Hydrodynamic interactions in solution

Having discussed the dynamics of polymer chains in the melt, in this section we will examine their motion in solution. As we shall see, here the diffusion coeflBicient cannot be described by either of the two equations for the melt but is given by 

We can analyse this force in more detail. To begin with, remember the simple case of a shear flow, created if two parallel plates are moved against each other with constant velocity Vx Here we find a linear velocity profile, like that included in Fig. 6.16. The power which has to be supplied is given 

The flow field v(r) produced by the particle if it is dragged with a constant velocity u through the liquid may generally be described as 

The velocity gradient tensor included in Eq. (6.134) follows by taking the derivative 

Here we introduced another tensor, of third rank, with components hijk- By using the distance Ar between r and the colloid at Vc as variable, it is expressed that the flow field moves together with the particle, i.e. is stationary in a particle fixed coordinate system. Importantly, the formulated dependence is of the linear type. Hence, it holds for Newtonian liquids which by definition obey linear laws. All low molar mass liquids behave in a Newtonian manner, at least within the limit of low velocities. 

---

Most descriptions of the dynamics of molecular or particle motion in solution require a knowledge of the frictional properties of the system. This is especially true for polymer solutions, colloidal suspensions, molecular transport processes, and biomolecular conformational changes. Particle friction also plays an important role in the calculation of diffusion-influenced reaction rates, which will be discussed later. Solvent multiparticle collision dynamics, in conjunction with molecular dynamics of solute particles, provides a means to study such systems. In this section we show how the frictional properties and hydrodynamic interactions among solute or colloidal particles can be studied using hybrid MPC-MD schemes. 

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. 

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)...

S. Kim, Singularity solutions for ellipsoids in low-Reynolds number flows With applications to the calculation of hydrodynamic interactions in suspensions of ellipsoids, Ini. J. Multiphase Flow 12, 469-91 (1986). 

Hsieh, C. C., and Larson, R. G., Modeling hydrodynamic interaction in Brownian dynamics simulations of extensional and shear flows of dilute solutions of high molecular weight polystyrene, J. RheoL, 48, 995-1021 (2004). 

Equations (3.99) and (3.102) are called the Stokes approximation, and are the basis of the hydrodynamic interactions in suspensions and polymer solutions. 

In Eq. (26), A v is the velocity of the solvent at the position of particle i due to the average effect of the hydrodynamic interactions of solute and solvent (electrophoretic effect) the other symbols were explained in the preceding section. Exchange reactions of the type... 

This quantity is measured in dynamic light-scattering technique, and its origin lies in the hydrodynamic interactions in the solution. 

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

The absence of any hydrodynamic interaction allows one directly to ask how the entanglement length scales as a function of density. Long-range hydrodynamic interactions in real solutions make this problem more complicated. One would like to know how Ng scales with the static excluded... 

It is sometimes hypothesized that hydrodynamic interactions in polymer solutions are screened, i.e., the interactions fall off with distance not as the /R of the Oseen tensor but instead decrease exponentially, e.g., as exp(—KR)/R. Analogies are sometimes drawn with another long-range potential, namely the Coulomb potential, which in electroneutral solutions and as a result of electroneutrality decreases not as 1// but instead as exp(—KR)/R. Corresponding analogies are not drawn between hydrodynamics and the Newtonian gravitational potential, because gravity is not screened. 

We have now examined three phenomenological behaviors, namely two-bead microrheology, leading-order concentration dependences of transport coefficients, and -dependent chain internal modes. These three lines of evidence converge to the same conclusion. Hydrodynamic interactions in polymer and colloid solutions are described by the Oseen and Kynch tensors they are not screened. 

The above results for the Rouse model are applicable to the experimental conditions where the hydrodynamic and excluded volume interactions and the entanglement effects can be completely ignored. We shall identify such an experimental regime later on. Now, we attempt to incorporate the hydrodynamic interaction in describing the chain dynamics in infinitely dilute solutions. The Rouse chain model incorporating the effect of hydrodynamic interaction is called the Kirkwood-Riseman modeF or Zimm model. These models differ from each other in certain subtle features and the numerical prefactors only the predicted molecular weight dependence of the longest relaxation time, viscosity of the solution, diffusion coefficient, etc. are the same. 

In Ref. 118 (Fig. 8), the intrinsic viscosity [g] is compared with the weight average molecular weight. It was found that the absolute value of [ ] and the slope of the curve (on the log-log scale) are much lower for solutions of branched than of linear polymers. This experimental result, together with the percolation expressions for the intrinsic viscosity (Eqs. (21, 22)) confirm that clusters undergo hydrodynamic interactions. In fact, the molecular weight exponent value of [ly] is much lower with than without hydrodynamic interactions. This result implies that a calculation of the viscosity of the reaction bath is correct only if hydrodynamic interactions are taken into account. 

---