Hydrodynamic interaction between beads

The Rouse model is the simplest molecular model of polymer dynamics. The chain is mapped onto a system of beads connected by springs. There are no hydrodynamic interactions between beads. The surrounding medium only affects the motion of the chain through the friction coefficient of the beads. In polymer melts, hydrodynamic interactions are screened by the presence of other chains. Unentangled chains in a polymer melt relax by Rouse motion, with monomer friction coefficient C- The friction coefficient of the whole chain is NQ, making tha diffusion coefficient inversely proportional to chain length ... 

This condition is guaranteed for the correct mobility matrix. However, the mobility matrix given by eqn (4.40) is an approximate one, and does not satisfy the inequality (4.1 ) in a certain configuration in which the beads are too close to each other. An improved formula which guarantees the inequality is proposed by Rotne and Prager. However, this correction is irrelevant for the asymptotic behaviour of N 1, which is determined by the hydrodynamic interaction between beads far apart from each other. Thus we shdl use eqn (4.40) for H, . 

Just as the Gaussian chain is the basic paradigm of the statistics of polymer solutions, so is its extension to the bead-spring model still basic to current work in the held of polymer dynamics. The two limiting cases of free draining (no hydrodynamic interaction between beads, characterized by the draining parameter A = 0) and non-free draining (dominant hydrodynamic interaction, A= CO, due to Rouse and Zimm, respectively, are sufficiently familiar that the approach is often known as the Rouse-Zimm model. ... 

I Hydrodynamic Interactions B. Zinun improved the Rouse model by taking into account hydrodynamic interactions between beads. He successfully obtained the expressions for the diffusion coefficient and the relaxation times that agree with experimental results. 

Fig. 47. Schematic representation of the hydrodynamic interaction (HI (1)) of two other beads 2 and 3 on a bead 1. The total HI may be written as HI (1) = h (1, 2) + h (1, 3) + [h (2, 3)] where h (1, 2) denotes the force of bead 2 on bead 1, etc. In the Oseen approach, the interaction between beads 2 and 3 is assumed to have no influence on bead 1...

The basic equipment consists of a column packed with stationary, solid beads in the size range 10 to 50 pm [41,42] (Figure 5.8). Means are provided for injecting about 0.2 cm of colloidal suspension, containing about 0.01% polymer by weight, into the flowing stream at the entrance of the column and monitoring the colloid in the column effluent. Particle separation occurs due to a hydrodynamic interaction between the particles and the velocity profile near solid surfaces. HDC is a fast technique but its resolution is low. 

Some authors (for example, Weill and des Cloizeaux [106]) aigue that because of the long-range nature of hydrodynamic interactions between paired beads the theoretically predicted asymptotic slope 0.8 of the relation between log [ ] and log M will be approached so slowly that it may not be observable experimentally. This implies that actually observed HMS relations with v < 0.8 are mere segments of a non-linear dependence of logf ] on logM. 

In accordance with the second approach the polymeric solutions viscosity anomaly is explained by the effect of the hydrodynamic interaction between the links of the polymeric chain, such links represent by themselves the beads into the necklace model. Accordingly to this effect the hydrodynamic flow aroimd the presented bead essentially depends on the position of the other beads into the polymeric ball. An anomaly of the viscosity was conditioned by the anisotropy of the hydrodynamic interaction which creates the orientational effect (prior work by Peterlin and Copic [7]). High values of the viscosity for the concentrated solutions and its strong gradient dependence cannot be explained only by the effect of the hydrodynamic interactioa... 

The transition from the Zimm regime [Eq. (17), [f]] N ] at dilute concentrations to Rouse-like behavior [Eq. (16), [rf N] at the intermediate concentrations is related to the hydrodynamic screening at high concentrations the hydrodynamic interactions between segments become negligible because the local velocity around the beads falls quickly. 

The Rouse model assumes unperturbed chains in the free-draining limit (the limit of no hydrodynamic interactions between monomers) that can be represented as a set of beads connected along the chain contour. The Zimm model, in addition to the Rouse model features, takes into account the hydrodynamic interaction of the beads. 

Since the hydrodynamic interaction decreases as the inverse distance between the beads (Eq. 27), it is expected that it should vary with the degree of polymer chain distortion. This is not considered in the Zimm model which assumes a constant hydrodynamic interaction given by the equilibrium averaging of the Oseen tensor (Eq. 34). 

Multiparticle collision dynamics provides an ideal way to simulate the motion of small self-propelled objects since the interaction between the solvent and the motor can be specified and hydrodynamic effects are taken into account automatically. It has been used to investigate the self-propelled motion of swimmers composed of linked beads that undergo non-time-reversible cyclic motion [116] and chemically powered nanodimers [117]. The chemically powered nanodimers can serve as models for the motions of the bimetallic nanodimers discussed earlier. The nanodimers are made from two spheres separated by a fixed distance R dissolved in a solvent of A and B molecules. One dimer sphere (C) catalyzes the irreversible reaction A + C B I C, while nonreactive interactions occur with the noncatalytic sphere (N). The nanodimer and reactive events are shown in Fig. 22. The A and B species interact with the nanodimer spheres through repulsive Lennard-Jones (LJ) potentials in Eq. (76). The MPC simulations assume that the potentials satisfy Vca = Vcb = Vna, with c.,t and Vnb with 3- The A molecules react to form B molecules when they approach the catalytic sphere within the interaction distance r < rc. The B molecules produced in the reaction interact differently with the catalytic and noncatalytic spheres. 

The most studied relaxation processes from the point of view of molecular theories are those governing relaxation function, G,(t), in equation [7.2.4]. According to the Rouse theory, a macromolecule is modeled by a bead-spring chain. The beads are the centers of hydrodynamic interaction of a molecule with a solvent while the springs model elastic linkage between the beads. The polymer macromolecule is subdivided into a number of equal segments (submolecules or subchains) within which the equilibrium is supposed to be achieved thus the model does not permit to describe small-scale motions that are smaller in size than the statistical segment. Maximal relaxation time in a spectrum is expressed in terms of macroscopic parameters of the system, which can be easily measured ... 

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