Zimm theory

Zimm s model (1956) is also a chain of beads connected by ideal springs. The chain consists of N identical segments joining + 1 identical beads with complete flexibility at each bead. Each segment, which is similar to a submolecule, is supposed to have a Gaussian probability function. The major difference between the two models lies in the interaction between the individual beads. In the Rouse model, such interaction is ignored in Zimm s model, such interaction is taken into consideration. 

According to Zimm s model, if a chain is suspended in a viscous liquid, each bead j encounters three different forces mechanical force. Brownian motion, and the motion of a fluid. 

Brownian Motion From the Brownian motion, the beads move, resulting in a force F that involves Hooke s law  

Motion of a Fluid Zimm adopted the Kirkwood-Riseman approximate form of the Oseen interaction formula to describe the force on the motion of a fluid  

The equation of motion is obtained by combining three equations representing three different forces  

Formulation. In this section, we will summarize the procedure for calculating dynamic mechanical properties of dilute polymer solutions by applying statistical mechanics to the model mentioned above. Basic equations of the theory consist of equations of motion for the polymer elements, an equation of motion for the fluid (hydrodynamics), a diffusion equation to describe the statistical nature of the problem and an equation of stress. 

For the Rouse model shown in Fig. 2.1, the equations of motion of frictional elements are given as 

the first three terms are the contributions due to the inertia of the elements, the frictional interaction of the solvent, and the tensile force of the elastic element, respectively. The last term is the contribution of the diffusion force corresponding to the Brownian motion of the elements, where / is the function of the coordinates of all the friction points and of time t and is called a distribution function. The statistical nature of the problem is introduced through /. Since rn is very small, the first term in Eq. (2.1) is very small unless the frequency is extremely high and so is neglected in the following calculations. 

The equation of motion of the solvent is given by macroscopic hydrodynamics as 

The last term in Eq.(2.4) is the perturbation of the flow due to the existence of frictional elements. The ij-th component of the Oseen 

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Zimm theory, multiparticle collision dynamics, polymers, 123-124... 

The theoretical prediction of these properties for branched molecules has to take into account the peculiar aspects of these chains. It is possible to obtain these properties as the low gradient Hmits of non-equilibrium averages, calculated from dynamic models. The basic approach to the dynamics of flexible chains is given by the Rouse or the Rouse-Zimm theories [12,13,15,21]. How-ever,both the friction coefficient and the intrinsic viscosity can also be evaluated from equilibrium averages that involve the forces acting on each one of the units. This description is known as the Kirkwood-Riseman (KR) theory [15,71 ]. Thus, the translational friction coefficient, fl, relates the force applied to the center of masses of the molecule and its velocity... 

The complex viscosity, i.e., the viscosity observed in the presence of an oscillatory shear rate, is a dynamic property that can be straightforwardly obtained from the Rouse, or Rouse-Zimm theory as the Fourier transform of the stress time-correlation function. Thus, these theories give [15]... 

Osaki,K., Schrag, J.L., Ferry, J.D. Infinite-dilution viscoelastic properties of poly(a-methylstyrene). Applications of Zimm theory with exact eigenvalues. Macromolecules 5,144-147 (1972). 

Bixon M, Zwanzig R (1978) Optimised Rouse-Zimm theory for stiff polymers. J Chem Phys 68(4) 1896-1902... 

Historically, the Zimm theory of A, was developed before the treatment of the excluded volume effect by Flory. 

B. H. Zimm, Theory of melting of the helical form in double chains of the DNA type. J. Chem. Phys. 33 1349-1356 (1960). 

Figure 4-13 contains the predictions of the Rouse theory on the left and of the Zimm theory on the right. As is to be expected, the predictions of the Zimm theory that takes the hydrodynamic interactions into account predicts well experimental data. 

Freely Draining Gaussian Chain (Rouse Theory) Dominant HI Theta Solvent (Zimm Theory) Dominant HI Good Solvent... 

Figure 3.13 Linear viscoelastic data (symbols) for polystyrene in two theta solvents, decalin and diocty Iphthalate, compared to the predictions (lines) of the Zimm theory with dominant hydrodynamic interaction, h = oo. The reduced storage and loss moduli and G are defined by = [G ]M/NAksT and G s [G"]M/A /cbT, where the brackets denote intrinsic values extrapolated to zero concentration, [G jj] = limc o(G /c) and [G j ] = limc +o[(G" — cor)s]/c), and c is the mass of polymer per unit volume of solution. The characteristic relaxation time to is given by to = [rj]oMrjs/NAkBT. For frequencies ro

Both the scaling with co and the proportionality constant, V3, are confirmed by experimental data (see Fig. 3-13). The lines in Fig. 3-13 are proportional to G and G" — computed in the nondraining limit. The agreement with data for a polystyrene of high molecular weight (M = 860,000) in theta solvents is excellent. In addition to its agreement with experimental data, the predictions of Zimm theory are supported by molecular dynamics simulations (Pierleoni and Ryckaert 1991 DUnweg and Kremer 1991). 

In a good solvent, where there are excluded-volume effects, G and G" can be fit to the Zimm theory simply by adjusting h downward, for finite Ns (Ferry 1980). Thus, as the solvent quality improves, the relaxation spectrum becomes more Rouse-like (since... 

Figure 3.15 The frequency-dependent in-phase and out-of-phase components of the dynamic viscosity, rj and rj in small-amplitude oscillatory shear, along with the shear-rate dependence of the first normal stress coefficient hi (y) for a 0.05 wt% solution of polystyrene of molecular weight 2.25 X 10 in a solvent of oligomeric styrene. The lines through the data show the predictions of the Zimm theory for r and 2r)"f(o and the Zimm theory for hi(y) modified to account for finite extensibility, as discussed in Section 3.6.2.2.I. The dashed lines are the contributions of the individual Zimm relaxation modes to 2rj"((o) /

The principal equations for elastic LS have been reviewed recently [270] and will be only summarized herein. In brief, following the Zimm theory [277], the intensity of the light scattering by a polymeric solution is in relation with the molar mass of the sample according to the following general equation ... 

Note. R-Z, Rouse-Zimm theory. EN, electrical network analogy. 

Calculation of Complex Modulus. In the Zimm theory, the Oseen tensor is approximated by its average value over the equilibrium configuration ... 

Peterlin-Tschoegl Method. The Zimm theory is constructed on the assumption that the polymer chain configurations follow a Gaussian distribution and hence the average distance between the i-th and the j-th elements satisfies a relation... 

A theory for the dynamic properties of ring polymers was proposed by Bloomfield and Zimm (54). This theory is another application of the method of the original Zimm theory (29) to a different geometry. Therefore, Eq. (2.1)—(2.10) are used again with a slight modification. Suppose the ring model consists of (N +1) beads. The (1, l)-element... 

Detailed comparison of dynamic properties predicted by various theories may be performed by comparing the plots of reduced intrinsic complex modulus [G j as a function of reduced angular frequency coR [see Eq. (1.18) for definition]. The effect of varying h in the Zimm theory as evaluated by Tschoegl (59) is shown in Fig. 2.2. It is obvious... 

The effect of excluded volume as predicted by the Tschoegl equation (39) is to shift the dynamic properties to more Rouse-like behavior. If one keeps h at infinity and takes = 1/3, then one obtains curves for [G ju and [G"]k which lie between the corresponding quantities for h = 1 and 25 in Fig. 2.2. Thus the effect of increasing e is qualitatively equivalent to decreasing h in the original Zimm theory (29). This effect of g diminishes as h decreases, and disappears at h = 0. 

Fig. 2.3. Relaxation spectrum of Ogasa-lmai theory (47) compared with those erf Rouse theory (27) and Zimm theory (29). Strength Gp of p-th relaxation mode is plotted against logarithm erf reduced relaxation time t,/tp. Strength of mode of the longest relaxation time at left side is nkTon each panel...

Fig. 2.5. Flory constant 0 from Zimm theory evaluated by various methods of calculation plotted against hydrodynamic interaction parameter h Circles from eigenvalues of difference equation (2.18) thin lines from Eq. (2.41) thick line result of Tschoegl from Eq. (2.20). Values of h are shown, and cross mark indicates N = 104 for a given value of h = hN 112 (72)...

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