI are screened in semidilute solutions. At higher concentrations and in bulk disentanglement by reptation and tube renewal dominate slow viscoelastic processes. 

The main results for the dynamics of dilute solutions reflect the importance of hydrodynamic interactions each moving monomer in the solvent creates a backflow field which decays very slowly with distance. In a semi-dilute solution, the interference between all these velocity fields induces a screening of the backflow field of a given monomer, which falls off exponentially after a characteristic distance A. This idea was originally proposed by Edwards and Freed we shall briefly summarize their theory for ideal Gaussian chains. 

The Zimm model predicts correctly the experimental scaling exponent xx ss M3/2 determined in dilute solutions under 0-conditions. In concentrated solution and melts, the hydrodynamic interaction between the polymer segments of the same chain is screened by the host molecules (Eq. 28) and a flexible polymer coil behaves much like a free-draining chain with a Rouse spectrum in the relaxation times. 

Under -conditions the situation is more complex. On one side the excluded volume interactions are canceled and E,(c) is only related to the screening length of the hydrodynamic interactions. In addition, there is a finite probability for the occurrence of self-entanglements which are separated by the average distance E,i(c) = ( (c)/)1/2. As a consequence the single chain dynamics as typical for dilute -conditions will be restricted to length scales r < (c) [155,156],... 

It is generally accepted that in semi-dilute solutions under good solvent conditions both the excluded volume interactions and the hydrodynamic interactions are screened owing to the presence of other chains [4,5,103], With respect to the correlation lengths (c) and H(c) there is no consensus as to whether these quantities have to be equal [11] or in general would be different [160],... 

The hydrodynamic interaction is introduced in the Zimm model as a pure intrachain effect. The molecular treatment of its screening owing to presence of other chains requires the solution of a complicated many-body problem [11, 160-164], In some cases, this problem can be overcome by a phenomenological approach [40,117], based on the Zimm model and on the additional assumption that the average hydrodynamic interaction in semi-dilute solutions is still of the same form as in the dilute case. 

If a residual hydrodynamic interaction over large distances does not exist (l/r H(c) = 0), the regime of screened Zimm relaxation vanishes, and only the crossover from unscreened Zimm to enhanced Rouse relaxation remains. 

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