Hydrodynamic interaction long-range

Hydrodynamic interaction is a long-range interaction mediated by the solvent medium and constitutes a cornerstone in any theory of polymer fluids. Although the mathematical formulation needs somewhat elaborate methods, the idea of hydrodynamic interaction is easy to understand suppose that a force is somehow exerted on a Newtonian solvent at the origin. This force sets the surrounding solvent in motion away from the origin, a velocity field is created which decreases as ... 

The Rouse model, as given by the system of Eq, (21), describes the dynamics of a connected body displaying local interactions. In the Zimm model, on the other hand, the interactions among the segments are delocalized due to the inclusion of long range hydrodynamic effects. For this reason, the solution of the system of coupled equations and its transformation into normal mode coordinates are much more laborious than with the Rouse model. In order to uncouple the system of matrix equations, Zimm replaced S2U by its average over the equilibrium distribution function ... 

The earliest and simplest approach in this direction starts from Langevin equations with solutions comprising a spectrum of relaxation modes [1-4], Special features are the incorporation of entropic forces (Rouse model, [6]) which relax fluctuations of reduced entropy, and of hydrodynamic interactions (Zimm model, [7]) which couple segmental motions via long-range backflow fields in polymer solutions, and the inclusion of topological constraints or entanglements (reptation or tube model, [8-10]) which are mutually imposed within a dense ensemble of chains. 

The dynamics of polymer solution is governed by the hydrodynamic interaction between moving segments mediated by the solvent [90,91]. This interaction is long range and couples the motions of the different segments strongly. In this... 

In fact, the diffusion constant in solutions has the form of an Einstein diffusion of hard spheres with radius Re. For a diffusing chain the solvent within the coil is apparently also set in motion and does not contribute to the friction. Thus, the long-range hydrodynamic interactions lead, in comparison to the Rouse model, to qualitatively different results for both the center-of-mass diffusion—which is not proportional to the number of monomers exerting friction - as well as for the segment diffusion - which is considerably accelerated and follows a modified time law t2/3 instead of t1/2. 

Collective hydrodynamical interactions. We have seen that the long-range hydrodynamical force is diverging at small wave numbers this suggests very much that one should consider... 

When long-range particle interactions and hydrodynamics effects are ignored, equation 5.22 becomes equivalent to the solution of von Smoutuowski 31 who obtained the collision frequency Is as ... 

In 1959, Zimm and Kilb (34) made some calculations of the intrinsic viscosities of certain branched polymer molecules, taking into account the hydrodynamic interaction between portions of the polymer chain, using a modification of the Rouse procedure. They carried out these difficult calculations for a quite restricted range of models, obtaining numerical results for equalarmed stars with 3, 4, and 8 branches, and for one modified star with 2 long branches and 8 shorter branches. They found that their numerical results for this set of structures could be approximately represented by ... 

The long-range, purely hydrodynamic interaction between two suspended spheres in a shear flow was first calculated by Guth and Simha (1936), yielding a value of kx = 14.1 via a reflection method. Saito (1950,1952) proposed two alternative modifications, obtaining kt = 12.6 and 2.5, respectively the latter value is obtained upon supposing a spatially uniform distribution of particles. 

This form is adopted from the molecular dynamics computer results of Levesque and Verlet. - The first term in Eq. (8.4) is the short-time colli-sional contribution characterized by the collisional time while the second term originates from long-range many-body interactions and is characterized by the hydrodynamic time t. From the discussion of Section VI it follows that in highly non-Markovian situations (such as large cuj, in barrier dominated processes) the collisional term in Eq. (8.4) makes the dominant contribution. (This observation is very significant for the analysis of the viscosity dependence of the rate. ) Within this model and using available informa-... 

In addition to the surface forces (see Section 5.4 above), two colliding particles in a liquid medium also experience hydrodynamic interactions due to the viscous friction, which can be rather long range (operative even at distances above 100 nm). The hydrodynamic interaction among particles depends on both the type of fluid motion and the type of interfaces. The quantitative description... 

Note also that for imaginary particles, which experience neither long-range surface forces (JJi j = 0) nor hydrodynamic interactions (P = 1), Equation 5.325a yields a collision efficiency = 1 and Equation 5.321 reduces to the von Smoluchowski " expression for the rate constant of the fast irreversible coagulation. In this particular case. Equation 5.319 represents an infinite set of nonlinear differential equation. If all flocculation rate constants are the same and equal to Up the problem has a unique exact solution " ... 

It has been established that Ko normally is independent of the solvent and the molecular weight of the polymer, though often dependent to some extent on the temperature. It is therefore possible to deduce values for the expansion factor in good solvents from intrinsic viscosities measured in them. From Eqs. (3.181) and (3.184) the linear expansion factor a.jj, which is a measure of long range interactions and pertains to hydrodynamic chain dimensions, is thus given by... 

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